Rethinking the IEEE e EDCA Performance Modeling Methodology Ilenia Tinnirello and Giuseppe Bianchi

Transcription

1 540 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 2, APRIL 2010 Rethinking the IEEE e EDCA Performance Modeling Methodology Ilenia Tinnirello and Giuseppe Bianchi Abstract Analytical modeling of the e enhanced distributed channel access (EDCA) mechanism is today a fairly mature research area, considering the very large number of papers that have appeared in the literature. However, most work in this area models the EDCA operation through per-slot statistics, namely probability of transmission and collisions referred to slots. In so doing, they still share a methodology originally proposed for the Distributed Coordination Function (DCF), although they do extend it by considering differentiated transmission/collision probabilities over different slots. We aim to show that it is possible to devise e models that do not rely on per-slot statistics. To this purpose, we introduce and describe a novel modeling methodology that does not use per-slot transmission/collision probabilities, but relies on the fixed-point computation of the whole (residual) backoff counter distribution occurring after a generic transmission attempt. The proposed approach achieves high accuracy in describing the channel access operations, not only in terms of throughput and delay performance, but also in terms of low-level performance metrics. Index Terms MAC model, quality of service, WLAN. I. INTRODUCTION I EEE wireless LANs [1] are experiencing impressive market success. Born in the early 1990s as niche technology for local area networking, WLANs have evolved to play a significant role in public access provisioning. In such an evolution, new requirements and challenges have soon emerged. Among those, an important issue is the ability to differentiate the level of performance experienced by different classes of users and applications. Two categories of approaches have been addressed by the e amendment [2] to differentiate the performance level experienced by different stations: centralized ones, where a channel access coordinator (the access point, or AP) schedules transmission opportunity grants, and distributed ones, where stations use different MAC parameters to alter their probability to access the channel while competing with other stations. The latter approaches fall under the Enhanced Distributed Channel Access (EDCA) extension specified in the e Manuscript received June 28, 2007; revised February 27, 2008; December 07, 2008; and July 17, 2009; approved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor A. Kumar. First published October 16, 2009; current version published April 16, I. Tinnirello is with the Università di Palermo, Palermo 90133, Italy ( ilenia.tinnirello@tti.unipa.it). G. Bianchi is with the Università di Roma Tor Vergata, Rome 00133, Italy ( giuseppe.bianchi@uniroma2.it). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TNET amendment. EDCA employs four access categories (ACs). Each AC contends for the medium using the same rules of the standard Distributed Coordination Function (DCF), but employs different channel access parameters: 1) a per-class arbitration inter-frame spaces (AIFS) after which a station is permitted to access the medium, instead of a common distributed inter-frame space (DIFS); 2) a different pair of minimum/maximum contention windows according to which backoff values are randomly extracted; and 3) a different maximum channel holding time (transmission opportunity TXOP) during which a station that has gained access to the channel may eventually deliver more than one frame. The dependency of the per-class performance on the specific per-class MAC parameter settings has been duly assessed in literature [3] [7], and the availability of analytical models has significantly contributed to the thorough understanding of the performance effectiveness of the EDCA mechanisms. The goal of this paper is not to study the performance of EDCA. Indeed, we believe that EDCA performance has been extensively addressed in prior literature, and little extra understanding is needed at this stage. Rather, this paper focuses on the methodology and the basic principles behind the EDCA analyses proposed up to now. As discussed in Section II, most of the stochastic models for the EDCA operation lay their roots in a modeling methodology first proposed in [8] for the original Distributed Coordination Function. Specifically, they specify per-slot transmission/collision statistics (in most cases differentiated in distinct slots or classes or slots) and compute such statistics through fixed-point iteration. In sight of this, we specifically aim at addressing the following questions. Are there methodological alternatives to the traditional per-slot modeling approaches? And what are the limitations of such approaches that can be overcome by completely different methodologies? In essence, we believe that, as is indeed apparent from analyzing the relevant literature (Section II), the extension of the original DCF modeling ideas to the more general EDCA scenario is definitely not straightforward. Moreover, we think that, in EDCA, a per-slot modeling methodology is no longer a convenient, nor a natural one, for at least three reasons. First, EDCA substantially differs from DCF, since distinct AIFSs are employed. Hence, it is no longer possible to define a reference time-scale common to all stations, thus losing a fundamental property exploited in [8]. Since access to some slot-times may be restricted to a subset of stations only, and access/collision probabilities may vary in different slot-times, it is necessary to differentiate the classes of slot-times analyzed, thus moving quite forward from the original DCF modeling approach /$ IEEE

