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    An investigative Model for WiMAX Networkswith Multiple Traffic summary

    AbstractIn that paper, we have introduced a simple and perfect investigative model forperformance evaluation of WiMAX networks with multiple traffic summaries. That capabletechnology has been intended to support many types of applications having different trafficcharacteristics. From which QoS parameters measured by the standard for traffic classes which isthe maximum sustained traffic rate (MSTR), that is an upper bound for user output. ConsideringMSTR implies the implementation of a throttling scheduling policy that regulates the user peakrate. Our model considered this policy and provides closed-form expressions giving all therequired performance parameters for each traffic profile at a click speed. The model is comparedwith extensive simulations that show its robustness and accuracy.

    I.INTRODUCTIONWiMAX stands for Worldwide Interoperability for Microwave Access, is a broadband wirelesstechnology that is totally based on IEEE standard 802.16. The effective version of IEEE802.16 is 802.16-2004 which is fixed/nomadic WiMAX [1]. It was pursued by a approval ofrevision IEEE 802.16e for mobile WiMAX in 2005 [2]. A new standard, 802.16m, is at presentunder description for providing advanced efficiency. Conversely, the group WiMAX discussionwas founded to identify profiles, technology options are chosen among those proposed by theIEEE standard, describe an uninterrupted architecture IEEE does not go beyond physical andMAC layer, and certificate products through inter-operability tests.

    In order to accommodate various traffic types, different service categories have beenintroduced for WiMAX networks. For example, best effort (BE) is a service category that could

    Handle web traffic. Each service category is characterized by its quality of service (QoS)parameters. One of the parameters associated with BE service is the maximum sustained trafficrate (MSTR). As defined in [2] (section 11.13.6), this is not the guaranteed rate but an upperbound. The procedure to implement this rate has been left open in the standard.

    Consider this limited achievable user data rate is an important challenge while dimensioning awireless network. Existing literature does not speak much about this subject. A detailed accountof simulation based BE traffic performance evaluation in WiMAX networks can be found in [3],[4] and [5]. However, MSTR has not been considered in these papers. Mean information rate(MIR), a notion similar to MSTR, has been introduced in [6]. Authors have studied theperformance of multi-profile internet traffic in presence of different MIR values for a WiMAXcell. They have used packet level simulations to evaluate the cell throughput performance for

    different number of users while considering possibility of multiple modulation schemes.

    As far as analytical modeling is concerned, multi-class processor-sharing queues based modelshave been proposed in [7][9]. In these models, variability of radio channel conditions, animportant property of wireless medium, has been taken into account. On the other hand, the ideaof MSTR has not been touched in these articles. Earlier we have presented analytical models formono-traffic [10], [11] and multi-traffic [12] BE without considering the parameter MSTR.

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    Three generic scheduling policies: slot sharing fairness, throughput fairness and opportunisticscheduling were taken into account. In contrast to our existing work, models proposed in thispaper are based on a fourth scheduling scheme, the throttling policy, in which maximumachievable user data rate is limited to MSTR. This in turn affects the resource utilization. Themodels proposed in this paper will offer the flexibility to network operators to dimension the

    WiMAX networks.The paper is as follows. Modeling assumptions are presented in section II. The analyticalmodels for mono/multi profile traffics are given in sections III and IV. Validation of models ispresented in section V. At the end, section VI gives a conclusion of this work.

    II. MODELING SUPPOSITIONIn this section, we discuss the assumptions that have been considered in development of themodel. Wherever required, related details of WiMAX system are also specified, variousNotations are also introduced in this section.A WiMAX time division duplex (TDD) frame comprises of slots that are the smallest unit ofresource 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 TF ofthis TDD frame is equal to 5 ms [2].

