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ANALYSIS OF M2/M2/1/R, N QUEUING MODEL FOR
MULTIMEDIA OVER 3.5G WIRELESS NETWORK DOWNLINK
SULEIMAN Y. YERIMA AND KHALID AL-BEGAIN
Mobile Computing, Communications and Networking RG
Faculty of Advanced Technology, University of Glamorgan
Pontypridd (Cardiff) CF37 1DL, Wales, UK
E-mail: {syerima,kbegain}@glam.ac.uk
ABSTRACT
Analysis of an M2/M2/1/R, N queuing model for the multimedia transmission over HSDPA/3.5G downlink is presented.
The queue models the downlink buffer with source multimedia traffic streams comprising two classes of flows: real-
time and non real-time. Time priority is accorded to the real-time flows while the non real-time flows are given buffer
space priority. An analytic evaluation of the impact of varying the buffer partition threshold on the QoS performance of
both classes of customers is undertaken. The results are validated with a discrete event simulation model developed in C
language. Finally, a cost function for the joint optimization of the traffic QoS parameters is derived.
KEYWORDS: 3.5G Wireless Networks, Performance Modelling, QoS optimization, Multimedia traffic, Stochastic
Models.
INTRODUCTION
3G WCDMA mobile cellular networks have been
widely deployed across the globe to support higher data
rate applications and services that previous cellular
networks were unable to provide. In recent times, new
standards and recommendations for the enhancement to
3G to meet insatiable demand for new wireless
broadband and broadcast services have been introduced.
In early 2002, Release 5 introduced improved support
for downlink packet data over WCDMA networks in the
form of High Speed Downlink Packet Access (HSDPA)
technology otherwise referred to as 3.5G (Van den berg
et al. 2005).
In addition to significantly reducing downlink latency,
HSDPA enables peak data rates of 14.4 Mbps and a
three-fold capacity increase in WCDMA networks. The
enhancement to WCDMA is achieved mainly through
the following techniques: Link adaptation, Hybrid ARQ,
and Fast Scheduling. To meet the demand on low
latency and rapid resource (re)allocation, the enhancing
functionalities have been located in the Node B, the
base station, as part of additions to the WCDMA MAC
layer. Furthermore, a shorter transmission time interval
(TTI) of 2ms, is employed which reduces overall delay
and improves tracking of fast channel variations
exploited by the link adaptation and channel-dependent
fast scheduling. See (3GPP 2001; Kolding et al. 2003;
Parkvall et al. 2006) for further details on HSDPA.
With the improved support for broadband and broadcast
services in WCDMA networks, the growth in demand
for multimedia-based applications and services is set to
escalate. In a HSDPA network carrying downlink
multimedia traffic destined for a user terminal in one of
its cells, QoS performance gains could be achieved from
differentiated consideration of the several flows
comprising the multimedia streams as they traverse the
Node-B downlink buffer. Based upon this idea, a novel
queuing model for performance evaluation of
multimedia traffic over 3.5G wireless downlink was
introduced in (Al-Begain et al. 2005). In both (Al-
Begain et al. 2005) and (Al-Begain et al. 2006), the
model has been investigated analytically with
BMAP/PH/1/R,N and M2/M2/1/N,∞ queues
respectively. While in (Yerima and Al-Begain, 2006), a
comparative analysis of the downlink buffer partitioning
schemes based on the model was carried out.
This paper extends the work presented in (Yerima and
Al-Begain, 2006) by further investigation of the impact
of varying the buffer partitioning threshold on the QoS
parameters of the multimedia traffic, by means of
analytical modelling. The results obtained are then
validated using Discrete Event Simulation of the system
in C language. Additionally, a cost function for joint
optimization of the investigated QoS performance
metrics is derived, and, finally it is shown that for a
given set of traffic and system parameters an optimum
buffer partition threshold is obtainable from the cost
function.
The rest of the paper is organized as follows. Section 2
describes the conceptual model of the system. Section 3
is devoted to the analytical evaluation of the model,
while numerical results of the experiments are presented
in section 4. Finally, the concluding remarks are given
in section 5.