2 TINNIRELLO AND BIANCHI: RETHINKING THE IEEE E EDCA PERFORMANCE MODELING METHODOLOGY 541 Second, it is often perceived that the characterization of a station in terms of per-slot (or per class of slots) transmission or collision probabilities is a direct consequence of the decoupling approximation made in [8]. As argued in this paper, this is not necessarily true. The model we are proposing in what follows still relies on a decoupling approximation, but performs the actual decoupling using parameters different from the per-slot access probabilities. Third, most EDCA models do not retain the original DCF analysis simplicity (ultimately two nonlinear equations). Indeed, they may envision more complex Markov chain models for the tagged stations, which mandate a numerical solution (versus a closed form one); furthermore, they may employ multiple parameters for characterizing the station operation, thus increasing the dimensionality of the nonlinear fixed-point system to be ultimately solved. As discussed in Section III-E, the extra modeling complexity introduced by our proposed approach is clearly high with respect to DCF models, but it is ultimately comparable with that of advanced EDCA models. The model proposed in this paper is, to the best of our knowledge, the very first approach that is not based on the specification of per-slot statistics. Though the model is still based on a decoupling (fixed point) appoximation, it is technically different from past approaches. As such, our proposed model cannot overcome the intrinsic limitations that fixed-point models do raise [9], nor can it be stretched to model extreme situations, such as backoff parameters that are likely to yield channel capture, where multiple stability emerges [10]. Finding a model that is able to properly operate in such scenarios therefore still remains an open issue. The remainder of this paper is structured as follows. Section II reviews some common modeling approaches in which EDCA is described as an extension the DCF model presented in [8]. In Section III, we propose a new modeling approach specifically devised for EDCA. Such modeling methodology significantly differs from prior work and leads (to the best of our knowledge and with the exception of our preliminary work [11], [12] where some of these ideas were drafted) to the first nonpersistent EDCA model. In Section IV, we derive low-level performance metrics and use such metrics to thoroughly understand the simplifications that emerge when persistent EDCA models are employed and the range of validity of persistent EDCA approximations. Finally, concluding remarks are drawn in Section V. II. EDCA MODELING APPROACHES In what follows, we assume the reader is familiar with the EDCA operation. Since TXOP differentiation does not affect the channel access mechanism operation, but only the amount of time a station holds the channel after winning a contention, most of the EDCA models focus on the differentiation mechanisms set forth by the usage of different values,, and for each access category AC. We recall that the values differ from each other for an integer number of backoff slots. In particular,, where is an integer greater than or equal to 2 for normal stations (as such it cannot be lower than a DCF, set as a plus two slot-times) and greater than or equal to 1 for APs. In each beacon frame, the AP broadcasts the values of these parameters chosen for each AC. In fact, they can be dynamically adapted according to the network conditions. We also recall that EDCA deploys separate queues at each station for different ACs, and each of these behaves as a single enhanced DCF contending entity. When the backoff counter of more than one AC expires, a virtual collision occurs, and the highest priority packet among the colliding ones is selected for actual transmission on the radio channel. For convenience, almost all the EDCA modeling papers do not account for virtual collisions, i.e., they assume that each station carries traffic for a single AC only. In this paper, we also rely on this convenient assumption. For a given number of competing ACs, this assumption provides a lower performance bound. In fact, when each AC acts in a separate station, all collisions are real and consume channel resources that would not have been consumed in case of virtual collision inside a station. Most of the previous work about EDCA modeling starts from the approach presented in [8], in which the DCF scheme is modeled as a -persistent slotted access protocol. By considering nonuniform slot scale, the protocol behavior is summarized into a unique parameter,, which represents the probability that a tagged station accesses the channel. This access probability, due to the fairness of the protocol, is equal for all the stations. Moreover, it is constant slot by slot. The derivation of is based on the assumption that the probability of experiencing a collision, i.e., to have simultaneous channel accesses originating from two or more stations: 1) does not depend on the number of already experienced collisions; and 2) does not depend on the time elapsed from the end of the last channel activity. A similar approach has been widely maintained for the EDCA analysis. Because of the prioritization mechanisms, a different parameter has been introduced for each access category ([4] [6]). ACs employing smaller contention windows and inter-frame times result in higher channel access probability, thus perceiving higher throughput fractions. The approach is pretty accurate whenever the only differentiation parameter is the contention window [4], but introduces approximations in case of inter-frame times differentiation. In fact, in such a case, lower priority ACs are prevented from accessing some channel slots, and the average channel access probability is not representative of the actual channel behavior. In other words, it is not possible to consider each AC as a -persistent CSMA process because ACs with longer AIFS have 0 probability to access the slots that immediately follow the channel transmissions. To account for this phenomenon, it has been proposed that the channel slots should be classified depending on the time that has elapsed from the previous transmission. In [13] [16], channel slots are grouped into different zones where different number of ACs are allowed to transmit. Obviously, in different zones, the same AC can experience a different collision probability since the contention level is not constant. For example, in the slots that are accessible by a restricted number of ACs, the collision probability is likely to be much lower than the average one. This problem is addressed in [17] by averaging the collision probability experienced zone by zone, according to the

3 542 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 2, APRIL 2010 Fig. 1. Channel access cycle. zone occurrence probability. An alternative approach [13], devised to more accurately model the correlation between consecutive channel slots, uses the per-slot differentiation of both the collision and the transmission probability. In this case, each th AC is described through a set of per-slot transmission probability, while its contention level is expressed through a set of collision probability, valid in consecutive slots. A different solution accounting for nonsynchronous backoff decrements is provided in [18], where the backoff decrement probabilities are introduced for each th AC. These probabilities are approximately evaluated by considering the event in which the highest priority stations extract a backoff value higher than. Although the zone modeling approaches take into account the correlation between consecutive idle slots, they ignore the correlation effects between consecutive channel accesses. In fact, in the presence of high-priority ACs employing very small values, the independence assumption among the statistics of consecutive transmissions can no longer be valid. To solve this limitation, some models have introduced a state variable [16], [19] that represents different events at which the channel statistics are regenerated. Specifically, in [16], a renewal analysis is presented on the basis of the channel zone where the last transmission is performed. In [19], the state variable accounts for the last transmission event since the transition probabilities of the backoff process depend on the length of the backoff freezing time, i.e., on the last transmission/collision event. Obviously, there is always a tradeoff between model accuracy and complexity. In specific ranges of the contention parameter settings, some correlation effects can be neglected and persistent models can lead to accurate results. Thus, some recent research has focused on model simplification, especially when the MAC model is coupled with admission control algorithms or transport protocol interactions, which add further complexity [20], [21]. III. ANALYTICAL FRAMEWORK In the following, we rely on the typical simplified description of an EDCA station as characterized by a single traffic flow of a given AC, thus avoiding the need to take into account the effect of the internal virtual collisions. We also develop the model in the assumption of ideal channel conditions and saturation load [8]. The first assumption implies that no transmission errors occur and the only cause of error is a collision with another frame. The second assumption means that stations always have a frame ready in the transmission buffer and are permanently in a contending state. A. Approach and Assumptions The DCF modeling approach proposed in [8] relies on a key idea frequently referred to as decoupling approximation. It involves modeling the operation of an individual tagged station, rather than in jointly modeling the evolution of all the stations competing for the access to the channel (this being infeasible because of space-state explosion with the number of stations). The remaining stations are then assumed to statistically behave as the tagged station and are further assumed to be independent. This allows us to write a set of nonlinear fixed-point equations that are then readily solved through ordinary numerical techniques. The asymptotic validity of the decoupling approximation has been formally justified through mean field techniques in [23]. Such a decoupling approach does not pose any constraint on how the effect of the competing stations should be accounted for in terms of statistical description. In [8] and subsequent works, the choice was to characterize each competing station only through its per-slot transmission probability and the resulting conditional collision probability. However, although this is the simplest approach, it is by no means the only possibility. Indeed, the model described hereafter characterizes each competing station through the steady-state distribution of the random process that describes the evolution in time of its backoff counter value. More precisely, let be the total number of stations. Each station is characterized at a generic instant of real system time by the continuous-time discrete-state random process, which keeps track of the value of the station s backoff counter. We conveniently introduce a discrete-time, for observing such a continuous backoff process. Specifically, each discrete-time represents a time instant in which a generic transmission/collision event, originating from any competing station, ends (this being made precise in Section III-B). The instant of times are denoted as integer values. In other words, each time identifies the start of a th cycle (see Fig. 1), where each cycle is composed of an initial random waiting time plus exactly one transmission or collision event. At each time, every station competing for the channel access is characterized by its constant AIFS setting and by its backoff counter random process, sampled at the start of a

