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IEEE 802.11ax Uplink Scheduler to Minimize
Delay: a Classic Problem with New Constraints
Dmitry Bankov*†, Andre Didenko* †, Evgeny Khorov*‡ , Vyacheslav Loginov*, Andrey Lyakhov*†
*Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia
†Moscow Institute of Physics and Technology, Moscow, Russia
‡Higher School of Economics, Moscow, Russia
Email: {bankov, dida, khorov, loginov, lyakhov}@iitp.ru
Abstract—In order to meet the continuously increasing de-
mands for high throughput in wireless networks, IEEE 802
LAN/MAN Standard Committee is developing IEEE 802.11ax:
a new amendment for the Wi-Fi standard. This amendment
provides various ways to improve the efficiency of Wi-Fi. The
most revolutionary one is OFDMA. Apart from obvious advan-
tages, such as decreasing overhead for short packet transmission
at high rates and improving robustness to frequency selective
interference, being used for uplink transmission, OFDMA can
increase power spectral density and, consequently, user data
rates. However, the gain of OFDMA mainly depends on the
resource scheduling between users. The peculiarities of OFDMA
implementation in Wi-Fi completely change properties of classic
schedulers used in other OFDMA systems, e.g. LTE. In the
paper, we consider the usage of OFDMA in Wi-Fi for uplink
transmission. We study peculiarities of OFDMA in Wi-Fi, adapt
classic schedulers to Wi-Fi, explaining why they do not perform
well. Finally we develop a novel scheduler, MUTAX, and evaluate
its performance with simulation.
Index Terms—Wi-Fi, IEEE 802.11ax, High Effiency WLAN,
OFDMA, Scheduling
I. INTRODUCTION
Nowadays, Wi-Fi has become the main technology for
wireless local area networks. High number of Wi-Fi devices,
as well as the number of deployed networks, leads to huge
interference. To improve efficiency of Wi-Fi networks in
existing and emerging indoor and outdoor scenarios, Wi-Fi
community is currently developing a new standard, namely
IEEE 802.11ax [1]. In contrast to 11ac, PHY layer data rate
of which tenfold excels the one of the preceding technology
(namely 11n), the expected quadruple growth of user through-
put in 11ax networks will be caused mostly by advanced
channel access techniques, rather than by PHY data rates
increased by just 37%.
The main feature of 11ax is Orthogonal Frequency Division
Multiple Access (OFDMA), which extends the legacy Wi-
Fi Carrier Sense Multiple Access with Collision Avoidance
(CSMA/CA) by introducing a possibility to divide channel
resources in frequency domain. Since 11a, Wi-Fi has been
using Orthogonal Frequency Division Multiplexing (OFDM).
However, with OFDM, at any time instant all tones (also
The research was done at IITP RAS and was supported by the Russian
Science Foundation (agreement No 16-19-10687).
referred to as subcarriers) are used to transmit data for one
user, while OFDMA allows assigning various tones to different
users. The efficiency of OFDMA significantly depends on how
the tones are scheduled between users. However the 11ax
standard will provide only a flexible framework, without any
predefined scheduling algorithms.
Fortunately, scheduling problem has been carefully inves-
tigated in cellular networks, like LTE, where OFDMA has
appeared much earlier. So at first sight, it is worth to use
one of the existing cellular schedulers and adapt it to Wi-
Fi peculiarities. This work itself is not easy, since OFDMA
fundamentals in Wi-Fi differ from that in LTE. Moreover, the
features of 11ax break assumptions used to derive schedulers
for LTE. Thus, nobody can guarantee that being applied to
11ax networks the LTE scheduler remain the best one.
In this paper, we compare OFDMA schemes in IEEE
802.11ax and LTE, and analyze problems that arise while
developing the schedulers for 11ax networks. We also make
the first step in this direction and consider a problem of
minimizing the average delivery time for uplink transmission.
Then we show why well-known schedulers designed for a
similar problem are not optimal for 11ax networks and develop
a new scheduler that outperforms them.
The rest of the paper is organized as follows. Section II
briefly describes the main features of OFDMA in 802.11ax,
reviews literature and states the problem. In Section III we
design a brand new scheduler for 11ax networks. We prove its
high efficiency in Section IV. Section V concludes the paper.
