Stochastic Modeling and Analysis of Feedback Control on the QoS VoIP Traffic in a single cell IEEE 802.16e Networks

Abstract

One of the major challenges in the next generation networks is maintaining the Quality of Service (QoS) for different users who use the services of such a technology. The problem increases when the flows that traverse the wireless link belong to classes with different requirements of QoS. In this case, the network services face several disadvantages caused by the unreliability of wireless channel and the channel sharing by several users. In this paper, we consider a single-cell IEEE 802.16e environment in which the base station allocates subchannels to the subscriber stations in its coverage area. Therefore, two systems are compared. In the first system, called system without Feedback Control, for each uplink subchannel, a separate single queue is used for buffering the VoIP packets from Subscriber Station (SS) to Base Station (BS). However, in the second system, called system with Feedback Control, an Active Queue Managment (AQM) is used to control the VoIP packets. Two thresholds are used in the queue in an effort to control the arrival rate of the VoIP packets. The VoIP arrivals are modeled by a two-state Markov Modulated Poisson Process (MMPP) process. A queuing analytical model is presented to evaluate the performance of both systems. Numerical results obtained show the positive impact of the AQM added to the second system on the performance parameters of VoIP packets compared to the first system.

Full text

Stochastic Modeling and Analysis of Feedback
Control on the QoS VoIP Traffic in a single cell
IEEE 802.16e Networks
Said El Kafhali, and Mohamed Hanini
Abstract—One of the major challenges in the next generation
networks is maintaining the Quality of Service (QoS) for
different users who use the services of such a technology. The
problem increases when the flows that traverse the wireless
link belong to classes with different requirements of QoS.
In this case, the network services face several disadvantages
caused by the unreliability of wireless channel and the channel
sharing by several users. In this paper, we consider a single-cell
IEEE 802.16e environment in which the base station allocates
subchannels to the subscriber stations in its coverage area.
Therefore, two systems are compared. In the first system, called
system without Feedback Control, for each uplink subchannel,
a separate single queue is used for buffering the VoIP packets
from Subscriber Station (SS) to Base Station (BS). However,
in the second system, called system with Feedback Control, an
Active Queue Managment (AQM) is used to control the VoIP
packets. Two thresholds are used in the queue in an effort to
control the arrival rate of the VoIP packets. The VoIP arrivals
are modeled by a two-state Markov Modulated Poisson Process
(MMPP) process. A queuing analytical model is presented to
evaluate the performance of both systems. Numerical results
obtained show the positive impact of the AQM added to the
second system on the performance parameters of VoIP packets
compared to the first system.
Index Terms—IEEE 802.16e, Active Queue Managment, VoIP
Traffic, Markov Chain, Queuing Theory, Quality of Service,
Performance Parameters.
I. INTRODUCTION
NOW there are many wireless communication innova-
tions occurring in telecommunications, it may be a part
of small incremental innovations or an important innovation
that can lead to technological breakthroughs. Consequently,
these innovations led to new concepts network referred to
next generation networks.
Generally, a next-generation network is a network based
on a new technology that, according to its supporters, will
be adopted across the board in the coming years. This term
is used frequently in the field of mobile services where it
has deployed networks, so called second generation networks
(2G) (Global System for Mobile communications (GSM) [1],
Code Division Multiple Access (CDMA) [2]), since the late
80s; and, third generation networks (3G) (Universal Mobile
Telecommunications System (UMTS) [3], Code Division
Multiple Access 2000 (CDMA2000) [4]) and new developed
Manuscript received June 22, 2016; revised October 30, 2016; accepted
November 10, 2016.
Said El Kafhali, Computer, Networks, Mobility and Modeling laboratory,
National School of Applied Sciences, Hassan 1st Univ, Morocco. (email :
kafhalisaid@gmail.com).
Mohamed Hanini, Computer, Networks, Mobility and Modeling labora-
tory, Faculty of Sciences and Technology, Hassan 1st Univ, Morocco. (email
: haninimohamed@gmail.com).
technologies such as WiMAX (Worldwide Interoperability
for Microwave Access) [5] and LTE (Long Term Evolution)
[6], considered as fourth generation networks (4G).
In this network evolution appeared WiMAX Network [5],
which envisioned supporting multiple multimedia services
such as internet browsing, voice telephony, interactive gam-
ing, email, video messaging, etc. These services demand
different Quality of Service requirements, such as average
packet delay, average Packet Drop Rate, bandwidth, through-
put, transmission delay, availability, jitter, and minimum
throughput requirements [7], [8]. To fit diverse QoS require-
ments is more difficult in wireless networks as compared
to the wired networks. This is due to the capacity of a
wireless channel varies randomly with time, the time-varying
channel conditions and resource conflict among multiple
users. Packet scheduling deals with radio resource allocation
and is directly related to the Quality of Service provision to
users demanding different applications [9].
