An Efficient Analytical Model for the Dimensioning of WiMAX Networks.
ABSTRACT This paper tackles the challenging task of developing a sim- ple and accurate analytical model for performance evaluation of WiMAX networks. The need for accurate and fast-computing tools is of primary importance to face complex and exhaustive dimensioning issues for this promising access technology. In this paper, we present a generic Marko- vian model developed for three usual scheduling policies (slot sharing fairness, throughput fairness and opportunistic scheduling) that provides closed-form expressions for all the required performance parameters at a click speed. This model is compared in depth with realistic simulations that show its accuracy and robustness regarding the dierent modeling assumptions. Finally, the speed of our analytical tool allows us to carry on dimensioning studies that require several thousands of evaluations, which would not be tractable with any simulation tool.
- [Show abstract] [Hide abstract]
ABSTRACT: 4G is promising a wireless broadband with data rates up to 1Gbps. The two candidate technologies for 4G are the Advanced Long Term Evolution (Advanced LTE) which is based on the 3GPP standards and the WiMAX 2.0 based on the IEEE 802.16 family of standards. The common feature of both technologies is that they will provide All-IP connectivity with flexible bit rates and quality of service guarantees for multiple classes of services including voice, mainly using voice over IP, data and video services. Most of the performance studies of 4G technologies use highly complex and sophisticated simulations due to the multiple complexity factors in investigating 4G technologies such as All-IP flexible bit rates, adaptive coding and modulation as well as the multi-services provided. These factors usually make any modelling attempt very difficult. This paper presents a numerical/analytical model for a 4GWiMAX cell based on a multi-dimensional Continuous-Time Markov Chain (CTMC) model. Performance measures were derived for the key performance indicators such as throughput and average bit rate per cell and per service class. By assuming minimum acceptable bit rates for certain quality of service guarantees, we derived measures for blocking probabilities. The model has been formulated and solved using MOSEL-2 (Modelling Specification and Evaluation Language) which captures the key features of a 4G system that affect services at session/call level. The results obtained from the model using sample parameters show shat, the model can provide very useful insight to system behavior and can give good first indication to the performance of such a complex system.Wireless Personal Communications 02/2013; · 0.43 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]
ABSTRACT: This paper presents a problem of queuing theoretic performance modeling and analysis of Orthogonal Frequency Division Multiple Access (OFDMA) under broad-band wireless networks. We consider a single-cell WiMAX environment in which the base station allocates sub channels to the subscriber stations in its coverage area. The sub channels allocated to a subscriber station are shared by multiple connections at that subscriber station. To ensure the Quality of Service (QoS) performances, two Connection Admission Control (CAC) schemes, namely, threshold-based and queue-aware CAC schemes are considered at a subscriber station. A queuing analytical framework for these admission control schemes is presented considering OFDMA-based transmission at the physical layer. Then, based on the queuing model, both the connection-level and the packetlevel performances are studied and compared with their analogues in the case without CAC. The connection arrival is modeled by a Poisson process and the packet arrival for a connection by a Markov Modulated Poisson Process (MMPP). We determine analytically and numerically different performance parameters, such as connection blocking probability, average number of ongoing connections, average queue length, packet dropping probability, queue throughput and average packet delay.04/2013; - SourceAvailable from: Malamati LoutaAntonios Sarigiannidis, Petros Nicopolitids, Georgios Papadimitriou, Panagiotis Sarigiannidis, Malamati Louta, Andreas Pomportsis[Show abstract] [Hide abstract]
ABSTRACT: Worldwide Interoperability for Microwave Access (WiMAX) family of standards have introduced a flexible, efficient, and robust wireless interface. Among other interesting features, WiMAX access networks bring into play a flexible determination of the ratio between the downlink and the uplink directions, allowing a relation width from 3:1 to 1:1 respectively. However, this promising feature is not properly utilized, since hitherto scheduling and mapping schemes proposed neglect it. In this work, this challenging issue is effectively addressed by proposing an adaptive model that attempts to adequately adjust the downlink-to-uplink sub-frame width ratio according to the current traffic conditions. In the context of a mobile WiMAX wireless access network, the Base Station is enhanced with an error-aware Learning Automaton in order to be able to identify the magnitude of the incoming and the outgoing traffic flows and in turn to suitably define the ratio on a frame-by-frame basis. The model designed is extensively evaluated under realistic and dynamic scenarios and the results indicate that its performance is clearly improved compared to schemes having predefined, fixed ratio values. K e y w o r d s -IEEE 802.16, Learning Automata, mapping, OFDMA, WiMAXIEEE Systems Journal 01/2014; · 1.27 Impact Factor
Page 1
An Efficient Analytical Model for the
Dimensioning of WiMAX Networks
Bruno Baynat1, Georges Nogueira1, Masood Maqbool2, and Marceau
Coupechoux2
1Universite Pierre et Marie Curie - Paris, France {firstname.lastname}@lip6.fr
2Telecom ParisTech - Paris, France {firstname.lastname}@telecom-paristech.fr
Abstract. This paper tackles the challenging task of developing a sim-
ple and accurate analytical model for performance evaluation of WiMAX
networks. The need for accurate and fast-computing tools is of primary
importance to face complex and exhaustive dimensioning issues for this
promising access technology. In this paper, we present a generic Marko-
vian model developed for three usual scheduling policies (slot sharing
fairness, throughput fairness and opportunistic scheduling) that provides
closed-form expressions for all the required performance parameters at a
click speed. This model is compared in depth with realistic simulations
that show its accuracy and robustness regarding the different modeling
assumptions. Finally, the speed of our analytical tool allows us to carry
on dimensioning studies that require several thousands of evaluations,
which would not be tractable with any simulation tool.