    System supposition1) We consider a single WiMAX cell and focus on the downlink part which is a critical portionof asymmetric data traffic.2) We assume that amount of overhead in the TDD frame is fixed. As a consequence, the totalnumber of slots available for data transmission in the downlink part is constant and will bedenoted by NS.3) The number of simultaneous mobiles that can be multiplexed in one TDD frame 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: different MCS allows adynamic adaptation of the transmission to the radio conditions. As the number of data subcarriersper slot is the same for all permutation schemes, the number of bits carried by a slot for a givenMCS is constant. The selection of appropriate MCS is carried out according to the value ofsignal to interference plus noise ratio (SINR). In case of outage, i.e., if the SINR is too low, nodata can be transmitted without error. We denote the radio channel states as: MCS k, 1 _ k _ K,where K is the number of MCS. By extension, MCS0 represents the outage state. The number ofbits transmitted per slot by a mobile station (MS) using MCS k is denoted mk.Channel assumption4) The coding scheme used by a given mobile can change very often because of the high

    variability of the radio link quality. We assume that each mobile sends a feedback channelestimation on a frame by frame basis, and thus, the base station can change its coding schemeevery frame. We thus associate a probability pk with each coding scheme MCSk, and assumethat, at each time-step TF, any mobile has a probability pk to use MCSk.

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    A transition out of a generic state n to a state n1 occurs when a mobile in ON periodcompletes its transfer. This departure transition is performed with a generic rate (n)corresponding to the total departure rate of the frame when n mobiles are active.

    Obviously, the main difficulty of the model resides in estimating the aggregate departure

    rates (n) that strongly depend on the chosen scheduling policy. Focusing in this paperon the throttling policy, we now explain how to do so when considering this particularpolicy. Note that our previous works [10][12] have considered full-capacity policies (slotsharing fairness, throughput fairness and opportunistic scheduling) and derived for them verydifferent expressions of the (n).

    2) Departure rates:In order to estimate the average departure rates (n), we first define thefollowing quantities.To compensate losses due to outage, we consider a slightly greater instantaneous bitrate than theMSTR, the Delivered BitRate:

    A mobile using MCSk needs a mean number of gk slots per frame to reach its DBR:

    Obviously, no slot is allocated to a mobile in outage so g0 = 0.From this, we then deduce g, the average number of slots per frame needed by a mobile to

    obtain its MSTR:

    Knowing g, we can now express the aggregate departure rates (n) as follows:

    The last part of this expression (n MSTR/ xon ) corresponds to the rate at which any of the nactive mobiles completes its transfer, assuming that there are always enough available slots inthe frames to satisfy the MSTR. The first part of this expression( NS/ max(ng ,NS) ) representsthe ratio of the global departure rate achieved by the n concurrent active transfers. Indeed, whenthere are n active mobiles, they need ng slots in average to obtain their MSTR.The steady-state probabilities _(n) can easily be derived from the birth-and-death structure of theMarkov chain (depicted in Fig. 1):

    Where p is given by:

    and _(0) is obtained by normalization.

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    The performance parameters of this system can be derived from the steady-state probabilities asfollows.The average number Q of active users is:

    The average numberD of departures (i.e., mobiles completing their transfer) per unit of time isgiven by:

    From Littles law, we can derive the average duration ton of an ON period (duration of anactive transfer):

    and compute the average throughput X obtained by each mobile in active transfer as:

    Finally, we can express the average utilization U of the TDD frame, as a weighted sum of theratios between the mean number of slots needed by the n mobiles to reach their MSTR and themean number of slots they obtain:

    Lastly, note that when max (Ng,NS) = NS, i.e., when the resources of the system are sufficientto satisfy the MSTR even if all the N mobiles of the cell are in active transfer, we can easilydemonstrate that the average throughput of each active mobile (obtained from relation 11)becomes X = MSTR

    IV. MULTI-TRAFFIC WINGWe now consider that users are divided into R classes of traffic with specific traffic profiles

    (MSTRr, xr on, tr off). Each mobiles of a given class r thus has a maximum instantaneousThroughput MSTRr and generates an infinite-length ON/OFF traffic, with an average ON size ofxron bits and an average OFF duration of tr off seconds.We assume that there is a fixed number Nr of mobiles belonging to each class in the cell. Finally,mobiles of different classes may have different channel models. A mobile of class r thus has aprobability pkr of using MCSk.We saw in the mono-traffic model, when expressing the steady-state probabilities pi(n), thatthese probabilities as well as all performance parameters mostly depend on the traffic profile

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    (MSTR, xon, toff ) through a single aggregated parameter given by relation 7. The keyassumption of this multi-traffic extension is to suppose that all the performance parameters of theresulting multi-class model are still dependent of the traffic profiles through a set of aggregatedparameters _r given by:

    As a consequence, we can transform any class-r profile (MSTRr, xron, troff ) into anequivalent profile(MSTR, xon, troff ), such that:

    By doing so for all classes, we transform the original system into an equivalent system where allclasses of traffic have the same maximum instantaneous throughput MSTR, the same averageON size xon, and different average OFF durations troff .