To avoid congestion in high-speed networks (such as
WiMAX network), due to increased traffic which transits
among them, we use buffers to handle the excess of traffic
when the debit outstrips the buffering capacity. But the
limited space of these queues causes the loss of packets
of information over time. The management mechanisms
queues have great utilities to avoid buffers congestion. The
traditional mechanisms such as, Passive Queue Management
(PQM) detects congestion only after a packet has been
dropped which can cause problems in the quality of service
in the network. The Active Queue Management (AQM) [10]
is an efficient tool to avoid saturation of the queue by
warning the transmitter that the queue is almost full to reduce
its speed before the queue is full. The buffer management
mechanisms focus on space management, in the other hand
scheduling priorities attempt to guarantee acceptable delay to
applications for which it is important that delay is bounded.
These mechanisms enable networks to improve the required
quality for multimedia application to end users. Hence, in this
work we make two mathematical models based on Queueing
Theory [11] to evaluate the performance of VoIP traffic in a
single cell IEEE 802.16e Networks. In the first model, for
each uplink subchannel, a separate single queue is used for
buffering the VoIP packets from Subscriber Station (SS) to
Base Station (BS). But in the second system, an AQM is used
to control the VoIP packets in order to reduce the speed of
arrivals VoIP packets and manage the loss of packets before
the queue is full. These models are compared in terms of
VoIP packets loss probability, mean number of VoIP packets
in the system, throughput and average VoIP packet delay.
The next section, Section II reviews the related work and
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introduces the problem statement. In section III, we introduce
the QoS of IEEE 802.16 MAC layer. In Section IV, we
give a description of proposed system models. The queueing
models and performance parameters have been described
in Section V. Section VI states numerical results. Finally,
section VII is devoted to the conclusion.
II. RE LATE D WOR K
Many researchers have been interested in performance
evaluation issues related to VoIP applications over different
technologies. The authors in [12] presented and compared
two queue management mechanisms with a time and space
priority mechanism for an end user in HSDPA. Those
mechanisms are used to manage access packets in the queue
giving priority to the Real Time packets and avoiding the
Non Real Time packet loss. A discrete time Markov chain
is formulated by considering MMPP as the traffic source of
RT packets. Numerical results obtained show the positive
impact of the AQM on the performance parameters of NRT
packets. In [13], the authors compared the performance of
Mixed Traffic Scheduler algorithms and identify one which
suitably trades off RT (Real-Time) as well as nRT (non
Real- Time) traffic capacity with user perceived Quality of
Service, for any ratio of VoIP and video users in a cell.
They proposed two MTS algorithms which treat voice and
video traffic. These are: dynamic packet scheduling and a
new limited Joint Time Frequency (JTF) scheduling. They
compared these algorithms with known ones like Modified
Largest Weighted Delay First [14] and SP-Dynamic [15].
These are compared for three different cases when number
of video users is very high, when number of Voice over IP
users is high, and when the number of video and VoIP users
are balanced.
The performance of WiMAX and UMTS for VoIP traffic is
evaluated by Jadhav et al. [16] using the OPNET simulator.
This evaluation is restricted to a low number of simultaneous
users. The main focus is the comparison between WiMAX
and UMTS, leaving the comparison between different QoS
service classes. They analyzed several performance metrics
such as jitter, delay and packet loss, but also the Quality
of Experience (QoE) perceived by the end user, through
the Mean Opinion Score (MOS). This study showed that
WiMAX is better than UMTS to support VoIP. The work
in [17] analyzes the efficiency of resource utilization and
VoIP capacity in IEEE 802.16e. It is stated that ertPS service
introduced by the IEEE 802.16e standard is more appropriate
than UGS and rtPS for VoIP services with variable data
rate and silence suppression. Authors in [18] analyzed the
QoS of VoIP services using three different traffic mod-
els: the cross-layer analytical model, M/G/1(m) model, and
the IEEE 802.16e/m scheduling approach to evaluate the
blocking probability and throughput of the VoIP services
of the cognitive radio. By comparing the VoIP performance
under the cross-layer analytical model, M/G/1(m), and the
IEEE802.16e/m traffic, they have proved that the relative
performance of the three traffic models is sensitive to the
queue length. In [19], the performance of VoIP application
is analyzed for a cognitive radio system using two state
MMPP model. Various numerical and simulation parameters,
such as packet dropping probability and VoIP capacity are
demonstrated. These results conclude that the VoIP capacity
is determined by bottleneck-link, which can be different
according to system parameters. In [20], the authors proposed
a Discrete-Time Markov Chain (DTMC) framework based on
a MMPP traffic model to analyze the performance of VoIP
traffic. Through the DTMC based on MMPP, they analyzed
and demonstrated various performance parameters, such as
average packet dropping probability, average throughput, and
average queue length. In the analytical model presented
in this paper, the signaling overhead is considered in the
evaluation of the performance of VoIP services in the IEEE
802.16e OFDMA system. The authors in [21], evaluated the
performance of WiMAX for VoIP by varying number of
nodes failure. The performance is analyzed by using OPNET
Modeller tool. The performance of VoIP is compared in
terms of end-to-end delay, throughput, and traffic sent. The
results showed that with increase in nodes failure end-to-end
delay increases and throughput and traffic Sent decreases.
Feng et al. [22] proposed a POMDP (Partially Observable
Markov Decision Process) based Sleep Window Determina-
tion (PSWD) approach for improving the performance of
sleep mode operation of IEEE 802.16m. Simulations results
show that the proposed PSWD scheme outperforms con-
ventional IEEE 802.16e and IEEE 802.16m corresponding
to various traffic demands and satisfies respective delay
constraints at the same time. A quality of service mechanism
based on concepts of fuzzy logic for scheduling different
traffic classes in WiMAX networks is introduced in [23].