1Introduction
The evolution of last-mile infrastructure for wired broadband networks faces
acute implications such as difficult terrain and high cost-to-serve ratio. Latest
developments in wireless domain could not only address these issues but could
also complement the existing framework. One of such highly anticipated tech-
nologies is WiMAX (Worldwide Interoperability for Microwave Access) based
on IEEE standard 802.16. The first operative version of IEEE 802.16 is 802.16-
2004 (fixed/nomadic WiMAX) [1]. It was followed by a ratification of mobile
WiMAX amendment IEEE 802.16e in 2005 [2]. On the other hand, the con-
sortium WiMAX Forum was found to specify profiles (technology options are
chosen among those proposed by the IEEE standard), define an end-to-end ar-
chitecture (IEEE does not go beyond physical and MAC layer), and certify
products (through inter-operability tests).
Some WiMAX networks are already deployed but most operators are still
under trial phases. As deployment is coming, the need arises for manufacturers
and operators to have fast and efficient tools for network design and performance
evaluation. Literature on WiMAX performance evaluation is constituted of two
sets of papers: i) packet-level simulations that precisely implement system details
and scheduling schemes; ii) analytical models and optimization algorithms that
derive performance metrics at user-level.
Page 2
In the former set, two interesting papers are [13] and [8] because they in-
vestigate QoS support mechanisms of the standard. Among the latter set of pa-
pers, [17] proposes an analytical model for studying the random access scheme
of IEEE 802.16d. Niyato and Hossain [15] formulate the bandwidth allocation of
multiple services with different QoS requirements by using linear programming.
They also propose performance analysis, first at connection level, and then, at
packet level. In the former case, variations of the radio channel are however not
taken into account. In the latter case, the computation of performance measures
rely on multi-dimensional Markovian model that requires numerical resolutions.
Not specific to WiMAX systems, generic analytical models for performance
evaluation of cellular networks with varying channel conditions have been pro-
posed in [6,7,14]. The models presented in these articles are mostly based on
multi-class processor-sharing queues with each class corresponding to users hav-
ing similar radio conditions and subsequently equal data rates. The variability
of radio channel conditions at flow level is taken into account by integrating
propagation models, mobility models or spatial distribution of users in a cell.
In order to use classical PS-queues results, these papers consider implicitly that
users can only switch class between two successive data transfers. However, as
highlighted in the next section, in WiMAX systems, radio conditions and thus
data rates of a particular user can change frequently during a data transfer. In
addition, capacity of a WiMAX cell may vary as a result of varying radio condi-
tions of users. As a consequence, any PS, DPS (discriminatory PS) or even GPS
(generalized PS) queue is not appropriate for modeling these channel variations.
In this paper, we develop a novel and generic analytical model that takes
into account frame structure, precise slot sharing-based scheduling and channel
quality variation of WiMAX systems. Unlike existing models [6,7,14], our model
is adapted to WiMAX systems’ assumptions and is generic enough to integrate
any appropriate scheduling policy. Here, we consider three classical policies: slot
sharing fairness, instantaneous throughput fairness, and opportunistic. For each
of them, we develop closed-form expressions for all performance metrics. More-
over, our approach makes it possible to take into account the so-called “outage”
situation. A user experiences an outage, if at a given time radio conditions are
so bad that it cannot transfer any data and is thus not scheduled. Once again,
classical PS-like queues are not appropriate to model this feature.