4 TINNIRELLO AND BIANCHI: RETHINKING THE IEEE E EDCA PERFORMANCE MODELING METHODOLOGY 543 Fig. 2. Backoff decrement operation: legacy DCF versus EDCA. cycle. The knowledge of such values allows for the determination of (Section III-B) the type of channel event (transmission or collision) that concludes the cycle, and the number of idle slots after which such event occurs. In order to quantify the statistics of such events, two key approximations are provided: 1) the decoupling approximation, where we assume that at time, the statistical behavior of a competing station is independently summarized only through the statistical distribution of its backoff counter process ; and 2) the stationarity of the backoff counter processes of the competing stations, meaning that the statistics seen by a tagged station at the beginning of a generic cycle (quantified in Section III-C) are independent of the specific cycle. In other words, subsequent cycles are assumed independent and the cycle statistics are regenerated at every cycle. Based on such statistics, the operation of the tagged station can be modeled as described in Section III-D. This leads to a fixed-point system that can be solved through numerical techniques (Section III-E). Finally, the analysis of the renewal cycle statistics (Section III-F) allows us to derive throughput (and other) performance metrics. B. Model Time-Scale For mapping the model time onto the real system time, we assume (nonrestrictively) that the station with the shortest possible AIFS value, is configured with, i.e.,. 1 All AIFSN values for the competing stations are expressed as the difference from the baseline value, i.e., for a generic station, we define Our model evolves, after each transmission or collision event, at the discrete-time immediately before the expiration of, as shown in Fig. 2. We recall that, in EDCA, the backoff counter is resumed one slot-time before the station s 1 The proposed model is not affected by the specific choice concerning the minimum deployed AIFSN value. We recall that the assumption AIF SN =2 is appropriate to model an ad hoc scenario because, in such cases, no station is permitted to have a lower AIFSN value (indeed AIF SN =1value is reserved to APs). class AIFS expiration. This means that at the end of the AIFS, the backoff counter is resumed to the frozen value decremented by unity. In Fig. 2, such a decrement is performed synchronously for stations B and C (which employ ) and one slot later for station A (which employs ). The figure also shows that a transmission is scheduled in the slot-time following the one where the backoff counter is decremented (station B) or resumed (station C) to 0. At time, it starts a new contention cycle, in which a generic station can be summarized by: 1) the nonnegative integer value representing its backoff counter value; and by 2) the number of initial consecutive slot-times whose station is prevented from accessing. Let be the nonnegative integer value representing the duration of the initial random time in cycle, expressed in slot-time units. Under the saturation assumption, all stations are persistently in competition and no new stations may asynchronously appear on the channel. Hence, can be computed as This relation is readily explained as follows. A station with that has not transmitted in the considered busy period will have, at time, the same backoff counter value it had at the beginning of the busy period, and this value is greater or equal than 0. If 0, the station will schedule transmission at the very first slot-time after its AIFS (namely, a DIFS for the case ), and as such no idle slots will elapse after the AIFS, i.e.,. Otherwise, if alone on the channel, the station would schedule transmission after empty slots. Conversely, a station with will wait for a number of idle slot-times before reaching the boundary of its AIFS value and scheduling its transmission after further idle slots. Since stations are competing, the number of elapsing idle-slots will be the minimum among the values across all the stations. The index for which the minimum is obtained represents the station that wins the contention. The contention is successful if such a minimum is unique; otherwise, a collision will occur between two or more stations that have the same minimum value. Since the backoff counter of each station (1)

5 544 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 2, APRIL 2010 is bounded by the maximum contention window, is therefore bounded by. C. Model of the Aggregated Station Behavior In this section, we describe the interaction model between a tagged station and all the other competing stations. Specifically, our interaction model is based on the knowledge of the steady-state distributions of the discrete random processes of all the stations. To this purpose, let be the steady-state distribution of the per-station backoff counter sampled at time As discussed in Section IV, the existence of such time-invariant steady-state distribution is guaranteed because the backoff Markov chain is irreducible and aperiodic. The distribution depends on the specific station index, and in particular, it has the same results for the stations belonging to a same EDCA class. Since, as discussed in Section III-E, the number of different EDCA classes is in general much lower than the total number of stations, this property can be exploited to simplify numerical computations. Note that for clarity of the presentation that follows, we prefer not to aggregate stations into classes, but to treat each station separately. We define the following convenient cumulative distribution: If we sequentially index the idle backoff slots that follow our model time as slot, represents the probability that station transmits before the occurrence of slot-time. For convenience, the definition is extended also to the case by considering for. Let us now focus on a generic station. For each slot index, we aggregate the behavior of the competing stations in terms of the probability that station observes the first transmission from other stations in that slot. Let be the probability that no competing station transmits before the occurrence of slot index Clearly, by definition, and decreases as grows. We can readily express the probability that station sees the first transmission in a cycle exactly at slot index as Note that is the probability that the cycle ends with a transmission/collision event, originating from one ore more competing stations, before the AIFS expiration for station. Thus, it also represents the probability that station does not update its backoff counter at the end of the cycle. (2) (3) (4) (5) D. Model of the Tagged Stations The statistical characterization of each class of stations depends on the specific MAC parameters settings. Thus, the analysis described in what follows is repeated for every station representative of a given class. Let us focus on a tagged station, i.e., a randomly chosen representative of a service class. Let be the tagged station index. The backoff operation of such a class requires to be modeled by two state variables. is the backoff stage value, defined as the number of retransmissions experienced by the head-of-line packet. Note that we rely on the generalized definition of backoff stage provided in [22] (number of retransmissions experienced), which differs from that used in [8] (where the backoff stage represents the stage in which a given contention window size is employed). is the backoff counter value assumed by the tagged station at a generic model time, i.e., at the beginning of a channel access contention cycle as defined in Section III-B. The tagged station is characterized by the following set of MAC parameters: : retry limit, i.e., maximum number of retransmissions a packet can undergo before being dropped. The immediate consequence being that the backoff stage assumes values in the range. : contention window value used in each backoff stage. The parameter thus implies that a station entering the backoff stage will extract a random value from the range. : the maximum possible value for the backoff counter. This parameter describes the maximum possible value that the backoff counter may assume, 2 and as such it is directly related with the dimension of the distributions derived in the previous Section III-C. : the AIFS parameter defined in Section III-B. Since at each stage the backoff counter may assume values in the range, the total number of bidimensional states is finite and equal to. Starting from a generic state, there is a nonzero probability to reach any other state in the future, including (, ). The time required to return to a given state is irregular because stage updates and random backoff extractions do not origin periodic state transitions. These conditions guarantee the existence of the steady-state distribution of the bidimensional process. Let be such a distribution where and, in most generality,. To simplify the notation, we also defined for. Thus, the backoff counter distribution defined in (2) is the corresponding marginal distribution 2 Based on the discussion in [22], this is not necessarily the contention window value assumed at the last backoff stage: The model may in fact include any arbitrary series of contention window values, i.e., not necessarily a monotonic one. (6)

