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Th`
ese de doctorat NNT : 2019SACLT043
Cloud-Radio Access Networks : Design,
Optimization and Algorithms
Th`
ese de doctorat de l’Universit´
e Paris-Saclay
pr´
epar´
ee `
a Institut Mines-T´
el´
ecom - T´
el´
ecom ParisTech
Ecole doctorale n◦580 Sciences et Technologies de l’Information et de la
Communication (STIC)
Sp´
ecialit´
e de doctorat : R´
eseaux, Information, Communications
Th`
ese pr´
esent´
ee et soutenue `
a Paris, le 10 Octobre 2019, par
NIE ZI MHARSI
Composition du Jury :
Djamal ZEGHLACHE
Professeur, Institut Mines-T´
el´
ecom - T´
el´
ecom SudParis Pr´
esident
Adlen KSENTINI
Professeur, Eurecom Rapporteur
Lina MROUEH
Maˆ
ıtre de conf´
erences - HDR, Institut Sup´
erieur d’Electronique de
Paris Rapporteur
Loutfi NUAYMI
Professeur, Institut Mines-T´
el´
ecom - IMT Atlantique Examinateur
Philippe MARTINS
Professeur, Institut Mines-T´
el´
ecom - T´
el´
ecom ParisTech Directeur de th`
ese
Makhlouf HADJI
Chercheur - HDR, Institut de Recherche Technologique - IRT
SystemX Co-directeur de th`
ese
Contents
List of Figures iii
List of Tables vi
List of Acronyms ix
Introduction 1
1 Context, motivations and contributions 3
1.1 Context and motivations : Cloud-Radio Access Network (C-RAN) . . 3
1.1.1 C-RAN architecture . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 C-RANbenefits.......................... 6
1.1.3 C-RANchallenges ........................ 7
1.2 Contributions ............................... 8
1.2.1 Low-complexity algorithms for constrained resource allocation
probleminC-RAN ........................ 9
1.2.2 Mathematical programming approach for full network cover-
age optimization in C-RAN . . . . . . . . . . . . . . . . . . . 10
1.2.3 Cost-efficient and scalable algorithms for BBU function split
placementinC-RAN....................... 11
1.3 Publications................................ 11
2 C-RAN optimization: mathematical background and state-of-the-
art 13
2.1 Introduction................................ 13
2.2 Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Combinatorial optimization : techniques and algorithms . . . 13
2.2.2 Simplicial homology background . . . . . . . . . . . . . . . . . 23
2.3 C-RAN optimization : state-of-the-art . . . . . . . . . . . . . . . . . 25
2.3.1 Constrained resource allocation problem . . . . . . . . . . . . 25
2.3.2 Network coverage optimization problem . . . . . . . . . . . . . 27
2.3.3 BBU functions split and placement problem . . . . . . . . . . 29
2.4 Conclusion................................. 32
i
ii
3 Constrained resource allocation in C-RAN 33
3.1 Introduction................................ 33
3.2 Problemstatement ............................ 34
3.2.1 Systemmodel........................... 34
3.2.2 Problem complexity . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Proposed algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1 Integer linear programming formulation . . . . . . . . . . . . . 38
3.3.2 Matroid-based approach . . . . . . . . . . . . . . . . . . . . . 40
3.3.3 b-Matching formulation . . . . . . . . . . . . . . . . . . . . . 42
3.3.4 Multiple knapsack-based approach . . . . . . . . . . . . . . . 44
3.4 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.1 Simulation settings and parameters . . . . . . . . . . . . . . . 46
3.4.2 Performance metrics . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.3 Performance analysis . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Conclusion................................. 55
4 Full network coverage optimization in C-RAN 57
4.1 Introduction................................ 57
4.2 Problemstatement ............................ 58
4.2.1 System model and problem description . . . . . . . . . . . . . 58
4.2.2 Problem complexity . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3 Branch-and-Cut formulation . . . . . . . . . . . . . . . . . . . . . . . 61
4.3.1 Convex hull characterization . . . . . . . . . . . . . . . . . . . 61
4.3.2 New valid inequalities . . . . . . . . . . . . . . . . . . . . . . 64
4.3.3 Complete mathematical formulation . . . . . . . . . . . . . . . 67
4.4 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4.1 Simulation parameters and settings . . . . . . . . . . . . . . . 69
4.4.2 Performance metrics . . . . . . . . . . . . . . . . . . . . . . . 69
4.4.3 Simulation results and performance analysis . . . . . . . . . . 70
4.5 Conclusion................................. 74
5 BBU function split placement in C-RAN 75
5.1 Introduction................................ 75
5.2 Problemstatement ............................ 76
5.2.1 BBU function split modeling . . . . . . . . . . . . . . . . . . . 76
5.2.2 Network topology description . . . . . . . . . . . . . . . . . . 77
5.2.3 Systemmodel........................... 78
5.2.4 Problem complexity . . . . . . . . . . . . . . . . . . . . . . . . 79
5.3 Exact mathematical approach . . . . . . . . . . . . . . . . . . . . . . 80
5.4 Approximation approaches : multi-stage graph algorithms . . . . . . 84
5.5 Performance evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.5.1 Simulation parameters and settings . . . . . . . . . . . . . . . 86
5.5.2 Performance metrics . . . . . . . . . . . . . . . . . . . . . . . 87
5.5.3 Simulation results and performance analysis . . . . . . . . . . 87
5.6 Conclusion................................. 91
iii
6 Conclusions and perspectives 93
Bibliography 95
iv
List of Figures
1.1 5G services and opportunities. Source: Cisco, 2018 [1] . . . . . . . . . 3
1.2 Traditional RAN to C-RAN architecture . . . . . . . . . . . . . . . . 5
1.3 C-RAN architecture and components . . . . . . . . . . . . . . . . . . 6
2.1 An example of a convex polytope . . . . . . . . . . . . . . . . . . . . 15
2.2 An example of linear inequality separating the convex polytope P
and the solution x∗............................ 17
2.3 Example of k-simplices . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Example of simplicial complex . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Delaunay triangulation. Source: [2] . . . . . . . . . . . . . . . . . . . 28
2.6 3GPP functional split options . . . . . . . . . . . . . . . . . . . . . . 30
2.7 Overview of considered functional split options . . . . . . . . . . . . . 31
3.1 System model for constrained resource allocation problem . . . . . . . 34
3.2 A solution example of the constrained resource allocation problem . . 35
3.3 Example of simulation scenarios for RRH-BBU assignment problem . 46
3.4 Matroid-based approach : rejection rate variation when increasing
number of edge data centers . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 SLA violations rate behavior of the matroid-based approach . . . . . 51
3.6 Resource utilization in different space dimensions . . . . . . . . . . . 52
3.7 Real trace : Orange 4G-LTE cell map in Paris. Source: [3] . . . . . . 53
4.1 System model: graph construction based on antennas positions and
interference ................................ 58
4.2 A graph solution example of the full coverage network problem . . . . 60
4.3 Each solid line edge (i, j) is necessary in the final graph/solution . . . 62
4.4 Example of edge intersection (interference) . . . . . . . . . . . . . . . 62
4.5 Example of graph solution containing a hole (v2, v4, v5, v6, v2) . . . . . 64
4.6 Example of a chordless cycle (v2, v3, v5, v8, v2) of size 4 . . . . . . . . . 65
4.7 Example of two connected components (triangulations) creating holes
inthefinalgraph ............................. 66
4.8 An Orange 4G-LTE cell map: before Branch-and-Cut optimization 73
4.9 An Orange 4G-LTE cell map: after Branch-and-Cut optimization 73
5.1 BBU function split modeling for each antenna demand . . . . . . . . 77
5.2 Physical network architecture . . . . . . . . . . . . . . . . . . . . . . 77
5.3 System model for BBU function split placement . . . . . . . . . . . . 78
v
vi
5.4 Example of Virtual Network Embedding problem. Source: [4] . . . . 80
5.5 A multi-stage graph example . . . . . . . . . . . . . . . . . . . . . . . 84
5.6 Algorithms’ convergence time using 20 edge cloud data centers . . . . 87
5.7 CPU residual resources behavior . . . . . . . . . . . . . . . . . . . . . 89
5.8 Latencybehavior ............................. 90
List of Tables
3.1 RRH-BBU assignment problem : variables and parameters . . . . . . 36
3.2 RRH-BBU assignment algorithms : simulation settings and parameters 47
3.3 Performance of the exact approach based on ILP formulation . . . . . 49
3.4 Heuristic algorithms’ performance assessment . . . . . . . . . . . . . 49
3.5 Performance evaluation using a real cellular network in Paris . . . . . 53
3.6 Algorithms’ scalability assessment . . . . . . . . . . . . . . . . . . . . 54
3.7 Algorithms’ qualitative comparison . . . . . . . . . . . . . . . . . . . 54
4.1 Network coverage optimization : simulation settings and parameters . 69
4.2 Exact algorithm performance: convergence time to the optimum . . . 70
4.3 Performance comparison : ILP vs Rips approach . . . . . . . . . . . . 71
5.1 BBU function split placement problem : variables and parameters . . 79
5.2 Algorithms’ performance comparison : ILP vs heuristic variants . . . 88
5.3 Scalability and convergence time comparison using Euclidean graphs . 90
vii
viii
ix
List of Acronyms
•BBU: BaseBand Unit
•BFD Best Fit Decreasing
•CAPEX: CAPital EXpenses
•COMP: Coordinated Multi-Point
•CPU: Central Processing Unit
•C-RAN: Cloud-Radio Access Network
•C-RoFN: Cloud-Based Radio over Optical Fiber Network
•DSP: Digital Signal Processing
•eMBB: enhanced Mobile BroadBand
•eNodeB: evolved Node B
•EPC: Evolved Packet Core
•FFD: First-Fit Decreasing
•FPGA: Field-Programmable Gate Array
•HARQ: Hybrid Automatic Repeat reQuest
•ICIC: Inter-cell Interference Coordination
•ILP: Integer Linear Programming
•IoT: Internet of Thing
•LP: Linear Programming
•MAC: Medium Access Control
•MILP: Mixed Integer Linear Program
•mMTC: massive Machine-Type Communications
•mmWave: millimeter Wave
•MWT: Minimum Weight Triangulation
•NGMN: Next Generation Mobile Networks
•NG-PoP: Next Generation Point of Presence
x
•OAI: OpenAirInterface
•OPEX: OPerating EXpenses
•PDCP: Packet Data Convergence Protocol
•PHY: Physical layer
•QoS: Quality of Service
•RAN: Radio Access Network
•RF: Radio Frequency
•RLC: Radio Layer Control
•RRH: Remote Radio Heads
•SA: Simulated Annealing
•SDN: Software Defined Network
•SDR: Software Defined Radio
•SLA:Service-Level Agreement
•TCO: Total Cost of Ownership
•UE: User Equipment
•URLLC: Ultra-Reliable and Low-Latency Communications
•VM: Virtual Machine
•VNE: Virtual Network Embedding
•VNR Virtual Network Requests
•V2X: Vehicle to anything
•WSN: Wireless Sensor Network
•3GPP: Third Generation Partnership Project
•4G: Fourth Generation
•4G-LTE: Fourth Generation-Long Term Evolution
•5G: Fifth Generation
1
Introduction
The main goal of this thesis is to investigate the deployment of Cloud Radio
Access Network (C-RAN) and its relevant and exciting research challenges such as
resource allocation and placement problems, network coverage optimization, etc.
In fact, with the exponential growth in data traffic demands, C-RAN is seen as a
key enabler for the next generation of mobile networks (5G) to handle the diverse
service requirements and reduce network costs, including CAPEX (CAPital EX-
penses) and OPEX (OPerating EXpenses). This architecture consists in decoupling
the BaseBand Units (BBU) from the Remote Radio Head (RRH) and centraliz-
ing the baseband processing into common data centers (a pool of BBUs) offering
resource utilization gains and cost savings.
In this thesis, we use combinatorial optimization techniques to propose new ap-
proaches that rapidly reach optimal or near optimal solutions for resource allocation
problems in the context of C-RAN when guaranteeing good Quality-of-Service (QoS)
for end-users’ demands.
Chapter 1 provides the main motivations of our research works. The context of
this thesis is then described by providing a brief introduction of C-RAN architecture,
its benefits and challenges. We also provide an overview of our novel contributions
and the list of scientific publications that we propose during the thesis.
Chapter 2 provides an overview of mathematical backgrounds used in this
manuscript, including two major domains : (i) combinatorial optimization and (ii)
simplicial homology and gives a survey on the most important challenges that we
address in this manuscript to enable the deployment of C-RAN architecture.
Chapter 3 discusses an exciting research challenge in the context of C-RAN which
is the problem of RRH-BBU assignment that aims to allocate limited computing re-
sources in common edge data centers to the heterogeneous antennas demands when
meeting strong latency requirements. We propose a complete mathematical formu-
lation based on Integer Linear Program (ILP) to describe the convex hull of the
RRH-BBU assignment problem and provide optimal solutions. For sake of scalabil-
ity, we introduce new approximation algorithms to rapidly find good strategies to
assign antennas demands to centralized data centers, under latency and processing
requirements. A performance evaluation of our proposed algorithms is discussed
using different simulation scenarios and performance metrics.
Chapter 4 presents an exact approach based on Branch-and-Cut methods to
reduce inter-cell interference, caused by the high density of cells in C-RAN, when
guaranteeing a full network coverage. We use simplicial homology-based approaches
to evaluate the ability of our approach in finding optimal solutions in acceptable
times.
Chapter 5 focuses on another major challenge to enable the deployment of C-
RAN which consists in finding optimal trade-offs between benefits of processing
centralization and strong latency requirements. In this chapter, we propose exact
and heuristic algorithms based on combinatorial optimization methods to determine
2
the optimal placement of baseband processing between RRHs and BBUs. The BBU
functions placement is based on different split configurations and transport net-
work characteristics. Two simulation scenarios and different performance metrics
are proposed to benchmark our proposed algorithms.
Finally, Chapter 6 concludes the manuscript and investigates future research
challenges.
Chapter 1
Context, motivations and
contributions
1.1 Context and motivations : Cloud-Radio Ac-
cess Network (C-RAN)
Mobile data traffic demands are exponentially increasing due to the rapid growth
in the number of connected terminals and mobile devices (smartphones, tablets,
etc). Furthermore, the promise of 5G networks is not only a simple evolution of
4G networks with higher peak throughput and larger spectrum bands, but also to
deal with new services and new business opportunities that are classified by 3GPP
[5] into three main families of use cases : (i) enhanced Mobile BroadBand (eMBB),
(ii) Ultra-Reliable and Low-Latency Communications (URLLC) and (iii) massive
Machine-Type Communications (mMTC).
Figure 1.1: 5G services and opportunities. Source: Cisco, 2018 [1]
As depicted in Figure 1.1, eMBB services gather the set of use cases requiring
high data rates across a wide coverage area, URLLC involve the applications that
3
4
have strict requirements on latency and reliability such as Vehicle to anything (V2X)
and industrial control applications and mMTC represent the need to support a very
large number of devices in small area forming an Internet of Things (IoT).
The deployment of such technologies is significantly increasing the network costs,
including CAPEX and OPEX, the amount of baseband processing as well as levels of
inter-cell interference because of the high density of cells. However, current network
architectures will no longer be able to deal with the demands for high data rates
and lower latencies and to support new 5G services. Therefore, network operators
need to investigate new network architecture for next generation of mobile networks
to meet these challenges and reduce CAPEX and OPEX. In this context, C-RAN
has been proposed as a promising network architecture to handle the diverse service
requirements and guarantee a good QoS for end-users. Unlike conventional mobile
networks where the baseband functions reside on the cell sites along with the an-
tennas, C-RAN decouples the traditional base station into RRHs and centralized
BBUs which will be pooled and used as shared resources, offering network resource
utilization gains and energy efficiency. C-RAN is expected to minimize network
costs (CAPEX and OPEX) by reducing the number of base stations needed to meet
antennas demands. Meanwhile, C-RAN will improve radio performance by facili-
tating various forms of multi-cell coordination to handle the inter-cell interference,
caused by the high density of cells.
In the following, we detail the fundamental aspects of C-RAN architecture and
its main components. Then, we discuss the main advantages of this architecture and
finally, we outline the most important challenges that we are facing while deploying
C-RAN architecture.
1.1.1 C-RAN architecture
Current generation mobile networks are using traditional Radio Access Network
(RAN) that consists in locating the radio and baseband processing functionalities in
the same base station. In fact, as depicted in Figure 1.2, the traditional base station
consists of two components, the antenna (RRH) and the BBU (data center), co-
located at the same macro site (eNodeB). Nevertheless, these networks are no longer
able to provide high data rates, meet strong latency requirements and guarantee
high QoS for end-users’ demands. In order to achieve these goals, C-RAN has been
proposed as a promising architecture for next generation of mobile networks (5G)
to handle the diverse service requirements. The main concept of C-RAN consists
in decoupling the BBUs from the antennas and pooling the computing resources
into centralized data centers, i.e. BBU pools. The BBU computation pools will be
shared among multiple base stations in order to achieve resource utilization gains
and network cost savings. Figure 1.2 illustrates the main difference between the
traditional RAN (the left part of Figure 1.2) and C-RAN architecture (the right
part of Figure 1.2) in terms of resource pooling and BBU centralization.
5
BBU BBU
BBU
RRH
RRH
RRH
(a) Traditional RAN
F ronthaul
Netw ork
BBU P ool (Centr alized Data Center)
RRH
RRH
RRH
(b) C-RAN architecture
Figure 1.2: Traditional RAN to C-RAN architecture
As mentioned above, the proposed C-RAN architecture consists in centralizing
the baseband processing of antennas demands from several cell sites in BBU pools
when the RRHs are connected to the centralized data centers through fronthaul
network. Figure 1.3 illustrates C-RAN architecture focusing on three main compo-
nents:
•RRHs or antennas: are located at the cell sites and they forward the base-
band signals received from User Equipments (UEs) to the BBU pools for cen-
tralized processing in the uplink while transmitting Radio Frequency (RF)
signals to UEs in the downlink. RRHs perform radio functions including RF
conversion, amplification, filtering, analog-to-digital conversion and digital-to-
analog conversion [6].
•BBU pool or centralized data center: is a centralized location of com-
puting and processing resources shared among multiple cell sites. In fact, each
BBU pool can serve 10 to 1000 RRHs [7] and it consists in centralizing the
processing of baseband signals of antennas demands and then optimizing the
allocation of computing resources.
•Fronthaul network: is a set of communication links between RRHs and
BBU pool. The fronthaul traffic exchanged between antennas and centralized
data centers can be transmitted using typical protocols including OBSAI [8]
and CPRI [9], which is the most widely used in cellular networks. Fronthaul
network can be released by different technologies such as optical fiber commu-
nications, standard wireless communications, or millimeter Wave (mmWave)
communications [10].
6
Centraliz ed Data Center Centr alized Data Center Centr alized Data Center
F ronthaul
Netw ork
RRH
RRH
RRH
Figure 1.3: C-RAN architecture and components
1.1.2 C-RAN benefits
C-RAN architecture comes with many advantages (see for instance [11], [12], [13]
and [14]) that will be detailed in the following :
•Cost savings in CAPEX and OPEX: centralization of computational re-
sources in C-RAN enables an efficient utilization of BBU resources by reducing
the total number of BBUs needed to meet end-user demands. This leads to
achieve lower energy consumption and potential cost reductions in CAPEX
and OPEX. In [11], authors conclude that C-RAN reduces CAPEX by 15%
and OPEX by 50% when compared to a traditional mobile network.
•Capacity and coverage improvement: in C-RAN architecture, centralized
data centers (BBU pools) are expected to host more end-user demands coming
from different cell sites. In [7], authors claim that each BBU pool should
support 10 to 1000 base station sites which enables to improve the network
capacity by covering a larger area and serving more users than traditional base
stations when guaranteeing high QoS.
•Resource utilization gains: since in C-RAN baseband processing of mul-
tiple cells is carried out in centralized BBU pools, resource sharing becomes
feasible and hence the resource allocation can be more flexible and on demand
unlike traditional networks. This enables to improve the efficiency of network
resource utilization.
•Inter-cell interference reduction: in C-RAN, the centralized processing en-
ables easy implementation of joint processing and scheduling algorithms, which
7
help to reduce inter-cell interference and improve spectral efficiency. Indeed,
efficient interference management techniques such as Coordinated Multi-Point
(CoMP) [15] and Inter-cell Interference Coordination (ICIC) [16] can be easily
implemented in the BBU pool which help to optimize the transmission from
many cells to multiple BBUs.
•Network flexibility and extensibility: With the centralization of comput-
ing resources in a common location, network operators will quickly deploy new
antennas (RRHs) and connect them to the BBU pool to cover more service ar-
eas and make upgrades to their network infrastructures. This will improve the
network scalability and flexibility and will facilitate the network maintenance.
1.1.3 C-RAN challenges
Despite all the benefits (as detailed above) brought by C-RAN architecture, a
number of challenges need to be addressed to enable and facilitate the deployment
of C-RAN. In this section, we focus on the main challenges of C-RAN that will be
addressed in this manuscript.
•Strong latency expectations on fronthaul network : in C-RAN archi-
tecture, the fronthaul network that connects RRHs to BBU pools must carry
a significant amount of data traffic demands in real time with high bandwidth
and strong latency requirements. In fact, according to [11] and [17], the trans-
mission delay on a link between RRHs and the centralized data centers should
be kept below 1 millisecond to meet HARQ1requirements which impose that
the maximum distance between RRH and BBU pool must not exceed 20 to 40
kilometers [11].
The fronthaul capacity and delay constraints can be alleviated by flexibly
splitting the baseband processing between BBU pools and RRHs. The envi-
sioned solution consists in moving a part of baseband processing functions in
centralized data centers in order to reduce the increasing data throughput and
the overheads of the signal transmitted through fronthaul network. The eval-
uation of different options of BBU functions split in C-RAN will be detailed
in Section 2.3.3 of Chapter 2.
•New resource allocation algorithms: another key challenge of C-RAN
is the assignment of computing and radio resources shared among multiple
cell sites. Network operators should investigate new approaches to determine
the best strategies to assign heterogeneous antennas demands to the available
edge data centers when satisfying hard latency requirements. An optimal
assignment between RRHs and BBUs enables to achieve resource utilization
gains by reducing the number of edge data centers used to satisfy antennas
demands. This will also minimize the network costs including CAPEX and
OPEX. This exciting research challenge will be addressed in Chapter 3.
1HARQ (Hybrid Automatic Repeat reQuest) is the process that poses the most stringent delay
requirement for cellular networks
8
•Inter-cell interference reduction: in C-RAN architecture, each BBU pool
should be able to support 10 to 1000 RRHs [7] and provide optimal and joint
baseband processing of antennas demands. Moreover, network operators need
to increase the density of cells by deploying more antennas in order to meet
the growing number of end-users’ demands. This requires to investigate new
algorithms to reduce inter-cell interference while guaranteeing the full network
coverage. We will address this challenge in Chapter 4.
•BBU functions placement: the centralization of RAN functions in BBU
pools allows to capitalize on computational gain and improve mobile network
functions, e.g. scheduling and flow control. Nevertheless, the processing of
RAN functionalities in a centralized location increases the fronthaul data rate
demands and requires strict latency requirements. Therefore, there is a need
for addressing the trade-off between RAN functions centralization and trans-
port requirements by finding the optimal placement of baseband functions
in C-RAN architecture. We will investigate in Chapter 5 new algorithms to
determine the optimal locations of BBU functions under different transport
requirements.
•Virtualization: In addition to the centralization of baseband processing,
network virtualization is an important technique for the realization of C-RAN
architecture. In fact, virtualization decouples the computing resources from
the physical hardware and creates new virtual entities called Virtual Machines
(VMs) which are responsible for handling BBU functions in each centralized
data center (for more details, see [18] and [19]). Recent efforts investigated new
software solutions to implement virtualized RAN. In fact, authors in [20] and
[21] propose a new software platform called OpenAirInterface (OAI), which
is an open-source Software Defined Radio (SDR) implementation for 4G/5G
networks including both RAN and Evolved Packet Core (EPC) functionalities.