The rest of paper is organized as follows. Modeling assumptions are presented
in Section 2. Section 3 presents the generic analytical model and its adaption to
the three considered scheduling policies. Validation and robustness of model are
discussed in Section 4. Section 5 finally gives an example of WiMAX dimension-
ing process.
2Modeling assumptions
The development of our analytical model is based on several assumptions related
to the system, the channel, the traffic and the scheduling algorithm. We present
here these assumptions. All of them will be discussed in Section 3.4, and, as
Page 3
will be developed in that section, most of them can be relaxed, if necessary,
by slightly modifying the model. Wherever required, related details of WiMAX
system are specified. Various notations are also introduced in this section.
A WiMAX time division duplex (TDD) frame comprises of slots that are the
smallest unit of resource and which occupies space both in time and frequency
domain. A part of the frame is used for overhead (e.g., DL MAP and UL MAP)
and the rest for user data. The duration TFof this TDD frame is equal to 5 ms [2].
System assumptions We consider a single WiMAX cell and focus on the
downlink part which is a critical portion of asymmetric data traffic.
1. Overhead in the TDD frame is assumed to be constant and independent of
the number of concurrent active mobile station (MS). As a consequence, the
total number of slots available for data transmission in the downlink part is
constant and will be denoted by NS.
2. We assume that the number of MS that can simultaneously be in active
transfer is not limited. As a consequence, any connection demand will be
accepted and no blocking can occur.
One of the important features of IEEE 802.16e is link adaptation: differ-
ent modulation and coding schemes (MCS) allows a dynamic adaptation of the
transmission to the radio conditions. As the number of data subcarriers per slot
is the same for all permutation schemes, the number of bits carried by a slot
for a given MCS is constant. The selection of appropriate MCS is carried out
according to the value of signal to interference plus noise ratio (SINR). In case of
outage, i.e., if the SINR is too low, no data can be transmitted without error. We
denote the radio channel states as: MCSk, 1 ≤ k ≤ K, where K is the number
of MCS. By extension, MCS0represents the outage state. The number of bits
transmitted per slot by a MS using MCSkis denoted by mk. For the particular
case of outage, m0= 0.
Channel assumption The MCS used by a given MS can change very often
because of the high variability of the radio link quality.
3. We assume that each MS sends a feedback channel estimation on a frame by
frame basis, and thus, the base station (BS) can change its MCS every frame.
Since we do not make any distinction between users and consider all MS as
statistically identical, we associate a probability pkwith each coding scheme
MCSk, and assume that, at each time-step TF, any MS has a probability pk
to use MCSk.
Traffic assumptions The traffic model is based on the following assumptions.
4. All users have the same traffic characteristics. In addition, we don’t consider
any QoS differentiation here.
5. We assume that there is a fixed number N of MS that are sharing the
available bandwidth of the cell.
Page 4
6. Each of the N MS is assumed to generate an infinite length ON/OFF elastic
traffic. An ON period corresponds to the download of an element (e.g., a
web page including all embedded objects). The downloading duration de-
pends on the system load and the radio link quality, so ON periods must be
characterized by their size. An OFF period corresponds to the reading time
of the last downloaded element, and is independent of the system load. As
opposed to ON, OFF periods must then be characterized by their duration.
7. We assume that both ON sizes and OFF durations are exponentially dis-
tributed. We denote by ¯ xonthe average size of ON data volumes (in bits)
and by¯toff the average duration of OFF periods (in seconds).
Scheduling assumption The scheduling algorithm is responsible for allocat-
ing radio resources to users. In wireless networks, scheduling may take into ac-
count their radio link quality. In this paper, we have considered three traditional
schemes. The slot fairness scheduling allocates the same number of slots to all
active users. The throughput fairness scheduling ensures that all active users
have the same instantaneous throughput. The opportunistic scheduling gives all
resources to active users with the best channel.
8. At any time and for all scheduling policies, if there is only one active user, we
assume that the scheduler can allocate all the available slots for its transfer.
3WiMAX Analytical model
3.1Markovian model
A first attempt for modeling this system would be to develop a multi-dimensional
Continuous Time Markov Chain (CTMC). A state (n0,...,nK) of this chain
would be a precise description of the current number nk of MS using coding
scheme MCSk, 0 ≤ k ≤ K (including outage). The derivation of the transitions
of such a model is an easy task. However the complexity of the resolution of this
model makes it intractable for any realistic value of K. In order to work around
the complexity problem, we aggregate the state description of the system into
a single dimension n, representing the total number of concurrent active MS,
regardless of the MCS they use. The resulting CTMC is thus made of N + 1
states as shown in Fig 1.
(N − n + 1)λ
01
n − 1
...
...