II. OFDMA IN IEEE 802.11A X NETWOR KS
A. Main Features
In contrast to LTE with a rigid traffic-independent time-
frequency numerology, in 11ax networks, OFDMA works at
the per-frame basis upon native to Wi-Fi Enhanced Distributed
Channel Access (EDCA), a sort of decentralized CSMA/CA
joint with automatic repeat request. OFDMA frames start with
a common preamble, which can be decoded by legacy devices.
Having received the preamble, a station (STA) learns the
duration of the frame and considers the medium as busy till
the end of the frame. The rest of the frame is understandable
only by 11ax STAs and can be formed according to OFDMA
concept, i.e. various tones of the frame can be assigned to
different STAs.978-1-5386-3531-5/17/$31.00 c
○2017 IEEE
484 tones
242 242
106 26 106 106 26 106
52 52 26 52 52 52 52 26 52 52
26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26
Figure 1: RU locations in 40 MHz channel
In 11ax networks, a group of tones assigned to a STA is
called Resource Unit (RU). The IEEE 802.11ax defines RUs
that consist of 26, 52, 106, 242, 484, 996 and 2×996 OFDM
tones. The set of available RUs depends on the channel width,
e.g., in a 40 MHz channel, the STAs can use RUs of up to
484 tones. A wide RU can be split into approximately twice
narrower RUs independently from others. The two exceptions
are as follows. 1) A 996-tone RU can be split into two 484-
tone RU and one 26-tone RU. 2) A 242-tone RU can be split
into two 106-tone RU and one 26 tone RU, see Fig. 1.
In LTE networks, a scheduler can allocate an arbitrary
subset of downlink (DL) RUs for a user or an arbitrary interval
of UL RUs for a user. 11ax imposes much stricter constrains
on RU selection, since both in DL and UL, a STA cannot be
assigned to more than one RU.
Another set of constraints limits the maximal MCS that
can be used in narrow RUs and the usage of MU MIMO.
Specifically, MU MIMO is only available in 106-tone and
greater RUs, while the novel 1024QAM can be used only
in 242-tone and greater RUs. Such constraints complicate
scheduling problem in 11ax networks.
An OFDMA frame may contain RUs with different number
of tones, however all RUs inside the frame must have the
same duration. For that, the STAs may use novel flexible
fragmentation or padding.
All transmissions inside one OFDMA frame must be syn-
chronized, i.e. start and finish at the same time instants.
This can be easily done in DL, where an OFDMA frame
is generated by the Access Point (AP). However, to provide
synchronization for uplink, the AP can use novel Trigger
Frame (TF). SIFS after TF reception, the STAs transmit their
parts of UL OFDMA frame. If needed, the AP acknowledges
reception of each part by sending a set of ACKs inside an
OFDMA frame or by sending Multi-STA Block Acknowledg-
ment (MSBA).
It is the AP which, for both DL and UL transmissions, de-
termines modulation and coding scheme (MCS), duration, RU
assignment and other OFDMA parameters. Such information
can be transmitted in frame headers (for DL) and in the Trigger
frame (for UL). An example of UL OFDMA transmission is
shown in Fig. 2.
As mentioned above, 11ax moves decision making logic
from the STAs to the AP that determines which STA transmits,
when, in which RU, how much data it transmits, etc. To
Time
Frequency
TF
STA 1
STA 2
...
STA 𝑥
MSBA
𝑆𝐼 𝐹 𝑆 𝑆𝐼 𝐹 𝑆
Figure 2: Frame handshake for UL OFDMA transmission.
make a correct decision, the AP needs to be aware of STAs’
buffered traffic and channel conditions. This information can
be periodically requested by the AP. Moreover, to notify the
AP about arrived packets, a STA can proactively send so-called
Buffer Status Report (BSR). For that, it can aggregate BSR
with a data frame sent in the dedicated RU. Another option
is legacy EDCA. Thanks to the recent change in the 11ax
draft standard, the AP can separately tune EDCA parameters
of scheduled STAs in such a way, that they do not contend for
the channel with unscheduled STAs. It means that if a STA
does not obtain UL RUs, it likely delivers a BSR at the first
transmission attempt because of the extremely low contention.
The third approach to send a BSR is using OFDMA Random
Access. For that, the AP can allocate one or several RUs for
random access and the contending STAs will randomly choose
one of such RUs. Anyway, thanks to these methods, the AP
can quickly obtain information about new frames buffered at
the associated STAs and waiting for UL RUs.