    With this transformation, the equivalent system can be described as a multi-class closed queuingnetwork with two stations (see Fig. 2):

    1) An IS (infinite-server) station that models mobiles in OFF periods. This station has class-dependent service rates lemda r given by:

    2) A PS (processor sharing) station that models active mobiles. This station has classindependent service rates (n) that in turn depend on the total number activemobiles (whatever their classes).

    Fig. 2. Closed-queueuing network.However, we cannot directly use the same expression of the average departure (n) obtained inthe mono-traffic case (Eq. 5). Indeed, if we look at the expression of the steady-stateprobabilities derived for the mono-traffic model (Eq. 6), we can see that they not only depend onthe traffic profile through the aggregated parameter _, but also through the parameter g that

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    represents the average number of slots per frame needed by a mobile to obtain its MSTR. Wethus propose to use a very similar expression for (n):

    in which MSTR and xon are the common values of the equivalent multi-class profiles, andg(n) is the average number of slots per frame needed by n mobiles to obtain their maximumthroughput.In order to derive an expression for g(n) that takes into account the different classes of traffic,we first express DBRr, the actual bit rate needed by a mobile of class r in order to reach itsMSTRr (while compensating losses due to outage):

    We then define gr, the mean number of slots needed by a mobile of class r to obtain its MSTRr,

    as:

    Secondly, we estimate the probabilities _r(n) that an active mobile belong to class r knowing thatn mobiles are active (i.e., n customers are in the PS station). These probabilities are obviouswhen n = N:

    and closely approximated when n = 1 by:

    Knowing alpha r (1) and alpha r (N), we then suppose that the _r(n) are a linear function of n:

    With

    And

    Lastly, we express the average parameter g(n) as:

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    Note that the alpha r(n) probabilities can alternately be obtained by considering a multi-dimensional Markov chain which states (n1, ..., nR) correspond to the detailed distribution of thecurrent active mobiles of each class in the system. From the numerical resolution of this chainwe can derive the exact values of the _r(n) probabilities. We have checked on several examplesthat the exact alpha r(n) probabilities are very well estimated by the linear approximation wepropose above. In addition, the impact of this approximation is very limited as it only mattersstates n such that ng(n) < NS (see Eq. 16).Finally it is important to emphasize that the use ofthis approximation enables to avoid the exponential complexity of solving a multi-dimensionalMarkov chain.A direct extension of the BCMP theorem [13] for stations with state-dependent rates can now beapplied to this closed queuing network. The population vector is denoted by N with arrow = (N1,...,NR). The detailed steady-state probabilities can then be expressed as follows:

    Where ni arrow = (ni1, ..., niR), nirbeing the number of class-rmobiles present in station i,

    And

    And G is normalization constant:

    All the performance parameters of interest can be derived from the steady-state probabilities asfollows. The average number of customers of class r in station 2, i.e., the average number ofclass-r active mobiles, denoted by Qr, is given by:

    Let Dr be the average number of class-r customers departing from station 2 by unit of time, i.e.,

    the average number of class-r mobiles completing their download by unit of time. Dr can beexpressed as:

    where r(arrow n2) is the departure rate of class-r mobiles when there are arrow n2 activemobiles:

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    With

    The average download duration of class-r mobiles, tron, is nothing but the average sojourn timeof class-r customers in station 2, and is obtained from Little law:

    We can then calculate the average throughput Xr obtained by customers of class r during theirtransfer as:

    Finally, we can express the utilization U of the TDD frame as:

    V. CONFIRMATION VIA SIMULATIONS

    In this section we discuss the validation of our analytical models through extensive simulations.We also study the effect of more complex traffic and channel models. We start with details ofsimulator followed by results for mono/multi-traffic models.