(N − n)λ
Nλλ
µ(1)
µ(n)
µ(n + 1)
µ(N)
n
n + 1
N
Fig.1. General CTMC with state-dependent departure rates.
– A transition out of a generic state n to a state n + 1 occurs when a MS in
OFF period starts its transfer. This “arrival” transition corresponds to one
MS among the (N −n) in OFF period, ending its reading, and is performed
with a rate (N −n)λ, where λ is defined as the inverse of the average reading
time: λ =
1
¯ toff.
Page 5
– A transition out of a generic state n to a state n−1 occurs when a MS in ON
period completes its transfer. This “departure” transition is performed with
a generic rate µ(n) corresponding to the total departure rate of the frame
when n MS are active.
Obviously, the main difficulty of the model resides in estimating the aggregate
departure rates µ(n). In order to do so, we first express µ(n) as follows:
µ(n) =¯ m(n)NS
¯ xonTF
,
(1)
where ¯ m(n) is the average number of bits per slot when there are n concurrent
active transfers. Obviously, ¯ m(n) depends on K, the number of MCS, and pk,
0 ≤ k ≤ K, the MCS vector probability. It also strongly depends on n, because
the number of bits per slot must be estimated by considering all possible distri-
butions of the n MS between the K + 1 possible MCS (including outage). It is
worthwhile noting that the parameters ¯ m(n) finally depend on the scheduling
policy, as it defines, at each time-step, the quantity of slots given to each of the
n MS with respect to the MCS they use.
In order to provide a generic expression of ¯ m(n), we define xk(j0,...,jK) the
proportion of the resource (i.e., of the NSslots) that is associated to a MS using
MCSk, when the current distribution of the n MS among the K + 1 coding
schemes is (j0,...,jK). The average number of bits per slot, ¯ m(n), when there
are n active users, can then be expressed as follows:
?K
k=1
¯ m(n) =
(n,...,n)
?
j0?= n
(j0,...,jK) = (0,...,0)|
j0+ ... + jK = n
?
mkjkxk(j0,...,jK)
??
n
j1,...,jK
? K
k=0
?
pjk
k, (2)
where?K
that takes into account all such possibles distributions.
k=0pjk
kis the probability of any distribution of the n MS such that the
number of MS using MSCk is jk, and
?
n
j0,...,jK
?
is the multinomial coefficient
3.2Scheduling policy modeling
We now present the adaptation of the model, for the three specific scheduling
policies we consider in this paper. For each of them we provide closed-form
expressions for the average number of bits per slots, ¯ m(n).
Slot sharing fairness Each time-step, the scheduler equally shares the NS
slots among the active users that are not in outage. If, at a given time-step,
there are n active MS, each of the MS that are not in outage receives a portion
NS
n−j0of the whole resource. As a consequence, the proportion of the resource that
is associated to a MS using MCSk, is thus given by: xk(j0,...,jK) =
any k ?= 0. By replacing these proportions in generic expression (2) we obtain:
1
n−j0for
Page 6
¯ m(n) =
(n,...,n)
?
j0?= n
(j0,...,jK) = (0,...,0)|
j0+ ... + jK = n
n!
n − j0
?
K
?
k=1
mkjk
?
K
?
k=0
pjk
k
jk!.
(3)
Instantaneous throughput fairness The resource is shared in order to pro-
vide the same instantaneous throughput to all active users that are not in outage.
This policy allows MS using MCS with a low bit rate per slot to obtain, at a
given time-step, proportionally more slots compared to MS using a MCS with
a high bit rate per slot. In order to respect instantaneous throughput fairness
between all active users that are not in outage, the xk(j0,...,jK) must be such
that: mkxk(j0,...,jK) = C for any k ?= 0, where C is a constant such that
?K
k=1jkxk(j0,...,jK) = 1. By replacing the proportions xk(j0,...,jK) in generic
expression (2), the average number of bits per slot ¯ m(n) becomes:
¯ m(n) =
(n,...,n)
?
j0?= n
(j0,...,jK) = (0,...,0)|
j0+ ... + jK = n
(n − j0)n!
K
?
jk
mk
k=0
pjk
k
jk!
K
?
k=1
.
(4)
Opportunistic scheduling All the resource is given to users having the high-
est transmission bit rate, i.e., the better radio conditions and then the better
MCS. Without loss of generality, we assume here that the MCS are classified in
increasing order: m0< m1< ... < mK. And even if it is still possible to derive
the average bit rates from generic expression (2), we prefer to give here a more
intuitive and equivalent derivation.