This challenge has been very well studied in the literature (see for instance
[22], [23] and [24]) and this is not in the scope of our work.
1.2 Contributions
This section highlights the main contributions of this thesis. Our contributions
are divided into three parts : in the first part, we propose new resource allocation
algorithms to efficiently assign antennas demands to centralized data centers un-
der strong latency expectations and limited computing resources’ constraints. The
optimal assignment will reduce the fronthaul throughput and maximize the compu-
tational gain. Such gains are strongly constrained by reducing the inter-cell inter-
ference levels that increase because of high density of cells. Thus, in second part, we
investigative an exact mathematical formulation to minimize inter-cell interference
when maintaining a full network coverage. In the third part, we focus on determin-
ing optimal locations of baseband functions based on different split configurations
and transport network characteristics.
9
1.2.1 Low-complexity algorithms for constrained resource
allocation problem in C-RAN
C-RAN is a novel mobile network architecture which consists in centralizing the
baseband processing in common edge data centers (sometimes referred to as Next
Generation-Point of Presence (NG-PoP)) and then sharing the computing resources
among different antennas (RRHs). This enables network operators to achieve effi-
cient network utilization and cost savings. However, such gains can only be achieved
by finding best strategies in the assignment of the edge data centers to the hetero-
geneous antennas demands while jointly reducing the resource utilization and the
communication latency on the fronthaul network between RRHs and edge data
centers. In this contribution, we propose a complete mathematical model based
on Integer Linear Programming (ILP) formulation to identify the most appropri-
ate strategies concerning antennas demands assignment to the available edge data
centers. The proposed ILP formulation optimizes jointly the latency (transmission
delay) on the fronthaul network and the resource consumption (expressed in terms
of active edge data centers). Our proposed optimization model provides optimal
solutions for small and medium problem instances. Thus, for larger problem in-
stances, we propose three approximation algorithms, based on exact theories and
approaches, that scale well, converge reasonably fast and provide good strategies of
RRH-BBU assignment :
•Matroid-based approach : we propose a new approximation algorithm
based on Matroid theory [25]. This approach models the constrained resource
allocation problem as a graphic matroid representation to find good strategies
to assign the edge data centers to the antennas demands. In fact, the objective
of this approach is to achieve an optimal assignment, when jointly minimizing
communication latency and resource consumption.
•b-Matching algorithm : we investigate an algorithm based on b-matching
approach [26] that aims to find the minimum weight matching between anten-
nas and edge data centers, with limited capacity of processing (CPU cores),
when satisfying the expected communication latency. The proposed b-matching
algorithm can reach optimal or near-optimal solutions for large network sizes.
•Multiple knapsack-based approach : another approximation algorithm
will be proposed using multiple knapsack formulation which is very used in the
literature to solve many variants of resource allocation problem (for instance
[27], [28], [29] and [30]). Hence, the proposed multiple knapsack formulation
will be used to evaluate the performance of the above heuristic algorithms.
A part of this work, in which we proposed an ILP formulation and matroid
algorithm, has been published in IEEE International Conference on Smart Com-
munications in Network Technologies 2018 (SaCoNet 2018) [31]. This publication
includes also a performance assessment of the proposed algorithms using different
simulation scenarios to quantify the scalability and the potential benefits of the
discussed approaches in the context of C-RAN. Next to that, this work has been
10
extended by proposing two other approximation approaches, b-matching formula-
tion and multiple knapsack-based algorithm, and a deep analysis to evaluate the
performance of the proposed algorithms in terms of efficiency, scalability and abil-
ity to find good solutions in acceptable times using simulations and a real 4G-LTE
network map. This part of work has been submitted in the International Journal of
Computer and Telecommunications Networking (Computer Networks) [32].
1.2.2 Mathematical programming approach for full network
coverage optimization in C-RAN
In C-RAN, network operators will increase the density of existing cells by de-
ploying more antennas in order to enhance the network capacity and coverage and
enlarge the network spectrum. However, cells densification comes with an increasing
of inter-cell interference which causes serious degradations of the provided networks’
QoS. Hence, network operators need new approaches to reduce inter-cell interference
and maintain a full network coverage jointly. Our contribution consists in proposing
a Branch-and-Cut algorithm to reach a good tradeoff between interference elimi-
nation/reduction and network coverage optimization in the context of C-RAN. In
fact, we propose a mathematical description modeling the problem according to the
RIPS approach based on simplicial homology [33] (more details can be found later
in Chapter 2). Our mathematical model describes the convex hull of the discussed
problem and allows to reach optimal solutions even for large problem instances. This
description is then enlarged by new valid inequalities and cutting planes to better
precise the polytope containing the optimal solution. This contribution based on
polyhedral approaches optimization is new and has never been addressed in the lit-
erature to cope with the full coverage hole problem in C-RAN. To reach the above
objectives, our contribution is described as follows :
•Minimize the number of coverage holes in the cellular network.
•Reduce or eliminate the inter-cell interference by adjusting the coverage radius
of antennas without creating coverage holes in the final network.
•Rapid (polynomial time) detection of coverage holes.
In addition, we provide a deep analysis of the performance of our Branch-and-Cut
algorithm using different simulation scenarios and a real network map to confront
our algorithm to different infrastructures and network topologies. This allows us to
evaluate the efficiency and reliability of our approach and the quality of the found
solutions through different performance evaluations and metrics.
This work has been published first in the IEEE Global Communications Con-
ference, GLOBECOM 2018 [34] and has been extended then to the International
Journal of Computer and Telecommunications Networking (Computer Networks)
[35].
11
1.2.3 Cost-efficient and scalable algorithms for BBU func-
tion split placement in C-RAN
As it was already mentioned, the main functionality of C-RAN consists in cen-
tralizing the baseband processing from multiple base stations into BBU pools (com-
mon data centers). This enables network operators to achieve many benefits in
terms of cost savings and capacity increasing. However, the deployment of C-RAN
requires very high capacity and low latency on the fronthaul links to connect anten-
nas (RRH) to the centralized data centers (BBUs). Thus, the challenge is to find
an optimal split of baseband processing between BBUs and RRHs in order to reach
good tradeoff between benefits of centralization and high transport requirements.
In this context, various functional splits have been proposed, each of which imposes
different throughput and delay requirements. This contribution considers 3GPP
RAN split [5] that is outlined as the best option in the 3GPP and it consists in
splitting the baseband functions into three components : i) PHY layer ii) MAC and
RLC layers and iii) PDCP layer. Accordingly and based on this split configuration,
we aim to propose new optimization algorithms to determine optimal locations of
baseband functions when considering strict transport requirements on the fronthaul
network in terms of latency.
We propose an exact approach based on ILP formulation to optimally deploy
BBU functions from multiple cells on the centralized data centers while jointly min-
imizing the network resource consumption and the end-to-end latency. The exact
approach provides the optimal solution for small and medium problem sizes. For
larger problem instances, we propose new heuristic algorithms based on the construc-
tion of an extended multi-stage graph to rapidly determine the optimal placement
of the baseband processing functions when jointly meeting their CPU and latency
requirements. These algorithms are benchmarked using different simulation scenar-
ios to evaluate the efficiency and scalability of our algorithms as well as their ability
to achieve optimal solutions in acceptable times.
This contribution has been published in IEEE Wireless Communications and
Networking Conference (WCNC 2018) [36].
1.3 Publications
The scientific publications during this thesis are summarized in the following:
•Journals
–N. Mharsi, M. Hadji, A mathematical programming approach for full
coverage hole optimization in Cloud Radio Access Networks, Computer
Networks, Volume 150, 2019, Pages 117 −126, ISSN 1389 −1286,
https://doi.org/10.1016/j.comnet.2018.12.015.
(http://www.sciencedirect.com/science/article/pii/S1389128618307928)
(Chapter 4)
12
–N. Mharsi, M. Hadji, Edge computing optimization for efficient RRH-
BBU assignment in Cloud Radio Access Networks, Computer Net-
works, submitted. (Chapter 3)
•International conferences
–N. Mharsi, M. Hadji, P. Martins Full Coverage Hole Optimization in
Cloud Radio Access Networks, Globecom 2018 : IEEE Global Com-
munications Conference, December 2018 Abu Dhabi, UAE. (Chapter
4)
–N. Mharsi, M. Hadji, Joint Optimization of Communication Latency and
Resource Allocation in Cloud Radio Access Networks, SacoNeT 2018
: IEEE International Conference on Smart Communications in
Network Technologies, October 2018 Algeria. (Chapter 3)
–N. Mharsi, M. Hadji, D. Niyato, W. Diego, R. Krishnaswamy, Scalable
and Cost-Efficient Algorithms for Baseband Unit (BBU) Function Split
Placement, WCNC 2018 : IEEE Wireless Communications and
Networking Conference, April 2018 Barcelona, Spain. (Chapter 5)
•Posters
–SDN DAYS 2017 and 2018 (Paris, France): Scalable and Cost-Efficient
Algorithms for Resource Allocation Problems in C-RAN. Poster and
Talk.
–CLOUD DAYS’2017 (Nancy, France): Resource allocation and BBU split
placement in Cloud Radio Access Networks Poster and Talk.
Chapter 2
C-RAN optimization:
mathematical background and
state-of-the-art
2.1 Introduction
Network operators are investigating new algorithms for resource allocation prob-
lems to enable and facilitate the deployment of C-RAN architecture. Combinatorial
optimization is considered as one of the most efficient techniques to address such
problems. In the first section of this chapter, we describe some combinatorial opti-
mization concepts and algorithms that we use to propose exact and heuristic algo-
rithms to optimally address resource allocation and network optimization problems
in the context of C-RAN. We introduce also the homology theory which provides
efficient algorithms to deal with coverage hole detection problems. In the second
section, we provide a deep analysis of the most relevant approaches in the literature
which have been proposed to address resource allocation problems in C-RAN.
2.2 Mathematical background
This section contains an overview of the most basic concepts that will be used in
the following chapters. We address some fundamental concepts concerning network
optimization with a focus on combinatorial optimization background and simplicial
homology approaches.
2.2.1 Combinatorial optimization : techniques and algo-
rithms
Combinatorial optimization techniques are very useful in modeling several types
of problems such as planning, routing, scheduling, assignment, and design that ap-
pear in many real life applications. Most of these problems are very complex and thus
13
14
very hard to solve. In this section, we introduce some combinatorial optimization
techniques and algorithms that we will use in this manuscript. We introduce linear
programming and different methods used to solve optimization problems. Then, a
class of approximation algorithms is presented to tackle scalability issues of some
optimization problems. Finally, we describe some classical optimization problems
and outline the most efficient algorithms proposed to optimally solve them.
2.2.1.1 Linear programming
Linear Programming (LP) is a powerful tool to model optimization problems by
means of mathematical relations, allowing to find the best solution from all possible
ones. In fact, LP is a mathematical formulation where problem decisions are repre-
sented by decision variables and problem constraints are expressed by mathematical
relations that describe conditions imposing the feasibility of the solutions. The aim
of LP is to find the best values of decision variables that satisfy all problem con-
straints and maximize (or minimize) an objective function. LP is a mathematical
programming model in which the objective function is a linear expression of the
decision variables and the constraints are given by a system of linear inequalities.
Each LP can be represented by the following standard form:
opt cTx
Subject to :Ax ≤b
x≥0
(2.1)
•x: vector of decision variables which are the quantities to be determined
in order to solve the problem.
•opt cTx:objective function is a function of decision variables that aims to
maximize or minimize some numerical value. This value can represent profit,
cost, revenue, distance, etc.
•Ax ≤b:constraints represent some mathematical relations to be respected
by the final solution which are expressed by linear inequalities.
•cT,Aand b: are respectively matrix transpose of coefficients, matrix of coef-
ficients and vector of coefficients.
Decision variables in the above LP (2.1) are continuous (real values). However,
in some cases, these variables could be integer. This leads to two other variants of
LP :
•Mixed Integer Linear Program (MILP) if only some of decision variables are
integer while some others can take real/continuous values.
15
•Integer Linear Program (ILP) if all decision variables are integer. This variant
is very well known and often used to model optimization problems that repre-
sent the integrity conditions of various real-life problems. The ILP formulation
can be represented as follows :
opt cTx
Subject to :Ax ≤b
x∈ {0,1}
(2.2)
Before providing an optimal solution, solving an ILP of (2.2) consists in exploring
all possible solutions that satisfy all problem constrains. These solutions represent
the set of feasible solutions and allow to constitute a convex polytope which is
the region obtained by the intersection of incident vectors, each of which represents
a constraint in the problem (as shown in Figure 2.1).
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•feasible
solution
optimal
solution
Figure 2.1: An example of a convex polytope
ILP models have many advantages, including :
•Solve optimization problems when guaranteeing optimum solution.
•Modeling very complex optimization problems by a simple mathematical for-
mulation where all problem constraints are represented by linear inequalities
and the goal of decision making is measured by cost (objective) function.
•A wide range of problems in real life can be modeled and solved easily using
different LP models.
In the following, we focus on ILP models that we mainly use to formulate and
solve the different addressed problems in this thesis.
2.2.1.2 Solving ILP : methods
Without loss of generalities, the simplest way to find an optimal solution for
an optimization problem is to explore and evaluate all possible solutions and then
select the best one. In this section, we introduce three methods that are used to
solve an ILP : (a) Branch-and-Bound, (b) cutting plane, (c) Branch-and-Cut.
16
(a) Branch-and-Bound
Branch-and-Bound is the simplest approach to obtain an optimum so-
lution of an ILP problem by exploring a complete enumeration of all possi-
ble solutions. Branch-and-Bound algorithm is based on the construction of
branching tree in which the ”root” node represents the original problem and
each ”son” node corresponds to a subproblem. This algorithm splits recur-
sively each problem into two sub problems, explores different branches of the
tree and calculates the solution value of the corresponding subproblem at each
node. The main two steps of Branch-and-Bound algorithm are :
•Branch: is the process of creating new subproblems from an original
problem. In fact, each problem (or subproblem) is partitioned into two
subproblems such that the union of feasible solutions of the subproblems
represents the feasible solutions of the original problem. This step is
executed recursively.
•Bound: is the process of evaluating each node. In fact, for each node, we
calculate the solution value obtained by the corresponding subproblem.
We stop branching this node if the obtained value of the corresponding
problem is above (in the case of minimizing problem) the best feasible
solution found so far.
Branch-and-Bound algorithm is an exact method allowing to find the opti-
mum solution. However, the number of nodes in the branching tree exponen-
tially increases with problem size. Therefore, to reduce the number of nodes
in the tree, some techniques such cutting plane are used to accelerate the
convergence time to solve the problem.
(b) Cutting plane
Cutting plane method is a technique used to accelerate the search of optimal
solutions for an ILP formulation. This approach considers the LP relaxation,
where the integrity constraints x∈ {0,1}are replaced by x∈R+. Cutting
plane method consists in solving the LP relaxation and verifying if the obtained
solution contains some fractional variables or not. If at least one variable is
fractional, new valid inequalities, that are violated by these fractional vari-
ables, are investigated and then added to the current LP relaxation. These
valid inequalities are called cutting planes.
The cutting plane method is always used with Branch-and-Bound algorithm
allowing to reduce the space solution (convex polytope) of the optimization
problem and thus find optimal solutions faster. The most well-known cutting
plane algorithm is the Gomory Cut algorithm [37] which consists in finding the
constraint to separate a fractional solution to any linear relaxation. Gomory
algorithm is extended then by Gomory-Chv˘atal algorithm.
Nevertheless, finding new valid inequalities violated by the obtained solution
(i.e. cutting planes) is not often very easy. In fact, the problem of finding such
inequalities is called separation problem which can be defined as follows:
17
Given a solution x∗∈Rnlying outside of the polytope P, a separation
problem consists in finding a linear inequality ax ≤αwhich is valid for the
polytope and violated by the solution x∗(see Figure 2.2).
P
ax ≤α
x∗
Figure 2.2: An example of linear inequality separating the convex polytope Pand the
solution x∗
(c) Branch-and-Cut
Branch-and-Cut is a method that combines Branch-and-Bound algorithm
with cutting plane technique to solve ILP models. In fact, Branch-and-Cut
considers the LP relaxation, where the integrity constraints of the ILP for-
mulation (2.2) are relaxed (the constraints x∈ {0,1}are replaced by x∈R+).
The LP relaxation is then solved and a lower (resp. upper) bound, in the
case of minimization (resp. maximization) problem, is found. If the obtained
solution contains fractional variables, new cutting planes are investigated (as
described above) and then added to the current LP relaxation. The whole
process is iterated until all variables in the obtained solution are integers. We
summarize in Algorithm 1 the Branch-and-Cut method.
2.2.1.3 Approximation algorithms
In addition to exact approaches based on ILP formulation, we will propose, later
in this thesis, new approximation algorithms based on exact theories and approaches
such as matroid and b-matching. The proposed heuristic algorithms can converge
faster and scale to larger problem instances.
(a) Matroid
Matroid is a discrete structure that generalizes the concept of independence
in linear algebra. There are several ways to define a matroid such as bases,
the rank function, independent sets and cycles.
In the following, we will use the definition based on independent sets. We
recall that in graph theory, an independent set is a set of edges in a graph that
do not contain a cycle[38]. The definition of a matroid is provided by [39] and
can be formulated as follows.
18
Algorithm 1 Branch-and-Cut algorithm
Input: An ILP formulation (2.2), denoted by ILP 0
Output: An optimum solution S∗
Set L:= {I LP 0}, set x:= −∞, and set x∗:= +∞
for each ILP lin Ldo
Step 1. Solve the LP relaxation of I LP land calculate the value of obtained
solution x
Step 2.
if x<x∗then
Update the value of the best known solution : x∗:= x
else
Cut the node which contains the current problem and go to Step 1
end if
Step 3.
if The obtained solution contains some fractional variables then
Find the cutting planes that are violated by the obtained solution (some may
be sufficient)
Add the violated inequalities to the current LP relaxation
Branch the current problem into two subproblems and put them into L
end if
end for
19
Definition A matroid M= (E, F) is a structure in which Eis a finite set
of elements and Fis a family of subsets of Everifying the following principal
properties:
(P1) ∅ ∈ F.
(P2) If A∈Fand B⊆A, then B∈F.
(P3) If A, B ∈F, and |B|>|A|thus ∃e∈B\A, such that A∪ {e} ∈ F.
If Fis only satisfying the properties (P1) and (P2), then we are invoking
an independent system. A maximal set of Eis said to be basis of matroid
and the rank r(A) of a subset of Ais the cardinality of a maximal independent
subset A. We note that all basis of a matroid have the same cardinality.
Other definitions and more details on matroid theory can be found in [25],
[40] and [41], for instance.
There are many examples of matroids such as uniform matroid, linear ma-
troid, graphic matroid. In the following, we introduce the graphic matroid
that we will use to efficiently solve resource allocation problems.
Graphic matroid (also known as cycle matroids of a graph) : is a matroid
whose independent sets are the forests in a given graph and can be represented
as follows:
Given a graph G= (V, E), a graphic matroid is M= (E, F), in which Fis a set
of trees’ forests in G:F={F⊆E|GM= (V, F )is the induced subgraph
of G such that F is a f orest}
In the case of the graphic matroid, the problem of finding the minimal
(maximal) forest can be optimally solved by a simple greedy algorithm (see [40]
and [42] for more details). This algorithm will be used with some modifications
when involving resource allocation problem in Chapter 3.
(b) b-Matching
Let G= (V, E ) be an undirected graph with edge weights u(e) for each edge
e∈Eand node capacities b(v) for each node v∈V. We denote by δ(v) the set
of incident edges of v. A b-matching is a generalization of ordinary matching
where all node capacities b(v) are equal to one. We note that a matching in
Gis a set of pairwise disjoint edges (i.e. the endpoints are all different) and
it is called a perfect matching if all vertices are covered. In the following, we
introduce the definition of capacitated, i.e. the edge weights u(e) are finite
numbers, b-matching provided by [39].
Definition Let Gbe an undirected graph with integral edge capacities u:
E→N∪ {∞} and numbers b:V→N. Then a b-matching in (G, u) is a
function f:E→Z+with f(e)≤u(e) for all e∈Eand Pe∈δ(v)f(e)≤b(v)
for all v∈V. In the case u= 1 we speak of a simple b-matching in G. A
b-matching is called perfect if Pe∈δ(v)f(e) = b(v) for all v∈V.
20
The b-matching polytope of (G, u) is the set of vectors x∈RE
+satisfying :
xe≤u(e),∀e∈E
X
e∈δ(v)
xe≤b(v),∀v∈V
X
e∈E(G[X])
xe+X
e∈F
xe≤ b1
2(X
v∈X
b(v) + X
e∈F
u(e))c,
∀X⊆V(G), F ⊆δ(X)
(2.3a)
(2.3b)
(2.3c)
where E(G(X)) represents a subset of edges in the subgraph G(X) generated
by a subset of vertices Xand δ(X) is a set of incident edges of X.
Constraints (2.3c) represent the blossom inequalities which are very used to
model several NP-Hard optimization problems and find their convex hulls. In
[43], authors gave a polynomial-time separation algorithm for b-matching poly-
topes by identifying the violated blossom inequalities. In [44], authors proved
that the separation problem for b-matching polytope (in the case of capaci-
tated b-matching) can be solved in polynomial time : O|n|2|m|ln (|n|2
|m|)
where nand mare the number of vertices and edges respectively in the graph
G.
2.2.1.4 Combinatorial optimization problems : some examples
We present in the following some combinatorial optimization problems that will
be used in this manuscript.
(a) Shortest path problem
One of the well-known combinatorial optimization problems is the problem
of finding a shortest path between two specified vertices in a graph Gsuch
that the total sum of the edges’ weights is minimum. Gcan be directed or
undirected graph and can contain negative weights. The shortest path problem
can be solved in polynomial time if there are no negative cycles in G[45].
There are three variants of shortest path problem :
•Two nodes shortest paths: given two nodes sand t, find the shortest path
between sand t
•Single source shortest paths : given a node s, compute the shortest paths
from a node sto all vertices in the graph.
•All pairs shortest paths: find a shortest path for all ordered pairs of
vertices (s, t) in the graph.
Many algorithms have been proposed to solve these variants of shortest
path problem. One of the most used algorithms is Dijkstra algorithm [46]
which consists in finding shortest paths in a graph (directed or undirected)
21
from a source sto all other nodes, instead of just a specific pair of source and
destination nodes. This is very useful in telecommunication networks where we
always need to compute the shortest path from the source to all destinations.
We note that the complexity of this algorithm is O(n2)where nis the number
of vertices in G.
Unlike Dijkstra algorithm that can be applied on a graph with only non-
negative edge weight values, Bellman-Ford algorithm [47] was proposed to find
shortest paths in a graph which may contain negative weights on edges. For
further reading about shortest path algorithms, see [48] for instance.
(b) Maximum flow problem
Let G= (V, E ) be a directed graph with edge capacities c:E→R+and two
specified vertices, ssource and tsink. The triple (G, s, t) is called a network.
The problem of maximum flow consists in transporting simultaneously as many
units as possible from sto tand can be defined as follows.
Definition A network flow of (G, s, t) is a function f:E→R+which satisfies
the following proprieties:
•f(x, y) = −f(y, x),∀(x, y)∈E
•f(x, y)≤c(x, y),∀(x, y)∈Ewhere c(x, y) is the capacity on the edge
(x, y)
•Py∈Vf(x, y)=0,∀x∈V\{s, t}
The problem of finding a maximum flow in Gcan be solved in polynomial
time [49] and many algorithms have been proposed such as Ford-Fulkerson
algorithm [46] which has a complexity of O(|m|2|n|)where nand mare the
number of vertices and arcs respectively in G.
(c) Knapsack problem and multiple knapsack formulation
Another well-known combinatorial optimization problem is the knapsack
problem which consists in finding an optimum subset from a set of items to
be filled into a knapsack with limited capacity. In fact, given a knapsack with
a maximal capacity cand a set of items j, each of which has a profit pjand
a weight wj. Knapsack problem aims to find a subset of items such that the
total profit of the selected items is maximized and the total weight does not
exceed the capacity of knapsack C. Alternatively, knapsack problem can be
formulated by the following ILP :
max Pn
j=1 pjxj
S.T. :
Pn
j=1 wjxj≤C;
xj∈ {0,1},∀j= 1, ..., n;
(2.4)
22
The knapsack problem is NP-Complete [50] but there is a pseudopoly-
nomial algorithm which can be quite efficient if the involved numbers (pjand
wj) are not too large [51].