OFDMA brings many benefits to 11ax networks. First
it makes transmission more reliable to frequency selective
interference and fading. This is especially important for wide
160 MHz channels introduced in 11ac. Second, with OFDMA
we can glue short packets destined for or originated from
various STAs, which significantly reduces overhead caused
by PHY headers and channel access time and interframe
spaces. Third, for UL transmissions by edge STAs it makes
sense to use narrow channels instead of wide ones. Indeed,
since the STAs are spatially separated they can simultaneously
transmit at the maximal allowed power without breaking legal
limitations on the emitted energy. Thus, by reducing RU width,
we increase power spectral density received from the STAs and
can use higher MCS. In other words, by allowing numerous
spatially separated STAs to transmit in parallel, we increase
the cumulative received power in comparison to the case when
only one STA transmits. For edge STAs, increasing power
spectral density leads to a higher MCS, so the average amount
of data received from an OFDMA frame is higher than that
from a legacy. Thus in contrast to LTE, in 802.11ax networks
the rates in UL RUs are non-additive, i.e. if a STA transmits
in twice wider RU, it is not guaranteed that it transmits twice
more data. This effect will be carefully investigated in the
paper.
B. Related Works
Despite the fact that 802.11ax amendment is expected to
be finished by 2019, it has been already studied in the
literature [2]–[5].
In [2], authors study the performance of the network con-
sisting of the legacy and 802.11ax STAs. Authors propose an
approach for optimal channel access parameter values selec-
tion, which guarantees fairness between legacy and 802.11ax
and significantly increases the number of the OFDMA trans-
missions for 802.11ax STAs. However, the model developed
in [2] allows only estimating the number of the OFDMA
transmissions in the network, but not the achievable data rate.
Many studies on 802.11ax performance were presented at
the meetings of the 802.11ax Task Group (TGax), e.g. [4],
[5]. However, although many network topologies have been
already studied, the only considered scheduler in TGax has
been the random scheduler with a static RU configuration.
The problem of scheduling in 11ax uplink has been con-
sidered in [3], which focuses on selection of the RU duration.
The authors initially assume that the AP does not know the
amount of data the STAs have and thus does not know the
correct duration of RUs that should be allocated. They propose
a scheme that can be used by the AP to obtain this information
and derive the best way to select the duration of STAs’
transmission in terms of throughput and energy consumption.
However, they do not consider any specific way to divide the
channel into RUs and to assign the RUs to the STAs. Neither
do they consider random access and the possibility of STAs
to deliver BSRs by aggregating them to the transmitted data.
Random access for UL OFDMA in 802.11ax has been
studied in [6]. The authors consider a scenario, when STAs
transmit saturated data flows and use only random access
for transmission. They describe a mathematical model of
transmission with OFDMA random access and use it for
analysis and optimization of the network performance in
terms of throughput and percentage of successful RUs. The
methodology introduced in the paper might be used to tune
the random access for OFDMA UL, however, usage of deter-
ministic access is more feasible in terms of channel efficiency
for the studied scenario.
The problem of resource allocation in cellular networks
is widely studied in literature [7]. In many papers, it is
formulated as the following optimization problem. Consider
a network with a base station (BS) and 𝑁associated users.
Each time unit, the BS runs scheduler which allocates 𝑀
RUs to some STAs in order to maximize some network-wide
utility function. The most widely used utility functions and
corresponding schedulers are listed below.
To maximize cumulative throughput 𝑆=𝑁
𝑖=1 𝑆𝑖(𝑡)at
time instant 𝑡, the BS uses the Max Rate (MR) scheduler,
which considers RUs one by one and assigns each RU to a
user with the highest nominal data rate in this RU (data rate of
user 𝑖in RU 𝑗is hereinafter denoted as 𝑟𝑗
𝑖). However, as shown
in many papers (e.g. [8]), under high load the MR scheduler
blocks users with low instant rates.
To avoid this problem, the Proportional Fair (PF) scheduler
has been proposed. It maximizes 𝑁
𝑖=1 log 𝑆𝑖(𝑡). For that, RU
𝑗shall be allocated to user ^
𝑖𝑃 𝐹 = arg max𝑖
𝑟𝑗
𝑖(𝑡)
𝑆𝑖(𝑡−1) .Authors
of [9] show that in long term the PF scheduler gives the same
amount of channel time for all users, therefore resulting in fair
resource allocation.