The proposed mechanism works out new weight value for
different queues adaptively by exploring three input param-
eters at simulation time: amount of real time and non real
time traffic in queues, throughput requirement for non-real
time flows, and latency requirement of real time data. The
proposed framework simplifies fair allocation of resources to
real as well as non real time in queues of SS together with
latency and throughput requirements.
Dai et al. [24] proposed a simple enhancement of the band-
width request messages in IEEE 802.16 for supporting voice
traffic. They put forwarding resource allocation and schedul-
ing schemes for use under real-time traffic conditions. The
obtained results show that the proposed bandwidth request
and scheduling systems achieve remarkably lower packet loss
probability than the standard IEEE 802.16 bandwidth request
with round robin scheduling. Capacity of VoIP traffic in a
CRN with imperfect spectrum sensing is studied in [25].
They modelled the VoIP traffic as MMPP; channel as two
state Markov chain. However, the authors have concentrated
more on finding the minimum target-detection and false
alarm probabilities. They demonstrated various analytical
and simulation parameters, such as the average throughput,
packet dropping probability, and VoIP capacity. In [26], the
authors compared QoS, throughput and mean waiting time of
traffic of IEEE 802.16e and IEEE 802.11 WLAN to observe
the impact of fading on the networks. The performance
parameters of IEEE 802.11 WLAN are evaluated employing
Giuseppe Bianchi state transition chain. A mathematical
model of VoIP application over wireless channel under IEEE
802.16e WLAN is analyzed under Rayleigh and Nakagami-
m fading cases with the help of MMPP and DTMC model.
Due to immunity of fading, the performance of video and
data integrated traffic of wired LAN is better than both
of IEEE 802.16e and IEEE 802.11. In [27], the authors
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proposed an M/M/n/n+K performance model of the voice
application of IEEE 802.16e under femto cellular network,
the QoS is measured based on throughput, mean queue length
and blocking probability. The steady probability states of
the proposed model are compared with the existing model
of WiMAX under Rayleigh fading environment. The profile
of blocking probability, normalized throughput and mean
queue length are shown against the length of packet. Authors
in [28] proposed and developed enhanced WiMAX uplink
traffic scheduling algorithm. The performance parameters
are average throughput, average delay, missed deadline ratio
and average queue size utilization ratio. The results acquired
from the performance analysis have proven the importance
of a check-and-balance system achieved across all computed
performance metrics. The authors in [29] proposed a graded
priority-based call admission control algorithm for LTE and
WiMAX networks. In this algorithm, the number of permissi-
ble connections shall be segregated into two classes and the
new connection shall be graded, prioritized, and admitted
based on the device requesting the connection. Simulation
results show that graded priority-based admission control
algorithm improves a higher connection admission rate for
select users compared to non-priority users.
In this paper, an analytical model is presented to eval-
uate the performance of VoIP packets in a single cell
IEEE 802.16e networks. A Continuous Time Markov Chain
(CTMC) is formulated by considering two-state MMPP as
the traffic source of VoIP packets. An AQM is used to en-
hance the performance of VoIP packets. Our main objective
is to compare two queue management mechanisms so as to
show the utility of the AQM mechanisms to enhance the QoS
for an end user in a single cell IEEE 802.16e networks.
III. QUAL IT Y OF SE RVICE SUPPORT IN IEEE 802.16E
In any mobile broadband network, users generate different
types of applications, including voice, video, email, ftp, and
browsing content. Each type comes with its own require-
ments for QoS. From the end users point of view, the QoS
of the service that the user has requested is perceived by
the users experience in relation to a particular application.
From the technical point of view, this duration results from a
complex interaction of factors like throughput, packet delay,
and residual bit error ratio. Similarly, the quality of a Voice
over IP (VoIP) call is perceived by the end users in terms of
delay and voice quality.
The IEEE 802.16e Media Access Control (MAC) layer
provides differential Quality of service (QoS) for various
classes of services, which are Unsolicited Grant Service
(UGS), Extended Real-Time Polling Service (ertPS), Real-
Time Polling Service (rtPS), Non-real-time Polling Service
(nrtPS), and Best Effort (BE) [30], [31]. The QoS parameters
and the supporting application types associated with each
classes of the IEEE 802.16e are presented as follows [32],
[33]:
•Unsolicited Grant Service (UGS): Supports the real-
time constant bit rate (CBR) applications such as T1/E1
and VoIP that generate fixed-size packets at periodic
intervals. Unsolicited applications grants are allocated to
eliminate the overhead and latency of the request/grant
process. VoIP without silence suppression is an example
of application that is cartegorized as UGS.
•real-time Polling Service (rtPS): Supports the real-time
services that generate variable-size packets on a periodic
basis, such as MPEG (Motion Pictures Experts Group)
video. An MPEG video categorised as rtPS traffic, needs
variable bandwidth at periodic intervals of time to avoid
jitter while viewing the video [34]. In this service, the
Base Station (BS) provides unicast polling opportunities
for the Mobile Station (MS) to request bandwidth.