6 TINNIRELLO AND BIANCHI: RETHINKING THE IEEE E EDCA PERFORMANCE MODELING METHODOLOGY 545 by is readily given (9) Fig. 3. Backoff evolution and slot index example, in the case =3. The evolution of the above-defined two-dimensional state during a cycle (i.e., the state transition probability between and ) depends on the occurrence of either of the five following cases. Case 1) The considered cycle ends before the backoff counter for the tagged station resumes. This occurs when at least one of the competing stations transmits in one of the AIFS slots in which the tagged station is not entitled to transmit. Fig. 3, case (a), shows an example of this situation for a tagged station with, when a transmission event occurs before the AIFS expiration (in the third slot). In such a case, the tagged station state (indicated in bold in the figure) at time is equal to that at time. The associated state transition probability is readily derived from the distributions introduced in Section III-C as Case 2) The backoff counter for station resumes, but the cycle ends with a transmission or collision event that does not involve the tagged station. Let the initial state be, where, otherwise the station would immediately transmit at backoff counter resumption, owing to the EDCA backoff counter decrement rules reviewed in Section III-B. If a transmission from one or more competing stations occurs at slot (with ), then the backoff counter for the tagged station will be decremented by. Based on (5), the probability that such a transmission occurs, and hence that the tagged station ends in the state,is An example is illustrated by case (b) in Fig. 3, where a transmission is shown to occur at the slot numbered as 3. This allows the tagged backoff counter to decrement by one unity (and precisely at the end of the AIFS expiration) before being frozen again during the busy channel period. This case corresponds to in (8). Case 3) The cycle ends with a collision involving the tagged station while that station has not yet reached its retry limit. If the initial state is, with, the transition probability to state (7) (8) where the collision is accounted for by the term, meaning that at least one of the remaining stations transmits during the slot where the tagged station also transmits. The new backoff counter value is uniformly drawn in the range [see Fig. 3, case (c)]. Case 4) The cycle ends with a collision involving the tagged station when it has reached its retry limit. In this case, the frame is dropped and a new frame is scheduled for transmission starting from backoff stage 0. Hence, for any initial state, the transition probability to state is (10) where, in such a case, the new backoff counter value is uniformly drawn in the range. Case 5) The cycle ends with a successful transmission from the tagged station. In such a case, the next backoff stage will be equal to 0, as a new frame will be scheduled for transmission, and the new backoff counter value will be uniformly drawn in the range. Starting from an arbitrary state, is the probability that station successfully transmits into the current cycle because all the other stations have a backoff expiration time greater than. Thus, the probability that the tagged station ends in state is (11) In summary, assuming the quantities provided in Section III-C are known, the steady-state distribution is computed by solving and normalizing the following linear system: For and (cases 1, 2, and 3): (12)

7 546 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 2, APRIL 2010 For and (cases 1, 2, 4, and 5): (13) Finally, in all the other cases ( and ). E. Numerical Resolution The proposed model is solved through a fixed-point iteration, using as fixed point the set of distributions, where the size of such distribution is (i.e., at most 1024 with standard parameters). More specifically, the solution is iteratively obtained as follows. Starting from an initial assumption on, for any station, we can readily compute the quantities (3), (4), and (5). These quantities are in turn used to compute the coefficients of the linear systems [(12) and (13); one system for every value ]. The solutions of the linear systems are normalized before computing the marginal distributions through (6). This cycle is repeated until a convergence criteria is satisfied (see below). The computation takes advantage of the fact that several stations may belong to a same EDCA class, and hence they are characterized by identical distributions,,,. Since the overall number of classes,, is generally much lower than (only four classes are standardized in e), we only need to compute different sets of distributions. Note that (4) may eventually be rewritten in a compacted form as (14) where and are the class indices in the range, and is the number of stations belonging to class. All the computations of the distributions,,, are straightforward. Hence, in principle, the computational complexity should be dominated by the resolution of the linear systems in the unknowns. We remark that this is also common in some other EDCA models, where (unlike simpler DCF models) a closed-form solution for the steady-state distribution of the multidimensional Markov chains describing the tagged stations may not be achievable. Conversely, the extra number of values upon which the fixed-point iteration is performed [namely, the distributions ] do not directly affect the computation, but may possibly affect the number of iterations necessary to reach convergence. However, with some attention in the implementation of an iterative resolution algorithm, the computational complexity may become marginal. In fact, an accurate solution of the steadystate distributions is only needed when the fixed-point iteration also reaches convergence. Hence, the computation of the linear system can be suitably merged with the fixed-point iteration. In particular, we have employed the following approach. The third term in (12) and the third and fourth terms in (13) are computed using the values obtained in the previous iteration and are hence considered constants in the system. The system is therefore reduced to a triangular one that is recursively solved in one step. Note that, even if we did not pay any particular attention to the initial conditions (by default chosen uniformly in the range of validity, and hence quite far from the final resulting distributions), we did not find convergence problems in practice. Actually, even in the worst case of maximum size, with (the number of EDCA classes), and using as convergence criteria an absolute error lower than over the probability of a successful transmission introduced in Section IV (16), we reached convergence in less than 100 iterations (order of 1 min in total over a 1.2 GHz Centrino laptop; this time reduced to just a few seconds for the case and ). F. Throughput In what follows, we focus on the explicit derivation of the throughput performance. Delay performances are in fact directly related to the throughput via the Little s Result, and as such there is no need to derive them explicitly (refer to [22] for the delay computation via the Little s Result in the somewhat tricky case of retry limits). From the renewal theory, we can express the throughput experienced by a generic station as the ratio between the average amount of information successfully transmitted in a cycle and the average duration of a cycle, i.e., (15) where is the probability that station successfully transmits in a cycle, is the size of the payload information contained in the frame (for simplicity we assumed the payload to be constant refer to [8] for a discussion about variable payload size), and is the average duration of a cycle. is readily computed as (16) In fact, is the probability that the station transmits during the -th slot inside the cycle, while is the probability that all other stations are scheduled to transmit after the occurrence of such a slot. The probability that the cycle ends with a generic successful transmission is simply the sum of the successful transmission probabilities for all competing stations, i.e.,. The average duration of a cycle,, is then computed as follows: (17) In this equation, and account for the duration of a successful transmission event and for a collision event, re-