In the following, we provide a knapsack algorithm based on dynamic pro-
gramming approach [52] which can find optimal solutions in O(nP ) time, where
nis the number of items and P=Pn
j=1 pj[51]. The algorithm 2 will be used
in Chapter 3 when involving a resource allocation problem.
Algorithm 2 Dynamic programming algorithm for knapsack problem
Input: Non-negative integers : nitems, a profit pjand a weight wjfor each item
j, a knapsack with capacity C
Output: An optimal subset of items to be filled into the knapsack
Step 1. Let Pbe an upper bound on the value of the optimum solution : P=
Pn
j=1 pj
Step 2. Set x(0,0) := 0 and x(0, k) := ∞for k= 1, ..., P
Step 3.
for j:= 1 to ndo
for k:= 0 to Pdo
Set s(j, k) := 0 and x(j, k) := x(j−1, k)
end for
for k:= pjto Pdo
if x(j−1, k −pj) + wj≤min{C, x(j, k)}then
Set x(j, k) := x(j−1, k −pj) + wjand s(j, k) := 1
end if
end for
end for
Step 4. Let k= max{i∈ {0, ..., P }:x(n, i)≤ ∞}, Set S:= ∅
for j:= ndown to 1 do
if s(j, k)=1then
Set S:= S∪ {j}and k:= k−pj
end if
end for
We introduce in the following the definition of multiple knapsack formula-
tion according to [30] that we use to propose new resource allocation algorithm
in Chapter 4.
Definition Given a set of nitems and a set of mknapsacks (m<n), with
pj= profit of item j,wj= weight of item j,ci= capacity of knapsack i,
find mdisjoint subsets of items with the total profit of the selected items is
a maximum, and each subset can be assigned to different knapsacks whose
capacity is less than the total weight of items on the subset.
23
The multiple knapsack problem is a generalization of the knapsack problem
from a single knapsack to mknapsacks, each of which has limited capacity.
The objective of multiple knapsack problem is to assign each item to at most
one of the knapsacks such that none of the capacity constraints are violated
and the total profit of the items putted into knapsacks is maximized.
In addition to combinatorial optimization techniques that we mainly use to pro-
pose new algorithms in this thesis, we briefly introduce in the following the homology
theory that provides powerful solutions to address network coverage hole detection
problems in Wireless Sensor Networks (WSN). In fact, we focus on two simplicial
homology-based approaches [33] : ˘
Cech complex and Rips complex, which are
very close to our proposal to model the full network coverage problem in the context
of C-RAN. Furthermore, we will use these approaches to benchmark our proposed
algorithm (see Chapter 4).
2.2.2 Simplicial homology background
One of the most efficient approaches to deal with coverage hole detection problem
is simplicial homology [33]. In fact, homology based approaches consist in analyzing
the topological properties of a domain or region by algebraic computations. Instead
of graphs, more generic objects are used, known as simplicial complexes. In this
section, we introduce the definition of a simplicial complex and we describe two
most useful approaches based on simplicial homology : ˘
Cech complex and Rips
complex (see [33] for more details about both approaches). These approaches will
be used in Chapter 4 to evaluate the performance of our proposed algorithm when
dealing with the full network coverage problem in C-RAN. For further details about
simplicial homology theory, see [53] for instance.
2.2.2.1 Simplicial complex and k-simplex
A simplicial complex is a combinatorial object composed by vertices, edges, tri-
angles, tetrahedra, and their n−dimensional counterparts, each of which represents
k−simplex. k−simplex is an unordered subset of k+ 1 vertices, where kis the
dimension of the simplex.
v0
0−simplex
v0
v1
1−simplex
v0
v1
v2
2−simplex
Figure 2.3: Example of k-simplices
24
v0v1
v2
v3
Figure 2.4: Example of simplicial complex
In fact, we show in Figure 2.3 some examples of k-simplex : 0-simplex is a
vertex, 1-simplex is an edge, 2-simplex is a triangle. To better understand these no-
tions, we present, in Figure 2.4, a simple example of a simplicial complex that con-
tains four 0-simplices : {v0},{v1},{v2}and {v3}, five 1-simplices {v0, v1},{v0, v3},
{v1, v2},{v1, v3}and {v2, v3}and one 2-simplex : {v0, v1, v3}while {v1, v2, v3}is not
2-simplex.
Homology techniques are very well used to model the topology of wireless sensor
networks and detect the existence of coverage holes. In the following, we introduce
two approaches, ˘
Cech complex and Rips complex, which use the simplicial homology
to verify the connectivity of the network and detect the existence of coverage holes.
In fact, for these approaches, the construction of simplicial complex enables to detect
and count the number of k−dimensional holes, called Betti numbers. Indeed, the
k−th Betti number, denoted by βk(for k−dimensional hole), counts the number of
cycles of k−simplices which are not filled with (k+ 1)−simplices. For connectivity
and coverage problems, only the first two Betti numbers are considered :
•β0: indicates the number of 1−dimensional holes, that is the number of con-
nected components in the simplicial complex (for example, β0= 1 in simplicial
complex in Figure 2.4).
•β1: indicates the number of 2−dimensional holes, that is the number of holes
in the simplicial complex (for example, β1= 1 in the simplicial complex of
Figure 2.4) .
2.2.2.2 ˘
Cech complex and Rips complex
We focus in this section on the two most useful approaches based on simplicial
homolgy, which are ˘
Cech complex and Rips complex. These approaches are based
on the verification of intersection between cells, to detect holes and connectivity
problems. ˘
Cech complex is introduced in [33] and can be defined as follows :
Theorem 2.2.1 ˘
Cech complex of a set of points U,˘
C(U), is the abstract simplicial
complex whose k−simplices correspond to nonempty intersections of k+ 1 distinct
elements of U, i.e. [v0, v1, ..., vk]is k−simplex if and only if (v0)∩(v1)∩...∩(vk)6=∅,
Where (vi)is the cell centered at vi.
25
Unfortunately, it is very difficult to compute ˘
Cech complex due to its high com-
plexity. Hence, another approach based on simplicial complex, named Rips complex,
is introduced to deal more easily with coverage and connectivity problems. It is de-
fined as follows.
Theorem 2.2.2 Rips complex of a set of points U,R(U), is the abstract simplicial
complex whose k−simplices correspond to unordered (k+ 1)−tuples of points in U
which pairwise intersect, i.e. [v0, v1, ..., vk]is k−simplex if and only if (vi)∩(vj)6=∅,
Where (vi)is the cell centered at vi.
Comparison between ˘
Cech complex and Rips complex
According to theorem 2.2.1, ˘
Cech complex provides the exact topology of the
network allowing to verify the connectivity of the network (β0) and compute the
number of existing holes (β1). However, it is very difficult to compute the ˘
Cech
complex due to the high complexity of determining k-simplices. Rips complex can be
constructed based on the connectivity graph of the network and gives an approximate
coverage by simple algebraic calculations. In [54], authors prove that Rips complex
provides a solution with an error between 0.5% and 3%. Hence, in Chapter 4, we will
use the Rips complex-based approach to model the full network coverage problem
in the context of C-RAN and to benchmark our proposed algorithm using different
simulation scenarios and performance metrics.
2.3 C-RAN optimization : state-of-the-art
Combinatorial optimization techniques and simplicial homology theory are the
main techniques that we will use to address three exciting research challenges in
the context of C-RAN. These challenges will be detailed in this chapter with a deep
analysis of the state-of-the-art on the most relevant researches in the literature.
2.3.1 Constrained resource allocation problem
The deployment of C-RAN architecture, where the infrastructure is shared across
multiple cell sites, is expected to reduce both capital and operating expenditures
(CAPEX and OPEX) as well as to improve the resource utilization efficiency [55].
Such gains can only be achieved by efficiently assigning the antennas (RRHs) de-
mands to the centralized data centers (BBU pool) when latency and processing
requirements are met. Therefore, to address this constrained resource allocation
problem, network operators are investigating new algorithms to determine the best
strategies to assign RRHs to BBUs (known as RRH-BBU assignment problem). The
proposed algorithms will jointly assign the processing and radio resources to anten-
nas demands taking advantage of the computing resource pooling in common edge
data centers. The optimal mapping between RRHs and BBUs, i.e. optimal RRH-
BBU assignment, is reached when jointly minimizing the communication latency on
the fronthaul network and computing resource consumption.
26
In this context, authors in [56] and [57] discussed new mathematical modeling to
cope with RRH-BBU assignment problem. They proposed a mathematical model
based on an ILP approach in which only BBUs processing capacity constraints are
considered. The proposed exact optimization model does not take into account
the transmission delay on the fronthaul network and the latency requirements of
antennas demands. To cope with scalability issues, both these references proposed
approximation algorithms that do not guarantee the convergence to an optimal
solution. In this thesis, we address the RRH-BBU assignment problem when jointly
meeting the strong latency requirements on fronthaul network and the edge data
centers’ limited capacity constraints. Our joint optimization is represented by an
exact formulation before investigating heuristic algorithms that converge to near-
optimal solutions in acceptable times.
Authors of reference [58] proposed a load-aware dynamic mapping between RRHs
and BBUs with the aim of minimizing the number of active BBUs required to pro-
cess the computational resource demands. The authors introduced a heuristic DRA
for Dynamic RRH Assignment to dynamically optimize the BBU pooling gain. They
claim that their approach delivers an almost optimal performance in terms of com-
putational resource gain and convergence time as compared to First-Fit Decreasing
(FFD)1algorithm. Similarly, another resource allocation algorithm was introduced
in [59] to minimize the number of active BBUs required to serve all users in the
network to save more energy. In this manuscript, and in addition to the proposed
ILP algorithm used as reference to benchmark other approaches, we propose three
heuristic approaches to guarantee the convergence of the constrained resource allo-
cation problem to optimal solutions in negligible times.
Another work addressing the RRH-BBU assignment and resource allocation
problem is proposed in [60]. Indeed, the authors of this reference proposed a greedy
algorithm to assign the aggregated demands of each cell to the BBU pool in such
a way that the power consumption of the physical resources is minimized. The au-
thors did not consider the latency requirements in their optimization model. Since
the latency and the transmission delay constraints are very strong in C-RAN archi-
tecture, we propose exact and heuristic algorithms based on a joint optimization of
communication latency and computing resource allocation.
In [61], the authors introduced a mathematical formulation based on ILP to
optimally assign antennas demands to different BBU pools. This work aims to
minimize the length of fiber while maximizing the statistical multiplexing gain for
each BBU pool hosting the baseband functions. Their approach shows that the
optimal assignment of RRHs to the BBU pools depends on the length of fiber and
BBU resources. In our work, we propose an exact formulation for the same problem
and to scale, our contribution consists in investigating new and rapid approaches
to guarantee the convergence to near optimal solutions when considering the same
parameters than those used in [61].
1FFD sorts all items in decreasing order of their sizes, and then puts each item into the first
bin that has sufficient remaining space.
27
Authors in [62] investigated new algorithms to determine the best strategies
for RRH-BBU mapping by finding the optimal clustering of existing RRHs. They
modeled as bin packing problem when considering two main constraints (i) the
radio resources of each active BBU must be enough to meet the demands of its
mapped RRHs and (ii) the set of antennas, that will be assigned to each BBU,
should be geographically adjacent. Exact and heuristic algorithms are provided
to reduce network power consumption when guaranteeing good QoS for end-users.
Nevertheless, the proposed formulation did not consider the communication latency
on the fronthaul network joining RRHs to BBU pools. In this manuscript, we
address the RRH-BBU mapping problem by proposing an exact approach based on
ILP model and approximation algorithms to find the best assignment of antennas
to centralized data centers when jointly considering the limited processing capacity
in BBU pools and the transmission delay on fronthaul links.
Similarly to [62], authors in [63] formulated the problem of RRH-BBU assignment
as a bin packing problem. In fact, after proposing an ILP model to address this
problem, authors in [63] used a simple Best Fit Decreasing (BFD)2algorithm to
assign RRHs to BBUs and then determine the number of active BBUs that should
be used to meet antenna demands (BFD is a well-know algorithm developed by [64]
to solve bin packing problem). In our work, in addition to an exact approach based
on ILP formulation, we propose new approximation algorithms to find near-optimal
solutions to deal with RRH-BBU assignment problem in acceptable times. These
algorithms will be benchmarked with the exact approach using different simulation
parameters and according to many performance metrics.
Some existing works (for instance [65], [66] and [67]) addressed the resource allo-
cation problem in C-RAN by only focusing on minimizing the energy consumption
in the BBU pool without taking into account the fronthaul latency constraints. In
this manuscript, we seek new algorithms to reduce the network costs, i.e. CAPEX
and OPEX, by jointly optimizing the resource consumption and the communication
latency in order to achieve optimal utilization of processing resources.
2.3.2 Network coverage optimization problem
In the context of C-RAN architecture, network operators are seeking to increase
the density of existing cells by deploying more antennas in order to enlarge the net-
work spectrum and enhance the network coverage and capacity [68]. Nevertheless,
this brings new challenges in inter-cell interference management and reduction when
maintaining a full network coverage. Indeed, to jointly cope with these goals, nu-
merous schemes have been proposed in different networks, such as Wireless Sensor
Networks (WSN), using different approaches.
One of the most used methods to detect coverage holes in WSNs is Delaunay
triangulation which consists in dividing the target field into triangles that have
2BFD sorts the items to be inserted in decreasing order of size, and puts each item into the
fullest bin in which is fits.
28
no other nodes inside. In Figure 2.5, we represent an example of graph based on
Delaunay triangulation method.
Figure 2.5: Delaunay triangulation. Source: [2]
Authors in [2] proposed a mathematical model based on Delaunay triangulation
to detect coverage holes in WSN and found the shortest paths for node movement
to heal the holes. The proposed algorithm used the Delaunay triangulation to find
a necessary and sufficient condition to determine the coverage of a triangle that
has no other nodes inside its circumcircle. The time complexity of this method is
close to O(bn) where nis the total number of nodes and brepresents the number
of adjacent nodes in the vicinity of each node. However, this proposal is based on
a simple mathematical formula that does not address all of the possible scenarios.
Furthermore, the proposed algorithm requires that cells should be identical with the
same coverage radius. In our considered C-RAN architecture, the antennas (cells)
have different coverage radius, depending on number of users supported by this
antenna. Therefore, we need to investigate new approaches to optimize the network
coverage when considering different cells’ radii.
Other works addressed the coverage problem using probability methods. In [69]
and [70], the authors used a probability method to eliminate all coverage holes de-
tected in the considered network by determining the smallest size of cells. Similarly
to [2], authors of these papers supposed that all cells have the same coverage radius
which is not realistic in the context of C-RAN. In our work, we propose new cutting
planes in Branch-and-Cut algorithm to rapidly detect coverage holes and reduce
inter-cell interference jointly.
Authors in [71] proposed an ILP model that maximizes the total coverage in the
context of WSN. The main limitation is that they discretize the area to be covered
into several cells, and each cell is discretized into several points. In the context of
C-RAN, we consider distributed antennas with different coverage radius and we aim
to optimize the total network coverage by adjusting the antennas radii.
Other works used simplicial homology (detailed in Section 2.2.2) to address the
coverage hole detection problem. In [72], the authors proposed an heuristic algo-
rithm to turn off the minimal number of cells without generating coverage holes. In
29
fact, authors used the simplicial complex which is introduced in [33] and [73] as a
representation of coverage topology. In this algorithm, at every time a cell is turned-
off, we need to compute the Betti numbers (β0and β1as defined in Section 2.2.2) to
ensure that the network coverage is maintained. However, turning off cells can not
optimize neither network coverage nor overlapping region. In this manuscript, we
propose an exact mathematical formulation to provide optimal solutions for network
coverage problem.
Similarly, another algorithm based on simplicial homology has been introduced
in [74]. This algorithm aims to minimize the total consumed power for wireless
networks. They used a heuristic approach based on Simulated Annealing (SA) to
find sub-optimal solutions instead of investigating rapid and efficient approaches to
attend optimal solutions. The SA algorithm adjusts the coverage radius of each cell
by building a complex graph based on RIPS complex and then computes the Betti
numbers to avoid the generation of coverage holes. In this manuscript, we propose
an optimization model considering multiple objectives such as inter-cell interference
reduction and network coverage holes elimination.
Other works tackled the network coverage problem in the context of C-RAN
when addressing multi-dimensional resource optimization issues. In fact, authors of
[75] proposed a Cloud-Based Radio over Optical Fiber Network (noted by C-RoFN)
architecture with multi-stratum resources optimization using Software Defined Net-
works (SDN) paradigm to better get a grasp of resource optimization problems
for C-RAN architecture. In the proposed architecture, optical spectrum and BBU
processing resources are optimized jointly to maximize radio coverage when meet-
ing quality of service. Authors in [76] provided a deep study on multi-dimensional
resources integration for service provisioning in cloud radio over fiber networks.
Indeed, they proposed a global optimization when considering together radio fre-
quency, optical network and processing resources leading to maximize radio cover-
age. A mathematical modeling is then provided and an experimental test bed is used
to confirm the efficiency and the feasibility of the proposed C-RoFN architecture.
2.3.3 BBU functions split and placement problem
The key obstacle in the adoption of C-RAN architecture, where the comput-
ing resources (BBUs) are decoupled from cell sites (RRHs), is the high-bandwidth
constraints and low-latency requirements on the fronthaul network which connects
the antennas (RRHs) to the edge data centers (BBU pools). In fact, authors in [7]
investigate the cloudification of RAN by characterizing baseband processing times
under different conditions. They underline that C-RAN architecture should take into
account the fronthaul capacity constraints, the latency requirements for baseband
processing and finally the execution environment (servers and operating systems).
The strong latency requirements on the fronthaul network could be reduced by split-
ting the processing of baseband functions between RRHs and BBUs. In this context,
several functional split options have been proposed by different organizations, e.g.
NGMN [77] and 3GPP [78], in order to find good trade-off between BBU functions
30
centralization and fronthaul network requirements [77].
Figure 2.6: 3GPP functional split options
The 3GPP split proposal is shown in Figure 2.6, in which each functional split
option represents a separation point between the layers that will be located in the
cell sites (RRHs) and those located in the BBU Pools (centralized data centers). We
recall that the main motivation to split the processing of BBU functions between
RRH and BBU is to achieve the largest possible extent of centralization that a
network architecture can allow.
Among the different functional split options depicted in Figure 2.6, our study
focuses on the following four options which are illustrated in Figure 2.7 for a better
understanding.
•Split 1 represents the traditional RAN architecture where PHY, RLC, MAC
and PDCP functions are co-located in the cell sites (option 1 in Figure 2.6).
•Split 2 is a partial centralization of BBU functions where all functions in
PHY, MAC and RLC layers are located in the cell sites and PDCP layer in
the centralized data center (Option 2 in Figure 2.6).
•Split 3 is a partial centralization where PHY layer are located in RRHs and
all functions in MAC, RLC and PDCP layers are incorporated in BBU pool
(Option 6 in Figure 2.6).
•Split 4 represents fully-centralized architecture in which PHY, MAC , RLC
and PDCP functions are moved to the BBU pools for centralized processing
(Option 8 in Figure 2.6).
In the split 1, all functions in PHY, RLC, MAC and PDCP layers are located in
the cell sites (RRH). Consequently, the latency constraints are less stringent which
decreases its dependency on expensive fronthaul technology. However, benefits of
virtualization and centralized multi-cell processing will be less. Contrary to split 1,
split 4 represents the fully-centralized architecture, considered in the first C-RAN
configuration, in which all functions in PHY, MAC, RLC and PDCP layers are
moved to the BBU pool for centralized processing. Authors in [13] showed that the
main benefit of this BBU function split option is its flexibility to support resource
sharing and its efficiency to reduce energy consumption, while the disadvantage is
the high data rate in fronthaul links expected for processing heterogeneous demands
from multiple RRHs. In [79], authors highlighted, using a real test bed, the efficiency
of the split option 4 in finding good trade-off between the processing of BBU func-
tions in centralized data centers and the fronthaul requirements. In [80], authors
31
RRH (antenna)
Centralized data center
(BBU pool)
PHY
MAC
RLC
PDCP
Split 1
PHY
MAC
RLC
PDCP
Split 2
PHY
MAC
RLC
PDCP
Split 3
PHY
MAC
RLC
PDCP
Split 4
Figure 2.7: Overview of considered functional split options
confirmed that the maximum multiplexing gain on BBU resources can be achieved
when considering the fully centralized C-RAN (split 4) after analyzing the differ-
ent function splits. Nevertheless, split 4 requires high fronthaul network capacity
compared to other split options such as Split 2. In fact, in Split 2, the PHY, MAC
and RLC layers are integrated into RRHs while the PDCP layer is moved to the
BBU pool to achieve increased multiplexing gains. This is feasible since functions
in PDCP layer are less capacity-intensive and not subject to real-time constraints.
Another BBU function split option that does not require high capacity on fronthaul
network is Split 3 that consists in locating the PHY layer in the cell sites and moving
other functions of upper layers to the centralized data centers. Obviously, both split
options 2 and 3 have less fronthaul throughput compared to the split 4 in which the
processing of baseband functions is centralized in BBU pool. However, the inter-cell
collaborative processing cannot be efficiently supported when considering split 3 due
to the the difficulty of interconnection between PHY layers and other layers.
Based on these different functional split options, many works investigated new
approaches to determine the optimal placement of baseband processing functions in
C-RAN architecture. In [81], authors proposed a low-cost solution by considering
dual-site processing where baseband functions are divided between the antenna (or
radio unit) and the data center (or digital unit). They provided an estimation of
processing and bandwidth requirements for each functional split option and the
total costs of ownership including equipment, civil work, commissioning as well as
operational costs such as electricity and maintenance. The authors of [81] claimed
that there is no ”one-size-fits-all” solution for functional split and they indicated that
hybrid RAN deployments contribute to Total Cost of Ownership (TCO) savings
when considering 5G configurations. Authors in [82] proposed a methodology to
derive guidelines for minimizing capital expenditure of C-RAN due to deployment of
fronthaul and baseband processing resources. This work minimizes the total length
of fiber while maximizing the multiplexing gain for each shared edge cloud hosting
the baseband functions. The authors in [83] proposed a model representing the
baseband processing functions as a directed graph where nodes represent different
32
processing functions and arcs represent connectivity between them. Then, they
introduced the placement of these functions on data centers located at different sites.
Computational costs and transport (link) costs are defined as well as constraints on
the delay (latency) incurred both for processing and transport. In this paper, the
problem of finding optimum locations to place baseband functions is equivalent to
finding an optimum clustering scheme for the graph nodes with the objective of
minimizing the total cost while ensuring that latency constraints are met.
2.4 Conclusion
In this chapter, we introduced some combinatorial optimization techniques and
algorithms which will be used to address optimization problems in the context of
C-RAN. First, we provided an overview of linear programming models which are
considered as powerful techniques to obtain optimal solutions. Then, we described
some optimization techniques that we used to deal with the addressed problems
in this thesis and propose new heuristic algorithms to have good solutions in rea-
sonable times even for large problem instances. Furthermore, we highlighted some
well-known optimization problems and different proposed approximation algorithms.
Another domain is represented in the second part which is the homology theory used
to propose new solutions for network coverage problem in the context of C-RAN.
In the second section, we presented a deep analysis of the challenges that we
will address in the following chapters and we highlighted the different approaches
proposed in the literature.
In the next chapter, we will address an exciting research challenge in the con-
text of C-RAN which is the assignment of antennas to edge data centers under
transport requirements and limited capacity constraints and we will investigate new
approaches to efficiently solve this problem.
Chapter 3
Constrained resource allocation in
C-RAN
3.1 Introduction
The deployment of C-RAN architecture, where computing resources in edge data
centers (BBU pool) are shared between multiple cell sites (RRHs), is expected to
reduce network costs, e.g. CAPEX and OPEX, and improve the resource utilization
efficiency. To achieve these goals, network operators need to investigate new efficient
resource allocation algorithms to assign the limited processing resources in edge data
centers to antennas (RRHs) demands when meeting strong latency requirements.