•extended real-time Polling Service (ertPS): Combines
features from UGS and rtPS service classes, supports
real-time service flows that generate variable-sized data
packets at periodic intervals. VoIP with silence suppres-
sion is an example traffic categorised as ertPS.
•non-real-time Polling Service (nrtPS): Is the best ap-
propriate for the delay tolerant applications. As in rtPS,
dedicated periodic slots are used for the bandwidth
request opportunity, but with much longer periods [35].
In nrtPS, it is allowable to have unicast polling op-
portunities, but the average duration between two such
opportunities is in the order of a few seconds, which is
large compared to rtPS. All the MSs which belong to the
group can also request resources during the contention-
based polling opportunity, which can often result in
collisions and additional attempts.
•Best Effort (BE): Provides very little QoS support and is
applicable only for services that do not have strict QoS
requirements. It is for the traffic with no minimum level
of service requirements such as Web browsing or email.
Like in nrtPS, contention slots are used for bandwidth
request opportunities as long as there is space available.
When a user generates packets, these packets are placed in
the queue designated for the type of packets. If a connection
does not exist for the service class, the MS sends a request to
establish a connection for the data. The connection request
contains the QoS parameters that the connection expects.
When the BS receives the connection request, it checks if
it can service the QoS requirement of the connection. If the
connection is accepted, the BS should be able to serve the
connection for its quality of service requires [29].
IV. SYS TE M MO DE L
A. Model description
We consider a system model based on a mobile TDD-
OFDMA system. Our system is composed on an UpLink
(UL) VoIP transmission from a Subscriber Station (SS) to
Base Station (BS) in an OFDMA access mode using single
carrier air-interface based WiMAX system. Each subscriber
station serves multiple users.The base station may allocate
different number of subchannels to different SS. For example,
a SS with higher priority could allocate more number of
subchannels. The TDD-based WiMAX system is operated on
a frame basis, where each frame consists of a downlink DL
subframe and an UL subframe. The DL subframe consists of
a preamble, a frame control header (FCH), data bursts, a DL-
MAP message, and an UL-MAP message. By broadcasting
a MAP message, the BS indicates the location, size, and
encoding of data bursts [35]. The duration of a frame is
denoted by Tf.
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B. System model without Feedback Control
In this system (Figure 1), for each uplink subchannel, a
separate single queue is used for buffering the VoIP packets
from SS to BS. The queueing discipline is First-In First-Out
(FIFO).The size of this queue is finite (i.e.,some packets will
be dropped if the queue is full upon their arrivals).
C. System model with Feedback Control
In this system (as shown in Figure 2), we add two
thresholds L1and L2(L1< L2< N such that L2=N−L1)
in the queue in order to control the arrival rate of the VoIP
packets [12], [36]. When the number of VoIP packets in the
buffer is less than the minimum threshold (L1), there is no
dropping and the source operates normally. If the number
of packets exceeds the maximum threshold (L2), then the
excess packets will be dropped. However, we assume that the
source stops sending packets when a full buffer is detected
(L2packets). Packet transmission can commence after the
next departure. In this way the packet loss due to buffer
overflow might be avoided. If the number of VoIP packets
in the system falls between the L1and L2, then the arrival
rate of VoIP packets is reduced. As in [37], the key idea
for the controller is to calculate the mean queue length and
the mean arrival rate over each frame time duration Tf. The
mean arrival rate of VoIP packets is measured by counting the
number of VoIP arrivals within each frame time and dividing
it over the frame time length. These measurements are used
to calculate the new position of the threshold L2for the
next frame time duration Tf+1 in order to maintain the mean
queue length at the required value.
D. Channel model
We consider a Nakagami-m channel model in which the
channel quality is determined by instantaneous received
Signal-to-Noise Ratio (SNR) γin each time slot [38]. With
adaptive modulation, the SNR at the receiver is divided into
N+ 1 non-overlapping intervals (i.e., N= 7 in the IEEE
802.16 standard) by threshold Γn(n∈0,1, ..., N )where
Γ0<Γ1< ... < ΓN=∞. The channel is said to be in
state n(i.e., rate IDnwill be used) if Γn< γ < Γn+1 . To
avoid possible transmission error, no packet is transmitted
when γ < Γ0. Note that, these thresholds correspond to the
required SNR specified for VoIP traffic as shown in Table I
[39].
With Nakagami-m fading, the probability of using IDn
(i.e. Pγ(n)) is given by:
Pγ(n) = Γ(m, mΓn
γ)−Γ(m, mΓn+1
γ)
Γm
(1)
where γis the average SNR, mis the Nakagami fading
parameter (m≥0.5), Γ(m)is the Gamma function which
equals Γ(m) = R∞
0tm−1exp(−t) dt, and Γ(m, γ)is the
complementary incomplete Gamma function which equals
Γ(m, γ) = R∞
γtm−1exp(−t) dt.
The Wireless channel is described by an M/M/1/N
finite state Markov chain taking the discrete AMC into
consideration [40]. The state transition probability of the
Modulation and Coding Scheme (MCS) level during the
frame duration is given for j∈ {0, ..., N}by [41]:
Pt(i, j) = 








Ni+1Tf
Pγ(i), if j =i+ 1,
NiTf
Pγ(i), if j =i−1,
1−Pt(i, i + 1) −Pt(i, i −1), if j =i,
0, otherwise.
