# 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.

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**ABSTRACT:**IEEE 802.16 OFDMA (Orthogonal Frequency Division Multiple Access) technology has emerged as a promising technology for broadband access in a Wireless Metropolitan Area Network (WMAN) environment. In this paper, we address the problem of queueing theoretic performance modeling and analysis of OFDMA under broad-band wireless networks. We consider a single-cell IEEE 802.16 environment in which the base station allocates subchannels to the subscriber stations in its coverage area. The subchannels allocated to a subscriber station are shared by multiple connections at that subscriber station. To ensure the Quality of Service (QoS) performances, a Connection Admission Control (CAC) scheme is considered at a subscriber station. A queueing analytical framework for these admission control schemes is presented considering OFDMA-based transmission at the physical layer. Then, based on the queueing model, both the connection-level and the packet-level 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 two-state 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.08/2013; - SourceAvailable from: Malamati LoutaAntonios G. Sarigiannidis, Petros Nicopolitids, Georgios I. Papadimitriou, Panagiotis G. Sarigiannidis, Malamati D. Louta, Andreas S. 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; · 1.75 Impact Factor -
##### Conference Paper: Performance measures of CAC mechanism in OFDMA system using shadow server approximation

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**ABSTRACT:**We study in this paper the problem of queueing theoretic of OFDMA-based WiMAX with two classes of connections, connections with High Priority (HP) and connections with Low Priority (LP). The key idea to solve the problem of the mixed priority strategy is the technique of the Shadow server approximation. Specifically, we present a single-cell WiMAX environment in which the base station allocates subchannels to the subscriber stations in its coverage area. The subchannels allocated to a subscriber station are shared by multiple connections at that subscriber station. To ensure the Quality of Service (QoS) performances, a Connection Admission Control (CAC) mechanism is considered at a subscriber station. Then, based on the queueing model, both the connection-level and the packet-level 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). Several performance measures are derived.Multimedia Computing and Systems (ICMCS), 2012 International Conference on; 01/2012

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-