8 TINNIRELLO AND BIANCHI: RETHINKING THE IEEE E EDCA PERFORMANCE MODELING METHODOLOGY 547 spectively. These durations are computed like all prior DCF or EDCA analytical models (see e.g., [8]), by adjusting the final inter-frame space to the assumption of the minimum AIFS after a successful transmission, and an EIFS after a collision. Finally, the term accounts for the fact that each cycle initially envisions a number of idle slots elapsing after a channel transmission or collision event, each lasting for a slot-time. The steady-state distribution of the process, defined by (1), can be obtained as follows. With slight, but convenient, abuse of notation, we can extend (4) also to the case of an external, inactive, observer, for convenience referred to with the station index (18) and thus we immediately derive Fig. 4. Per-class throughput in the case of AIFS-based differentiation. (19) The expectation is then straightforward from the distribution and can be conveniently expressed as (20) Finally, we remark that from the per-station throughput we can immediately compute the total network throughput and the per-class throughput by appropriately summing the individual throughput contributions. IV. PERFORMANCE ANALYSIS As noted before, the goal of this work is not to discuss in detail the performance evaluation of the EDCA differentiation mechanisms that have been the subject of extensive research work. Rather, after validating our model with simulation results, we seek to analyze the low-level performance metrics and verify a number of common assumptions made in EDCA models relying on persistent approaches. Indeed, we found that the accuracy of -persistent models tends to reduce when: 1) small and window values are employed; 2) the difference in the AIFS setting between stations increases. In what follows, unless otherwise specified, we consider only two AC classes: high priority (HP) and low priority (LP). Although the model allows for the simultaneous use of more than two classes, bringing this into consideration would only complicate the plots but without adding further insights. Each contending station has a single traffic queue, corresponding to a single priority class; hence we use the terms priority class and priority station as synonyms. HP stations are configured as follows:,, and. The settings for LP stations is varied by either increasing the value and/or increasing the AIFS parameters. In all cases, the Retry Limit parameter has been set to 7, the data channel rate is set to 11 Mbps, and the basic rate is set to 1 Mbps. All the parameters that are not explicitly reported are those of the b PHY specification with long PLCP preamble. A. Model Validation We compared our analytical results with simulation results obtained by means of a custom-made C++ simulator. Since our performance evaluation is focused on the MAC performance only, we preferred our simulator, which has been extensively validated, to NS-2. For each network scenario, we ran a simulation lasting 105 s and we collected statistics at each 10 s after a transient phase of 5 s. In all the cases, the confidence of our results was lower than 3%. Fig. 4 shows the per-class throughput performance when HP and LP stations are differentiated only by their AIFS setting. Results were obtained for a same number of HP and LP stations competing for the channel access; this number is varied in the x-axis. Two LP station configurations are considered: (i.e.,, and (i.e.,. The LP stations use the same and values of the HP stations. Fig. 5 shows results when only differentiation is employed. Two LP station configurations are reported: and. The other LP parameters are not varied with respect to the HP stations, i.e., and. In both cases, the throughput is reported on the -axis using a logarithmic scale, to highlight the throughput of the LP stations better (whose evaluation is often less accurate). The two curves show that, for both AIFS differentiation and differentiation, the model results (lines) are in perfect agreement with the simulation ones (symbols). As is obvious, the curves still confirm the well known different behavior of the two differentiation mechanisms (see, e.g., the extensive discussion in [7]). In case of AIFS differentiation, a greater number of competing stations does not affect the HP throughput. It however, results in a reduced LP throughput thereby proving that AIFS differentiation is effective in protecting HP traffic from the LP traffic. Conversely, in the case of differentiation, an increased network load results in reduced stations throughput, with station performance being almost unaffected. This phenomenon occurs because in case of high collision rates it is more likely

9 548 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 2, APRIL 2010 values ). In fact at each slot index corresponding to a window update, the curves show an change of slope. These results are very different from persistent or zone models where the backoff residual distribution is geometric and the decaying parameter is constant. Finally, the figure shows that the backoff distribution shapes of the LP stations change significantly depending on the parameter ( or AIFS) used for the performance differentiation. Fig. 5. Per-class throughput in the case of CW-based differentiation. B. Per-Slot Performance Metrics As discussed in Section II, all the previous EDCA models rely on the notions of transmission probabilities and/or conditional collision probabilities, defined on a per-slot basis. In the more advanced models, these metrics are further distinguished on the basis of the zone where the slot is considered. Conversely, our approach does not introduce either of these two notions. Hence, to compare our model with the others and to assess the validity of the assumption of constant (or per-zone constant) transmission or conditional collision probabilities, we here derive such metrics as a by-product of the proposed analysis. In fact, per-slot metrics are derived starting from the backoff distributions and the companion and distributions computed for each station or priority class. First, let us define the following equivalent per-slot transmission probabilities: at slot prior slots idle (21) Fig. 6. Backoff distributions in the case of i) AIFS-based ( =3, =0) and ii) CW-based (CW = 64, CW = 16) differentiation for n = n. Analysis (line) vs. Simulation (points). that the HP stations increase their contention window, thus reducing the performance differentiation with LP stations. As anticipated in the introduction, we believe that a model validation in terms of throughput performance only, while acceptable for practical purposes, is not sufficient to assess the theoretical validity of a proposed model. To this purpose, we focus on the backoff counter distributions that are the distributions upon which the model s fixed-point iteration is based. Fig. 6 compares the model prediction (lines), for the case, in terms of backoff counter distributions and, with being the slot index after a generic busy channel period, versus the ones measured at the end of our simulations (symbols). From this figure, we can draw some interesting conclusions. First, the model accuracy is impressive especially if we consider that matching a distribution is harder than matching a summarizing parameter such as the throughput. Accuracy is also maintained for very low probability values in the range (note the logarithmic -scale). Second, the backoff distributions are strongly affected by the contention window updating law (i.e., by the set of The definition focuses on the probability that station transmits at a generic slot, in the assumption that the station has reached such slot, i.e., that no other transmission has happened in the slots prior to the occurrence of slot. Slots are counted sequentially after the end of each busy channel period, starting from slot 0. In a model such as [8], the transmission probability is by construction a constant value for each index, and all the slots are considered independent. In our model this is no longer true, hence the transmission probability depends upon the specific slot index under consideration. Since the unconditional probability that the tagged station transmits at slot is given by, and since a station is prevented from accessing any slot before its AIFS expiration, it readily follows that (22) where the denominator in the second equation accounts for the probability of the conditioning event, i.e., that no transmission from both the tagged station and any other other competing stations occur prior to the occurrence of slot. Fig. 7 shows the per-slot transmission probabilities, computed like in 22 from the backoff distributions, for the AIFSbased and CW-based differentiation cases under consideration, and with the same parameters used in Fig. 6. In this figure, simulation results are not plotted to improve readability. In all cases, the results predicted by the model agree very well with those obtained through simulation, as is indeed expected from the very