(2)
where iis the MCS level in the current frame and jis the
MCS level in the next frame. The level crossing rate, Ni, is
defined as [42]:
Ni=r2πmγi
γ
fd
Γ(m)mγi
γm−1exp −mγi
γ(3)
where fdis the maximum Doppler shift given in hertz.
E. Arrival VoIP traffic model
VoIP traffic has low data rates (in the order of tens of
Kbit/s) and exhibits low burntness. Because these strin-
gent requirements and particular characteristics, VoIP traffic
should be treated differently than other traffic in the network.
VoIP traffic at source level is characterized by an active
period followed by inactive period. During the active period,
the source sends packets at regular intervals. In WiMAX
networks, the VoIP packets are assumed to be scheduled
from the uplink queue in accordance with the FIFO policy for
every frame [43]. Each VoIP packet uses the AMC scheme at
the physical layer. The uplink MCS levels are determined by
the BS in relation to the quality of the signal received from
each SS. In [44], the arrivals process of the VoIP traffic has
been modeled as an exponentially distributed on-off model
with a mean on-time of α−1= 352 ms and a mean off-time
of β−1= 650ms. We use the two-state MMPP to model the
aggregate VoIP traffic requested frome active voice users, as
shown in Figure 2. The MMPP processes are very suitable
for formulating the multiuser VoIP traffic and capturing the
interframe dependency between consecutive frames [45]. The
two-state MMPP are characterized by the arrival Poisson
rates and the transition rates between them. The probability
that the status of the users is inactive (= off) in the simple
on-off model can be obtained by poff =β−1/(α−1+β−1),
and pon = 1 −pof f . The transition rate matrix (R)and
the Poisson arrival rate matrix (A)of the two-state MMPP
process can be expressed as follows:
R=−δ0δ0
δ1−δ1, A =λ00
0λ1(4)
The average arrival rate for the VoIP traffic at the queue
during the frame duration Tf, denoted λ, is given by:
λ=p0λ0+p1λ1=δ1λ0+δ0λ1
δ0+δ1
(5)
where pmis the probability that the process is in phase
m, (m= 0,1).
For simplicity, we use the Index of Dispersion for Counts
(IDC) matching technique to determine the four parameters
of the two-state MMPP as follows [46]:
λ0=A
Mv
P
j=0
jπj
Mv
P
i=0
πi
, λ1=A
Nv
P
j=Mv+1
jπj
Nv
P
i=Mv+1
πi
(6)
IAENG International Journal of Computer Science, 44:1, IJCS_44_1_04
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sŽ sŽ
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>E >E
K&D
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ŚĂŶŶĞů^ƚĂƚĞ
µ
YƵĞƵĞ^ƚĂƚĞ
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Fig. 1. System model without Feedback Control.
TABLE I
MODULATION AND CODING SCHEMES FOR VOIP TRA FFIC
Rate ID Modulation Level (Coding) Information Bits/Symbol Required SNR (db)
0 BPSK (1/2) 0.5 6.4
1 QPSK (1/2) 1 9.4
2 QPSK (3/4) 1.5 11.2
3 16QAM (1/2) 2 16.4
4 16QAM (3/4) 3 18.2
5 64QAM (2/3) 4 22.7
6 64QAM (3/4) 4.5 24.4
where πj=Nv
jpj
on(1 −pon )Nv−j,Mv=bNv.ponc,
and A, which is the emission rate in the active state, equals
1/Tf. The transition rates are given as follows:
δ0=2(λ1−λavg )(λavg −λ0)2
(λ1−λ0)λavg (IDC(∞)−1) (7)
δ1=2(λ1−λavg )2(λavg −λ0)
(λ1−λ0)λavg (IDC(∞)−1) (8)
where λavg =Nv.A.pon and IDC(∞)is given as follows
[47].
IDC(∞) = 1 + 2(λ0−λ1)2δ0δ1
(δ0+δ1)2(λ0δ1−λ1δ1)(9)
V. QUEUEING MODEL AND PERFORMANCE PARAMETERS
A. System without Feedback Control
To model this system, we use a queueing system
M M P P −2/M/1/N characterized by a two states MMPP
arrival process with parameters λ0,λ1,δ0, and δ1, and
exponentially distributed service time with parameter µ. The
inter-arrival times are exponential and all these variables are
mutually independent between them, the corresponding state
transition diagram is shown in Figure 3.