The optimal mapping between RRHs and BBUs (RRH-BBU assignment) is achieved
while jointly minimizing the communication latency on the fronthaul network and
the resource consumption by reducing the number of active edge data centers needed
to meet antennas demands.
For this purpose, an exact mathematical model based on Integer Linear Program-
ming (ILP) is formulated to address the constrained resource allocation problem and
provide the most appropriate strategies for RRH-BBU assignment when processing
and latency requirements are met. Then, we seek new approximation algorithms
that scale well, converge reasonably fast and find good solutions for RRH-BBU
assignment problem.
The remainder of this chapter is organized as follows. In Section 3.2, we de-
scribe the system model that will be used to define the constrained resource allo-
cation problem and then we study the complexity of the addressed problem. In
Section 3.3, an exact mathematical formulation is provided to meet the RRH-BBU
assignment problem for small and medium size networks and then we introduce
three approximation algorithms with significantly less complexity to deal with large
problem instances. Numerical results are presented in Section 3.4 to highlight the
performance of our proposed algorithms using several simulation scenarios and a
real cellular network. Section 3.5 concludes this chapter.
33
34
3.2 Problem statement
In this section, we describe the system model that we consider to address the
constrained resource allocation problem (RRH-BBU assignment problem1) and we
introduce all variables and parameters used in the description of the problem. Then,
and before providing our proposed algorithms, we discuss the complexity of the
RRH-BBU assignment problem when considering all constraints that will be defined
below.
3.2.1 System model
We consider the system model, as shown in Figure 3.1, to define the constrained
resource allocation problem that aims to efficiently assign the antennas demands
to the most appropriate edge data centers when strict latency and processing re-
quirements are met. Our system model represents a C-RAN network where RRHs
(antennas) and BBU pools (edge data centers) are deployed in a large area. As de-
picted in Figure 3.1a, our network architecture contains a set of antennas, denoted
by I, each of which is defined by a position on the plane. These antennas i∈Ihave
variable expected latencies liand processing requirements in terms of CPU cores ci,
depending on aggregated end-users’ demands. The RRHs are served by a finite set
of available edge data centers denoted by J. Each edge data center j∈Jhas a
limited computing processing capacity Cjexpressed as number of CPU cores.
ED1(C1)
ED2(C2)
A1
A2
A3
A4(c4;l4)A5
A6
A7
L21
(a) Network topology example
ED1(C1)
ED2(C2)
A1
A2
A3
A4(c4;l4)
A5
A6
A7
L21
(b) bipartite graph construction
C1(resp. C2) : total number of available CPU cores in BBU pool ED1(resp. ED2)
c4: number of CPU cores requested for processing the demands of antenna A4
l4: expected latency for processing the demands of antenna A4
L21 : communication latency on the fronthaul link between A2and ED1
Figure 3.1: System model for constrained resource allocation problem
The antennas are connected to the edge data centers via fronthaul network which
is represented by a set of communication links. Each fronthaul link between an
1We use ”constrained resource allocation problem” and ”RRH-BBU assignment problem” in-
terchangeably throughout this chapter
35
antenna i∈Iand an edge data center j∈Jhas a transmission delay Lij that should
be kept below 1 millisecond in order to meet HARQ2requirements (see [11], [17]
and [84]). This requires that the maximum distance dij between RRH iand BBU
pool jmust not exceed 20 to 40 kilometers (see for instance [11] and [85]). The
data traffic on the fronthaul network can be transmitted using different protocols,
most commonly CPRI [9], or in some cases OBSAI [8]. In our system model, and
according to [11] and [86], the transmission delay on the fronthaul network is 5
microseconds per Kilometer and thus the communication (fronthaul) latency
between RRHs and BBU pools vary between 100 and 200 microseconds at the
most.
As depicted in Figure 3.1, our network topology (Figure 3.1a) can be modeled
by a weighted bipartite graph G= (I∪J, E) containing a set of antennas Iin
one side, a set of edge data centers Jin the other side and a set of fronthaul links
represented by the set of edges E. The weight value, denoted by Lij , on each edge
in the graph Grepresents the communication latency between the antenna i∈I
and the edge data center j∈J. The bipartite graph G= (I∪J, E ) will be used to
efficiently assign each antenna to exactly one edge data center when meeting
the processing and latency requirements.
For sake of clarity, we give in Figure 3.2 a simple example of C-RAN network
which is composed by 6 RRHs (antennas), 2 edge data centers (BBU pools) and
a fronthaul network represented by a set of communication links. The constrained
resource allocation problem consists in determining the optimal strategies to assign
the antennas demands to the available edge data centers under strict processing and
latency requirements.
(a) Initial graph (b) Final graph : a feasible solution
Figure 3.2: A solution example of the constrained resource allocation problem
2HARQ (Hybrid Automatic Repeat reQuest) is the process that poses the most stringent delay
requirement for cellular networks
36
Hence, we aim to select, in the bipartite graph of Figure 3.2a, the optimal match-
ing of all considered antennas with the available edge data centers when jointly
meeting processing and latency requirements. The optimal assignment of all consid-
ered antennas to the BBU pools is achieved when latency and resource consumption
(number of used edge data centers) are minimized. The right graph (Figure 3.2b)
represents a feasible solution of the RRH-BBU assignment problem.
For sake of clarity, we summarize in Table 3.1 all variables and parameters that
will be used, in the following, to model the constrained resource allocation problem.
Table 3.1: RRH-BBU assignment problem : variables and parameters
G= (I∪J, E) : weighted bipartite graph
I: set of antennas/RRHs
J: set of edge data centers/BBU pools
E: set of communication links between Iand J
dij : distance between an antenna i(with coordinates
(xi, yi)) and an edge data center j(with coordinates
(xj, yj))
ci: total number of CPU cores requested for processing the
aggregated demands of antenna i
Cj: available computing resources (CPU cores) in each edge
data center j
li: expected latency for processing the aggregated de-
mands of antenna i
Lij : transmission delay (latency) on the communication link
between an antenna iand an edge data center j
xij : binary decision variable
xij =
1,if the antenna i∈Iis assigned
to the edge data center j∈J
0,otherwise
yj: binary decision variable
yj=(1,if the edge data center jis used
0,otherwise
Before investigating new algorithms to deal with the constrained resource allo-
cation problem, we address in the following the problem’s complexity.
37
3.2.2 Problem complexity
We provide a theorem and a proof confirming the NP-Hardness of the RRH-BBU
assignment problem.
Theorem 3.2.1 Finding the optimal assignment of the antennas (RRHs) demands
to the available edge data centers (BBU pools) is an NP-Hard problem.
Proof As it is described above, the constrained resource allocation problem con-
sists in finding the optimal assignment of antennas demands to the available edge
data centers with the aim of satisfying latency and processing requirements and
minimizing network resource utilization. Our problem is close to the Generalized
Assignment Problem (GAP) (see [87] for more details), which is a classical gen-
eralization of both multiple knapsack problem [88] and bin packing problem [89].
Indeed, GAP consists in finding a feasible packing of the items (each item is defined
by a size and a profit) into the bins (each bin has a limited capacity) that maximizes
the total profit.
Our constrained resource allocation problem is very similar to GAP in which the
antennas can be considered as items and edge data centers are the bins. Further-
more, compared to GAP, our constrained resource allocation problem has additional
constraints concerning the latency requirements on the communication links joining
the antennas and edge data centers. Hence, the relaxation of these constraints give
an instance of GAP which means that the optimal solution of GAP is a feasible
(not necessarily optimal) solution for RRH-BBU assignment problem.
Authors in [88] and [90] have proven the NP-Hardness of GAP. Therefore, by
using the previous linear reduction from our problem to GAP, we deduce that our
RRH-BBU assignment problem is also NP-Hard which means that finding the op-
timal assignment of the antennas demands to the available edge data centers is an
NP-Hard problem.
The aim of the RRH-BBU assignment problem is to find an optimal mapping
of all antennas demands on the available edge data centers when satisfying the
processing and latency requirements. To achieve this objective, we discuss, in the
following, an exact approach and heuristic algorithms to attend optimal and near-
optimal solutions, respectively.
3.3 Proposed algorithms
In this section, we provide an exact approach based on ILP formulation to deter-
mine the optimal assignment of antennas demands to the edge data centers. Since
our addressed problem is NP-Hard, we propose three approximation algorithms to
efficiently deal with large instances of RRH-BBU assignment problem. It is worth
noting that an ILP formulation is proposed to provide optimal (best) solutions for
the RRH-BBU assignment problem for small and medium size networks and will
be used to benchmark the performance of our proposed heuristic algorithms using
several metrics.
38
3.3.1 Integer linear programming formulation
In this section, we formulate the RRH-BBU assignment problem by a mathe-
matical formulation based on ILP approach. This allows to find optimal solutions
to deal with small and medium problem instances. The decision variables and all
parameters, that we will use in the mathematical formulation, are defined in Table
3.1.
Objective function
The objective of our constrained resource allocation problem is to efficiently
assign the antennas demands to the most appropriate (”best”) edge data centers
when jointly satisfying the processing and latency requirements. This objective will
be reached by finding the best trade-off between transport requirements on the fron-
thaul network and the number of active edge data centers. In fact, similarly to [56],
[65], [82] and [91], our objective function in (3.1) contains two terms : the first
denotes the total assignment cost in terms of transmission delay on the fronthaul
network and the second term represents the total network resource utilization ex-
pressed by the number of used edge data centers. This is equivalent to select, in the
final graph (as shown in the example of Figure 3.2), the optimal matching between
the set of antennas and the available edge data centers when jointly optimizing the
latency on the fronthaul network and the computing resource consumption.
min F=X
j∈JX
i∈I
Lij ×xij +X
j∈J
yj(3.1)
Our constrained resource allocation problem has to comply with a number of
constraints which will be summarized and mathematically expressed in the following.
Constraints
Constraints (3.2) guarantee that each antenna ishould be connected to exactly
one edge data center j. These constraints are considered in the graph solution of
Figure 3.2 where each antenna is mapped on exactly one edge data center.
X
j∈J
xij = 1,∀i∈I(3.2)
Constraints (3.3) ensure that the assignment of antennas demands to the BBU
pools does not violate the edge data centers’ limited capacity constraints. In fact, as
mentioned in Section 3.2.1, each edge data center jhas a limited processing capacity
Cjin terms of CPU cores and thus the total number of CPU cores requested for
processing all antennas must not exceed the available computing resources of the
selected edge data center.
X
i∈I
ci×xij ≤Cj×yj,∀j∈J(3.3)
39
Our optimization will select the most appropriate fronthaul links that satisfy
the strict latency requirements of the antennas demands. In fact, constraints (3.4)
impose that the transmission delay on the selected communication link between the
antenna and the edge data center must not exceed the expected latency. Thus, as
shown in Figure 3.2, only expected latencies will be kept in the final solution. This
is guaranteed by the following inequalities:
Lij ×xij ≤li,∀i∈I,∀j∈J(3.4)
Constraints (3.5) ensure that if there exists at least one antenna assigned to the
edge data center j(i.e. Pi∈Ixij ≥1), then this edge data center is activated (i.e.
yj= 1) and can be used to host other antennas as long as its processing capacity
is not exceeded. We recall that the optimal assignment of antennas demands to the
edge data centers is reached when the number of used edge data centers is minimized.
This will help network operators to reduce their network costs including CAPEX
and OPEX.
yj≤X
i∈I
xij ,∀j∈J(3.5)
Complete mathematical formulation
Our mathematical model is hence characterized by the following ILP formulation
(3.6). Using an exact method, i.e. Branch-and-Bound (see Section 2.2.1.1 of Chapter
2), the proposed mathematical formulation explores all feasible solutions for the
RRH-BBU assignment problem and selects the best one allowing to find the optimal
strategies to assign the limited processing resources in the available edge data centers
to the antennas demands. The solution provided by the ILP formulation (3.6) is
optimum (”best” solution) and thus resource utilization gains can be achieved when
the number of used edge data centers and the fronthaul latency are minimized.
min F=Pj∈JPi∈ILij ×xij +Pj∈Jyj
S.T. :
Pj∈Jxij = 1,∀i∈I
Pi∈Ici×xij ≤Cj×yj,∀j∈J
Lij ×xij ≤li,∀i∈I,∀j∈J
yj≤Pi∈Ixij ,∀j∈J
xij , yj∈ {0,1},∀i∈I,∀j∈J;
(3.6)
Since the NP-Hardness of our constrained resource allocation problem (see the
proof in Section 3.2.2), the necessary convergence time to obtain optimal solutions
40
using an exact approach based on ILP formulation exponentially increases with the
increase of number of antennas demands. Thus, we need to investigate new ap-
proximation algorithms that converge rapidly and provide optimal or near-optimal
solutions for large problem instances. In the following, we introduce three heuristic
algorithms (i) matroid-based approach, (ii) b-matching formulation and (iii) mul-
tiple knapsack-based algorithm. We recall that the obtained solution by the exact
approach based on ILP formulation (3.6) is optimum (”best” solution) and will be
used to evaluate the quality of solutions provided by the proposed heuristic algo-
rithms.
3.3.2 Matroid-based approach
In addition to the exact approach based on the above description of the convex
hull of the addressed problem, we seek a polynomial time algorithm that can scale
to larger number of antennas and edge data centers. Since the optimal solution
provided by ILP formulation in (3.6) is efficiently optimizing the latency and the
resource allocation jointly, we propose new algorithm based on matroid theory [40]
with similar properties and criteria.
Using the weighted bipartite graph G= (I∪J, E) constructed according to
Figure 3.1b, the optimal solution of the constrained resource allocation problem
consists in hosting each antenna demand in exactly one edge data center. Similarly,
in the bipartite graph G, each vertex i∈Iwill be assigned to exactly one vertex
j∈J, and each vertex j∈Jcan be a neighbor of different vertices in Ias each edge
data center can host more than one antenna. This yields a solution as presented in
Figure 3.2b, showing a forest of trees optimally linking antennas Iand edge data
centers J. Thus, we propose the following theorem that defines our matroid for
RRH-BBU assignment problem. We note that this matroid is well known in the
literature and it is noted by the graphic matroid (see [25] for instance).
Theorem 3.3.1 Let G= (I∪J, E)be a simple bipartite graph as shown in Figure
3.1b. By relaxing data centers’ limited capacities constraints, M= (E, F)is a
matroid, with F={A⊆E, A is a f orest of trees}.
In the following, we provide the proof of theorem 3.3.1 (we will use some concepts
and definitions which are detailed in Section 2.2.1.3 of Chapter 2).
Proof Based on the bipartite graph G= (I∪J, E), we investigate trees decom-
position with a minimum cost. In other words, and after the relaxation of data
centers’ limited capacity constraints, we seek to find an optimal basis of the graphic
matroid.
In this proof, we use the definition of matroid that we introduced in Chapter 2
(Section 2.2.1.3). Thus, the proof is given as follows:
•The first condition (P1) of the definition concerning matroids, is trivial.
41
•The second condition (P2) of the matroid definition : Suppose we have A∈F,
and according to the definition of F,Ais a forest of trees. Thus, if B⊆A,
then the connected components of Bare also trees even by deleting one or
multiple edges in A. This leads to easily conclude that B∈F.
•To prove the last condition (P3) of the matroid definition, we note by A=
∪k
i=1Aiwhich represents the connected components (trees) of A. Then, for all
i= 1, . . . , k, we suppose Gi= (Ti, Ai), where Giis a tree with |Ti|vertices
and |Ai|edges. This leads to deduce the number of vertices of Agiven by
nA=
k
X
i=1
|Ti|=|A|+k. (3.7)
We also suppose B=∪t
j=1Bj, we note by G0
i= (T0
i, Bi), where G0
iis a tree
with |T0
i|vertices and |Bi|edges. The number of nodes of Bis then given by :
nB=
t
X
j=1
|T0
i|=|B|+t. (3.8)
By using |B|>|A|, two cases are discussed:
1. If nB> nA(t>k) : We suppose that Breaches more vertices than A, so
there exists a vertex xcovered by Band not by A. Suppose that e∈Bis
an edge which contains xas one of its two extremities, we finally deduce
that A∪ {e} ∈ F.
2. If nB< nA: We suppose that the edges of Bconnects each couple of
nodes in Ain the same connected component (tree) Ai. Using the absurd
reasoning, we suppose that there is no edge e∈B\A, leading to get
A∪ {e} ∈ F. This means that:
–The edge e∈B, relies two vertices in the same component (tree) Ai
and forms a cycle.
In this case, the number of edges of Bwill verify |B|≤| V1|+|
V2|+. . . +|Vk|, then |B|≤| A|which contradicts our hypothesis
|B|>|A|.
As mentioned in Section 2.2.1.3 of Chapter 2, the above matroid formulation,
defined by theorem 3.3.1, can be optimally solved by a simple greedy algorithm.
However, this algorithm does not consider the constraints of limited processing ca-
pacity in the edge data centers. In fact, these constraints are strong in our RRH-BBU
assignment problem because they influence the choice of which edge data center, the
antenna will be assigned. Hence, we introduce some modifications in the matroid
formulation to consider the edge data centers’ limited capacity constraints. The
complete matroid-based algorithm is illustrated in Algorithm 3.
42
Algorithm 3 Matroid-based algorithm for RRH-BBU assignment problem
Put A=∅;
le1≤le2≤. . . ≤lem;
for i= 1 to mdo
if A∪ {ei} ∈ Fthen
if cI(ei)≤CT(ei)then
A:= A∪ {ei}
CT(ei)−=cI(ei)
end if
end if
end for
leiis the communication latency on the edge ei;
I(ei) (resp. T(ei)) represents the initial (resp. terminal) extremity of the edge ei;
cI(ei)represents the number of CPU cores requested for processing the antenna demand I(ei);
CT(ei)represents the available amount of CPU in an edge data center T(ei).
Matroid-based algorithm’s complexity
It is important to evaluate the complexity of our proposed matroid-based al-
gorithm. We note that the addressed problem is NP-Hard, and we need rapid and
cost-efficient approaches to cope with this complexity. In fact, our proposed matroid-
based algorithm has a global complexity (in the worst case) of O(mln(m) + m),
where the first term mln(m) is the complexity of sorting a set of medges according
to their weights (latency in our case), and the second term mis the number of times,
the ”For” loop indicated in Algorithm 3 has been executed.
In addition to the matroid-based algorithm, we introduce in the following an-
other heuristic algorithm based on b-matching approach. This proposal aims to
find the optimal mapping between RRHs and BBUs, when satisfying all antennas
demands. Using b-matching algorithm, we seek to rapidly reach optimal or near-
optimal solutions for large instances of RRH-BBU assignment problem. This may
not be feasible with matroid-based approach, especially when the number of an-
tennas demands becomes important (more than 100 antennas) and the computing
resources in available edge data centers are limited.
3.3.3 b-Matching formulation
In this section, we propose a new heuristic approach based on b-matching theory
[39] to address larger problem instances and to attend optimal or near optimal
solutions in negligible times. The b-matching algorithm uses the bipartite graph, as
described in Figure 3.1b, to find the minimum weight b-matching between antennas
and edge data centers when jointly reducing the number of used edge data centers
and the latency requested for processing the aggregated demands.
According to the definition of b-matching approach (see Section 2.2.1.3 of Chap-
ter 2), we introduce new algorithm that solves the constrained resource allocation
43
problem by finding the minimum weight b-matching between antennas and edge data
centers in the bipartite graph G= (I∪J, E). This algorithm will jointly consider
the strong latency requirements of antennas demands and the limited processing
capacity constraints of the edge data centers.
Proposition 3.3.2 Let G= (I∪J, E)be a weighted bipartite graph. The con-
strained resource allocation problem defined above can be solved by finding the min-
imum weight b-matching while considering the following parameters:
•The integral edge capacities : u= 1.
•b(i)=1,∀i∈I (I is a set of antennas).
•b(j) = min{|Ij|,bCj
c(j)c},∀j∈J (J is a set of edge data centers).
where :
•Ijis a subset of antennas that can be assigned to the edge data center j∈J
when satisfying the expected latency and CPU cores amount requested for
each antenna demand : Ij={i∈I|(li≥Lij )∧(ci≤Cj)}.
•c(j) is the average number of CPU cores of antennas demands that can be
assigned to the edge data center j∈J:c(j) = Pi∈Ijci
|Ij|.
In addition and in order to help our optimization to find optimal solution with
integer variables, we add the blossom inequalities given by the following formula
(3.9).
X
e∈E(G[X])
xe+X
e∈F
xe≤ b1
2(X
v∈X
b(v) + |F|)c,∀X⊆I∪J, F ⊆δ(X) (3.9)
where E(G(X)) represents a subset of edges in the subgraph G(X) generated by a
subset of vertices Xand δ(X) is a set of incident edges of X(more details can be
found in Section 2.2.1.3 of Chapter 2).
Finally, we use the obtained result of Proposition 3.3.2 to define a new minimum
weighted b-matching formulation to polynomially solve the constrained resource
allocation problem. The mathematical formulation is given by the following model:
min F=Pe∈ELe×xe
S.T. :
Pe∈δ(i)xe= 1,∀i∈I;
Pe∈δ(j)xe≤min{|Ij|,bCj
c(j)c},∀j∈J;
Pe∈E(G[X]) xe+Pe∈Fxe≤ b1
2(Pv∈Xb(v) + |F|)c,∀X⊆I∪J, F ⊆δ(X);
xe∈R+,∀e∈E;
(3.10)
44
b-Matching algorithm’s complexity
To assess the ability of our b-matching algorithm to find good solutions for
large-scale graph instances in reasonable times, we analyze in this section the com-
plexity of the proposed algorithm. We note that the objective of this algorithm is to
rapidly assign antennas demands to available edge data centers under strict latency
requirements and limited processing capacity constraints. The complexity of our
proposed b-matching approach based on linear programming is O|V|2|E|ln(|V|2
|E|)
where V=I∪Jand Eis the set of weighted links between Iand J. This approach
is a simple linear program with a negligible complexity. For interested readers, more
details can be found in [43] and [44].
In the following, we provide another heuristic algorithm using the multiple knap-
sack formulation. The multiple knapsack approach has been very well used in the
literature (see for instance [27], [92] and [93]) to address resource allocation problems
in different contexts. In our context of C-RAN, we propose a heuristic algorithm
based on multiple knapsack formulation to solve the RRH-BBU assignment prob-
lem. The obtained solutions by this algorithm will be benchmarked with matroid
and b-matching algorithms to better evaluate the performance of our algorithms
using different metrics.
3.3.4 Multiple knapsack-based approach
In addition to the exact mathematical formulation and heuristic algorithms pro-
posed above, we propose a new algorithm using the well known multiple knapsack
formulation to address the RRH-BBU assignment problem when considering the
limited capacities of edge data centers. In fact, the multiple knapsack formulation
is a generalization of the classical knapsack problem from a single knapsack to m
knapsacks with different capacities. The objective of multiple knapsack algorithm
is to assign each item to at most one of the knapsacks such that none of the capac-
ity constraints are violated and the total profit of the items put into knapsacks is
maximized.
According to the definition of multiple knapsack formulation (see Section 2.2.1.3
of Chapter 2) and by considering the bipartite graph G= (I∪J, E), we obtain the
following equivalence between our constrained resource allocation problem and the
multiple knapsack formulation :
•The knapsacks are the edge data centers (j∈J).
•The antennas demands (i∈I) are the items to be inserted in the knapsacks
(data centers).
•The weight wjis the amount of CPU cores cirequested for processing the
antenna demand i.
•The profit pjdoes not vary between different antennas demands and can be
set to 1 (pj= 1).
45
The previous formulation addresses the constrained resource allocation problem
by only focusing on the processing capacity of the edge data centers when relaxing
the latency requirements of antennas demands. This relaxation influences the choice
of which edge data center will host the antennas demands. Hence, in order to
consider these constraints in the final solution, we introduce a simple modification in
the multiple knapsack algorithm which consists in checking if the expected latency
is guaranteed before assigning the antenna demand to the edge data center. We
illustrate our multiple knapsack formulation in Algorithm 4.