10 TINNIRELLO AND BIANCHI: RETHINKING THE IEEE E EDCA PERFORMANCE MODELING METHODOLOGY 549 Fig. 7. Per-slot transmission probability (j) and (j) for: 1) AIFSbased differentiation ( =3, =0); and 2) CW-based differentiation (CW =64, CW =8); n = n =5. accurate analysis/simulation matching in the backoff distributions reported in the previous Fig. 6. Even if the figure reports only the values of the first 16 slots occurring after a generic busy channel event, a number of interesting considerations do emerge. First, generally, the values and are not constant in either the AIFS or CW based differentiation cases, and they only seem to be almost constant in the LP case with CW-based differentiation. Second, even if these values may be reasonably approximated, for low slot indices with a constant value, (with an error in the order of 10% 15%), such approximation would no longer be reasonable for in high slot indices. For example, for both differentiation cases under consideration, the value reduces by more than 40% starting from slot 8. Note that the abrupt change in the per-slot transmission probability occurs when the value is reached (and a similar discontinuity appears also for LP stations in the case of AIFS differentiation). This clearly implies that any -persistent model, hence including the original DCF model [8], is in error when transmissions occur after more than consecutive idle slots. However, the occurrence probability of long bursts of empty slots is low, and this justifies a posteriori why this finding has in most cases marginal impact on the numerical results. These considerations are quantitatively supported by Fig. 8, which reports the distribution providing the occurrence probability of inter-frame empty slots. The distribution is plotted for since the probability that a transmission occurs after a burst of more than 10 empty slots is lower than 1%. From the comparison between simulation and analytical results plotted in the figure, we confirm that our model is able to correctly track the distribution, despite the fact that it cannot be approximated by a geometrical distribution. The other quantity that plays a fundamental role in all previous DCF and EDCA models is the conditional collision probability, namely the probability that, upon transmission, a station will collide. Similarly, just like what has been done for the case of, we can now define the following equivalent per-slot conditional collision probabilities: from collides at slot (23) Fig. 8. Slot occupancy distribution for: i) AIFS-based differentiation ( = 3, = 0); and 2) CW-based differentiation (CW = 64, CW =8); n = n =5. Fig. 9. Per-slot conditional collision probability p (j) and p (j) for: 1) AIFS-based differentiation ( = 3, = 0); and 2) CW-based differentiation (CW =64, CW =8); n = n =5. Since is the probability that station sees the first transmission in a cycle at slot, and since is the probability that station transmits in a cycle at slot, it readily follows (24) Fig. 9 plots the per-slot conditional collision probability for the same two scenarios considered in Fig. 7. Note that in this figure we have decided, unlike the previous figure, to report the whole set of possible slot indexes. A logarithmic scale is used on the -axis for convenience, and slots have therefore been renumbered from 1 to 256. The key hypothesis of every persistent model is the assumption that the collision probability does not depend on the specific access slot. The figure clearly shows that this should not be generalized as being always the case. First, widely different collision probabilities are encountered in the two regions separated by the specific LP AIFS setting. In

11 550 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 2, APRIL 2010 this figure s case, since, an abrupt change in the curve shape occurs with the occurrence of the fourth slot. This is an obvious consequence of the fact that a different number of stations may access in these two regions and, specifically, that only HP stations may access the first three slots. The same reason, as discussed in Section II, has driven the definitions of zone-based EDCA models, which are able to account for different per-zone collision probabilities. Second, and less expected, the assumption of a constant conditional collision probability fails as long as the slot index exceeds the value. Also in this case, the impact of such collisions is not remarkable due to the low probability of encountering a long burst of idle slots after a busy channel period (see again Fig. 8). More generally, the slope of the curves appears to change in correspondence with the per-stage CW values adopted (8, 16, 32, 64, 128, 256). From the figure, we see that the collision probability initially decreases as the slot index grows. This can be intuitively explained by considering that a collision at a large slot index occurs only when all the stations are using a large backoff window value. In such a case, the probability of a collision reduces. Similarly, since all the stations must transmit within the maximum number of possible idle slots, the collision probability tends to 1 as the slot index grows toward the maximum window size, 256. C. On the Independence of Collisions From the Backoff Stage Most of the prior papers in the WLAN modeling area relied on the assumption that the conditional collision probabilities are independent of the backoff stage. This is a direct consequence of the fact that such probabilities lay at the basis of their underlying fixed-point iteration. Since our model uses a different decoupling approximation approach, it is interesting to assess if such an independence is also present in our case. The extraction of per-stage performance metrics is indeed trivial in our approach. Let be the per-slot conditional collision probabilities further conditioned to the specific backoff stage in which the tagged station is considered. can be computed by using, in the expressions given in (24), the backoff counter distribution conditioned to the fact that the tagged station is accessing the channel when in backoff stage This yields (25) (26) Hence, according to our model, the per-slot collision probability is independent of the backoff stage. However, this independence does not imply that the average conditional collision probability, which is the parameter used in all the persistent models, can also be considered as constant. More specifically, we define the average per-stage collision probability by weighting the per-slot values with the actual probability (indeed TABLE I PER-STAGE AVERAGE CONDITIONAL COLLISION PROBABILITY, E[p (jjs)], FOR BOTH AIFS AND CW DIFFERENTIATION; REFERENCE HP PARAMETERS: =0; CW =8; CW =256 stage-dependent) that the tagged station transmits in the considered slot. In formulas, this results in (27) where we have accounted for the fact that station is entitled to transmit only starting from slot, and we have made use of (25) and (26) in the simplification. Table I shows three scenarios: one HP station against five LP, same number of HP and LP stations (5), and five HP stations against one LP. Generally, we can see from the table that the per-stage average conditional collision probabilities are basically constant and fairly independent of the stage. However, this no longer holds true for the AIFS differentiation case in highly unbalanced scenarios such as (, ). In fact, for such a case, the average conditional collision probability encountered by HP stations shows a relative increase with the stage value of almost 30% with respect to the initial value (from 12.6% to 16.6%). A slight decrement in the average conditional collision probability can also be noted for the LP station that is competing alone against five HP stations. D. Comparison With Zone-Based Models The per-slot performance analysis carried out in Section IV-B has provided several insights on the validity of some common modeling assumptions. In summary, our analysis has shown that: 1) the collision probability abruptly varies when slots elapse (see Fig. 9); and 2) the transmission probability exhibits significant variations when the slot index exceeds the minimum contention window (see Fig. 7). Based on 1), we trivially conclude that persistent EDCA models, which do not explicitly take into account that the initial slots in a cycle are accessed only by the HP stations and not by the LP ones, are deemed to produce inaccurate results. This is fully confirmed by numerical results not presented here for reasons of space.

