On the throughput performance of multirate ieee 802 11 networks with variable loaded stations...

1594 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 5, MAY 2010

On the Throughput Performance ofMultirate IEEE 802.11 Networks with

Variable-Loaded Stations: Analysis, Modeling, anda Novel Proportional Fairness Criterion

M. Laddomada, F. Mesiti, M. Mondin, and F. Daneshgaran

Abstract—This paper focuses on multirate IEEE 802.11 Wire-less LAN employing the mandatory Distributed CoordinationFunction (DCF) option. Its aim is threefold. Upon startingfrom the multi-dimensional Markovian state transition modelproposed by Malone et.al. for characterizing the behavior of theIEEE 802.11 protocol at the Medium Access Control layer, itpresents an extension accounting for packet transmission failuresdue to channel errors. Second, it establishes the conditionsunder which a network constituted by 𝑁 stations, each stationtransmitting with its own bit rate, 𝑅

(𝑠)𝑑 , and packet rate,

𝜆𝑠, can be assumed loaded. Finally, it proposes a modifiedProportional Fairness (PF) criterion, suitable for mitigating therate anomaly problem of multirate loaded IEEE 802.11 WirelessLANs, employing the mandatory DCF option. Compared to thewidely adopted assumption of saturated network, the proposedfairness criterion can be applied to general loaded networks.

The throughput allocation resulting from the proposed algo-rithm is able to greatly increase the aggregate throughput of theDCF, while ensuring fairness levels among the stations of thesame order as the ones guaranteed by the classical PF criterion.

Simulation results are presented for some sample scenarios,confirming the effectiveness of the proposed criterion for opti-mized throughput allocation.

Index Terms—DCF, Distributed Coordination Function, fair-ness, IEEE 802.11, MAC, multirate, non-saturated, proportionalfairness, rate adaptation, saturation, throughput, traffic, un-loaded, unsaturated.

I. INTRODUCTION

CONSIDER the IEEE802.11 Medium Access Control(MAC) layer [1] employing the DCF based on the Carrier

Sense Multiple Access Collision Avoidance CSMA/CA accessmethod. The scenario envisaged in this work considers 𝑁contending stations; each station generates data packets withconstant rate 𝜆𝑠 by employing a bit rate, 𝑅(𝑠)

𝑑 , which dependson the channel quality experienced. In this scenario, it isknown that the DCF is affected by the so-called performanceanomaly problem [2]: in multirate networks the aggregate

Manuscript received September 5, 2008; revised May 8, 2009 and Novem-ber 3, 2009; accepted February 4, 2010. The associate editor coordinating thereview of this paper and approving it for publication was D. Wu.

This work was supported by PRIN 2007, prot. 2007FYETBY.M. Laddomada is with the Electrical Engineering Dept., Texas A&M

University-Texarkana (e-mail: [email protected]).F. Mesiti and M. Mondin are with DELEN, Politecnico di Torino, Italy

(e-mail: [email protected]).F. Daneshgaran is with the ECE Dept., California State University, LA,

USA.Digital Object Identifier 10.1109/TWC.2010.05 081191

throughput is strongly influenced by that of the slowestcontending station.

After the landmark work by Bianchi [3], who providedan analysis of the saturation throughput of the basic 802.11protocol assuming a two dimensional Markov model at theMAC layer, many papers have addressed almost any facet ofthe behaviour of DCF in a variety of traffic loads and channeltransmission conditions.

Contributions proposed in the literature so far can beclassified into two main classes, namely DCF Modelling andDCF Throughput and Fairness Optimization.

DCF modelling. This is the topic that received the mostattention in the literature since the work by Bianchi [3].Papers [4]-[6] model the influence of real channel conditionson the throughput of the DCF operating in saturated trafficconditions, while [7]-[9] thoroughly analyze the influence ofcapture on the throughput of wireless transmission systems.Paper [10] investigates the saturation throughput of IEEE802.11 in presence of non ideal transmission channel andcapture effects. The behavior of the DCF of IEEE 802.11WLANs in unsaturated traffic conditions has been analyzed in[11]-[18]. In [19], the authors look at the impact of channelinduced errors and of the received Signal-to-Noise Ratio(SNR) on the achievable throughput in a system with rateadaptation, whereby the transmission rate of the terminal ismodified depending on either direct or indirect measurementsof the link quality.

Multirate modeling of the DCF has received some attentionquite recently [20]-[24] as well. In [20] an analyticalframework for analyzing the link delay of multirate networksis provided. In [21]-[22], authors provide DCF modelsfor finite load sources with multirate capabilities, while in[23]-[24] a DCF model for networks with multirate stations isprovided and the saturation throughput is derived. Remediesto performance anomalies are also discussed. In both previousworks, packet errors are only due to collisions among thecontending stations.