Algorithm 4 Modified Multiple Knapsack Algorithm
Input: G= (I∪J, E), Antenna demands, Edge data centers.
Output: A joint mapping (CPU, Latency) of all antennas demands on the avail-
able edge data centers.
This is summarized formally in steps:
Step 1: Sort the edge data centers (j∈J) in increasing order of their CPU
capacities Cj;
Step 2: Select the antennas demands that can be assigned to the selected edge
data center jby checking if :
•The expected latency of the antenna demand is provided by the communi-
cation link joining it to the selected edge data center j;
•The available computing resources in the selected edge data center jare
greater than the number of CPU cores requested by the antenna demand;
Step 3: Pick as many antennas demands as possible to the selected edge data
center using the dynamic programming approach (see Algorithm 2);
Step 4: Update the total number of available CPU cores in the selected edge data
center;
Step 5: Repeat Steps 2, 3 and 4 until all considered antennas demands are as-
signed to the edge data centers;
3.4 Performance evaluation
The simulation and experiments use the optimization solver Cplex [94] for the
linear programming approaches, the exact approach based on ILP formulation in
(3.6) and the b-matching formulation in (3.10). We evaluate the performance of
the exact algorithm and then we compare the obtained solutions (optimum) with
those found by our heuristic algorithms in terms of convergence time, scalability and
optimality. Each simulation scenario is run 100 times using different parameters.
46
3.4.1 Simulation settings and parameters
The performance evaluation of our algorithms is conducted using a 2.40 GHz PC
with 8 GB RAM. The number of antennas is generated following a Poisson process
with a parameter Λ = λ×space dimensions, where λis varying in the range
[0.1; 1], and space dimensions in the range [5; 20]. In Figure 3.3, we illustrate four
examples of simulation scenarios when considering a cellular network in a region of
space dimensions space dimensions = 10 ×10 and varying the density of antennas
λ∈ {0.3; 0.5; 0.8; 1}.
(a) space dimensions = 10 ×10,
λ= 0.3
(b) space dimensions = 10 ×10,
λ= 0.5
(c) space dimensions = 10 ×10,
λ= 0.8
(d) space dimensions = 10 ×10,
λ= 1.0
Figure 3.3: Example of simulation scenarios for RRH-BBU assignment problem
Each antenna comprises a random number of demands (from end-users) pre-
sented in terms of an amount of CPU cores in the [5;10] interval (some papers such
as [60] and [95] are considering allocation of Physical Resources Blocks PRBs, this
is not changing our mathematical modeling and the convergence of our algorithms
to good solutions). The number of edge data centers is set to 20 each of which has
random computing resources or number of available CPUs drawn in the [50; 200]
47
CPU cores range. The workloads (i.e. aggregated amount of end-users demands in
terms of equivalent CPU cores) of the antennas demands are expecting a latency
to not exceed 1 millisecond and this is drawn randomly in the [0.1; 1] milliseconds
range. For sake of clarity, we summarize the simulation settings and parameters in
Table 3.2.
Table 3.2: RRH-BBU assignment algorithms : simulation settings and parameters
Parameters Values
Density of antennas λ∈[0.1; 1]
Space dimensions 10 ×10; 20 ×20;...
Poisson parameter Λ = λ×space dimensions
Number of antennas Poisson distribution: P(Λ)
Antenna coordinates Uniform distribution:
U(0, space dimensions)
Number of edge data centers 20
Latency between antenna iand
edge data center j
5µs/km
Expected latency of antenna i li∈[0.1ms; 1ms]
Number of CPU cores required
by each antenna i
ci∈[5; 10]
Number of CPU cores in each
edge data center j
Cj∈[50; 200]
3.4.2 Performance metrics
The metrics used for the performance assessment of our algorithms (exact and
heuristics) are detailed in the following :
•Convergence time: is the time needed by the algorithms to converge to their
best solutions.
•Resource utilization rate: is defined as the percentage of edge data centers
that are used to host the aggregated antennas demands and it can be expressed
as follows :
Resource utilization rate(%) = Pj∈Jyj
|J|×100 (3.11)
where |J|is the total number of available edge data centers.
•Gap: is used to benchmark the proposed heuristics with the exact ILP al-
gorithm used as “reference and optimal solution”. With no loss of generality,
we focus on the comparison of CPU resource consumption (expressed by the
percentage of edge data centers used to host all antennas demands). We note
48
that the quality of the solution provided by the heuristic algorithms is better
when the cost gap value is smaller (optimum when the gap is equal to
0). This metric is formally expressed as:
Gap(%) =|Utilization rate(ILP)−U tilization rate(Heuristic)|(3.12)
•Rejection rate: is the average of the percentage of antennas demands that
cannot be assigned to each edge data center. This metric, can be expressed as
a function of the decision variables and parameters described in Table 3.1 :
Rejection rate(%) = |I| − Pj∈JPi∈Ixij
|I|×100 (3.13)
where |I|is the total number of antennas.
•SLA violations rate: is the average of over-used edge data centers in terms
of CPU cores. This metric will be mainly used to evaluate the ability of the
matroid-based approach in finding optimal solutions that do not violate the
edge data centers’ limited capacity constraints (which is defined, in the ILP
formulation, by constraints (3.3)). We only focus on matroid-based algorithm
(as defined by theorem 3.3.1) because there are no SLA violations with ILP,
b-matching and multiple knapsack appraoches. The average of SLA viola-
tions rate can be expressed as a function of decision variables and parameters
(described in Table 3.1).
SLA violations rate(%) = 1
|J|×X
j∈JPi∈Ici×xij −Cj×yj
Cj×yj
×100 (3.14)
where |J|is the total number of available edge data centers.
3.4.3 Performance analysis
3.4.3.1 Performance evaluation of ILP based approach
Table 3.3 depicts the performance results in terms of convergence time and rejec-
tion rate of the exact algorithm based on ILP formulation. This algorithm explores
all feasible solutions before finding the optimum. This causes an exponential in-
crease of the convergence time when increasing the number of antennas. Indeed,
the ILP approach needs more than 4 minutes (4.39 minutes) to converge to optimal
solutions for an instance of 400 antennas and 20 available edge data centers. This is
expected since the addressed problem is NP-Hard. Thus, the ILP approach can be
used for small or medium instances with a number of antennas not exceeding 100.
Furthermore, the rejection rate is always equal to 0 which means that the exact
approach based on ILP formulation is always able to assign all antennas demands
to the available edge data centers.
49
Table 3.3: Performance of the exact approach based on ILP formulation
Space λ #Antennas Convergence time Rejection rate
10 ×10
0.3 30 9.63s 0
0.5 50 10.92s 0
0.8 80 11.87s 0
1 100 12.58s 0
20 ×20
0.3 120 62.09s 0
0.5 200 86.56s 0
0.8 320 2.87min 0
1 400 4.39min 0
3.4.3.2 Performance evaluation of heuristic algorithms
In Table 3.4, we consider different simulation scenarios by varying the dimensions
of the considered space area as well as the density of deployed antennas (see the
examples in Figure 3.3). Using these simulations, we would like to evaluate the
performance of our proposed approximation algorithms: matroid-based algorithm
(Algorithm 3), b-matching formulation given by (3.10) and the Multiple knapsack-
based approach (Algorithm 4).
Table 3.4: Heuristic algorithms’ performance assessment
Space λ #Antennas Heuristic algorithm Convergence time Gap(%) Rejection rate(%)
10 ×10
0.3 30
matroid 0.28ms 7 0
b-matching 0.34s 7 0
multiple knapsack 0.57ms 8 0
0.5 50
matroid 0.38ms 5 0
b-matching 0.36s 5 0
multiple knapsack 1.01ms 11 0
0.8 80
matroid 0.51ms 6 0
b-matching 0.26s 5 0
multiple knapsack 1.69ms 15 0
1 100
matroid 0.88ms 6 0
b-matching 0.39s 4 0
multiple knapsack 3.35ms 15 0
20 ×20
0.3 120
matroid 0.94ms - 1
b-matching 0.4s 4 0
multiple knapsack 4.35ms - 1
0.5 200
matroid 1.02ms - 4
b-matching 0.39s 6 0
multiple knapsack 7.71ms - 1
0.8 320
matroid 1.75ms - 17
b-matching 0.34s 6 0
multiple knapsack 25.44 - 3
1 400
matroid 2ms - 19
b-matching 0.33s 5 0
multiple knapsack 39.89ms - 4
50
As shown in Table 3.4, our heuristic algorithms are benchmarked with the ILP
approach, that provides optimum solutions, using three performance metrics : the
convergence time, the gap (3.12) to compare with optimal solutions provided by the
exact approach and the rejection rate (3.13). We note that we calculate the gap
only if the rejection rate is equal to 0, otherwise it is not really significant.
Table 3.4 highlights clearly the efficiency of the matroid-based algorithm in find-
ing near optimal solutions faster than the exact approach based on ILP formulation.
Indeed, the matroid approach provides good solutions with an average gap not ex-
ceeding 7% in worst cases and needs 2 milliseconds to converge when considering
large graphs of 400 antennas and 20 available edge data centers. Thus, the matroid-
based approach can be used to cope with large problem instances. However, the
matroid approach comes with some drawbacks such as it cannot assign all antennas
demands for large problem instances. This is shown by the rejection rate metric of
which its value can reach 19% for an instance of 400 antennas and 20 available edge
data centers.
To better evaluate the performance of our matroid-based algorithm, we calculate
the rejection rate when increasing the number of considered edge data centers. For
that, we consider two network instances of 320 and 400 antennas and we varied the
number of edge data centers from 20 to 60. The obtained results of these simulations
are represented by Figure 3.4.
20 25 30 35 40 45 50 55 60
0
10
20
30
40
50
60
70
80
90
100
Number of edge data centers
Rejection rate (%)
320 Antennas
400 Antennas
Figure 3.4: Matroid-based approach : rejection rate variation when increasing number of
edge data centers
The simulation results in Figure 3.4 show that the rejection rate depends on the
amount of available computing resources and thus decreases when the number of
available edge data centers increases. In fact, for the first simulation scenario (320
antennas), matroid-based algorithm attends a rejection rate equal to 0 when there
are at least 40 available edge data centers, while for the second simulation scenario
(400 antennas), the rejection rate vanishes when there are at least 50 available edge
data centers. This means that the matroid-based algorithm becomes more efficient
when more resources (edge data centers) are considered.
In addition and in order to get a better grasp of the relative performance of the
51
matroid-based approach, we illustrate in Figure 3.5 the SLA violations rate behavior
according to different network sizes. In fact, we consider 4 simulation scenarios :
50,100,200,320 antennas to be efficiently assigned to a number of edge data centers
ranging from 20 to 100. We recall that, for this simulation, we consider the matroid-
based algorithm (as defined in theorem 3.3.1) when relaxing the edge data centers’
limited capacity constraints and we calculate the SLA violations rate as defined by
Formula (3.14).
20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
70
80
90
100
Number of edge data centers
SLA violations rate (%)
50 Antennas
100 Antennas
200 Antennas
320 Antennas
Figure 3.5: SLA violations rate behavior of the matroid-based approach
Simulation results in Figure 3.5 confirm that the SLA violations rate decreases
when more processing resources (edge data centers) are considered. This confirms
that the efficiency of the matroid-based algorithm depends on the amount of the
available processing resources and attends good solutions when more resources (edge
data centers) are used.
3.4.3.3 Resource utilization behavior
Figure 3.6 depicts the percentage of resource utilization (in terms of number of
used edge data centers) obtained by the exact approach based on ILP formulation,
which provides optimal solutions, and by the three approximation algorithms (ma-
troid, b-matching and multiple knapsack). We recall that the value of the resource
utilization rate is calculated according to Formula (3.11). With a weak advantage
of the ILP method which consists in investigating all the feasible solutions before
keeping the optimal one, the matroid-based approach and b-matching algorithms
can find an efficient assignment of antennas demands to the available edge data cen-
ters while the solution obtained by multiple knapsack algorithm consumes a larger
number of edge data centers (as shown in Figure 3.6a).
It is important to mention that for larger problem instances (Figure 3.6b), b-
matching algorithm always provides good solutions in terms of resource utilization,
close to the optimum solution by ILP, with a rejection rate equal to 0%. How-
ever, for matroid and multiple knapsack algorithms, the resource utilization rate
52
λ= 0.3λ= 0.5λ= 0.8λ= 1
0
10
20
30
40
50
60
70
80
90
100
Resource utilization rate (%)
ILP
b-matching
matroid
multiple knapsack
(a) Space dimensions = 10 ×10
λ= 0.3λ= 0.5λ= 0.8λ= 1
0
10
20
30
40
50
60
70
80
90
100
r= 1%
r= 1%
r= 4%
r= 1%
r= 17%
r= 3%
r= 19%
r= 4%
Resource utilization rate (%)
ILP
b-matching
matroid
multiple knapsack
(b) Space dimensions = 20 ×20
ris the rejection rate calculated according to Formula (3.13)
Figure 3.6: Resource utilization in different space dimensions
depends on the rejection rate (negligible but different from zero) in the case of large
network size (see the legends appended to the resource utilization rate for matroid
and multiple knapsack algorithms in Figure 3.6b). Therefore, we deduce that b-
matching algorithm can easily scale when large problem instances are
considered and thus can be used by network operators to efficiently reduce their
network costs (CAPEX and OPEX) and achieve network utilization gains.
3.4.3.4 Algorithm’s performance evaluation using real traces
To better evaluate the performance of our proposed algorithms, we consider a
real trace from a 4G-LTE cell map of the network operator Orange, in a small area
in Paris [3]. As shown in Figure 3.7, this topology represents a cellular network
containing 50 antennas with their given geographical positions (coordinates). Then,
according to [11] and [17], we place 20 edge data centers on the cell map such that
the distance separating the antennas and the edge data centers is limited between
20 and 40 Kilometers. Similarly to the simulation parameters described in Table
3.2, we consider that each edge data center have a limited capacity of processing
in terms of CPU cores while the antennas demands have variable processing and
latency requirements.
In this experimentation, we apply our exact approach based on ILP formulation
(3.6) and the three proposed approximation algorithms, including matroid-based al-
gorithm (Algorithm 3), b-matching formulation (3.10) and multiple knapsack-based
approach (Algorithm 4), on the 4G-LTE cell map of Figure 3.7. The solutions pro-
vided by these algorithms are benchmarked according to three performance metrics :
convergence time, resource utilization rate given by (3.11) and rejection rate defined
by (3.13).
53
Figure 3.7: Real trace : Orange 4G-LTE cell map in Paris. Source: [3]
Table 3.5 shows that both matroid-based approach and b-matching formulation
provide optimal solutions (the same solution provided by the ILP approach) in
negligible times. In fact, with a weak advantage of the matroid-based approach
which converges to the optimum in 23.52 ms, the b-matching algorithm can also
find an efficient assignment of antennas demands to the available edge data centers in
0.58 ms. However, the solution obtained by multiple knapsack algorithm consumes
a larger number of edge data centers, with a resource utilization rate equal to 25%.
Regarding the rejection rate metric, all proposed algorithms can assign all considered
antennas demands to the the available edge data centers and satisfy their latency
and processing requirements without SLA violations.
Table 3.5: Performance evaluation using a real cellular network in Paris
Convergence Resource utilization Rejection
Algorithm time (ms) rate (%) rate (%)
ILP formulation 334.21 15 0
b-Matching algorithm 23.52 15 0
Matroid-based approach 0.58 15 0
Multiple knapsack algorithm 1.7 25 0
3.4.3.5 Scalability evaluation
The performance assessment would not be complete without addressing the scal-
ability for very large problem instances. In fact, we propose a simulation scenario
with an instance of 400 antennas and number of edge data centers in {60,80}
which are both generated according to the parameters detailed in Table 3.2. Sim-
ulation results in Table 3.6 confirm the efficiency of matroid-based approach and
54
b-matching algorithm in finding good solutions in negligible times compared to ILP
approach. Indeed, the matroid algorithm provides near optimal solutions (gap not
exceeding 2%) in less than 28 milliseconds and the b-matching algorithm can op-
timally solve the assignment problem in less than 4 seconds (with gap value not
exceeding 3%). However, the ILP approach is not converging in more than 1 hour
due to the exploration of all feasible solutions.
Regarding the multiple knapsack algorithm, the gap is a bit high compared to
matroid and b-matching algorithms and reaches 19% for an instance of 400 antennas
and 80 edge data centers.
Table 3.6: Algorithms’ scalability assessment
#Antennas1#Edge2ILP b-matching matroid multiple knasapck
Time Time Gap Time Gap Time Gap
400 60 34.28min 1.47s 3 6.42ms 2 82.84ms 18
80 1.02hour 3.97s 2 27.16ms 2 107.4ms 19
This simulation is executed 100 times with different parameters.
1Antennas are generated as described in Table 3.2.
2Edge data centers.
3.4.3.6 Comparative analysis of proposed algorithms
In this section, we present a comprehensive comparison of the proposed algo-
rithms for the constrained resource allocation (RRH-BBU assignment) problem. A
taxonomy of these approaches in terms of: i) computational complexity ii) cost sav-
ings (including OPEX and CAPEX), iii) scalability, iv) implementation difficulty
are highlighted in Table 3.7. As shown in this table, the matroid and b-matching
algorithms are globally more efficient in finding good solutions in negligible times
and in scaling larger problem instances. However, we note that it is not easy to im-
plement the b-matching algorithm (described by mathematical formulation (3.10))
due to the complex implementation of the blossom inequalities (3.9).
Table 3.7: Algorithms’ qualitative comparison
Algorithm Complexity Cost savings Scalability Implementation
difficulty
ILP formulation Exponential
b-Matching algorithm Polynomial
Matroid-based algorithm Logarithmic
Multiple knaspack algorithm Linear
55
3.5 Conclusion
In this chapter, we addressed the constrained resource allocation problem (RRH-
BBU assignment problem) with the objective of determining the best strategies to
assign antennas demands to available edge data centers when jointly optimizing
communication latency and resource consumption. Hence, we proposed an exact
algorithm based on ILP formulation to find optimal solutions for small and medium
size networks. The exact algorithm optimizes the resource consumption (in terms
of active edge data centers) and communication latency associated for assigning an-
tennas demands to the most appropriate edge data centers. However, this algorithm
is known to not to scale for large problem instances. Therefore, we proposed three
approximation algorithms based on exact theories and approaches : matroid-based
approach, b-matching algorithm and multiple knapsack-based algorithm to meet
larger number of antennas demands in negligible times.
The performance evaluation has been conducted using different simulation sce-
narios and five performance metrics. The simulation results have revealed the effi-
ciency of the matroid-based approach and b-matching algorithm in terms of optimal
solutions and convergence time even for large problem instances. This is confirmed
by the numerical results when considering a real trace from a 4G-LTE cell map.
Nevertheless, the optimal assignment of antennas to the edge data centers, where
many RRHs share common BBU computational resources, could be achieved when
the inter-cell interference are reduced. Thus, in the next chapter, we will investigate
new optimal approach to reduce interference between antennas when guaranteeing
full network coverage.
56
Chapter 4
Full network coverage
optimization in C-RAN
4.1 Introduction
The significant gain in terms of latency, resource utilization and cost savings,
achieved by the optimal RRH-BBU assignment (detailed in the previous chapter), is
largely constrained by the increasing of inter-cell interference that severely decreases
the end-users’ QoS. In fact, network operators are investigating new solutions to in-
crease the density of existing cells by deploying more antennas in order to enlarge
network spectrum and fulfill end-users requirements. Network densification is con-
sidered as a key method in C-RAN architecture to enhance network coverage and
capacity and achieve higher data rates. However, this brings a variety of challenges
including the management and reduction of inter-cell interference, generated by the
dense deployment of antennas, and the optimization of network coverage by detect-
ing and eliminating coverage holes. Consequently, it is crucial for network operators
to investigate new approaches that enable to find a good tradeoff between inter-cell
interference elimination/reduction and network coverage optimization.
In this chapter, we seek new approaches that enable to consolidate and re-
optimize the antennas radii in order to reduce inter-cell interference when main-
taining the full network coverage in C-RAN. We propose a new mathematical model
based on ILP formulation to describe the convex hull of full network coverage opti-
mization problem. Then, we enlarge this description by adding new valid inequalities
and cutting planes to accelerate the convergence time and reach optimal solutions
even for large number of antennas. Finally, we propose a deep Branch-and-Cut
algorithm based on these cutting planes to efficiently solve the large-scale problem
instances and we evaluate the efficiency and usefulness of our proposed approach
compared to those proposed in the state of the art.
The rest of the chapter is organized as follows. In Section 4.2, we describe the
problem statement and the network topology that will be used and then we dis-
cuss the complexity of our problem. In Section 4.3, we provide a Branch-and-Cut
approach describing the convex hull of the joint interference and coverage optimiza-
tion problem and we reinforce this formulation by proposing new families of valid
57
58
inequalities to accelerate the convergence time to the optimum. Numerical results
and performance assessment can be found in Section 4.4 followed by a conclusion in
Section 4.5.
4.2 Problem statement
In this section, we present the system model that we use to define the full network
coverage problem in the context of C-RAN. Then, we provide a complexity study
of the addressed problem.
4.2.1 System model and problem description
We consider a cellular network deployed in a large area represented by a set of
antennas denoted by A. Each antenna iis defined by its position on the plane and its
coverage radius riwhich varies in the range [rmin
i;rmax
i], where i= 1,...,|A|. The
coverage area of each antenna is modeled by circles with variable coverage radius.
v1
v2v3
v4v5
v6
v7
v8v9
v10
v11
v1
v2
v3
v4v5
v6
v7
v8v9
v10
v11
Figure 4.1: System model: graph construction based on antennas positions and
interference
59
In fact, in real life, cells have random shape of coverage area which depends on
geographic, environmental and network parameters (base station location, transmis-
sion power, terrain and artificial structures properties, . . .). In the literature and for
sake of representation and analytical simplicity, approximate approaches are often
adopted to design and model the cells’ coverage area in cellular networks. In par-
ticular, [96], [97] and [98] used hexagons to model the cells coverage area with no
overlap between cells. This approximation is frequently employed in planning and
analyzing wireless networks due to its flexibility and convenience. However, since
the hexagons are only an idealization of the irregular cell shape, a simpler approach,
called circular-cell approximation, is used to model the cell coverage area by circles
(see [99], [100] and [101] for example). The circular-cell approximation is reasonable
and very used in the modeling of cellular networks due to its low computational
complexity. Hence, some references (see [102] and [103]) are using this approach
to address the network coverage problem, and the authors proposed methods and
algorithms that do not converge in acceptable times and do not provide good solu-
tions. Our optimization is using circles to represent antennas coverage areas, and
we propose an exact formulation that always provide optimal solutions in negligible
times.
As depicted in Figure 4.1, we represent our network using an undirected graph
denoted by G= (A,E) where Aand Eare the sets of available nodes and edges,
respectively. There is an edge (i, j) between two antennas iand jif the following
condition (4.1) is met :
ri+rj≥dij (4.1)
where riand rjare the radii of antennas iand jrespectively, and dij is the Euclidean
distance between i(with the coordinates (xi, yi)) and j(with the coordinates (xj, yj))
and provided by:
dij =q(xj−xi)2+ (yj−yi)2(4.2)
Let δij be the overlapping (inter-cell interference) caused by two antennas iand
j.
δij =ri+rj−dij (4.3)
This overlapping is causing interference that should be reduced or totally elim-
inated when considering full network coverage and connectivity. Thus, similarly to
the simplicial homology and Delaunay triangulation approaches (detailed in chapter
2), we aim to extract, from a given graph G(lower part of Figure 4.1), a subgraph
G∆composed by adjacent triangles, each of which represents a complete coverage of
the area around three antennas (according to the definition of Rips complex which
can be found in Section 2.2.2 of Chapter 2). These graphs are illustrated in Figure
4.2. We note that the triangulated graph in the right part of Figure 4.2 represents
a total covered network with minimum inter-cell interference.