The state of the system is described by the two dimen-
sional process Yt= (St, Xt), where Stis the state (phase)
of an irreducible Continuous Time Markov Chain (CTMC),
Xtis the number of VoIP packets in the queue at the end of
every frame. Thus, the state space of the system is given by:
E={(s, x)/s ∈ {0,1},0≤x≤N}(10)
From the Figure 3 we observe that if lexicographic or-
dering of the states is used, then the infinitesimal generator
2N×2Nmatrix is Q= [Qi,j ], where iand jare two-
dimensional vectors, is given by [48]:
Q=











D0A10 0 0 0 . . . 0 0 0
A2A0A10 0 0 . . . 0 0 0
0A2A0A10 0 . . . 0 0 0
0 0 A2A0A10. . . 0 0 0
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
0 0 0 0 0 . . . 0A2A0A1
0 0 0 0 0 0 . . . 0A2D1











(11)
where
D0=−(λ0+δ0)δ0
δ1−(λ1+δ1)(12)
IAENG International Journal of Computer Science, 44:1, IJCS_44_1_04
(Advance online publication: 22 February 2017)
______________________________________________________________________________________

/WEĞƚǁŽƌŬ
^ƵďƐĐƌŝďĞƌ^ƚĂƚŝŽŶ;^^Ϳ
ĂƐĞ^ƚĂƚŝŽŶ;^Ϳ
^ƵďƐĐƌŝďĞƌ^ƚĂƚŝŽŶ;^^Ϳ
sŽ sŽ
sŽ/W sŽ/W
>E >E
K&D
ƚƌĂŶƐŵŝƚƚĞƌ
DDWWͲϮ
ƌƌŝǀĂůƐ
ŚĂŶŶĞů^ƚĂƚĞ
µ
>>
dŚĞĐŽŶƚƌŽůůĞƌ
;> ĂĚũƵƐƚŵĞŶƚͿ
ƌƌŝǀĂůƌĂƚĞ
ĂĚũƵƐƚŵĞŶƚ
Fig. 2. System model with Feedback Control
0
δ
Ϭ͕EͲϮ
͙
0
λ
0
λ
0
λ
0
λ
0
λ
0
λ
µµ
µ
µµµ
Ϭ͕ϮϬ͕ϭϬ͕Ϭ Ϭ͕EͲϭ Ϭ͕E
ϭ͕EͲϮ
͙
1
λ
1
λ
1
λ
1
λ
1
λ
1
λ
µ
µ µ
µ µ µ
ϭ͕Ϯϭ͕ϭϭ͕Ϭ ϭ͕EͲϭ ϭ͕E
0
δ
0
δ
0
δ
0
δ
0
δ
1
δ
1
δ
1
δ
1
δ
1
δ
1
δ
Fig. 3. State transition diagram for system model without Feedback Control
D1=−(δ0+µ)δ0
δ1−(δ1+µ)(13)
A0=−(λ0+δ0+µ)δ0
δ1−(λ1+δ1+µ)(14)
A1=λ00
0λ1(15)
A2=µ0
0µ(16)
The steady state probability πof the system is expressed
as follows:
π= [π0,0, π1,0, π0,1, π1,1, ..., π0,N−1, π1,N −1, π0,N , π1,N ]
(17)
where πs,x values can be obtain by solving the following
finite set of steady-state equations [49]:
πQ = 0,
1
X
s=0
N
X
x=0
πs,x = 1 (18)
Using stochastic balance equations, we obtain the steady-
state probability as :



















π0,0=π0,1µ+π1,0δ1
λ0+δ0
π1,0=π1,1µ+π0,0δ0
λ1+δ1
π0,i =π0,i+1µ+π1,i δ1+π0,i−1λ0
λ0+δ0+µ, i = 1, ..., N −1
π1,i =π1,i+1µ+π0,i δ0+π1,i−1λ1
λ1+δ1+µ, i = 1, ..., N −1
π0,N =π0,N−1λ0+π1,N δ1
µ+δ0
π1,N =π1,N−1λ1+π0,N δ0
µ+δ1
(19)
We then calculate the performance parameters as follows.
First, to obtain the loss probability Ploss we again notice
IAENG International Journal of Computer Science, 44:1, IJCS_44_1_04
(Advance online publication: 22 February 2017)
______________________________________________________________________________________

that πN=π0,N +π1,N is the proportion of time that the
buffer is full. The proportion of VoIP packets that are lost
is therefore, the ratio of the number of VoIP packets arrive
during the frame time duration Tfthat the buffer is full to
the total number of VoIP packets that arrive. Therefore, the
loss probability can be obtained as follows:
Ploss =λ0π0,N +λ1π1,N
λ(20)
We compute E[N], the mean number of current VoIP
packets in the system as:
E[N] =
1
X
s=0
N
X
x=0
xπs,x (21)
The throughput measures the number of packets transmit-
ted in one frame. It can be obtained from:
φ=λ(1 −Ploss)(22)
Finally, the average VoIP packet delay is defined as the
number of frames that a packet waits in the buffer (queue)
since its arrival before it is transmitted. From Littles law, we
can obtain the average delay as follows:
D=E[N]
λ(1 −Ploss)(23)
B. System with Feedback Control
The queueing model used in this system can be considered
as a modification of a M M P P −2/M/1/N queue. The
state transition diagram for the proposed system is shown
in Figure 4. The first threshold L1is fixed and the second
threshold L2can be adjusted to any position in the queue.
Let kbe the total number of VoIP packets in the queue at
time t.
•If 0≤k < L1, then the arrival rate of the VoIP packets
is λ0in state 0(MMPP is in state 0) and λ1in state 1
(MMPP is in state 1).
•If L1≤k < L2, then the arrival rate of the VoIP packets
is reduced to λ0L1=λ0/2in state 0(MMPP is in state
0) and λ1L1=λ1/2in state 1(MMPP is in state 1).