    A. Simulation Details1)System Parameters:System bandwidth is assumed to be 10 MHz. The downlink/uplink ratioof the WiMAX TDD frame is considered to be approximately 2/3 (32 OFDM symbols on thedownlink and 15 on the uplink).We assume for the sake of simplicity that the protocol overheadis of fixed length (2 symbols) although in reality it is a function of the number of scheduledusers. Considering subcarrier permutation PUSC on the downlink, there are 30 sub-channels anda slot is defined as 2 OFDM symbols and one sub channel. With above parameters, the totalnumber of data slots (excluding overhead) per TDD downlink sub-frame is NS = 30(32 2)/2 =450.2) Traffic Parameters:Values of MSTR, mean ON data volume (main page and embedded

    objects) and OFF period (reading time) for both mono and multi-traffic are given inTab. I.In the validation study, we assume that the ON sizes are exponentially distributed as it is the casein 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 considertruncated Pareto distributions in the robustness study. Recall that the mean value of the truncatedPareto distribution is given by:

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    Where alpha is the shape parameter, b is the minimum value of pareto variable and q is the cutoffvalue for truncated pareto distribution. Two values of q are considered: lower and higher. Thesehave been taken as hundred times and thousand times the mean value respectively. The mean

    value in both cases (higher and lower cutoff) is 3 M bits for the sake of comparison with theexponential model. The value of alpha = 1.2 has been adopted from [14]. The correspondingvalues of parameter b for higher and lower cutoff are calculated using Eq. 36. Pareto parameters,used in simulations for xon = 3 Mbps, are summarized in Tab. II.

    3) Channel Models:A generic method for describing the channel between the BS (Base Station)and a MS is to model the transitions between MCS by a finite state Markov chain (FSMC). Thechain is discrete time and transitions occurs every L frames, with L TF < tcoh, the coherencetime 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) 0i,jK. Note that anadditional state (state 0) is introduced to take into account outage (SINR is below the minimum

    radio quality threshold). Stationary probabilities pk provide the long term probabilities for a MSto receive data with MCS k.In analytical study, channel model is assumed to be memory less, i.e., MCS are independentlydrawn from frame to frame for each user, and the discrete distribution is given by the

    (pi) 0i,jK. This corresponds to the case where pij = pj for all i. This simple approach, referredas the memory less channel model, is the one considered in the validation study. Let PT (0) bethe transition matrix associated to the memory less model.

    In the robustness study, we introduce two additional channel models with memory. In thesemodels, the MCS observed for a given MS in a frame depends on the MCS observed in theprevious frame according to the FSMC presented above. The transition matrix is derived from

    the following equation:

    Where I is the identity matrix and parameter a is a measure of the channel memory. A MS indeedmaintains its MCS fora certain duration with mean tcoh = 1/(1 a). With a = 0, the transitionprocess becomes memory less. On the other extreme, with a = 1, the transition process will haveinfinite memory and MS will never change its MCS. For simulations we have taken a equal to0.5, so that the channel is constant in average 2 frames. This value is consistent with thecoherence time given in [15] for 45 Km/h at 2.5 GHz. We call the case where all MS have thesame channel model with memory (a = 0.5), the average channel model. Note that the stationaryprobabilities of the average channel model are the same as those of the memory less model.

    TABLE ITRAFFIC PARAMETERS

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    TABLE IIPARETO PARAMETERS FOR xon = 3 MBPS

    TABLE IIISTATIONARY PROBABILITIES FOR THREE CHANNEL MODELS

    TABLE IVCHANNEL PARAMETERS.

    As the channel depends on the BS-MS link, it is possible to refine the previous approach byconsidering part of the MS to be in a bad state, and the rest in a good state. Bad and goodstates are characterized by different stationary probabilities but have the same coherence time. Inthe so called combined channel model, half of the MS are in a good state, the rest in a bad state,

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    (a)Average number of active users. (b) Average utilization. (c) Stationary probabilities forN = 20.

    Fig. 3. Validation study with MSTR = 512 Kbps, xon = 3 Mbits and toff = 3 s.

    a b c

    (a) Average throughput per user for different values of MSTR (512 and 2048 Kbps).(b) Average throughput per user for MSTR=2048 Kbps and different traffic distributions.(c) Average throughput per user for MSTR=2048 Kbps and different channel models.

    Fig. 4. Robustness study with xon = 3 Mbits and toff = 3 s.

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