12 TINNIRELLO AND BIANCHI: RETHINKING THE IEEE E EDCA PERFORMANCE MODELING METHODOLOGY 551 However, those findings alone do not allow for the assessment of the accuracy of the so-called zone models (e.g., [16] and [17]). As discussed in Section II, those models provide different per-slot metrics on the basis of the region such slots belong to. For instance, in the case of two priority classes, they distinguish the slots where only HP stations may access (hereafter referred to as HP zone), from those (LP zone) where both HP and LP stations compete. As shown in Fig. 8, the probability that a transmission occurs in a specific slot index rapidly decreases, meaning that high-indexed slots marginally contribute to the high-level performance metrics (such as throughput). As such: 1) it is reasonable to approximate the collision probability with a constant value across each zone; and 2) it is also reasonable to approximate the transmission probability with a constant value. 3 To understand in which conditions zone models may eventually lead to inaccurate results, we have implemented a zone model mainly based on the approach presented in [17]. It approximates the EDCA operation by assuming that each contending station is characterized by a constant transmission probability. This convenient assumption is indeed satisfactory in sight of the results plotted in Fig. 7. The model takes into account the different zones by differentiating the collision probabilities experienced by the tagged station in different zones, based on the actual number of competing stations. In formulas, considering for convenience only the case of two classes HP and LP, and denoting with and the number of HP and LP competing stations, respectively (28) The collision probabilities are then averaged to produce a summary average collision probability. In the case of the LP stations, only one zone is involved, and hence. In the case of HP, it is convenient to introduce a distribution that represents the probability that a randomly chosen slot is the th slot in a cycle. This can be obtained by renormalizing [17] the terms computed through the following recursion: where represents the zone (namely, HP or LP) the slot belongs to, and (29) (30) The average collision probability for the HP stations is then computed by taking the weighted average: (31) 3 We remark that some zone models do further differentiate the transmission probability experienced in different zones. However, Fig. 7 suggests that such a differentiation is not expected to add significant accuracy to the resulting model. The above-described model is computationally extremely efficient, as its complexity is similar to the original DCF model (in contrast with many other proposed EDCA approaches). In fact, it is ultimately solved by independently expressing each as a function of the relevant and of the contention window settings, this being achieved through the closed-form expressions provided in [8]. The key difference of a zone model (actually any model based on per-slot persistent probabilities, not only the specific one described) and our proposed approach is the different way in which correlation among subsequent slots within a cycle is accounted for. While zone models are able, by construction, to account for the fact that slots in different zones are accessed by a different number of stations, they must assume that subsequent slots are independent. This assumption may become critical when small congestion windows are employed. In such case, as the slot index grows, it is more likely that stations will transmit due to the expiration of their residual backoff counters, and this will result in an increased level of contention. The effect of such correlation is quantified in Table II. We consider a somewhat extreme, but still realistic, scenario in which both HP and LP stations are characterized by. The table compares the per-class throughput 4 and the average collision probability predicted by our approach to that obtained by the zone model (superscript ) and that measured through simulation (superscript ). Three cases are considered: one LP station against five HP stations, same number (5) of LP and HP stations, and one HP station against five LP ones. Different AIFS settings for the LP station (, 3, 7) are considered. First, the table shows that in such conditions the zone model fails to reach the accuracy level provided by our approach. More specifically, the throughput performance for the LP stations becomes less accurate, especially as the setting grows. In fact, for higher settings, the correlation among adjacent slots in the HP zone becomes more critical. Second, both the simulation results as well as our model show that the collision probability for the LP stations grows with the setting, up to the case of, in which a 100% collisions probability is measured (as it is indeed expected, given the setting of the HP stations that prevents LP traffic from ever transmitting 4 An important issue, frequently overlooked in past EDCA modeling work, is the way in which the throughput should be specifically computed for a zone model. Assume for simplicity that T = T = T (this holds in the case of basic access and EIFS = ACK Timeout). A rigorous approach would involve the computing of the throughput by analyzing a whole transmission cycle. Specifically, let X(j) be the probability that a transmission occurs at the jth slot. For the considered zone model X(j) =[10 P (z(j))] P (z(i)) where, for the special case j = 0, the second product term is assumed to be equal to 1. The throughput is now readily computed as S = PSIZE n [1 0 p (z(j))] 1 (j 1 + T )X(j) where the denominator takes into account that a cycle comprising j empty slots lasts for a time equal to j 1 + T, or, in other words, by clearly accounting for the fact that slot durations are correlated within a cycle.