60
v1
v2
v3
v4v5
v6
v7
v8v9
v10
v11
(a) Initial Graph
v1
v2
v3
v4v5
v6
v7
v8v9
v10
v11
(b) Final Graph
Figure 4.2: A graph solution example of the full coverage network problem
4.2.2 Problem complexity
As mentioned before, our full network coverage problem consists in constructing
a graph composed only by adjacent triangles. This triangulation method can be
assimilated to the Minimum Weight Triangulation (MWT) problem which consists
in finding in a graph Ga set of edges of minimum total weight that triangulates
the total nodes of G. This problem has been proven to be NP-Hard [104]. In the
following, we discuss the complexity of the full network coverage problem when the
reached optimum solutions do not violate any of the problem constraints that will
be defined below.
Theorem 4.2.1 For an instance of the optimal full network coverage problem de-
fined above, deciding whether a solution with no violations exists is NP-Complete.
Proof To prove this theorem, we will proceed according to the following steps:
It is important to recall that to cope efficiently with our problem, we investigated a
polyhedral approach describing a set of valid inequalities to attend optimal solutions
using a Branch-and-Cut strategy. This polyhedral approach will lead to find an
optimal minimum weight triangulation when eliminating totally network interference
represented by the set of edges intersections. This family of valid inequalities is
illustrating the main difference between the full network coverage problem that we
are addressing in our work and the MWT problem.
For sake of clarity, and for a given instance of a weighted graph G= (A,E),
let ϕ∗
mwt be the optimal value of the MWT problem, and ψ∗
hole cov the optimum
found when solving the full network coverage problem. As our problem is more
constrained compared to the MWT problem, then we deduce that ϕ∗
mwt ≤ψ∗
hole cov.
This implies that, the relaxation of the constraints which consists in eliminating
existing interference (edges intersections) will hold to retrieve an instance of the
MWT problem. Indeed, the optimal solution of the minimum weight triangulation
problem is a feasible (not necessarily optimal) solution in the full network coverage
problem instance.
In addition, in 2006, W. Mulzer and G. Rote (see [104]) have proven the NP-
Hardness of the MWT problem. Thus, by using the previous linear reduction from
61
our problem to the MWT problem, we conclude that the full network coverage
problem is also NP-Hard. This implies that the decision formulation concerning
the existence of no violation solutions of the full network coverage problem is NP-
Complete.
Our problem is then NP-Complete, and we need rapid approaches to attend
optimal solutions in acceptable times. Our proposal is based on the construction of
a complete description of the convex hull of the incidence vectors characterizing the
optimal solution of the full network coverage problem.
4.3 Branch-and-Cut formulation
To cope with the full network coverage problem, we propose a Branch-and-Cut
algorithm based on the description of the convex hull of the problem’s incidence
vectors. This description consists in various families of valid inequalities leading to
attend the optimal solution in acceptable times.
4.3.1 Convex hull characterization
Before introducing our mathematical formulation, we start by providing the vari-
ables and parameters that will be used our formulation.
•We consider our initial graph G= (A,E) representing the network topology
as illustrated by the lower part of Figure 4.1. Ais the set of antennas and E
is the set of edges between antennas. According to formula (4.1), we populate
the graph G(see lower part of Figure 4.1).
•Each antenna i∈Acan operate with its own coverage radius riwhich varies
in the range [rmin
i;rmax
i].
•Let xij be a binary variable indicating if the edge (i, j) of Gis considered in
the final solution (xij = 1), or not (xij = 0).
•Let N(i) be the set of neighborhood nodes/antennas of i. A node jis a
neighbor of ionly if the condition (4.1) used with the maximum radii values,
is verified.
•Let I(i, j) be the set of all edges (i, j) that don’t intersect with any other edge
(k, l), where (i, j ) and (k, l) do not have common extremities.
The objective of the full network coverage problem is to detect rapidly holes
in the network and reduce considerably inter-cell interference represented by the
overlapping regions measured mathematically by formula (4.3). These objectives
will be reached by optimizing the radii values of the antennas. This is equivalent
to select in the final solution, only couple of antennas with a minimum Euclidean
62
distance guaranteeing the graph connectivity and the full network coverage which
allows to intuitively reduce the inter-cell interference. This objective is given by:
min Γ = X
i∈AX
j∈N(i)
dij ×xij (4.4)
The full network coverage problem has to comply with a number of constraints
which will be summarized and mathematically expressed in the following.
Constraints (4.5) guarantee that each node ihas at least two neighbors in the
graph (we recall that the objective is to obtain a triangulation meeting connectivity
and reduced interference).
X
j∈N(i)
xij ≥2,∀i∈A(4.5)
Constraints (4.6) impose that if an edge (i, j) do not have any intersection in the
initial graph (see solid line edges in Figure 4.3), then xij = 1 in the final graph (the
solution graph). These constraints are mathematically provided by:
xij = 1,∀i∈A,∀j∈N(i),∀(i, j)∈I(i, j ) (4.6)
v1
v2
v3
v4v5
v6
v7
v8v9
v10
v11
Figure 4.3: Each solid line edge (i, j) is necessary in the final graph/solution
In order to avoid any intersection in the final graph between any two edges
(i, j) and (k, l) (see Figure 4.4 in which i, j, k, l can be represented by v1, v6, v4, v5
respectively), we propose the following nonlinear inequality:
xij ×xil ≤xjl +X
k∈N(i)
xik,∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j) (4.7)
v1
v2
v3
v4
v5v6
Figure 4.4: Example of edge intersection (interference)
63
Constraints (4.7) are nonlinear as we used the product of two decision variables.
We replace them (constraints (4.7)) by new family of linear inequalities when intro-
ducing a new binary variable zijl, such that zijl =xij ×xil ,∀i∈A,∀j∈N(i),∀l∈
N(i)∩N(j). Thus, we obtain the constraints in (4.8), (4.9) and (4.10).
zijl ≤xij (4.8)
zijl ≤xil (4.9)
zijl ≥xij +xil −1 (4.10)
By summing (4.8) and (4.9), we obtain:
zijl ≤1
2(xij +xil)
We finally have three new valid inequalities for the full network coverage problem,
and they are provided by:
zijl ≤xjl +X
k∈N(i)
xik,∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j) (4.11)
zijl ≤1
2(xij +xil),∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j) (4.12)
zijl ≥xij +xil −1,∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j) (4.13)
Our mathematical model is hence characterized by the following Integer Linear
Programming:
min Γ = Pi∈APj∈N(i)dij ×xij
S.T. :
Pj∈N(i)xij ≥2,∀i∈A
xij = 1,∀i∈A,∀j∈N(i),∀(i, j)∈I(i, j )
zijl ≤xjl +Pk∈N(i)xik,∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j)
zijl ≤1
2(xij +xil),∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j)
zijl ≥xij +xil −1,∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j)
xij , zijl ∈ {0,1},∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j);
(4.14)
64
4.3.2 New valid inequalities
To address larger problem instances and to better describe the convex hull of
the full network coverage problem, we need to investigate new valid inequalities and
facets allowing to accelerate convergence time and to find optimal solutions jointly.
Thus, we propose to investigate new families of inequalities that are valid for our
problem.
4.3.2.1 Chordless cycle inequalities
Solving the mathematical formulation provided in (4.14) allows to obtain optimal
solutions for the full network coverage problem. Nevertheless, and for some initial
graph instances, the described convex hull in (4.14) is missing some solutions that
do not contain holes. In Figure 4.5, we show a simple example of cellular network
composed by 6 antennas with various coverage radii that can be represented by the
left graph. The solution obtained (the right graph) by the ILP formulation in (4.14)
has a coverage hole (v2, v4, v5, v6, v2).
v3
v4
v5
v6
v1
v2
(a) Initial Graph
v3
v4
v5
v6
v1
v2
(b) Final Graph
Figure 4.5: Example of graph solution containing a hole (v2, v4, v5, v6, v2)
Therefore, to integrate holes detection constraints in our mathematical formu-
lation, we investigate a new facet or valid inequality based on holes detection that
should be added to our optimization. These inequalities are based on detecting
Chordless Cycles, that will be defined in the following.
Definition 4.3.1 [105] Let Gbe an undirected graph and let v0, v1, . . . , vk−1be a
sequence of kdistinct vertices such that there is an edge between viand v(i+1) mod k
(∀i= 0, . . . , k −1), and no other edge between any two of these vertices. Then, this
sequence is a chordless cycle on kvertices. A hole may be a chordless cycle on four
or more vertices.
According to the previous definition, we would like to optimally solve the full
network coverage problem when detecting all the existing holes in the initial graph.
For this, we propose the following result.
Theorem 4.3.1 For any initial connected graph G, and for each chordless cycle
Cin G, such that |C| ≥ 4, the following inequality (4.15) is valid for the global
65
coverage hole problem (E(C)is the set of edges of the chordless cycle C):
x(E(C)) ≤3 (4.15)
Proof Let Gbe an undirected graph and v0, v1, . . . , vk−1(with k≥4) the set of
vertices making a chordless cycle (i.e. a hole) noted by C. Our objective is to detect
chordless cycles (holes) and then eliminate them using our optimization.
Using the absurd reasoning, we suppose that x(E(C)) = k−2
P
i=0
xvi,vi+1 ≥4 which
means that our optimization should keep at least 4 edges in Cleading to obtain one
of the two following cases:
1. A solution with a chordless cycle noted by {v0, v1, . . . , vk−1, v0}which repre-
sents a hole
2. A solution with at least two intersecting edges that represent an interference
in the final solution
The case 1 is not feasible as our optimization is focusing on eliminating all the
existing holes. The second case 2 cannot hold thanks to constraints (4.7) eliminating
intersections and interference efficiently.
v1
v2v3
v4
v5
v6
v7
v8
Figure 4.6: Example of a chordless cycle (v2, v3, v5, v8, v2) of size 4
To better understand the proof, we propose a simple example of cellular net-
work represented by the graph of Figure 4.6 that contains a chordless cycle C=
{v2, v3, v5, v8, v2}of size 4 . We can easily remark that (v2, v3), (v3, v5) and (v5, v8)
do not intersect with any other edge in the graph. So, by applying constraints (4.6),
we obtain a solution with xv2v3= 1, xv3v5= 1 and xv5v8= 1. Hence, we discuss two
cases on the status of the edge (v2, v8) of the chordless cycle C:
•xv2v8= 1 : In order to eliminate intersections in the final graph/solution, the
edges (v1, v3), (v1, v5), (v3, v7) and (v5, v7) will be removed (xv1v3= 0, xv1v5= 0,
xv3v7= 0 and xv5v7= 0) using constraints (4.7). This means that our final
solution has a coverage hole, and this is not desirable.
•xv2v8= 0 : There is no coverage hole in the final graph/solution (full coverage)
while totally eliminating network interference. In this case xv2v3+xv3v5+xv5v8+
xv8v2≤3, leading to x(E(C)) ≤3.
66
4.3.2.2 Separation of chordless cycles inequalities (4.15)
Thanks to inequalities (4.15), we guarantee the non existence of holes in our final
and optimal solution. This is due to the generation and implication of (4.15) in the
mathematical model (4.14). Nevertheless, as the number of chordless cycles can be
exponential, then we cannot explore all of the existing chordless cycles as this can
be time consuming for our optimization.
The separation of inequalities (4.15) consists in finding chordless cycles C∗vio-
lating constraints (4.15). This is a well known NP-Hard problem [45] and there is
an exponential number of possibilities. Thus, we only explore finding few number
of chordless cycles to eliminate holes in the final graph.
In order to find sufficient number of chordless cycles, we use an approximation
algorithm that recursively executes Depth First Search (DFS) method. In fact, the
main idea of this algorithm is to generate, for each vertex, a set of expanding paths
using DFS strategy until a chordless cycle is found (i.e. the selected path should
respect the conditions in Definition 4.3.1). This algorithm provides a set of violated
chordless cycles constraints in acceptable times (in order of O(n+m2), where n
and mare respectively the number of vertices and edges in the graph G). We add
these facets to our final optimization to eliminate possible holes in the final graph.
For further details about the heuristic algorithm that we use to enumerate some
chordless cycles, see [105] and [106].
4.3.2.3 Connectivity inequalities
In addition to the constraints eliminating chordless cycles (4.15), the new math-
ematical formulation (4.14) + (4.15) can lead to find optimal triangulations in a
non connected graph (see Figure 4.7 representing a triangulation of two connected
components).
v1
v2
v3
v4
v5
v6
v7
v8
v9
v10
v11
v12
Figure 4.7: Example of two connected components (triangulations) creating holes in the
final graph
Moreover and by definition, two connected components in the final graph are cre-
ating a hole. Thus, we investigate new valid inequalities (facets) to obtain a unique
optimal triangulation without holes. This consists to guarantee the connectivity of
67
the final graph. We introduce the following constraints that will be integrated to
our mathematical formulation.
Theorem 4.3.2 For any initial graph G= (A,E), and for each subset S⊆A,
the following inequality (4.16) is valid to guarantee the network connectivity for the
global coverage hole problem:
x(δ(S)) ≥1 (4.16)
where δ(S)represents the set of edges with exactly one extremity (or end-point) in
Sand the other one in the complement set of S(i.e. S).
Proof Let aand btwo nodes or antennas in the final graph that contains at least
two connected triangulations components. For instance, we can imagine that a=v3
and b=v8as illustrated in Figure 4.7. Then, it is clear that the maximum flow or the
minimum cut between aand bin the graph of Figure 4.7 is zero, as they are separated
into two connected components. The objective in our problem is to construct one
triangulation with a minimum weight when guaranteeing the connectivity. Thus,
we impose aand bto be in the same component. To do this, we simply have to
impose that the maximum flow or the minimum cut between this couple of nodes
should be greater than 1 (the value 1 is selected to guarantee that there exists at
least one edge between aand b). Recall that the considered weights in this graph
are the actual solution xe, e ∈Eof the full network coverage problem. By applying
connectivity constraints (4.16) we guarantee that all separated couples of nodes will
be jointly on the same and unique connected component.
4.3.2.4 Separation of connectivity inequalities (4.16)
The separation problem of (4.16) consists in finding the optimal set of nodes
(antennas) S∗that violates these constraints. Thus, we investigate all of the possible
couples of nodes aand bthat are not in the same connected component in the final
graph. Next to that, we identify a minimum cut (set of edges) separating aand
b, and then impose that the value of this minimum cut (using the weights x) will
not exceed 1. Exploring all of the possible sets Sviolating (4.16) is NP-Hard as
there is an exponential number of possibilities. Thus, we propose to explore only
few number of sets that can be found polynomially when solving the minimum cut
or maximum flow problem using a well known algorithm such Ford-Fulkerson [107].
In fact, few generations of (4.16) can be sufficient to guarantee the connectivity of
the final graph.
4.3.3 Complete mathematical formulation
To summarize, and by considering all of the described constraints leading to find
optimal solutions for the full network coverage problem, our mathematical formula-
68
tion is then provided by:
min Γ = Pi∈APj∈N(i)dij ×xij
S.T. :
Pj∈N(i)xij ≥2,∀i∈A
xij = 1,∀i∈A,∀j∈N(i),∀(i, j)∈I(i, j )
zijl ≤xjl +Pk∈N(i)xik,∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j)
zijl ≤1
2(xij +xil),∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j)
zijl ≥xij +xil −1,∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j)
x(E(C)) ≤3,∀C⊆A,|C| ≥ 4, C is a chordless cycle
x(δ(S)) ≥1,∀S⊆A
xij , zijl ∈ {0,1},∀i∈A,∀j∈N(i),∀l∈N(i)∩N(j);
(4.17)
Finally, we provide, in Algorithm 5, a summary of our proposed algorithm based
on branch-and-Cut formulation to cope with the full network coverage problem.
The final Algorithm 5 describes all steps, including graph construction and trans-
formation and the execution of the complete mathematical formulation provided by
(4.17).
Algorithm 5 Full network coverage algorithm : Branch-and-Cut ap-
proach
Input: A real telecommunications network (cells with antennas) with given in-
terference and coverage holes
Output: A full coverage network (no holes) with no interference
•Graph transformation of the real network G: each antenna is a node
•There is an edge between two nodes (antennas) iand jif ri+rj≥dij
•An interference is represented by an intersection of two edges
•Run the Branch and Cut optimization model (4.17)
•The optimized/obtained network has no holes and no interference
69
4.4 Performance evaluation
4.4.1 Simulation parameters and settings
The performance evaluation of our algorithm, coded in Java, is conducted using
an Intel Core CPU at 2.40 GHz with 8 GB RAM. Each initial network represented
by a graph comprises a random number of antennas following a Poisson process with
a parameter Λ = λ×space dimensions, where λis varying in the range [0.1; 1], and
space dimensions are generated according to two essential spaces (5×5 and 10×10).
Each antenna, represented by a vertex of this graph, has a radius value initialized to
rmax = 1km. The simulation considers the generation of 100 feasible instances for
each run. For sake of clarity, we summarize the simulation settings and parameters
in Table 4.1.
Table 4.1: Network coverage optimization : simulation settings and parameters
Parameters Values
Density of Antennas λ∈[0.1; 1]
Space Dimensions 5 ×5; 10 ×10;...
Poisson Parameter Λ = λ×space dimensions
Number of Antennas Poisson Distribution: P(Λ)
Antenna Coordinates Uniform Distribution:
U(0, space dimensions)
Min Coverage Radius rmin = 0.1km
Max Coverage Radius rmax = 1km
For our simulation and experiments, we use the optimization solver Cplex [94]
to solve the exact mathematical model (4.17), and we also compare and benchmark
our algorithm to other existing approaches.
4.4.2 Performance metrics
The algorithm performance assessment is based on the following metrics:
•Convergence time: is the time needed by the proposed exact algorithm to
find an optimal solution.
•Interference elimination rate: is the rate of eliminated interference such
that, and with no loss of generality, we consider an interference as an intersec-
tion of two edges in the final graph (i.e. the triangulation graph).
70
•Coverage hole: is the network coverage in terms of existing holes in the final
solution. Hence, zero holes leads to a full coverage hole.
To assess performance of the proposed approach of the full network coverage
problem using the described metrics, we considered a real trace and random in-
stances described as follows:
1. Random instances: networks with an average number of antennas ranging
in [7; 100] interval, and an average number of edges in the [18; 456] interval
according to the formula (4.1).
2. Real trace: we used a real trace from a coverage cell 4G-LTE of the network
operator Orange, in a small area in Paris [3]. This topology is in an area
containing 26 4G-LTE antennas with their given geographical positions (coor-
dinates), 94 edges and a maximum radius value of 0.4 km for each antenna.
4.4.3 Simulation results and performance analysis
4.4.3.1 Algorithm’s performance comparison with the state-of-the-art
Our performance evaluation starts by assessing the execution time needed by the
Branch-and-Cut algorithm to find the optimal solution to the full network coverage
problem.
Table 4.2: Exact algorithm performance: convergence time to the optimum
Space λ #Antennas #Edges #(4.15)*#(4.16)** Convergence Time (s)
5×5
0.3 7.5 18.81 1 1 0.016
0.5 12.5 34.63 1.13 1 0.0677
0.8 20 70.21 5.79 3.45 9.43
1.0 25 105.4 11.92 10.25 34.31
10 ×10
0.3 30 96.16 6.08 5.81 4.28
0.5 50 186.21 17.79 12.64 64.47
0.8 80 298.24 32.64 5.71 67.13
1.0 100 456.35 63 14.59 139.4
*#(4.15) is the average number of chordless cycle constrains (4.15) added to the mathematical
formulation.
** #(4.16) is the average number of connectivity constrains (4.16) added to the mathematical
formulation.
In Table 4.2, the average convergence time to the optimum remains bellow 35
seconds and 140 seconds in the worst case for the scenario with average networks size
of 100 antennas. The Branch-and-Cut algorithm scales reasonably well with problem
size (number of antennas and edges) and thanks to the efficiency of the added cutting
71
planes provided by formulas (4.15) and (4.16) guaranteeing an optimal result with
β0= 1 and β1= 0 (a covered network without holes and with a unique connected
component). Moreover, this average execution time depends on the average number
of added constraints ((4.15) and (4.16)) to our global optimization. As illustrated in
Table 4.2, running and adding these constraints is requiring negligible times thanks
to the heuristics approaches deployed to separate them in polynomial time.
In the following, we assess the convergence time and interference elimination rate
of our algorithm and benchmark them with the solution provided by Rips complex
which is a well known method to cope with the network coverage hole problem.
Recall that Rips complex is a simplicial homology-based approach that consists in
verifying the intersections between cells to detect holes and connectivity problems.
For further details about this approach, we provided in Section 2.2.2 of Chapter 2
an overview of simplicial homology approaches, including Rips complex.
As our approach is based on an exact mathematical formulation leading to always
attend the optimum, then Rips-based approach can be considered as an upper bound
to our algorithm.
Table 4.3: Performance comparison : ILP vs Rips approach
Space Density #Antennas #Edges Convergence Time (s) Interference elimination (%)
ILP Rips*ILP Rips*
5×5
0.3 7.5 18.81 0.016 0.67 100 88.89
0.5 12.5 34.63 0.0677 2.51 100 96.42
0.8 20 70.21 9.43 12.36 100 97.91
1.0 25 105.4 34.31 34.76 100 98.01
10 ×10
0.3 30 96.16 4.28 11.83 100 96.15
0.5 50 186.21 64.47 57.38 100 95.61
0.8 80 298.24 67.13 63.91 100 94.96
1.0 100 456.35 139.4 74.9 100 97.97
*Rips-based approach is one of the most efficient algorithms that use Simplcial homology tech-
niques to deal with the full network coverage problem (for more details, see Section 2.2.2 of Chapter
2).
For small graphs or networks (at most 25 antennas in Table 4.3), the Branch-and-
Cut algorithm has negligible average convergence time compared to the necessary
convergence time required by Rips method. The worst case for these graphs con-
cerns the scenario with 25 antennas in which the ILP necessitates 34.31 seconds (to
converge to the optimal solution) compared to 34.76 seconds for Rips method to
converge to a feasible solution.
For larger graphs (between 50 and 100 antennas in Table 4.3) our algorithm is
spending little more time compared to the convergence time of Rips method, as
72
we spent time to reach the optimum in the contrary of Rips method looking only
for a feasible solution, and not necessary optimal ones. This is confirmed by the
interference elimination rates provided in Table 4.3.
The interference elimination rate metric is reported in Table 4.3 and confirms
that the Branch-and-Cut algorithm performs better than Rips-based approach even
for large networks. In fact, our approach is eliminating totally the interference
for all the considered scenarios compared to Rips-based approach that proposes
final solutions with remaining interference and holes. Thus, our proposed approach
guarantees the optimality of the found solution in terms of interference elimination
and full network coverage jointly. Rips method is providing weak network coverage
hole and partial interference elimination (98.01% as the best result when considering
small graphs). Hence, by considering jointly the expected convergence time, the total
interference elimination and the full network coverage solutions, the Branch-and-Cut
approach can be used online by network providers offering connectivity and mobile
services to end-users.
In other words, constraints (4.7) are dedicated to eliminating intersections and
they are violated when two edges in the new graph, have a common point of intersec-
tion in the graph representation. Constraints (4.7) are stronger when combined with
the other valid inequalities described in the mathematical model (4.17) which finds
an optimal graph with only adjacent triangles (a full covered zone according to the
Delaunay definition). Thus, the joint optimization leads to eliminate totally these
intersections (interference) as the used Branch and Cut approach is guaranteeing
the optimality (zero interference and no holes).
In the following, we evaluate the performance of our proposed Branch-and-Cut
algorithm when considering a real cellular network and we analyze its scalability
when addressing very large instances of the full network coverage problem.
4.4.3.2 Algorithm’s performance evaluation using real traces
To better evaluate the performance of our exact algorithm based on Branch-and-
Cut method, we consider a real cellular network as shown in Figure 4.8. This network
is in a small area in Paris, containing 26 antennas, 94 edges and an interference
rate equivalent to 80.85%. Note that in our work, and with no loss of generality,
an interference is the intersection of two edges in the graphic representation of
the network. In this experimentation, we would like to apply our Branch-and-Cut
algorithm on the map of Figure 4.8 when assessing the three metrics cited above
(i.e. coverage hole, interference elimination, and convergence time).