Both MR and PF have been designed for saturated (infinite)
flows. From the practical point of view, it is worth to consider
unsaturated scenario with finite flows of given length. Such
flows can represent HTTP requests or files transmitted in a
wireless network. For such traffic, it is worth to consider
upload time which affects user perceived quality of experience
(QoE). If the channel properties do not change with time, the
rate in different RUs is the same and additive, the Shortest
Remaining Time First (SRTF) scheduler is proven to provide
minimal average upload time. Simplistically, SRTF allocates
all RUs to a user for which 𝐷𝑖(𝑡)
𝑟𝑖is the minimal one, where
𝐷𝑖(𝑡)is the remaining amount of data of user 𝑖and 𝑟𝑖is its
rate in the whole channel. As explained in Section II-A, the
assumptions made to derive such a scheduler are not met for
UL OFDMA transmission in 11ax networks, which brings us
to the following problem statement.
C. Problem Statement
Consider a scenario with an AP and a group of STAs
associated to it. From time to time, STAs generate finite data
flows to be delivered to the AP. To notify the AP about new
flow, the STAs use EDCA to transmit BSR. Thus the whole
channel can be allocated to those STAs, which are known by
the AP as having traffic.
In the paper, we consider the following problem: to design a
scheduler for UL OFDMA data transmission in 11ax networks
that minimizes the average upload time.
III. DES IGN OF TH E SCH ED ULER
In this section, we design a novel scheduler called MUTAX,
Minimizing Upload Time in 11AX networks. While deriving
formulae we neglect the effects related to packetization (in-
cluding aggregation and fragmentation) overhead. Apart from
that, for shortness we consider only 𝑛STAs with flows and
assume that each STA has only one flow. Both STA and flow
are denoted as 𝑖, 𝑖 = 1, ..., 𝑛.
Let slot be a time interval between two consequent TFs. It
should be noted that slots may have different duration. The
maximal one is related to the standard limit of 5484 µsfor
the physical protocol data unit (PPDU) duration. A slot can
be shorter, if all STAs that transmit in this slot have no more
data.
The MUTAX algorithm has two steps. At the first step, we
order existing flows and calculate the sum upload time of the
flows, as if we used exhaustive service. At the second stage,
we try to improve the sum upload time by serving some flows
in parallel. Let us consider the steps in detail.
At the first step, we serve each flow exhaustively. Except for
waiting, the time needed to upload flow 𝑖equals 𝑡𝑖=𝐷𝑖
𝑟𝑖. The
first STA finishes delivering its flow by time 𝑡1. The second
STA starts right after the first one and delivers its flow by time
𝑡2+𝑡1, etc. As the result, the total upload time for existing
flows is
𝑇𝑠𝑡𝑒𝑝1=
𝑛
𝑖=1
(𝑛−𝑖)𝑡𝑖.(1)
Obviously, to minimize the sum upload time we have to sort
the STAs in the ascending order by 𝑡𝑖.
At the second step, we divide the channel into several RUs.
Let 𝑚be the number of RUs and 𝑗, 1≤𝑗≤𝑚be the
index of RU in the considered RU configuration. 𝑥𝑗
𝑖is an
indicator which equals 1if STA 𝑖is assigned to RU 𝑗, and 0,
otherwise. For shortness, 𝑋is the two dimensional matrix of
𝑥𝑗
𝑖representing RU assignment to the STAs.
With defined notation, the total upload time 𝑇(𝑋)of
existing flows differs from 𝑇𝑠𝑡𝑒𝑝1in the following way. First,
the upload time of each of 𝑛flows increases by the current slot
duration 𝜏. Second, if 𝑥𝑗
𝑖= 1, the remaining amount of data
of flow 𝑖decreases by the amount of data the STA transmits
in RU 𝑗of the current slot: Δ𝐷𝑗
𝑖= min 𝐷𝑖, 𝜏 ×𝑟𝑗
𝑖. Thus
𝑇(𝑋) = 𝑛𝜏 +
𝑛
𝑖=1
(𝑛−𝑖)𝐷𝑖−𝑚
𝑘=1 𝑥𝑗
𝑖Δ𝐷𝑗
𝑖
𝑟𝑖
.(2)
Since both 𝜏and Δ𝐷𝑗
𝑖depend on the resource allocation 𝑋,
minimization of 𝑇(𝑋)requires exhaustive search by possible
ways to allocate the RUs to the STAs, and to simplify the task
we propose an heuristic approach based on two assumptions.