MODEL DESCRIPTION
A single-server priority queue with a finite capacity T =
N + R, is considered for the buffer model. From the
multimedia source, two classes of traffic, real-time (RT)
and non real-time (NRT), arrive according to the
stationary Poisson process with mean rates λrt and λnrt
respectively. Service times are assumed to be
exponentially distributed with mean rates µrt and µ nrt
with priority service given to real-time customers
whenever both types are present in the buffer. A non
pre-emptive priority access to service is accorded to the
real-time flows. Scheduling of service is FIFO (First-In-
First-Out) for both customers. The maximum numbers
of real-time customers that can be admitted into the
buffer is restricted to R, R > 0, and on arrival, are placed
ahead of the non real-time customers to be scheduled
for transmission.
Thus, real-time customers have ‘time priority’ over non
real-time customers, but, due to restriction of the real-
time customers’ access to the buffer, non real-time
customers have a ‘space priority’. Limiting the number
of admissible real-time customers to a constant, R,
effectively partitions the buffer, with R representing the
threshold as shown in Figure 1. Thus, the QoS
performance metrics such packet loss probability and
mean waiting time, of both customers are determined by
R, the threshold position which is also the number of
admissible real-time customers. Dynamic control of the
QoS performance metrics to meet optimization goals or
application requirements can potentially be achieved by
adaptively varying the threshold R.
Figure 1: HSDPA Downlink Buffer Model
ANALYTICAL EVALUATION
The system state is described by the stochastic process
S(t) = (R(t); N(t)), t ≥ 0; where R(t) is the number of
real-time customers and N(t) represents the number of
non real-time customers. All random variables are
exponentially distributed and hence the underlying
stochastic process is a two-dimensional continuous-time
Markov chain (CTMC) with finite state space as
depicted in Figure 2. The steady-state probabilities,
P( i ,j), of the system states are defined by:
( , ) lim ( ( ) , ( ) )
t
P i j P R t i N t j
→∞
= = =
,
0, , 0,
i R j N
= =
If the steady-state probability vector of all the possible
states P(i, j )of the CTMC is denoted by P, then the
steady-state probabilities can be obtained by solving the
following set of equations:
PG=0 and Pe=1 (1)
Figure 2: CTMC for the M2/M2/1/R, N queuing system
Where G is the transition probability matrix and e is a
column vector of the appropriate dimension consisting
of ones. If we denote by Pij →i’ j’ the steady-state
probability of transition from a given state S = (i, j) to
another state S = (i’, j’), then from observation of the
CTMC, the entries of the matrix G are defined by:
Pij →i’ j’ =
The system performance measures are calculated from
the following set of equations. Mean number of real-
time customers is given by:
0 0
( , )
R N
i j
rt
N i P i j
= =
=∑ ∑ (3)
Mean number of non real-time customers is given by:
0 0
( , )
R N
i j
nrt
N j P i j
= =
=∑ ∑ (4)
Loss probability of real time customers is given by:
0
( , )
N
j
rt
L P R j
=
=∑ (5)
Loss probability of non real time customers is given by:
Transmission to
user terminal
RT flows
NRT flows
R
Buffer
partition
threshold
Service
process
Multimedia
traffic
source
T
Maximum
buffer
capacity
λnrt i’ = i, j’ = j+1 j ≠ N
λrt i’ = i+1, j’ = j, i ≠ N
µnrt i’ = i, j’ = j-1, j ≠ 0
µrt i’= i-1, j’ = j, i ≠ 0
(λrt + λnrt) - (µnrt + µ rt) i’ = i = 0, j’ = j = 0
λrt - µrt i’= i = 0, j’ = j (1≤ j ≤ N-1)
(λrt +µnrt) - (λnrt + µrt) i = i = 0, j’ = j = N
λnrt - µnrt i’ = i (1≤ i ≤ R-1), j’= j=0
(λnrt +µrt) - (λrt + µ nrt) i’ = i = R, j’= j = 0
µrt - λrt i’ = i = R, j’ = j (1≤ j ≤ N-1)
(µnrt + µrt) - (λrt + λnrt) i’ = i = R, j= j = N
µnrt - λnrt i’ = i (1≤ i ≤ R-1), j’ = j=N
0, otherwise
(2)
λrt
λrt
λnrt
λrt
λrt
λrt
λnrt λnrt λnrt
µrt
µrt
µrt
λrt
λrt
µrt
λnrt
µnrt
λnrt
0, 1 0, 0 0, N 0,N-1 0, 2
µnrt
λnrt
µnrt µnrt
λnrt λnrt
µnrt
λnrt
µnrt
λnrt