We consider a system with n current active MS. We denote by αi(n) the
probability of having at least one active user (among n) using MCSiand none
using a MCS giving higher transmission rates (i.e., MCSj with j > i). As a
matter of fact, αi(n) corresponds to the probability that the scheduler gives at
a given time-step all the resource to MS that use MCSi. As a consequence, we
can express the average number of bits per slot when there are n active users as:
¯ m(n) =
K
?
i=1
αi(n)mi.
(5)
In order to calculate the αi(n), we first express the probability that there are no
MS using a MCS higher than MCSias: p≤i(n) =
calculate the probability that there is at least one MS using MCSiconditioned
by the fact that there are no MS using a better MCS: p=i(n) = 1−
αi(n) can thus be expressed as: αi(n) = p=i(n)p≤i(n).
?
1 −?K
j=i+1pj
?n
1−
. Then, we
?
pi
j=0pj
Pi
?n
.
Page 7
3.3Performance parameters
The steady-state probabilities π(n) can easily be derived from the birth-and-
death structure of the Markov chain (depicted in Fig. 1):
π(n) =
N!
(N − n)!
Tn
?
Fρn
n
Nn
S
i=1
¯ m(i)
π(0),
(6)
where ρ is given by relation (7) and plays a role equivalent to the “traffic inten-
sity” of Erlang laws [9], and π(0) is obtained by normalization.
ρ =¯ xon
¯toff
(7)
The performance parameters of this system can be derived from the steady-
state probabilities as follows. The average utilization¯U of the TDD frame is
given by:
N
?
The average number of active users¯Q is expressed as:
¯U =
n=1
π(n)min
?
n
¯ xon
NS¯ m(n),1
?
.
(8)
¯Q =
N
?
n=1
nπ(n).
(9)
The mean number of departures¯D (MS completing their transfer) by unit of
time, is obtained as:¯D =?N
We finally compute the average throughput¯ X obtained by each MS in active
transfer as:
¯ X =¯ xon
n=1π(n)µ(n). From Little’s law, we can derive the
average duration¯tonof an ON period (duration of an active transfer):¯ton=
¯ Q
¯ D.
¯ton.
(10)
3.4Discussion of the modeling assumptions
Our Markovian model is based on several assumptions presented in Section 2.
We now discuss these assumptions one by one (item numbers are related to the
corresponding assumptions), evaluate their accuracy, and provide, if necessary
and possible, extensions and generalization propositions.
1. DL MAP and UL MAP are located in the downlink part of the TDD frame.
They contain the information elements that allow MS to identify the slots
to be used. The size of these MAPs, and as a consequence the number NS
of available slots for downlink data transmissions, depends on the number
of MS scheduled in the TDD frame. In order to relax assumption 1, we can
express the number of data slots, NS(n), as a function of n, the number
of active users. This dependency can be easily integrated in the model by
replacing NSby?n
i=1NS(n) in relation (6), and NSby NS(n) in relation (1).
Page 8
2. A limit nmaxon the total number of MS that can simultaneously be in active
transfer, can be introduced easily if required. The corresponding Markov
chain (Fig. 1) has just to be truncated to this limiting state (i.e., the last
state becomes min(nmax,N)). As a result, a blocking can occur when a new
transfer demand arrives and the limit is reached. The blocking probability
can be derived easily from the Markov chain [4].
3. Radio channel may be highly variable or may vary with some memory. Our
analytical model only depends upon stationary probabilities of different MCS
whatever be the radio channel dynamics. This approach is authenticated
through simulations in Section 4.
4. More complex systems with multiple-traffic or differentiation between users
would naturally result into more complex models. This is left for future work.
5. Poisson processes are currently used in the case of a large population of users,
assuming independence between the arrivals and the current population of
the system. As we focus in this paper on the performance of a single cell
system, the potential population of users is relatively small. The higher the
number of on-going data connections, the less likely the arrival of new ones.
Poisson processes are thus a non-relevant choice for our models. Note however
that if Poisson assumptions have to be made for connection demand arrivals,
one can directly modify the arrival rates of the Markov chain (i.e., replace
the state-dependent rates (N − n)λ by some constant value, and limit the
number of states of the Markov chain as explained above in point 2).
6. Each MS is supposed to generate infinite length ON/OFF session traffic.
In [3], an extension to finite length sessions is proposed in the context of
(E)GPRS networks, where each MS generates ON/OFF traffic during a ses-
sion and does not generate any traffic during an inter-session. This work
shows that a very simple transformation of traffic characteristics that in-
creases OFF periods by a portion of the inter-session period, enables to
derive the average performance from the infinite length session model. The
accuracy of this transformation is related to the insensibility of the average
performance parameters with regards to the traffic distributions (see next
point). A similar transformation can be applied to our WiMAX traffic model.