Firstly, we neglect the change of 𝑛𝜏 for different allocations, as
a slot cannot be too long due to standard limitations. Secondly,
we sort the STAs in the ascending order by 𝑡𝑛only once,
before considering different ways to assign RUs to the STAs.
Under these assumptions, to minimize 𝑇(𝑋), we have to max-
imize the following expression 𝑚
𝑘=1 𝑛
𝑖=1 (𝑛−𝑖)𝑥𝑗
𝑖Δ𝐷𝑗
𝑖
𝑟𝑖.
Let us denote 𝜆𝑗
𝑖= (𝑛−𝑖)Δ𝐷𝑗
𝑖
𝑟𝑖and define the following
optimization problem:
max
𝑖
𝑗
𝑥𝑗
𝑖𝜆𝑗
𝑖(3)
subject to
𝑖
𝑥𝑗
𝑖≤1,∀𝑗(4)
𝑗
𝑥𝑗
𝑖≤1,∀𝑖(5)
𝑖
𝑗
𝑥𝑗
𝑖≤𝑚(6)
This problem1is known as the assignment problem which
can be solved in polynomial time using the Kuhn-Munkres
algorithm [10]. Its solution is the assignment ^
𝑋that yields
minimal 𝑇(𝑋).
Note that assignment ^
𝑋is found for a specific configuration
of RUs. To minimize upload time, we consider different ways
1If we define 𝜆𝑗
𝑖=
𝑟𝑗
𝑖
𝑆𝑖, where 𝑆𝑖is the amount of data transmitted by STA
𝑖, we obtain the optimization problem for the adaptation of the PF scheduler
to 11ax.
to split the channel into RUs. For each configuration of RUs
we solve the optimization problem and thus find the best
assignment of the channel resources. The examination of RU
configuration can be hastened by excluding configurations
that are obviously worse than the known alternatives, e.g.,
if we have two STAs and a 20 MHz channel, we consider
configuration with one 106-tone, one 52-tone and three 26-
tone RUs, but exclude configuration with one 106-tone RU
and five 26-tone RUs.
IV. NUMERICAL RES ULTS
To evaluate the developed scheduler, we use the well-known
NS-3 simulator [11]. We have implemented the MUTAX
scheduler, as well as 11ax adaptations of PF, MR and SRTF.
Since the performance of PF significantly depends on how the
channel is divided into RUs, we find the best RU configuration
with exhaustive search.
We run simulation in a scenario described in Section II-C.
The network operates in a 40 MHz channel at 5 GHz. The
flow sizes are drawn from truncated lognormal distribution
with minimal, average and maximal values of 1 KB, 500 KB,
and 5 MB, respectively. When a flow is delivered, the next
flow is generated after a random delay drawn from truncated
exponential distribution with minimal, average and maximal
values of 0.1 s,0.3 s and 0.6 s, respectively. We consider
channel with no fading and the 802.11ax path-loss model for
the residential scenario [12]. All STAs transmit with power of
15 dBm. The AP has a set of bounds on receive power for
each MCS, and assigns each STA the maximal MCS that fits
the specified bounds. As mentioned in Section II-A, the MCS
assigned to a STA can differ depending on the RU width.
In first set of experiments, the STAs are located uniformly
within a small circle of radius 𝑅= 5 m around the AP. In
such a case, the channel quality is so high that in any RU the
STAs use the maximal permitted MCS. Obviously, in this case
OFDMA cannot bring any profit against SRTF, as the division
of the channel without changing the MCS cannot increase the
data rate, provided that the MCS can cope with the noise. This
result is supported by simulation, see Fig 3, which shows that
if the channel for all STAs is perfect, MUTAX yields the same
upload time, as SRTF, and they both outperform the PF and
MR schedulers by up to 30%. As we consider traffic model
of a closed loop system, along with the reduction of upload
time MUTAX provides the increase of goodput compared to
the other schedulers.
The second set of experiments corresponds to the case when
STAs are located in a larger circle of radius 𝑅= 20 m. Such
conditions provide variety of MCSs among STAs and among
RU widths, so it becomes feasible to split the channel between
different users. According to simulation results, see Fig. 4, in
this case designed to minimize upload time the classic SRTF
works much worse than even 11ax adaptation of PF. At the
same time, MUTAX shows 20% lower upload time than PF.
The results show that in this case the schedulers based on
the exhaustive service (MR and SRTF) are much less efficient
than channel-splitting schedulers (MUTAX and PF) and the
Figure 3: Upload time, busy channel time ratio and goodput vs the number of STAs in the small circle.