•If k≥L2, then no VoIP packets arrives in the queue.
For this system model, the state of the system is described
at time t(t≥0) by the stochastic process Zt= (Ut, Vt),
where Utis the phase of the MMPP and Vtis the number
of the VoIP packets in the queue at time t. The state space
of Ztis given by:
F={(u, v)/u ∈ {0,1},0≤v≤L2}(24)
Due to the use of two thresholds in this system model,
the queue needs to be considered in two parts in order
to calculate the steady-state probabilities. Using the same
analysis, the steady-state probabilities can be expressed as a
solution of the balance equations as follows:









































π0,0=π0,1µ+π1,0δ1
λ0+δ0
π1,0=π1,1µ+π0,0δ0
λ1+δ1
π0,i =π0,i+1µ+π1,i δ1+π0,i−1λ0
λ0+δ0+µ, i = 1, ..., L1−1
π1,i =π1,i+1µ+π0,i δ0+π1,i−1λ1
λ1+δ1+µ, i = 1, ..., L1−1
π0,L1=π0,L1−1λ0+π1,L1δ1+π0,L1+1µ
λ0L1+δ0+µ
π1,L1=π1,L1−1λ1+π0,L1δ0+π1,L1+1µ
λ1L1+δ1+µ
π0,k =π0,k+1µ+π1,k δ1+π0,k−1λ0
λ0L1+δ0+µ, k =L1+ 1, ..., L2−1
π1,k =π1,k+1µ+π0,k δ0+π1,k−1λ1L1
λ1L1+δ1+µ, k =L1+ 1, ..., L2−1
π0,L2=π0,L2−1λ0L1+π1,L2δ1
µ+δ0
π1,L2=π1,L2−1λ1L1+π0,L2δ0
µ+δ1(25)
The performance parameters are derived from the steady-
state probabilities as follows. The loss probability of VoIP
packets Ploss
CF is defined as the flow rate of a VoIP packet lost
on the total flow rate and is given by the following formula:
Ploss
CF =λ0L1π0,L2+λ1L1π1,L2
λeff
(26)
Where λeff is the effective arrival rate of VoIP packets in
this system. It is computed by the following formula:
λeff =
1
X
s=0
λs
L1
X
x=0
xπs,x +
1
X
s=0
λs
2
L2
X
n=L1+1
nπs,n (27)
We can compute ECF [N], the mean total number of VoIP
packets in the system is given by:
ECF [N] =
1
X
s=0
L1
X
x=0
xπs,x +
1
X
s=0
L2
X
n=L1+1
nπs,n (28)
The throughput can be obtained from:
φCF =λef f (1 −Ploss
CF )(29)
Finaly, from Little’s law formula, we can obtain the
average delay of VoIP packets in the system as follows:
DCF =EC F [N]
λef f (1 −Ploss
CF )(30)
VI. NUMERICAL RE SU LTS AN D AN ALYS IS
In this section we present the numerical results of both
systems. We use the Matlab software to numerically solve
and evaluate various performance parameters. The perfor-
mance parameters are measured respectively under different
traffic intensities with channel SNR in the range of rate ID =
0, and under different channel qualities with constant traffic
intensity.
According to the data reported in this work, the maximum
data rate capacity of the WiMAX channel was 20Mbps.
Average SNR on each subchannel is 5dB. The carrier
frequency was 2.6GHz with a bandwidth equal to 12M H z,
number of sub-carriers was 2048 and the modulation Level
was QP SK (1/2). The VoIP traffic can be modeled by a two-
state MMPP model, namely transition rate δ0, from state 0
to state 1, is 0.17, while the reciprocal transition rate δ1,
from state 1to state 0, is 0.08. The VoIP arrival rates λ0
IAENG International Journal of Computer Science, 44:1, IJCS_44_1_04
(Advance online publication: 22 February 2017)
______________________________________________________________________________________

0
δ
Ϭ͕>
͙
0
λ
0
λ
0
λ
0
λ
µµ µµµµ
Ϭ͕ϮϬ͕ϭϬ͕Ϭ Ϭ͕> нϭ Ϭ͕> нϮ
ϭ͕>
͙
1
λ
1
λ
1
λ
1
λ
µ
µ µ µ µ µ
ϭ͕Ϯϭ͕ϭϭ͕Ϭ ϭ͕> нϭ ϭ͕> нϮ
0
δ
0
δ
0
δ
0
δ
0
δ
1
δ
1
δ
1
δ
1
δ
1
δ
1
δ
͙
µ
͙
µ
Ϭ͕>
ϭ͕>
µ
0
δ
1
δ
µ
1
1L
λ
1
1L
λ
1
1L
λ
1
1L
λ
1
0L
λ
1
0L
λ
1
0L
λ
1
0L
λ
Fig. 4. State transition diagram for system model with Feedback Control
and λ1, associated to state 0and 1, are equal, respectively,
to 22.1and 7.16. Bandwidth demand for VoIP application
was uniformly distributed in the range of 128 to 512Kbps.