7. Memoryless traffic distributions are strong assumptions that are validated by
several theoretical results on PS-like queues. Several works on insensitivity
have shown that the average performance parameters are insensitive to the
distribution of ON and OFF periods [5,11,12]. In its generic form, our model
is no longer equivalent to any PS-like queue, but we show in Section 4 by
comparing our model to extensive simulations (using Pareto distributions),
that insensibility still holds or is at least a very good approximation.
8. In some cellular networks (e.g. (E)GPRS), MS have limited transmission
capabilities because of hardware considerations. This constraint defines a
maximum throughput the network interface can reach or a maximum number
of resource units that can be used by the MS. This characteristic has been
introduced in the case of (E)GPRS networks [4] and consists in reducing the
departure rates of the first states of the Markov chain. The same idea can
be applied to our WiMAX model.
Page 9
4 Validation
In this section we discuss the validation and robustness of our analytical model
through extensive simulations. For this purpose, a simulator has been developed
that implements an ON/OFF traffic generator and a wireless channel for each
user, and a centralized scheduler that allocates radio resources, i.e., slots, to
active users on a frame by frame basis. We start with details of simulation before
presenting the simulations results for the validation and robustness studies.
4.1Simulation Models
System Parameters System bandwidth is assumed to be 10 MHz. The down-
link/uplink ratio of the WiMAX TDD frame is considered to be 2/3. We assume
for the sake of simplicity that the protocol overhead is of fixed length (2 sym-
bols). Considering subcarrier permutation PUSC, the total number of data slots
(excluding overhead) per TDD downlink sub-frame is NS= 450.
Traffic Parameters In our analytical model, we consider an elastic ON/OFF
traffic. Mean values of ON data volume (main page and embedded objects) and
OFF period (reading time), are 3 Mbits and 3 s respectively.
In the first phase (validation study), we assume that the ON data volume is
exponentially distributed as it is the case in the analytical model assumptions.
Although well adapted to Markov theory based analysis, exponential law does
not always fit the reality for data traffic. This is the reason why we consider
truncated Pareto distributions in the second phase (the robustness study). Re-
call that the mean value of the truncated Pareto distribution is given by equation
¯ xon=
α−1
value of Pareto variable and q is the cutoff value for truncated Pareto distribu-
tion. Two values of q are considered: lower and higher. The mean value in both
cases (q = 300 Mbits and b = 611822 bits for the higher cutoff and q = 3000
Mbits and b = 712926 bits lower cutoff) is 3 Mbits for the sake of comparison
with the exponential model. The value of α = 1.2 has been adopted from [10].
αb
?1 − (b/q)α−1?, where α is the shape parameter, b is the minimum
Channel Models A generic method for describing the channel between the
BS and a MS is to model the transitions between MCS by a finite state Markov
chain (FSMC). The chain is discrete time and transitions occurs every L frames,
with LTF<¯tcoh, the coherence time of the channel. In our case, and for the sake
of simplicity, L = 1. Such a FSMC is fully characterized by its transition matrix
PT = (pij)0≤i,j≤K, where state 0 represents outage. Stationary probabilities pk
provide the long term probabilities for a MS to receive data with MCS k.
In our analytical study, channel model is assumed to be memoryless, i.e., MCS
are independently drawn from frame to frame for each user, and the discrete
distribution is given by the (pi)0≤i,j≤K. This corresponds to the case where
pij = pj for all i. This simple approach, referred as the memoryless channel
model, is the one considered in the validation study. Let PT(0) be the transition
matrix associated to the memoryless model.
Page 10
Table 1. Stationary probabilities for channel models.
Channel
model
Memoryless AverageCombined
goodbad
50% MS 50% MS
0.5a00.50.5
p0
p1
p2
p3
p4
0.225
0.110
0.070
0.125
0.470
0.225
0.110
0.070
0.125
0.470
0.020
0.040
0.050
0.140
0.750
0.430
0.180
0.090
0.110
0.190
Table 2. Channel parameters.
ChannelMCSBits per
state
{0,...,K}
0
1
2
3
4
andslot
mk
outage
Outage
QPSK-1/2
QPSK-3/4
16QAM-1/2 m3 = 96
16QAM-3/4 m4 = 144
m0 = 0
m1 = 48
m2 = 72
In the robustness study, we introduce two additional channel models with
memory. In these models, the MCS observed for a given MS in a frame depends
on the MCS observed in the previous frame according to the FSMC presented
above. The transition matrix is derived from equation PT(a) = aI+(1−a)PT(0)
given that 0 ≤ a ≤ 1. In this equation, I is the identity matrix and parameter
a is a measure of the channel memory. A MS maintains its MCS for a certain
duration with mean¯tcoh= 1/(1−a). With a = 0, the transition process becomes
memoryless. On the other extreme, with a = 1, the transition process will have
infinite memory and MS will never change its MCS. For simulations we have
taken a equal to 0.5, so that the channel is constant in average 2 frames. This
value is consistent with the coherence time given in [16] for 45 Km/h at 2.5 GHz.