1, 1 1, 0 1, N 1,N-1 1, 2
µnrt
λnrt
µnrt µnrt
λnrt
λnrt
µnrt
µnrt
λnrt
R-1,1 R-1,0 1, N
R-1,N-1
R-1,2
µnrt
λnrt
µnrt µnrt µnrt
µnrt
λnrt
R, 1 R, 0 R, N
R, N-1
R, 2
µnrt
λnrt
µnrt µnrt
λnrt
λnrt
µnrt
µrt λrt µrt λrt µrt λrt µrt λrt µrt
µrt λrt µrt λrt µrt λrt µrt λrt µrt
µrt
µrt λrt λrt
µrt λrt µrt λrt
λrt µrt
µrt
0
( , )
R
i
nrt
L P i N
=
=∑ (6)
With the equations (3) – (6), the mean delay or waiting
time can be calculated using Little’s theorem (Bolch et
al. 1998) thus:
Mean delay for real-time customers is given by:
r
(1 )
rt
rt
rt t
N
D
L
λ
=× − (7)
Similarly, mean delay for non real-time customers is
given by:
r
(1 )
nrt
nrt
nrt n t
N
DL
λ
=× − (8)
Joint Optimization of the System Qos Parameters
As mentioned earlier, the threshold of the buffer can be
varied to meet QoS requirements or optimize the QoS
parameters according to some given criteria. In (Al-
Begain et al. 2006), an economic criterion γ called the
Weighted Grade of Service (WGoS) was formulated for
the investigated queuing network model. An adapted
form of the WGoS criterion suited to the M2/M2/1/R, N
buffer queuing model being investigated in this paper is
assumed to be of the following form:
[ ]
[ ]
(1 )
(1 )
rt
rt rt
rt nrt
nrt
nrt nrt
rt nrt
CLrt L Lrt CDrt D
CLnrt L Lnrt CDnrt D
λ
γλ λ
λ
λ λ
= × + − × ×
+
+ × + − × ×
+
(9)
Where CDrt is the penalty for the mean delay of RT
customers; CDnrt is the penalty for the mean delay of
NRT customers. Likewise, CLrt is the penalty for the
loss of RT customers while CLnrt is the cost penalty for
the loss of NRT customers. We use γ, the WGoS
function, to determine the optimum operating threshold
position for a given set of traffic and system parameters,
in the next section.
NUMERICAL RESULTS
In this section, the results of two sets of experiments are
presented. The first investigates the impact of the buffer
partition threshold on traffic performance. The second
employs the optimization cost function, γ, to determine
the optimum buffer partition threshold for a given set of
traffic parameters.
Impact of Buffer Partition Threshold on Traffic
Performance
From the analytical model developed earlier, we study
the effect of varying the buffer partition threshold R, on
traffic performance. Performance measures are obtained
from the formulae shown earlier, using the analytical
modelling tool MOSEL (Al-Begin et al. 2002; Beutel
2003). In order to validate the analytical study, an
equivalent simulation model was developed in C
language and the results from both models are
illustrated side-by-side. Figures 3 to 6 show the results
of the first scenario assuming the following parameters:
λnrt = 6, µnrt = 10, µ rt = 20. The buffer partition threshold
R is varied from 2 to 16 whilst buffer capacity T= R+N
is fixed at 20. The loss and delay performances for both
classes of traffic are shown for λrt = 2, 12, and 18
corresponding to low, medium and high RT traffic loads
respectively.
In Figure 3 we see that the loss probability of NRT
traffic increases with R which is because less buffer
space becomes available to NRT packets as R increases.
Notice also the deteriorating loss performance with
increase in λrt for a given buffer partition threshold,
which is as a result of the priority access to service
enjoyed by the RT traffic. Figure 4 shows how R affects
the mean NRT delay. It is clear that at medium and high
λrt load, the mean delay increases, peaks and then
decreases again as R is increased. The mean NRT delay
increases at first because more RT traffic is buffered as
R increases. The drop in the mean delay with further
increase in R can be attributed to less NRT traffic being
retained in the buffer, as the corresponding NRT buffer
space N decreases correspondingly. Figure 5 shows RT
loss probability dropping with increase in R as
expected, while in Figure 6 mean RT delay is seen to
increase with R and vice versa. Notice also from
Figures 4 to 6 that low intensity of RT traffic seems to
have only marginal effect on traffic performance for this
scenario.