13 552 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 18, NO. 2, APRIL 2010 TABLE II THROUGHPUT (IN Mbps) AND COLLISION PROBABILITY COMPARISON BETWEEN SIMULATION (S ;p ), OUR MODEL (S ;p ), AND ZONE MODEL (S ;p ) Fig. 10. Slot occupancy distributions for AIFS-based differentiation ( =3, = 0) and CW settings CW = 8, CW = 256. Comparison between our model, simulation and zone-based results for the case of one HP station and five LP stations. successfully on the channel). Conversely, the zone model may only assume this probability to be constant and independent of the setting. In the more general case of large values, the zone model results are quite accurate in predicting the throughput performance. Even in the more critical case of one HP station against five LP stations, when,, and, the HP and LP throughput values predicted by the zone model are somewhat accurate, and specifically Mbps and Mbps versus the simulation results Mbps and Mbps. However, this does not necessarily imply that the zone model also accurately predicts more critical performance metrics. In fact, thanks to the large value of the frame transmission time with respect to an empty slot-time, even a nonsatisfactory prediction of the number of empty slots inside a cycle may still lead to a satisfactory throughput prediction. Indeed, Fig. 10 compares the distribution obtained through the zone model, our approach and simulation, for the considered scenario and shows that a nonnegligible error is shown to appear in correspondence of the low-indexed slots. From a methodological point of view, these issues should be kept in mind when considering the application of an EDCA-like model to a different system (e.g., a slotted MAC protocol) where the empty slot size may not be much smaller with respect to the duration of a transmission/collision slot. V. CONCLUSION In this paper, we propose a novel methodology for the analytical modeling of the e EDCA operation. Similar to prior models, we also rely on a decoupling-type approximation. However, unlike prior work in this area, our approach does not base the resulting fixed-point iteration on the per-slot transmission and collision probabilities. Rather, our model characterizes each competing station through the steady-state distribution of the random process that describes the evolution in time of its backoff counter value. In so doing, it succeeds in capturing the correlation among subsequent slots in a transmission or collision cycle. The proposed model results are extremely accurate in predicting not only high-level performance metrics such as throughput performance, but also low-level metrics such as backoff counter distribution, per-slot transmission and collision probabilities, statistical distribution of the number of idle slots between two consecutive transmissions, etc. The modeling complexity, although greater than that of traditional DCF models (and than that of a few EDCA models), is in any case not critical, with a worst-case numerical resolution time in the order of 1 min over an ordinary laptop. The approach presented in this paper assumes independence among subsequent transmission cycles. Model extensions may target the goal of further including correlation among subsequent transmission/collision cycles, thus taking into account effects such as short-term unfairness and capture. REFERENCES [1] Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications, IEEE Std , Nov. 1999,. [2] IEEE e supplement to Part 11: Wireless medium access control (MAC) and physical layer specification: Medium access control (MAC) enhancements for quality of service (QoS), [3] I. Aad and C. Castelluccia, Differentiation mechanisms for IEEE , in Proc. IEEE INFOCOM, 2001, vol. 1, pp [4] Y. Xiao, Performance analysis of IEEE e EDCF under saturation conditions, in Proc. IEEE ICC, Paris, France, 2004, pp [5] J. Gosteau, M. Kamoun, S. Simoens, and P. Pellati, Analytical developments on QoS enhancements provided by IEEE EDCA, in IEEE ICC, Paris, France, 2004, pp [6] L. Zhao and C. Fan, Enhancements of QoS differentiation over IEEE WLAN, IEEE Commun. Lett., vol. 8, no. 8, pp , Aug [7] G. Bianchi, I. Tinnirello, and L. Scalia, Understanding e contention-based prioritization mechanisms and their coexistence with legacy stations, IEEE Network, vol. 19, no. 4, pp , Jul. Aug [8] G. Bianchi, Performance analysis of the IEEE distributed coordination function, IEEE J. Sel. Areas Commun., vol. 18, no. 3, pp , Mar

14 TINNIRELLO AND BIANCHI: RETHINKING THE IEEE E EDCA PERFORMANCE MODELING METHODOLOGY 553 [9] M. Benaim and F. Le Boudec, A class of mean filed interaction models for computer and communication systems, Perform. Eval., vol. 65, no , pp , [10] V. Ramaiyan, A. Kumar, and E. Altman, Fixed point analysis of single cell IEEE e WLANs: Uniqueness, multistability and throughput differentiation, in Proc. ACM Sigmetrics, Banff, AB, Canada, Jun. 2005, pp [11] G. Bianchi and I. Tinnirello, Analysis of priority mechanisms based on differentiation frame spacing in CSMA-CA, in Proc. IEEE VTC, Orlando, FL, Oct. 2003, vol. 3, pp [12] G. Bianchi and I. Tinnirello, On the accuracy of some common modeling assumptions for EDCA analysis, presented at the CCCT, Orlando, FL, Jul [13] J. Zhao, Z. Guo, Q. Zhang, and W. Zhu, Performance study of MAC for service differentiation in IEEE , in Proc. IEEE Globecom, Taipei, Taiwan, 2002, pp [14] Z. Kong, D. H. K. Tsang, B. Bensaou, and D. Gao, Performance analysis of IEEE e contention-based channel access, IEEE J. Sel. Areas Commun., vol. 22, no. 10, pp , Dec [15] Z. Tao and S. Panwar, Throughput and delay analysis for the IEEE e enhanced distribute channel access, IEEE Trans. Commun., vol. 54, no. 8, pp , Apr [16] J. Hui and M. Devetsikiotis, A unified model for the performance analysis of IEEE e EDCA, Trans. Commun., vol. 53, no. 9, pp , Sep [17] J. W. Robinson and T. S. Randhawa, Saturation throughput analysis of IEEE e enhanced distributed coordination function, IEEE J. Sel. Areas Commun., vol. 2, no. 5, pp , Jun [18] H. Zhu and I. Chlamtac, Performance analysis for IEEE e EDCF service differentiation, Trans. Wireless Commun., vol. 4, no. 4, pp , Jul [19] T. Tsai and M. We, An analytical model for IEEE e EDCA, in Proc. IEEE ICC, Seoul, Korea, 2005, pp [20] X. Chen, H. Zhai, X. Tian, and Y. Fang, Supporting QoS in ieee e wireless LANs, Trans. Wireless Commun., vol. 5, no. 8, pp , Aug [21] V. A. Siris and C. Courcoubetis, Resource control for the EDCA mechanisms in multi-rate IEEE e networks, in Proc. IEEE WoWMoM, Buffalo, NY, Jun. 2006, pp [22] G. Bianchi and I. Tinnirello, Remarks on IEEE DCF performance analysis, IEEE Commun. Lett., vol. 9, no. 8, pp , Aug [23] C. Bordenave, D. McDonald, and A. Proutiere, Random multi-access algorithms, a mean field analysis, in Proc. Allerton Conf. Commun., Control Comput., 2005, pp Ilenia Tinnirello received the Laurea degree in electronic engineering and the Ph.D. degree in communications from the University of Palermo, Palermo, Italy, in April 2000 and February 2004, respectively. She has been an Assistant Professor with the University of Palermo since January In 2004, she was a Visiting Researcher with Seoul National University, Seoul, Korea. In 2006, she was a Visiting Researcher with Nanyang Technological University, Singapore. Her research activity has mainly focused on wireless networks in particular, on multiple access algorithms with quality of service provisioning, cross-layer interactions between access solutions and physical layer, and mobility management and load balancing in wireless packet networks. Giuseppe Bianchi received the Laurea degree in electrical engineering from Politecnico di Milano, Milan, Italy, in 1990, and the specialist degree in information technology from the CEFRIEL Research Center of Milan, Milan, Italy, in He has been a Full Professor of Telecommunications with the School of Engineering, University of Roma Tor Vergata, Rome, Italy, since January Before his current appointment, he was a Research Consultant for CEFRIEL ( ), an Assistant Professor with Politecnico di Milano ( ), and an Associate Professor with the University of Palermo, Palermo, Italy, ( ) and the University of Roma Tor Vergata ( ). His research activity spans several areas, among which include design and performance evaluation of broadband networks, multiple access in wireless local area networks, and network security and privacy.