We call the case where all MS have the same channel model with memory (a =
0.5), the average channel model. Note that the stationary probabilities of the
average channel model are the same as those of the memoryless model.
As the channel depends on the BS-MS link, it is possible to refine the previous
approach by considering part of the MS to be in a “bad” state, and the rest in
a “good” state. Bad and good states are characterized by different stationary
probabilities but have the same coherence time. In the so called combined channel
model, half of the MS are in a good state, the rest in a bad state, and a is kept
to 0.5 for both populations.
Three models are thus considered: the memoryless, the average, and the com-
bined channel models. Wireless channel parameters are summarized in Tab. 2.
Considered MCS are given including outage, and for each of them, the num-
ber of bits transmitted per slot. Channel stationary probabilities are given in
Tab. 1. The probabilities for the combined model are obtained by averaging
corresponding values of good and bad model stationary probabilities.
4.2
In this section, we first present a comparison between the results obtained
through our analytical model and scheduling simulator. The output parameters
in consideration are¯U,¯ X, and π(n) (see Section 3.3).
Simulation Results
Validation Study In this study, simulations take into account the same traffic
and channel assumptions as those of the analytical model. However, in simulator
Page 11
05 1015 20
0.2
0.4
0.6
0.8
1
Average utilization
Number of users in the cell
(a) Average utilization.
Slot fair sim
Slot fair model
X fair sim
X fair model
Opp sim
Opp model
01020304050
0
2
4
6
8x 10
6
Average throughput per user (bit/s)
Number of users in the cell
Slot fair sim
Slot fair model
X fair sim
X fair model
Opp sim
Opp model
(b) Average throughput per
user.
1020304050
0
0.05
0.1
Steady state probability
Number of active users
Slot fair sim
Slot fair model
X fair sim
X fair model
Opp sim
Opp model
(c) Stationary probabilities
for N = 50.
Fig.2. Performance validation for the three scheduling policies with ¯ xon = 3 Mbits
and¯toff = 3 s.
01020304050
0
2
4
6
8x 10
6
Average throughput per user (bit/s)
Number of users in the cell
xon=1Mbit sim
xon=1Mbit model
xon=3Mbit sim
xon=3Mbit model
xon=5Mbit sim
xon=5Mbit model
Fig.3. Average throughput
per user for different loads.
01020304050
0
2
4
6
8x 10
6
Average throughput per user (bit/s)
Number of users in the cell
Model
Sim (exponential)
Sim (Pareto low)
Sim (Pareto high)
Fig.4. Average throughput
per user for different traffic
distributions.
01020304050
0
2
4
6
8x 10
6
Average throughput per user (bit/s)
Number of users in the cell
Model
Sim (memoryless channel)
Sim (Average channel)
Sim (Combined channel)
Fig.5. Average throughput
per user for different chan-
nel models.
MCS of users are determined on per frame basis and scheduling is carried out
in real time, based on MCS at that instant. The analytical model on the other
hand, considers stationary probabilities of MCS only.
Fig. 2(a, b) show respectively the average channel utilization (¯U) and the
average instantaneous throughput per user (¯ X) for the three scheduling schemes.
It is clear that simulation and analytical results show a good agreement: for both
utilization and throughput, the maximum relative error stays below 6% and the
average relative error is less than 1%. Fig. 2(c) further proves that our analytical
model is a very good description of the system: stationary probabilities π(n)
are compared with those of simulations for a given total number N = 50 of
MS. Again results show a perfect match between two methods with an average
relative error below 9%. At the end, Fig. 3 shows the validation for three different
loads (1, 3 and 5 Mbps). Our model shows a comparable accuracy for all three
load conditions with a maximum relative error of about 5%.
Robustness Study In order to check the robustness of our analytical model
towards distribution of ON data volumes, simulations are carried out for ex-
ponential and truncated pareto (with lower and higher cutoff). The results for
this analysis are shown in Fig. 4. The average relative error between analytical
results and simulations stays below 10% for all sets. It is clear that considering
a truncated Pareto distribution has little influence on the design parameters.