Figure 4.9 reveals for the topology in Figure 4.8 of reasonable size, the obtained
covered network when applying our Branch-and-Cut algorithm which has the advan-
tage of exploring the entire network space at once during optimization. The obtained
triangulation in Figure 4.9 is optimal and with no holes leading to a network with
a full network coverage. Our exact algorithm has totally eliminated the existing
interference and reached the optimal solution in less than 1 sec (or exactly in 0.006
sec). This real network instance is in fact ”easy” to solve using our Branch-and-Cut
approach.
73
Figure 4.8: An Orange 4G-LTE cell map: before Branch-and-Cut optimization
Figure 4.9: An Orange 4G-LTE cell map: after Branch-and-Cut optimization
74
4.4.3.3 Scalability analysis
To discuss the scalability analysis of our approach, we propose a network instance
of 1000 antennas generated as described in Table 4.1. We apply our mathematical
formulation provided by (4.17) and the obtained convergence time is close to 116.88
seconds for an optimal solution with no holes (full coverage) and no interference.
Note that the selected network/graph instance do not contain chordless cycles and
the obtained result is a connected graph (i.e. without many connected components).
This allows to avoid generating chordless cycles inequalities (4.15) and connectivity
constraints (4.16) which can be time consuming when added to our optimization.
Indeed, the generation of these cutting planes can be time consuming even if their
used separation algorithms are converging in polynomial time, as it is depicted in the
previous simulations in Table 4.2. This explains the necessary convergence time (for
a network of 1000 antennas) which is less than the necessary time for a network with
100 antennas in which 63 chordless cycles constraints and 15 connectivity constraints
are used to attend the optimum (see Table 4.2).
4.5 Conclusion
In this chapter, we proposed an exact mathematical formulation based on Branch-
and-Cut methods to deal with the increase of inter-cell interference caused by high
density of cells in C-RAN when maintaining the full network coverage. Then and
in addition to classical mathematical modeling, we investigated new valid inequal-
ities, i.e. chordless cycles and connectivity constraints, for our optimization model
in order to scale with large number of antennas.
We evaluated the performance of our proposed approach using several simulation
scenarios and a real network map. The simulation results reveal the efficiency of
our approach that performs consistently well across all scenarios and performance
metrics. This confirms the ability of our algorithms in providing good solutions that
jointly optimize the full network coverage and minimize the inter-cell interference
caused by network densification.
However, the dense deployment of antennas in C-RAN has another consequence
which is the significant increase of the baseband processing amount needed to meet
antenna demands. Since in C-RAN the baseband processing of antennas is carried
out in centralized BBU pools, this imposes excessive bandwidth constraints and low
latency requirements on the fronthaul network connecting RRHs to the centralized
data centers. These requirements can be reduced by considering a more flexible
split of baseband processing between RRHs and centralized BBU pool. The next
chapter will focus on proposing new algorithms to optimally deploy BBU functions
in C-RAN architecture when considering different split configurations and transport
network characteristics.
Chapter 5
BBU function split placement in
C-RAN
5.1 Introduction
With the growth in mobile data traffic demands, network operators will have to
add significant amounts of spectrum as well as to increase the density of cells by
deploying more antennas. This will not only increase the inter-cell interference levels
(discussed in the previous chapter), but also will significantly increase the amount
of baseband processing needed to meet the growing number of antenna demands
and thus, increasing the amount of data traffic demands, with strict latency and
bandwidth constraints, between antennas and centralized data centers. In fact,
while we investigated in the previous chapter new solutions to reduce inter-cell
interference when guaranteeing a full network coverage, this chapter discusses how
to overcome the tradeoff between benefits of baseband processing centralization in
BBU pools and strong latency requirements on fronthaul links.
In this chapter, we investigate new approaches to find best tradeoffs between cen-
tralization and transport requirements on fronthaul network. Indeed, to meet these
requirements, more flexible distribution (or split) of baseband processing functions
will be considered between RRHs and the centralized data centers. In this context, a
range of BBU function splits is being introduced and studied, each of which presents
different needs for capacity, latency and bandwidth on the fronthaul network (see
chapter 2 for more details). In our work, we will consider 3GPP RAN split option
[108] to model the aggregated antenna demands and we seek to efficiently determine
the optimal placement of BBU functions in the centralized data centers based on
the considered split configuration and transport network characteristics. Resource
utilization gains can be achieved by determining the optimal placement of BBU
functions when jointly meeting processing and latency requirements of the antennas
demands.
To cope with this problem, we propose a mathematical model based on integer
linear programming approach to optimally deploy the baseband processing function
in the network when jointly minimizing the resource consumption and the expected
latency. Then, for sake of scalability, we investigate new heuristic algorithms based
75
76
on the construction of a multi-stage graph to obtain good solutions for larger network
size in acceptable times. Finally, we evaluate, using several simulation scenarios,
the efficiency of the proposed algorithms in finding optimal solutions with reduced
complexity.
This chapter will be organized as follows. In Section 5.2, we introduce the system
model used to address the BBU function split placement problem and we provide a
brief discussion on the problem complexity. Section 5.3 describes a complete math-
ematical formulation of the addressed problem, based on an ILP approach for small
problem size while in Section 5.4, we introduce four heuristic algorithms based on
graph theory to accelerate the convergence time when guaranteeing good solutions.
Section 5.5 reports the numerical results of performance evaluation highlighting the
efficiency and scalability of the proposed algorithms. Finally, Section 5.6 concludes
this chapter.
5.2 Problem statement
The BBU function split placement problem consists in determining the optimal
locations of baseband functions in C-RAN network when considering the 3GPP RAN
split option (a detailed overview of this functional split option with a description
of considered BBU functions can be found in Section 2.3.3 of Chapter 2). The
optimal placement of BBU functions can be achieved by finding the best tradeoff
between BBU computation centralization and fronthaul network requirements while
the network resource consumption is minimized.
In the following, we introduce the modeling of antennas demands by directed
chains when considering 3GPP RAN split option and we describe the network topol-
ogy which is considered to deploy BBU functions. Then, we present the system
model considered to optimally solve the BBU function split placement problem and
study the complexity of the addressed problem.
5.2.1 BBU function split modeling
We consider 3GPP RAN split to model our BBU function split placement prob-
lem. This split configuration is outlined as the best option in the 3GPP standards
[108]. As depicted in Section 2.3.3 of Chapter 2, this split option consists in separat-
ing the baseband processing of antennas demands into three connected components :
the first component is the PHY layer, the second contains the MAC and RLC layers
and the third component for PDCP layer. In fact, PHY, RLC and MAC functions
require lower fronthaul latency, and their processing requirement is dependent on
the traffic. Thus, co-locating these functions on the infrastructure, shared across
multiple cell sites, is expected to yield high pooling gains and also enable advanced
techniques to manage inter-cell interference. On the contrary, the PDCP layer is less
capacity-intensive and not subject to real-time constraints. Therefore, the PDCP
layer can be placed flexibly to achieve increased multiplexing gains. Figure 5.1 shows
the modeling of antennas demands according to this split option.
77
Figure 5.1: BBU function split modeling for each antenna demand
3GPP RAN split option is used to model the aggregated antenna demands which
are represented, in Figure 5.1, by a set of directed chains. Each function chain is
composed by three nodes representing four connected layers : PHY, MAC+RLC
and PDCP. For sake of simplicity, each node has computing resource requirements,
expressed in terms of CPU cores. These nodes are connected by two weighted arcs
which indicate the sequencing between BBU functions with expected communication
latency.
5.2.2 Network topology description
Figure 5.2 shows the physical network architecture that we will consider in the
BBU function split placement problem. In fact, we model our physical network
as an hierarchical architecture, which is a promising approach to achieve flexible
deployment strategies of BBU functions across shared data centers.
Figure 5.2: Physical network architecture
Our network architecture is composed by a set of embedded cloud data centers, a
limited number of edge cloud data centers, and a centralized core cloud data center,
78
each of which has its own computing processing resources represented by number of
available CPU cores. The embedded clouds are located close to the antennas and
can be supported by accelerators (DSP and FPGA). Hence, the embedded cloud will
be probably employed for executing PHY functions. We note that each embedded
cloud is assigned to exactly one antenna and thus the number of antennas is equal
to the number of embedded clouds. The edge cloud may be located further away
from the antennas, and it is typically used for aggregating and processing traffic
of multiple cell sites. Baseband functions above the cell-level PHY layer as well as
functions for inter-site basedband coordination can be located at the edge cloud.
The centralized core cloud is used for non real-time functions such as PDCP. For
the best of our knowledge, this network modeling is very similar to Orange network
topology, i.e. Next Generation Point of Presence (NG-PoP)[109].
5.2.3 System model
The aim of the BBU function split placement problem is to optimally deploy the
requested chains (Figure 5.1) on the network architecture (Figure 5.2) when jointly
meeting the CPU and latency requirements. Figure 5.3 illustrates our considered
system model for the BBU function split placement problem.
Figure 5.3: System model for BBU function split placement
Our system model contains a set of antennas/RRHs, denoted by K, each of
which has an aggregated demand represented by a directed chain (the left part
of Figure 5.3). In fact, each chain has exactly 3 virtual nodes representing PHY,
MAC+RLC and PDCP layers, and 2 virtual arcs to represent the BBU functions
chaining. Each virtual node iof antenna demand khas variable processing require-
ments expressed in terms of the number of CPU cores denoted by ck
i. Each virtual
arc (i, i + 1) connecting two consecutive virtual nodes iand i+ 1 has a latency re-
quirements, denoted by lk
i,i+1. On the right part of Figure 5.3, we model the physical
network as an undirected graph G= (J,E) where Vand Eare the sets of available
physical nodes and arcs, respectively. Each physical node j, in which jcan be an
edge node, embedded node, or core node in the corresponding cloud, has a limited
CPU capacity denoted by Cj. There is an arc between two physical nodes jand
79
j0of two different levels with a fronthaul latency denoted by Lj,j0. Note that if the
physical node j0is not a neighbor of j, then the latency Lj,j0is the sum of latency
values on the shortest path between nodes jand j0. We denote by P(j), the set of
all physical nodes j0such that there exists a shortest path between nodes jand j0.
For sake of clarity, we summarize in Table 5.1 all parameters and variables that
are used to define our system model. These notations will also be used later in
Section 5.3.
Table 5.1: BBU function split placement problem : variables and parameters
G= (J,E) : weighted directed graph (the right part of Figure
5.3, for instance)
J: set of physical nodes jincluding embedded
clouds, edge clouds and one core cloud
E: a set of communication links between different
physical nodes j
Cj: Number of CPU cores available in the physical
node j(embedded, edge and core clouds)
L(j,j0): Latency on the link joining two physical nodes
jand j0
j0∈P(j) : set of all physical nodes that joins the physical
node jto j0
K: set of directed chains kthat represents the ag-
gregated demands (the left part of Figure 5.3)
i∈ {1,2,3}: set of virtual nodes iof each chain kwhere 1 rep-
resents the PHY layer, 2 represents the second
node which contains the MAC and RLC layers
and 3 represents the PDCP layer
ck
i: Number of CPU cores requested for process-
ing the virtual node i(PHY, MAC + RLC and
PDCP layers) of chain k
lk
(i,i+1) : Expected latency between two consecutive vir-
tual nodes iand i+ 1 of the same chain k
5.2.4 Problem complexity
Before investigating new algorithms, we discuss the complexity of the BBU func-
tion split placement problem. We note that the aim of this problem is to determine
the optimal mapping of the requested chains on the multi-stage network when jointly
meeting strong latency and CPU requirements of the aggregated demands. We in-
troduce the following theorem for the complexity of the addressed problem:
80
Theorem 5.2.1 The BBU function split placement problem is NP-Hard.
Proof The aim of BBU function split placement problem is to optimally deploy the
baseband processing functions of antenna demands on the network topology (see
Figure 5.3) when jointly meeting latency and CPU requirements. By considering
the system model described in Section 5.2.3, our problem consists in finding the
optimal mapping of a set of directed chains (BBU functions) on multi-stage graph
(physical network topology). This problem formulation is very close to the well
known Virtual Network Embedding (VNE) problem (described in Figure 5.4).
Figure 5.4: Example of Virtual Network Embedding problem. Source: [4]
In fact, the VNE problem consists in finding an optimal mapping of Virtual
Network Requests (VNR) on a physical network when both computational and net-
work requirements are met, which is equivalent to find a mapping of two undirected
graph. This problem has been very well studied in the literature and its complexity
has been investigated. In fact, authors in [110] proposed a general polynomial-time
reduction which implies the NP-hardness of the VNE problem even when restricting
the problem to very specific subclasses of graphs (for more details on VNE problem,
see also [111] and [112]). Since our problem has additional constraints compared
to VNE problem, such as the chaining of BBU functions, we deduce that the BBU
function split placement problem is also NP-Hard.
5.3 Exact mathematical approach
In this section, we investigate a new mathematical formulation based on ILP
approach to address the BBU function split placement. This problem consists in
determining the optimal mapping of the requested chains on the physical network
when optimizing the resource consumption in terms of CPU cores and the expected
latency.
81
Decision variables
We start our problem’s modeling by introducing three decision variables as fol-
lows:
•xk
i,j is a binary variable, the value of which is 1 if the virtual node, i.e. BBU
function, iof the antenna kis placed on the physical node j, and 0 otherwise.
•yk
(i,i+1);(j,j0)is a binary variable, the value of which is 1 if a virtual edge (i, i+1)
for a link between two BBU functions of the antenna demand kis placed on a
physical path joining two physical nodes jand j0, and 0 otherwise.
•zjis a binary variable, the value of which is 1 if a physical node jis used, and
0 otherwise.
We recall that all parameters, which will be used in the ILP formulation, are
summarized in Table 5.1 in the previous section.
Objective
The objective of the BBU function split placement is to map jointly all the
connected chains to the physical network while minimizing the total CPU core con-
sumption and the end-to-end latency. It is given by:
min F=−X
j∈J
Cjzj−X
k∈KX
i∈{1,2,3}
ck
ixk
i,j
+X
k∈KX
j∈JX
j0∈P(j)X
i∈{1,2}
L(j,j0)yk
(i,i+1);(j,j0)
(5.1)
Formula (5.1) is the mapping from the left part of Figure 5.3 to the right part of
Figure 5.3. The first term of (5.1) denotes the residual computing (CPU) resources
in different data centers, i.e. embedded, edge and core cloud data centers while the
second term represents the total costs in terms of latency provided for processing the
BBU functions chains (the aggregated antenna demands). The optimum solution of
the BBU function split placement problem will jointly reduce the expected latency
on the fronthaul network and maximize the residual computing resources, in terms
of CPU cores, in each node of physical network. This solution should respect a set
of constraints of the addressed problem which will be detailed in the following.
Constraints
The optimization problem for the BBU function split placement has a certain
set of constraints as follows:
•Constraints (5.2) avoid the placement of the physical layer of the chain kon
the other embedded nodes that are not assigned to the antenna k.
X
j∈J1
1(k,j)xk
1,j = 0,∀k∈K(5.2)
where 1(k,j)is equal to 1 if the antenna kis not assigned to the embedded
node j, and 0 otherwise and J1represents the set of embedded clouds.
82
•Constraints (5.3) guarantee that each virtual node, i.e., a BBU function, is
deployed on exactly one physical node.
X
j∈J
xk
i,j = 1,∀k∈K,∀i∈ {1,2,3}(5.3)
•Constraints (5.4) ensure that the placement of BBU functions cannot consume
more resources than that available on the selected physical node.
X
k∈KX
i∈{1,2,3}
xk
i,j ×ck
i≤Cj,∀j∈J(5.4)
•Constraints (5.5) are used to guarantee the chaining of the BBU functions.
For example, if the PHY layer is deployed on the physical node i, then the
virtual node that contains the MAC and RLC layers should be deployed on j
such that there is a physical path P(j) starting from node ito node j.
xk
i,j ≤X
j0∈P(j)
xk
i+1,j0,∀k∈K,∀i∈ {1,2},∀j∈J(5.5)
•Constraints (5.6) and (5.7) are used to ensure that if a virtual node iis de-
ployed on a physical node j, i.e. xk
i,j = 1, and the virtual node i+ 1 is
hosted by a physical node j0, i.e. xk
i+1,j0= 1, then the virtual arc (i, i + 1)
should be deployed on the physical path starting from node jto node j0, i.e.
yk
(i,i+1);(j,j0)= 1.
X
j∈V
yk
(i,i+1);(j,j0)=xk
i+1,j0,∀k∈K,∀i∈ {1,2},∀j0∈P(j) (5.6)
X
j0∈P(j)
yk
(i,i+1);(j,j0)=xk
i,j ,∀k∈K,∀i∈ {1,2},∀j∈J(5.7)
•Constraints (5.8) guarantee that each virtual arc (i, i+1) is deployed on exactly
one physical path.
X
j∈JX
j0∈P(j)
yk
(i,i+1);(j,j0)= 1,∀k∈K,∀i∈ {1,2}(5.8)
•Constraints (5.9) impose that the fronthaul latency on the selected path in
the physical network must not exceed the latency requirements of the BBU
function chains.
L(j,j0)×yk
(i,i+1);(j,j0)≤lk
(i,i+1),∀k∈K,∀i∈ {1,2},∀j∈J,∀j0∈P(j) (5.9)
•Constraints (5.10) indicate that if there exists at least one BBU function de-
ployed on a physical node j, then the former should be used to host other
virtual nodes if necessary.
xk
i,j ≤zj,∀j∈J,∀k∈K,∀i∈ {1,2,3}(5.10)
83
Complete mathematical formulation
We summarize our mathematical model in the following ILP formulation (5.11).
This ILP model uses Branch-and-Bound method (see Section 2.2.1.1 of Chapter 2)
to provide for the BBU function split placement problem, the optimum solution
from all possible ones.
min F=−Pj∈JCjzj−Pk∈KPi∈{1,2,3}ck
ixk
i,j +
Pk∈KPj∈JPj0∈P(j)Pi∈{1,2}L(j,j0)yk
(i,i+1);(j,j0)
S.T. :
Pj∈J11(k,j)xk
1,j = 0,∀k∈K
Pj∈Jxk
i,j = 1,∀k∈K,∀i∈ {1,2,3}
Pk∈KPi∈{1,2,3}xk
i,j ×ck
i≤Cj,∀j∈J
xk
i,j ≤Pj0∈P(j)xk
i+1,j0,∀k∈K,∀i∈ {1,2},∀j∈J
Pj∈Jyk
(i,i+1);(j,j0)=xk
i+1,j0,∀k∈K,∀i∈ {1,2},∀j0∈P(j)
Pj0∈P(j)yk
(i,i+1);(j,j0)=xk
i,j ,∀k∈K,∀i∈ {1,2},∀j∈J
Pj∈JPj0∈P(j)yk
(i,i+1);(j,j0)= 1,∀k∈K,∀i∈ {1,2}
L(j,j0)×yk
(i,i+1);(j,j0)≤lk
(i,i+1),∀k∈K,∀i∈ {1,2},∀j∈J,∀j0∈P(j)
xk
i,j ≤zj,∀j∈J,∀k∈K,∀i∈Jv
xk
i,j ∈ {0,1},∀i∈ {1,2,3},∀j∈J;
y(i,i+1);(j,j0)∈ {0,1},∀i∈ {1,2},∀j∈J;
zj∈ {0,1},∀i∈ {1,2},∀j∈J;
(5.11)
The obtained solution by the ILP approach will be used as reference and
optimal solution to evaluate the performance of the approximation algorithms
that we will introduce, in the next section, to address large problem instances in
acceptable times. This is not feasible with the exact approach based on ILP model
(5.11) due to the NP-Hardness of our addressed problem.
84
5.4 Approximation approaches : multi-stage graph
algorithms
To deal with large problem sizes, we propose efficient heuristic algorithms that
converge to optimal or near-optimal solutions. These heuristics are based on the
construction of an extended multi-stage graph Gm= (Nm,Em,3) by focusing on the
number of available physical nodes, and the three BBU function layers. Emis the
set of arcs of Gmthat will be clearly identified in the following.
1
4
2
4
K
20
PHY MAC+
RLC PDCP
l12 l23
11
#CPU #CPU #CPU
2
Figure 5.5: A multi-stage graph example
To populate the multi-stage graph Gm, we suppose that each physical node,
in the edge cloud and core cloud, is able to host different BBU functions (PHY,
MAC+RLC, and PDCP) of different antennas and meeting the node’s limited ca-
pacity in terms of CPU cores. Thus, from the multi-stage graph construction (Fig-
ure 5.5), the set of arcs between nodes jand j0in two different levels are weighted
by the value of a shortest path Pj,j0in terms of latency values on the links between
physical nodes (using Dijkstra algorithm which is detailed in Section 2.2.1.4 of Chap-
ter 2). We consider three levels in our multi-stage graph and each level corresponds
to a BBU function.
Figure 5.5 represents the extended multi-stage graph that we obtain for an ex-
ample of 4 available physical nodes in the first level, and two available nodes in the
second and third levels. As shown in the multi-stag graph, we can find the same
physical node at different levels depending on their available CPU capacity. We
recall that each physical node can host more than one BBU function of the same
BBU function chain.
Based on this graph, we propose four strategies of multi-stage approach to deploy
BBU function chains on the available data centers.
•MIN-MIN: considers the BBU function chain that has the minimum total
amount of CPU resources, i.e. the sum of CPU cores in each BBU function,
85
and deploys each BBU function, i.e. virtual node, on the physical node, i.e.
data center, which has the minimum amount of available computing resources
(CPU cores). This will be repeated until all BBU function chains are deployed
on the physical network, otherwise, the problem has no complete solution using
this strategy.
•MIN-MAX: selects the BBU function chain which has the minimum total
amount of requested CPU resources and starts the placement of each virtual
node by contacting the data center with the maximum amount of avail-
able CPU resources. This will be repeated until all BBU function chains are
deployed on the physical network, otherwise, the problem has no complete
solution using this strategy.
•MAX-MIN: considers the BBU function chain which has the maximum
amount of requested CPU resources, and then places each BBU function on
the physical node which has the minimum amount of available CPU resources.
This will be repeated until all BBU function chains are deployed on the physical
network, otherwise, the problem has no complete solution using this strategy.
•MAX-MAX: selects the BBU function chain which has the maximum total
amount of requested CPU resources, and deploys each BBU functions on the
physical which has the maximum amount of available CPU resources. This
will be repeated until all BBU function chains are deployed on the physical
network, otherwise, the problem has no complete solution using this strategy.
According to the previous strategies, we summarize the multi-stage approach in
Algorithm 6.
Algorithm 6 Multi-stage graph algorithm
Input: Aggregated BBU functions chains, Physical network.
Output: A joint mapping of all the requested chains on the physical network.
Step 1: Select a strategy (MIN-MIN, MIN-MAX, MAX-MIN or MAX-MAX) to
find the optimal or near-optimal mapping of BBU function chains;
Step 2: Sort all the requested chains according to the total amount of requested
CPU for each chain c;
Step 3: Create the multi-stage graph Gmaccording to the description given above;
Step 4: If the selected chain cis deployed : repeat until all chains are deployed,
otherwise : the problem has no complete*solution.
∗There is a solution when all the BBU functions chains are deployed successfully.
86
Multi-stage based algorithm’s complexity
It is important to evaluate the complexity of our proposed multi-stage algorithm.
We note that the addressed problem is NP-Hard, and we need rapid and cost-efficient
approaches to cope with this complexity. As described in Algorithm 6, we detail
below the complexity of our algorithm :
•Step 1 is just used to select which strategy we will follow among MIN-MIN,
MIN-MAX, MAX-MIN and MAX-MAX strategies.
•Step 2 consists in sorting the aggregated demands according to the total
amount of requested CPU cores. For that, we used the well known ”Quicksort
method” with a complexity of nln(n) (in our case, nrepresents the number of
antennas or aggregated demands).
•In Step 3, we construct a multi-stage graph for each demand, thus its com-
plexity in the worst case is equal to the number of physical nodes which is
negligible in our case.
•Finally, we need to iterate ntimes to deploy all aggregated demands on phys-
ical network.
Therefore, our proposed algorithm has a global complexity of O(nln(n) + n) in
the worst case. This complexity is negligible to address large network sizes of BBU
function split placement problem.