The queue size is 150 VoIP packets (i.e. N= 150).
For system with Feedback Control, the value of the
threshold L1is fixed to 50 VoIP packets (i.e. L1= 50).
Note that, we vary some of these parameters based on the
evaluation scenarios whereas the others remain fixed.
0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Traffic I nte nsity (p ackets/frame)
Packet Loss Probability
System wihout Fee dback Control
System wih Fee dback Control
Fig. 5. Packet Loss Probability under Traffic Intensity.
0 10 20 30 40 50
0
50
100
150
Traffic I nte nsity (p ackets/frame)
Mean number of VoIP packets in queue
System wihout Fee dback Control
System wih Fee dback Control
Fig. 6. Mean Number of VoIP Packet in queue under Traffic Intensity.
The performance parameters under different traffic in-
tensity are shown in Figures 5, 6, 7 and 8 for packet
loss probability, mean number of VoIP packets in queue,
0 10 20 30 40 50
0
5
10
15
Traffic I nte nsity (p ackets/frame)
Average VoIP packet delay (packets/frame)
System wihout Fee dback Control
System wih Fee dback Control
Fig. 7. Average VoIP Packet Delay under Traffic Intensity.
0 10 20 30 40 50
2
3
4
5
6
7
8
9
10
Traffic I nte nsity (p ackets/frame)
Throughput(packets/frame)
System wihout Fee dback Control
System wih Fee dback Control
Fig. 8. Throughput under Traffic Intensity.
average VoIP packet delay, and throughput, respectively.
These performance parameters are significantly impacted by
traffic intensity. When the traffic intensity varies and shows
that the second system is more effective, especially when
traffic intensity is higher. We remark that with lower traffic
intensity, both systems have the same performance. Indeed,
with lower values the control has no effect as the number of
packets in the queue does not exceed the first threshold.
Variations in packet loss probability and average packet
delay under different channel qualities are shown in Figures 9
and 10. We remark that when the SNR increases, the packets
IAENG International Journal of Computer Science, 44:1, IJCS_44_1_04
(Advance online publication: 22 February 2017)
______________________________________________________________________________________

4 5 6 7 8 9 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
SNR (db)
Packet Loss Probability
System wihout Fee dback Control
System wih Fee dback Control
Fig. 9. Packet Loss Probability under different channel qualities.
4 5 6 7 8 9 10
0
10
20
30
40
50
60
70
SNR (db)
Average VoIP packet delay (frames)
System wihout Fee dback Control
System wih Fee dback Control
Fig. 10. Packet Loss Probability under different channel qualities.
loss probability and the average delay of the VoIP packets are
lower in the second system than in the first system and when
SNR is lower, the second system is clearly more effective.
We remark that the second system where the system
is combined with an AQM achieves a gain on the loss
probability and average delay of VoIP packets. The results
show that when the SNR level is high, the mechanism with
control and the mechanism without control have the same
behavior as the quality of channel is good and all packets
can be processed.
VII. CONCLUSIONS
In this paper, we have analyzed and compared the per-
formance of VoIP traffic in a single cell IEEE 802.16e
using two system models, called system without Feedback
Control model, and system with Feedback Control model.
The performance parameters of these system models are
analyzed in terms of packet loss probability, mean number
of VoIP packets in queue, average VoIP packet delay, and
throughput. We have considered a WiMAX system model in
which a base station serves multiple subscriber stations and
each of the subscriber stations is allocated with a certain
number of subchannels by the base station. Mathematical
tools are used in this study, we have used two-state MMPP
processes to model the arrival of VoIP packets in the system,
and performance parameters are analytically deducted. By
comparing the VoIP performance parameters under the sys-
tem without Feedback Control model, and the system with
Feedback Control model, we showed that the second system
model outperforms than the first one.
.
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Said El Kafhali is an assistant professor of
computer sciences at National School of Applied
Sciences, Hassan 1st University, Morocco. He
joined the National School of Applied Sciences
in January 2014. He obtained his PhD in com-
puter science and networks in 2013 from Hassan
1st University. He received the B.Sc. degree in
Computer Sciences from Sidi Mohamed Ben Ab-
dellah University, in 2005, and a M.Sc. degree
in Mathematical and Computer engineering from
Hassan 1st University, in 2009. He is an IAENG
member and a member of the IAENG Society of Internet Computing and
Web Services and the IAENG Society of Wireless Networks. His current
research interests queuing theory, Performance Modeling and Simulation,
Cloud Computing, and Networks Security.
Mohamed Hanini is currently a professor at the
department of Mathematics and computer science
in the Faculty of Sciences and techniques, Set-
tat, Morocco. He obtained his PhD degree in
mathematics and computer in 2013 and a master
degree of mechanical engineering and scientific
computing in 2007 from the Faculty of Sciences
and Techniques, Settat Morocco, and a Bachelor
degree from Cadi Ayyad University, Marrakesh,
Morocco. He is a member of e-NGN Africa and
Middle East research group, and an IAENG mem-
ber. His research interests include stochastic processes, queuing theory, mod-
eling, evaluation and analysis of computer systems and telecommunication
networks performances.
IAENG International Journal of Computer Science, 44:1, IJCS_44_1_04
(Advance online publication: 22 February 2017)
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