Page 12
Next we evaluate the robustness of our analytical model with respect to the
channel model. We compare the analytical results with simulation for the three
pre-cited channel models: memoryless, average and combined (with stationary
probabilities given in Tab. 1). If we look at the plot of Fig. 5, we can say that
even for a complex wireless channel, our analytical model shows considerable
robustness with an average relative error below 7%. We can thus deduce that for
designing a WiMAX network, channel information is almost completely included
in the stationary probabilities of the MCS.
5 Network design
In this section we provide some examples to demonstrate application of our
model while considering throughput fairness scheduling. However, results can
be obtained in the same manner for other scheduling schemes by using their
respective average bits per slot ¯ m(n).
5.1 Performance graphs
We first draw 3-dimensional surfaces where performance parameters are function
of, e.g., N, the number of users in the cell and ρ, the combination of traffic
parameters. For each performance parameter, the surface is cut out into level
lines and the resulting 2-dimensional projections are drawn. The step between
level lines can be arbitrarily chosen.
The average radio resource utilization of the WiMAX cell¯U, and the aver-
age throughput per user¯ X for any MS in the system are presented in Fig. 6
and 7 (corresponding to the radio link characteristics presented in Section 4).
These graphs allow to directly derive any performance parameter knowing the
traffic load profile, i.e., the couple (N,ρ). Each graph is the result of several
thousands of input parameter sets. Obviously, any simulation tool or even any
multi-dimensional Markov chain requiring numerical resolution, would have pre-
cluded the drawing of such graphs.
5.2Dimensioning study
In this section, we show how our model can be advantageously used for dimen-
sioning issues. Two examples, each respecting a certain QoS criterion, are given.
In Fig. 8 we find minimum number N of MS in the cell to guarantee that
the average radio utilization is over 50%. This kind of criterion allows operators
to maximize the utilization of network resource in comparison with the traffic
load of their customers. For a given traffic load profile and a given set of system
parameters, the point of coordinates (NS,ρ) in the graph is located between two
level lines, and the level line with the higher value gives the optimal value of N.
The QoS criterion chosen for second example is the user throughput. We
have taken 50 Kbps, an arbitrary value of minimum user throughput. Next we
find the maximum number Nmaxof users in the cell to guarantee the minimum
Page 13
10000
1000
100
0.1%
1%
5%10%20%
50%90%
ρ
trafc load
N
number of users
Average radio utilization
Fig.6. Average utilization¯U.
ρ
trafc load
N
number of users
1000
100
10000
5e04
1e05
2e05
1e06
5e06
1e07
Average throughput per user
Fig.7. Average throughput per user¯ X.
ρ trafc load
number of slots
NS
1000
100
10000
10
50
100
200
500
600
25
Minimum number of mobiles Nmin
55
Fig.8. Dimensioning the minimum value of
N for having¯U ≥ 50%.
throughput threshold. In Fig. 9, a given point (NS,ρ) is located between two
level lines. The line with the lower value gives Nmax. As explained before, the
average throughput per user is inversely proportional to N.
The graphs of Fig. 9 and8 can be jointly used to satisfy multiple QoS
criteria. For example, if we have a WiMAX cell configured to have NS = 450
slots and a traffic profile given by ρ = 300 (e.g., xon= 1.2 Mbits and toff= 20 s),
Fig. 8 gives Nmin= 55, and Fig. 9 gives Nmax= 200. The combination of these
two graphs recommend to have a number of users N ∈ [55;200] to guarantee a
reasonable resource utilization and a minimum throughput to users.
ρ trafc load
number of slots
NS
1000
100
50
100
200
500
350
1000
Minimum number of mobiles Nmax
225
Fig.9. Dimensioning the maximum value
of N for having¯ X ≥ 50 Kbps per user.
6Conclusion
As deployment of WiMAX networks is underway, need arises for operators and
manufacturers to develop dimensioning tools. In this paper, we have presented
novel analytical models for WiMAX networks and elastic ON/OFF traffic. The
models are able to derive Erlang-like performance parameters such as throughput
per user or channel utilization. Based on a one-dimensional Markov chain and
the derivation of average bit rates, whose expressions are given for three main
scheduling policies (slot fairness, throughput fairness and opportunistic schedul-
ing), our model is remarkably simple. The resolution of model provides closed-
form expressions for all the required performance parameters at a click-speed.
Page 14
Extensive simulations have validated the model’s assumptions. The accuracy of
the model is illustrated by the fact that, for all simulation results, maximum
relative errors do not exceed 10%. Even if the traffic and channel assumptions
are relaxed, analytical results still match very well with simulations that shows
the robust nature of our model.
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Dynamics of IP traf-