5.5 Performance evaluation
In this section, we assess the performance of the proposed algorithms using two
simulation scenarios. The four heuristic algorithms are benchmarked with the ILP
solution for small and medium problem sizes. We also discuss the scalability of our
heuristic algorithms when considering large network sizes.
5.5.1 Simulation parameters and settings
For our simulations, we consider a physical network topology similar to that
in [77]. Specifically, we consider a random number of antennas ranging in [20,500], a
number of edge clouds in the interval [10,20], and 1 core node according to Orange
network topology (see [109] for more details). We then use the two following
scenarios:
•Scenario 1: Random Graphs: For each BBU function, we randomly generate
a CPU requirement from [1,9] CPU cores, and a random amount of available
CPU cores ranging in [10,50] CPU cores for the embedded cloud, [30,80] CPU
cores for the edge cloud, and {50,100}CPU cores for the core cloud. Moreover,
the latency requirements on each arc of each BBU function chain as well as
the fronthaul latency values are randomly generated according to [108] and
[80].
87
•Scenario 2: Euclidean Graphs: The same parameters as described above are
adopted except for the latency on the physical arcs which is generated based
on the Euclidean distance1.
For our simulation and experiments, the performance evaluation of the proposed
algorithms is conducted using an Intel Core CPU at 2.40 GHz with 8 GB RAM. We
used the IBM optimization solver Cplex [94] to solve the exact mathematical model
in (5.11) and we implemented our algorithms in Java.
5.5.2 Performance metrics
In order to evaluate the performance of our heuristics compared to the ILP
formulation algorithm used as ”the reference and optimal solution”, we define two
performance metrics as follows :
•Convergence time: is the time needed by the algorithms to converge to their
best solutions.
•Gap: is used to benchmark the proposed heuristics compared with the exact
ILP formulation algorithm used as “the reference and optimal solution”. This
metric can be expressed as follows:
Gap(%) = Costheur −Costoptimum
Costoptimum
(5.12)
where Costheur and Costoptimum are the objective functions (according to (5.1))
of the proposed heuristic approaches and the ILP solution, respectively.
5.5.3 Simulation results and performance analysis
Figures 5.6 depicts the convergence time of the ILP model and the four heuristic
algorithms when considering Random and Euclidean graphs, respectively.
100 200 300 400 500
0
300
600
900
1,200
1,500
1,800
2,100
2,400
Number of Antennas
Convergence Time (s)
ILP
MIN-MIN Heuristic
MAX-MIN Heuristic
MAX-MAX Heuristic
MIN-MAX Heuristic
(a) Euclidean graphs
100 200 300 400 500
0
300
600
900
1,200
1,500
1,800
2,100
Number of Antennas
Convergence Time (s)
ILP
MIN-MIN Heuristic
MAX-MIN Heuristic
MAX-MAX Heuristic
MIN-MAX Heuristic
(b) Random graphs
Figure 5.6: Algorithms’ convergence time using 20 edge cloud data centers
1the distance between two points iand jwith the coordinates (xi, yi) and (xj, yj) is provided by
p(xj−xi)2+ (yj−yi)2) where the coordinates xand yare generated according to the positions
of the nodes in Figure 5.3
88
The ILP solution is obtained based on the branch and bound algorithm that
explores the convex hull of the BBU function split placement problem and then
enumerates all the feasible solutions. This causes an exponential increase in terms
of convergence time, which is undesirable.
Table 5.2 evaluates the quality of solutions obtained by heuristic algorithms by
calculating the cost gap (according to Formula 5.12) for Euclidean and Random
graphs. We note that the heuristic algorithm provides an optimum solution when
the value of the gap is equal to 0 in Table 5.2.
Table 5.2: Algorithms’ performance comparison : ILP vs heuristic variants
#Antennas #Edge Clouds Heuristic Variant Gap (Euclidean)*% Gap (Random)** %
60
10
MIN-MIN 0 3.07
MIN-MAX 0 0
MAX-MIN 2.96 2.62
MAX-MAX 0 0
15
MIN-MIN 0 2.79
MIN-MAX 0 0
MAX-MIN 2.24 2.22
MAX-MAX 0 0
20
MIN-MIN 1.88 2.11
MIN-MAX 0 0
MAX-MIN 2.05 1.94
MAX-MAX 0 0
80
10
MIN-MIN 2.49 3.08
MIN-MAX 0 0
MAX-MIN 2.64 2.56
MAX-MAX 0 0
15
MIN-MIN 2.92 2.76
MIN-MAX 0 0
MAX-MIN 2.3 2.32
MAX-MAX 0 0
20
MIN-MIN 2.55 2.28
MIN-MAX 0 0
MAX-MIN 1.87 2.0
MAX-MAX 0 0
*The average gap (as described in 5.12) using scenario 2 : Euclidean graph.
** The average gap (as described in 5.12) using scenario 1 : Random graph.
As shown in Table 5.2, the MAX-MAX algorithm can provide optimal solutions
(when the Gap = 0) of the problem in few seconds. This algorithm focuses on select-
ing chains according to the maximum amount of required CPU, and then choosing
the available physical nodes with the maximum amount of CPU cores. Neverthe-
less, this algorithm is not able to explore much more alternatives or solutions in the
space of feasible solutions when applied to large number of antennas. In fact, the
MAX-MAX algorithm exploration is limited and can only address the cases with
400 antennas with 20 edge nodes as illustrated by Figure 5.6.
The MIN-MAX algorithm provides also optimal solutions (see Table 5.2 when
the Gap = 0) for the BBU function split placement even in the case of a large
89
network size when we consider sufficient available resources. Unfortunately, this
algorithm is not able to deeply explore the convex hull of the problem especially for
the case of small number of edge nodes (see Figure 5.6).
The MAX-MIN and MIN-MIN algorithms provide near-optimal solutions. In
fact, Table 5.2 shows that a maximum gap of 2.96% for Euclidean graphs, and
3.08% for Random graphs are achieved when considering algorithms MAX-MIN
and MIN-MIN respectively. These algorithms are able to explore deeply the space
of feasible solutions. For these reasons, MAX-MIN and MIN-MIN can address
larger instances of BBU function split placement problem, compared to MAX-MAX
and MIN-MAX approaches. In fact, Figure 5.6 depicts that MAX-MIN and MIN-
MIN can provide solutions for our BBU function split placement problem when the
number of antennas demands reaches 500 on a physical network containing 20 edge
nodes while the MAX-MAX and MIN-MAX algorithms is able to solve problem
instances of up to 400 antennas when considering 20 edge nodes.
In the following, we define residual resources (CPU) as the available and unused
amount of servers’ CPU using Euclidean graphs to assess the resource allocation of
our heuristic algorithms. These algorithms are benchmarked by the ILP solution.
We observe from Figure 5.7 that the solutions in terms of CPU residual resources
obtained from the heuristic algorithms are close to those of the ILP solution. The
physical nodes allocated by the heuristic algorithms are similar to those of the
ILP solution. This is represented by negligible difference/gap between the curves
illustrated in Figure 5.7.
20 40 60 80 100
1,000
1,200
1,400
1,600
1,800
2,000
2,200
2,400
2,600
2,800
3,000
3,200
3,400
3,600
Number of Antennas
CPU Residual Resources
ILP
MIN-MIN Heuristic
MAX-MIN Heuristic
MAX-MAX Heuristic
MIN-MAX Heuristic
Figure 5.7: CPU residual resources behavior
Figure 5.8 illustrates the total and incurred latency when increasing the num-
ber of antennas. We observe that this latency, for the ILP,MAX-MAX and
MIN-MAX solutions is in general close to zero and this is due to the placement
of the BBU functions at the same physical node. In fact, we assume that if two
or more BBU functions are placed in the same physical node, then the necessary
latency between these functions is negligible (close to zero). Nevertheless, the solu-
tion provided by the MIN-MIN and MAX-MIN algorithms generate some latency
(capped by 60µsec in the worst case) to guarantee the chaining of the deployed BBU
functions.
90
20 40 60 80 100
0
10
20
30
40
50
60
70
80
90
100
Number of Antennas
Latency (µsec)
ILP
MIN-MIN Heuristic
MAX-MIN Heuristic
MAX-MAX Heuristic
MIN-MAX Heuristic
Figure 5.8: Latency behavior
The performance assessment would not be complete without addressing the scal-
ability of our proposed algorithms for large problem instances. We illustrate in
Table 5.3 the obtained simulation results regarding the scalability of the exact ap-
proach based on ILP formulation and the heuristic algorithms evaluated on Eu-
clidean graphs.
Table 5.3: Scalability and convergence time comparison using Euclidean graphs
#Antennas #Edge Clouds Heuristic Variant Heuristic Time (s) ILP Time
100
10
MIN-MIN 2.25
29.11min
MIN-MAX 2.49
MAX-MIN 4.35
MAX-MAX 2.87
15
MIN-MIN 2.98
33.47min
MIN-MAX 3.26
MAX-MIN 6.17
MAX-MAX 4.00
20
MIN-MIN 4.19
42.85min
MIN-MAX 3.98
MAX-MIN 7.99
MAX-MAX 4.86
200
10
MIN-MIN 42.51
>12h
MIN-MAX No solution*
MAX-MIN 82.15
MAX-MAX 35.43
15
MIN-MIN <1min
>15h
MIN-MAX No solution*
MAX-MIN <2min
MAX-MAX 43.01s
20
MIN-MIN <1min
>16h
MIN-MAX <1min
MAX-MIN <2min
MAX-MAX <1min
*BBU functions chains cannot be all deployed.
Table 5.3 shows a significant gap between the convergence times of the four
heuristic algorithms and the ILP solution. This is expected as the ILP explores all
91
feasible solutions. Clearly, our approximation algorithms can reach the optimal or
near-optimal solutions with substantially less computation time.
In some cases as shown in Table 5.3, for example, the case of 200 antennas and 10
edge clouds, the MIN-MAX algorithm is not able to find a solution and this is due to
two reasons: (i) there is not enough resources to host the demands and (ii) the MIN-
MAX algorithm cannot explore certain feasible solutions. Moreover, we observe that
the MAX-MIN algorithm requires slightly more computation time than those of the
other heuristic algorithms. This is due to the fact that the MAX-MIN algorithm is
deploying the different BBU functions on different servers necessitating the usage of
a shortest path before reaching a solution meeting the latency requirements.
5.6 Conclusion
In this chapter, we studied the BBU function split placement problem represented
as a mapping of BBU function chains on a hierarchical network topology modeled
as a multi-stage graph. We proposed an exact formulation based on integer linear
programming approach to describe the convex hull of the addressed problem. How-
ever, this optimization is known to not to scale for large problem instances due to
the NP-Hardness of the BBU function split placement problem. For a large network
size, we proposed new approximation algorithms based on the construction of an
extended multi-stage graph to optimally deploy the requested chains in the network
architecture when taking into account the high latency requirements and the limited
processing capacity of physical nodes (in the centralized data centers).
The evaluation results revealed the efficiency of the MAX-MAX and MIN-MAX
algorithms in terms of optimal solutions and convergence time even in the case of a
large network size. The MIN-MIN and MAX-MIN algorithms are also favorable in
terms of near-optimal solutions and convergence time.
The next chapter will be dedicated to summarize all our contributions of the
previous chapters, and will highlight the important research challenges that we would
like to address in the future.
92
Chapter 6
Conclusions and perspectives
In this chapter, we summarize our contributions for C-RAN optimization and
propose some open future research topics.
Conclusions and main contributions
We addressed in this manuscript resource allocation problems in the context of C-
RAN by proposing new scalable and cost-efficient algorithms based on combinatorial
optimization techniques. In fact, C-RAN is considered as a promising network
architecture for 5G mobile networks to meet diverse service requirements and to
deal with new business opportunities, e.g. eMBB, mMTC, URLLC, etc. C-RAN
is expected to reduce network costs, e.g. CAPEX and OPEX, and improve the
resource utilization efficiency.
To achieve these goals, we investigated in Chapter 1 new algorithms to assign
antennas (RRHs) demands to available edge data centers when latency and process-
ing requirements are met. Our proposal aims to optimally address the RRH-BBU
assignment problem by jointly optimizing the resource utilization and the commu-
nication latency on the fronthaul network. We modeled this problem using an exact
ILP formulation to determine the most appropriate strategies in RRH-BBU assign-
ment. This exact algorithm optimizes the resource consumption (in terms of active
edge data centers) and communication latency associated for assigning antennas de-
mands to available edge data centers. We proved also that the addressed problem
is NP-Hard. Thus, in order to address large problem instances, we proposed three
approximation algorithms : matroid-based approach, b-matching-based formulation
and multiple knapsack-based algorithm to meet a larger number of antennas de-
mands in negligible times. The performance evaluation revealed that the matroid
and b-matching algorithms can rapidly find good RRH-BBU assignment solutions
when achieving resource utilization gains.
However, such gains can be achieved only when reducing the inter-cell interfer-
ence caused by the high density of cells in C-RAN. To consider these constraints, we
proposed a complete mathematical formulation based on Branch-and-Cut methods
to jointly minimize the levels of inter-cell interference and maintain the full network
93
94
coverage. We added new valid inequalities, i.e. chordless cycles and connectivity
constraints, for our optimization model in order to reduce the space solution, i.e.
convex hull, of the addressed problem and then accelerate the necessary convergence
time to obtain optimum solutions. We evaluated the performance of our proposed
approach by comparing with Rips-based approach which is considered as one of the
most efficient algorithms to address the full network coverage problem. We con-
sidered different simulation scenarios and a real cellular network, e.g. small area
in Paris, to evaluate the performance of our proposal. In both cases, the obtained
results showed the efficiency of our approach that performs consistently well across
all scenarios and performance metrics and proved its ability in providing good so-
lutions that jointly optimize the full network coverage and minimize the inter-cell
interference caused by network densification.
In addition to the increase of inter-cell interference levels, improving the existing
density of cells by deploying more antennas will significantly increase the amount of
baseband processing needed to meet the growing number of antennas demands. This
leads also to increase the amount of data traffic demands, with strict latency and
bandwidth constraints, between antennas and centralized data centers. To address
these issues, we discussed in Chapter 5 how to find best trade-offs between benefits
of C-RAN in terms of baseband processing centralization and strong latency require-
ments on fronthaul links, which represent the main obstacle for the deployment of
C-RAN. For that, we investigated in Chapter 5 new algorithms to determine the
optimal locations of BBU processing functions between cell sites (RRHs) and BBUs
when considering 3GPP solution, outlined as the best split option proposal, to relax
the latency and bandwidth requirements on the fronthaul network. We proposed
an exact formulation based on ILP modeling and heuristic algorithms to provide
optimal or near optimal solutions. Then, we highlighted the performance of each
proposed algorithm in terms of optimal solutions, convergence time and scalability.
Future research directions
In the following, we propose some open research challenges that we would like
to address in the future :
•For sake of simplicity, we only considered the communication latency on the
fronthaul network joining antennas (RRHs) and centralized data centers (BBU
pools) to model our resource allocation problems in the context of C-RAN. It
would be very interesting to consider also the BBU processing time (compute
latency) required to perform different BBU functions co-located in the edge
data centers. This can lead to nonlinear objective functions that should be
efficiently optimized. The problem becomes more complex and requires depth
studies relying on Lagrangian relaxations, for instance. Furthermore, the data
traffic on the fronthaul network, which connects the antennas and the cen-
tralized data centers, can be transmitted using different protocols including
CPRI and OBSAI. The fronthaul network can be realized by different technolo-
gies, such as optical fiber communication, standard wireless communication,
95
or mmWave communication. The impact of these protocols and technologies
can be investigated to better evaluate the performance of our proposed models
and algorithms.
•In Chapter 4, we discussed how to jointly reduce inter-cell interference and
guarantee the full network coverage for C-RAN. In order to achieve these con-
flicting goals, we modeled the coverage of each antenna by circular area with
various coverage radii. However, in real life networks, antennas do not have
regular shapes and their coverage area depends on geographic, environmen-
tal and network parameters. As future work, our problem modeling can be
extended by taking into account the irregularity of antennas’ coverage area
to better evaluate the performance of our approach when considering real life
constraints. To reach these objectives, new mathematical modeling of these
constraints should be investigated.
•The performance of our proposed exact ILP formulation for full network cover-
age problem (Chapter 4) has been evaluated using different network topologies.
Among these networks, we identified some network variants, which are only
composed by cliques with at most four edges, that can be solved optimally in
negligible times when the integrity constraints are relaxed. Hence, new mathe-
matical formulation with valid inequalities can be investigated to characterize
polynomial time variants of the NP-Complete full network coverage problem.
•Machine learning approaches can be used to address resource allocation prob-
lems in the context of C-RAN. In fact, a large amount of numerical results
has been collected from our different simulation scenarios. These simulation
results can be exploited using machine learning algorithms to improve the
quality of the solutions founded by our proposed optimization algorithms. In
fact, machine learning techniques have been recently used in many mathemat-
ical optimization to accelerate the necessary convergence time to find optimal
solutions (see for instance [113], [114] and [115]). Hence, hybrid machine-
learning and optimization methods can be used to improve the performance
of our proposed algorithms for resource allocation problems in the context of
C-RAN.
•Network slicing is another challenge that has the merit to be addressed to en-
able the deployment of next generation mobile networks (5G). In fact, network
slicing is considered as one of the key enablers to enhance the flexibility of C-
RAN and to meet new 5G services and opportunities. It consists in deploying
multiple logical networks over a shared physical infrastructure, and then pro-
viding as a service or slice. In Chapter 5, exact and heuristic algorithms are
proposed to address the problem of the optimal placement of BBU function
chains on a shared network infrastructure. This problem is very similar to
the problem of how to allocate the shared resources to slices. Similarly to our
proposed algorithms, we can propose new optimization approaches to address
the network slicing challenges in the context of C-RAN.
96
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Titre : Cloud-Radio Access Networks : Conception, Optimisation et Algorithmes
Mots cl´
es : C-RAN, 5G, Optimisation Combinatoire, Allocation des Ressources
R´
esum´
e : Cloud-Radio Access Network (C-RAN) est une archi-
tecture prometteuse pour faire face `
a l’augmentation exponentielle
des demandes de trafic de donn´
ees et surmonter les d ´
efis des
r´
eseaux de prochaine g´
en´
eration (5G). Le principe de base de C-
RAN consiste `
a diviser la station de base traditionnelle en deux
entit´
es : les unit´
es de bande de base (BaseBand Unit, BBU) et
les tˆ
etes radio distantes (Remote Radio Head, RRH) et `
a mettre
en commun les BBUs de plusieurs stations dans des centres de
donn´
ees centralis´
es (pools de BBU). Ceci permet la r ´
eduction des
coˆ
uts d’exploitation, l’am´
elioration de la capacit´
e du r´
eseau ainsi
que des gains en termes d’utilisation des ressources. Pour at-
teindre ces objectifs, les op´
erateurs r´
eseaux ont besoin d’investi-
guer de nouveaux algorithmes pour les probl´
emes d’allocation de
ressources permettant ainsi de faciliter le d´
eploiement de l’archi-
tecture C-RAN. La plupart de ces probl `
emes sont tr`
es complexes
et donc tr`
es difficiles `
a r´
esoudre. Par cons´
equent, nous utilisons
l’optimisation combinatoire qui propose des outils puissants pour
adresser ce type des probl`
emes.
Un des principaux enjeux pour permettre le d ´
eploiement du C-RAN
est de d´
eterminer une affectation optimale des RRHs (antennes)
aux centres de donn´
ees centralis´
es (BBUs) en optimisant conjoin-
tement la latence sur le r´
eseau de transmission fronthaul et la
consommation des ressources. Nous mod´
elisons ce probl`
eme `
a
l’aide d’une formulation math´
ematique bas´
ee sur une approche de
programmation lin´
eaire en nombres entiers permettant de determi-
ner les strat´
egies optimales pour le probl`
eme d’affectation des res-
sources entre RRH-BBU et nous proposons ´
egalement des heu-
ristiques afin de pallier la difficut ´
e au sens de la complexit´
e algo-
rithmique quand des instances larges du probl´
eme sont trait´
ees,
permettant ainsi le passage `
a l’´
echelle. Une affectation optimale
des antennes aux BBUs r´
eduit la latence de communication at-
tendue et offre des gains en termes d’utilisation des ressources.
N´
eanmoins, ces gains d´
ependent fortement de l’augmentation des
niveaux d’interf´
erence inter-cellulaire caus´
es par la densit´
e´
elev´
ee
des antennes d´
eploy´
ees dans les r`
eseaux C-RANs. Ainsi, nous
proposons une formulation math´
ematique exacte bas´
ee sur les
m´
ethodes Branch-and-Cut qui consiste `
a consolider et r´
e-optimiser
les rayons de couverture des antennes afin de minimiser les in-
terf´
erences inter-cellulaires et de garantir une couverture maximale
du r´
eseau conjointement. En plus de l’augmentation des niveaux
d’interf´
erence, la densit´
e´
elev´
ee des cellules dans le r´
eseau C-
RAN augmente le nombre des fonctions BBUs ainsi que le trafic
de donn´
ees entre les antennes et les centres de donn´
ees centra-
lis´
es avec de fortes exigences en terme de latence sur le r´
eseau
fronthaul. Par cons´
equent, nous discutons dans la troisi`
eme partie
de cette th`
ese comment placer d’une mani´
ere optimale les fonc-
tions BBUs en consid´
erant la solution split du 3GPP afin de trouver
le meilleur compromis entre les avantages de la centralisation dans
C-RAN et les forts besoins en latence et bande passante sur le
r´
eseau fronthaul. Nous proposons des algorithmes (exacts et heu-
ristiques) issus de l’optimisation combinatoire afin de trouver rapi-
dement des solutions optimales ou proches de l’optimum, mˆ
eme
pour des instances larges du probl`
eme.
Title : Cloud-Radio Access Networks : Design, Optimization and Algorithms
Keywords : C-RAN, 5G, Combinatorial Optimization, Resource Allocation
Abstract : Cloud Radio Access Network (C-RAN) has been propo-
sed as a promising architecture to meet the exponential growth in
data traffic demands and to overcome the challenges of next ge-
neration mobile networks (5G). The main concept of C-RAN is to
decouple the BaseBand Units (BBU) and the Remote Radio Heads
(RRH), and place the BBUs in common edge data centers (BBU
pools) for centralized processing. This gives a number of benefits in
terms of cost savings, network capacity improvement and resource
utilization gains. However, network operators need to investigate
scalable and cost-efficient algorithms for resource allocation pro-
blems to enable and facilitate the deployment of C-RAN architec-
ture. Most of these problems are very complex and thus very hard
to solve. Hence, we use combinatorial optimization which provides
powerful tools to efficiently address these problems.
One of the key issues in the deployment of C-RAN is finding the
optimal assignment of RRHs (or antennas) to edge data centers
(BBUs) when jointly optimizing the fronthaul latency and resource
consumption. We model this problem by a mathematical formula-
tion based on an Integer Linear Programming (ILP) approach to
provide the optimal strategies for the RRH-BBU assignment pro-
blem and we propose also low-complexity heuristic algorithms to
rapidly reach good solutions for large problem instances. The op-
timal RRH-BBU assignment reduces the expected latency and of-
fers resource utilization gains. Such gains can only be achieved
when reducing the inter-cell interference caused by the dense de-
ployment of cell sites. We propose an exact mathematical formula-
tion based on Branch-and-Cut methods that enables to consolidate
and re-optimize the antennas radii in order to jointly minimize inter-
cell interference and guarantee a full network coverage in C-RAN.
In addition to the increase of inter-cell interference, the high density
of cells in C-RAN increases the amount of baseband processing
as well as the amount of data traffic demands between antennas
and centralized data centers when strong latency requirements on
fronthaul network should be met. Therefore, we discuss in the third
part of this thesis how to determine the optimal placement of BBU
functions when considering 3GPP split option to find optimal tra-
deoffs between benefits of centralization in C-RAN and transport
requirements. We propose exact and heuristic algorithms based on
combinatorial optimization techniques to rapidly provide optimal or
near-optimal solutions even for large network sizes.
Universit´
e Paris-Saclay
Espace Technologique / Immeuble Discovery
Route de l’Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France