Figure 3: NRT loss Vs R for λrt = 2, 12, and 18
Figure 4: NRT delay Vs R for λrt = 2, 12, and 18
Figure5: RT loss Vs R for λrt = 2, 12, and 18
Figure 6: RT delay Vs R for λrt = 2, 12, and 18
Similarly, Figures 7 to 11 illustrate the results of the
second scenario investigated with the following
parameters: λrt = 12, µrt = 20, µnrt = 10 and T=20. R is
again varied from 2 to 16 and λnrt assumes the values 2,
6, and 9 corresponding to low, medium and high NRT
traffic. Figure 7 shows similarity to figure 3, i.e. NRT
loss probability increasing with larger R and also with
higher NRT traffic. In Figure 8, the mean NRT delay
increases, peaks and then drops again with increase in R
as in Figure 4. Thus, the same reasons given earlier that
account for the behavior in Figure 4 also apply here.
Finally, the RT traffic performance for the second
scenario is depicted in Figures 9 and 10. RT loss
probability is seen to decrease with R as expected (c.f.
Figure 9), but, increasing NRT intensity from medium
to high loads seem to have very little effect on RT loss.
This could also be attributed to the prioritized access to
service accorded to RT traffic which to some extent
shields it from effects of λnrt variation. From Figure 10
we also see that as R increases so does mean RT delay.
Again, due to RT service priority access, high and
medium NRT have only marginal effect on the mean
RT delay performance. The results of the experiments
suggest that varying the buffer partition threshold affect
the QoS performance metrics of the multimedia traffic
differently and, thus, an optimum threshold can be
found by trading off the QoS performance metrics
against each other.
Figure7: NRT loss Vs R for λnrt = 2, 6, and 9
Figure8: NRT delay Vs R for λnrt = 2, 6, and 9
Figure9: RT loss Vs R for λnrt = 2, 6, and 9
Figure10: RT delay Vs R for λnrt = 2, 6, and 9
Deriving Optimum Buffer Partition Threshold from
Traffic Parameters
From the WGoS equation (9), it is clear that the
optimum buffer capacity threshold can be determined
for a given set of traffic parameters since the
performance metrics are dependent on those parameters.
For this experiment, the cost values are taken as
follows: CLrt = 300, CLnrt =50; CDrt=1000, and
CDnrt=1. From Figure 11 we see that for λrt = 12, λrt =
18 the optimum buffer capacity threshold, satisfying the
WGoS economic criterion with the above given cost
parameters, is the minimum value of 3. Other traffic
parameters are taken as: λnrt = 6, µnrt = 10, µrt = 20 and T
= 20 respectively. Likewise, Figure 12 shows that for
both λnrt = 6, λnrt = 9 and traffic parameters: λrt = 12, µrt
= 20, µnrt = 10, T=20, the buffer threshold which
minimizes γ, the WGoS, is 3.
Figure 11: WGoS Vs R for λrt = 12, and 18
Figure 12: WGoS Vs R for λnrt = 2, 6, and 9
CONCLUDING REMARKS
In this paper analysis of the M2/M2/1/R, N queuing
model for multimedia downlink transmission in 3.5G is
presented. The results of the experiments illustrate
clearly the impact of varying the downlink buffer
partition threshold on the traffic performance at various
load levels. Results of both analytic evaluation and
simulation are shown to be in very close agreement. In
addition, an economic criterion i.e. the WGoS, for the
joint optimization of the QoS parameters is derived for
the model. Finally, it is shown that for a given set of
traffic and system parameters an optimum buffer
partition threshold which jointly optimizes the
multimedia traffic QoS performance can be obtained
from the WGoS. In our future work, other arrival and
service distributions will be considered for the queuing
model. Furthermore, we aim to incorporate the model
into holistic simulated HSDPA communication
scenarios and then carry out performance evaluation.
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