Figure 4: Upload time, busy channel time ratio and goodput vs the number of STAs in the large circle.
gap between them increases with the number of client STAs.
In a large circle scenario the gain in goodput of MUTAX
against SRTF and MR is almost 100%.
V. CONCLUSION
In the paper, we have studied scheduling problem in IEEE
802.11ax networks, the standard of which is currently under
development. To our best knowledge, this paper is the first one
dedicated to scheduling in IEEE 802.11ax networks, which
shows that this field of research has a lot of open issues to be
solved. In the paper we show that because of 11ax OFDMA
peculiarities the existing schedulers cannot be directly applied
to the new technology. Specifically, we have considered the
problem of uplink scheduling which aims to minimize average
upload time in a scenario with high number of active users.
We showed that depending on the scenario sometimes it is
worth to use classic SRTF scheduler, while in other cases
the channel should be split between several STAs in order to
minimize upload time. We develop a novel scheduler, called
MUTAX, which adaptively chooses the best strategy and
significantly outperforms existing popular solutions. For future
research, we plan to consider a more strict set of constraints
for the scheduler, related to the recently added to the standard
statement that the AP should equalize the receive power from
various STAs. Such a constraint is also important from the
receiver implementation point of view.
REFERENCES
[1] E. Khorov, A. Kiryanov, and A. Lyakhov, “IEEE 802.11 ax: How to
Build High Efficiency WLANs,” in 2015 International Conference on
Engineering and Telecommunication (EnT). IEEE, 2015, pp. 14–19.
[2] E. Khorov, V. Loginov, and A. Lyakhov, “Several edca parameter sets
for improving channel access in ieee 802.11 ax networks,” in Wireless
Communication Systems (ISWCS), 2016 International Symposium on.
IEEE, 2016, pp. 419–423.
[3] M. Karaca, S. Bastani, B. E. Priyanto, M. Safavi, and B. Landfeldt,
“Resource management for OFDMA based next generation 802.11
WLANs,” in Wireless and Mobile Networking Conference (WMNC),
2016 9th IFIP. IEEE, 2016, pp. 57–64.
[4] S. Kim, H. Park, J. Kim, K. Ryu, and H. Cho, 11ax
PAR Verification through OFDMA, January 2017. [Online].
Available: https://mentor.ieee.org/802.11/dcn/16/11-16-1363-01-00ax-
11ax-par-verification-through-ofdma.ppt
[5] H. Su, J. Pang, J. Zhu, J. Zhang, and C. Ma,
Preliminary 11ax PAR Verification, September 2016. [Online].
Available: https://mentor.ieee.org/802.11/dcn/16/11-16-1363-01-00ax-
11ax-par-verification-through-ofdma.ppt
[6] L. Lanante, H. Ochi, T. Uwai, Y. Nagao, M. Kurosaki, and C. Ghosh,
“Performance Analysis of the 802.11ax UL OFDMA Random Access
Protocol in Dense Networks,” in Communications, 2017. ICC’17. IEEE
International Conference on. IEEE, 2017.
[7] E. Khorov, A. Krasilov, and A. Malyshev, “Radio resource scheduling
for low-latency communications in lte and beyond,” in Quality of Service
(IWQoS), 2017 IEEE/ACM 25th International Symposium on. IEEE,
2017, pp. 1–6.
[8] F. Capozzi, G. Piro, L. A. Grieco, G. Boggia, and P. Camarda, “Down-
link packet scheduling in lte cellular networks: Key design issues and a
survey,” IEEE Communications Surveys & Tutorials, vol. 15, no. 2, pp.
678–700, 2013.
[9] R. Kwan, C. Leung, and J. Zhang, “Proportional fair multiuser schedul-
ing in lte,” IEEE Signal Processing Letters, vol. 16, no. 6, pp. 461–464,
2009.
[10] F. Bourgeois and J.-C. Lassalle, “An extension of the munkres algorithm
for the assignment problem to rectangular matrices,” Communications
of the ACM, vol. 14, no. 12, pp. 802–804, 1971.
[11] The ns-3 Network Simulator. [Online]. Available: http://www.nsnam.org/
[12] TGax Simulation Scenarios. https://mentor.ieee.org/802.11/ dcn/14/11-
14-0980-16-00ax-simulation-scenarios.docx.