DCF throughput and fairness optimization. This is perhapsthe issue most closely related to the problem dealt with inthis paper. The main reason for optimizing the throughputallocation of the 802.11 DCF is the behaviour of the basic

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LADDOMADA et al.: ON THE THROUGHPUT PERFORMANCE OF MULTIRATE IEEE 802.11 NETWORKS WITH VARIABLE-LOADED . . . 1595

DCF in heterogeneous conditions, with stations transmittingat multiple rates: the same throughput is reserved to anycontending station regardless of its bit rate, with the unde-sired consequence that lowest bit rate stations occupy thechannel for most time with respect to high rate stations [25].Furthermore, the optimization of the aggregate throughputwhen different stations contend for the channel with differentbit rates cannot be done without considering an appropriatefairness approach; the reason is that the optimum throughputwould be achieved when only the highest rate stations accessthe channel [25]. In order to face this problem, a varietyof throughput optimization techniques, which account forfairness issues, have been proposed in the literature. Paper [25]proposes a proportional fairness throughput allocation criterionfor multirate and saturated IEEE 802.11 DCF by focusing onthe 802.11e standard. In papers [26]-[29] the authors proposenovel fairness criteria, which fall within the class of the time-based fairness criterion. Time-based fairness guarantees equaltime-share of the channel occupancy regardless of the stationdata rate.

Paper [30] investigates the fairness issue in 802.11 multiratenetworks by analyzing various time-based fairness criteria.It demonstrates that with equal time-share of the channeloccupancy among multirate stations, the throughput achievedby a reference station in a multirate scenario with 𝑁 con-tending stations is equal to the throughput that the samereference station would achieve in a single rate scenario whencontending with other 𝑁 − 1 stations with its same rate.Furthermore, the authors prove that the proportional fairnesscriterion corresponds to fair channel time allocation in amultirate scenario.

The effect of the contention window size on the perfor-mance of the DCF have been also investigated in [31]-[33] ina variety of different scenarios. Finally, papers [34]-[39] havebeen devoted to the throughput optimization of the underlinedDCF by optimizing a number of key parameters of the DCF,such as the minimum contention window size or the packetsize.

A common hypothesis employed in the literature regardsthe saturation assumption, which sometimes does not fit quitewell to real network traffic conditions. In real networks, trafficis often non-saturated, different stations usually operate withdifferent loads, i.e., they have different packet rates, whilethe transmitting bit rate can also differ among the contendingstations. Channel conditions are far from being ideal andoften packet transmission has to be rescheduled until the datais correctly received. Due to Rayleigh and shadow fadingconditions, a real scenario presents stations transmitting atdifferent bit rates, because of multirate adaptation foreseenat the physical layer of WLAN protocols such as IEEE802.11b. In all these situations the common hypothesis, widelyemployed in the literature, that all the contending stations havethe same probability of transmitting in a randomly chosen timeslot, does not hold anymore.

The aim of this paper is to investigate the behaviour ofthe DCF in the most general scenario of a multirate network,when all the previous effects act jointly, as well as to presenta proportional fairness criterion which accounts for generalloading conditions as exemplified by the packet rate 𝜆𝑠 of

the contending stations. Contrary to the aforementioned worksavailable in the literature, we assume that the 𝑠-th stationgenerates data packets with its own size, 𝑃𝐿(𝑠), with its ownconstant rate 𝜆𝑠 by employing a bit rate, 𝑅(𝑠)

𝑑 , which dependson the channel quality experienced, and it employs a minimumcontention window with size 𝑊

(𝑠)0 . Hence, each station is

in a proper load condition, which is independent from theloading conditions of the other contending stations. Notice thatthese hypotheses make the model proposed in this work quitedifferent from the ones available in the literature, where thesaturated condition is mostly adopted. One consequence of theproposed analysis is that unloaded, heterogeneous networksdo not need any throughput allocation among stations. Wepropose a theoretical framework in order to identify whethera tagged station is saturated, given the traffic conditions ofthe remaining stations. As a starting point for the derivationsthat follow, we consider the bi-dimensional Markov modelproposed in [12], and present the necessary modifications inorder to deal with multirate stations, non ideal transmissionchannel conditions, and different packet sizes among thecontending stations.

Compared to our previous work [14] where all the stationsin the network could transmit at an equal bit rate, this workis much more general in that 1) we investigate a multirateheterogeneous network of 𝑁 contending stations, 2) we pro-pose a framework for identifying the loading condition ofthe multirate network, and 3) we propose an optimizationframework for generalized proportional fairness in multiratenetworks.

The main contributions of this work compared to the worksin the literature rely on the unique framework for analyzing,modeling, and optimizing multirate networks with variable-loaded contending stations. To the best of our knowledge,such a generalized scenario where packet errors at the physicallayer and variable-loaded and multirate stations are consideredall together has never been investigated in the literature. Wenotice in passing that the very common assumption of satu-rated network does not always reflect real network scenarios.Therefore, this work is very useful to analyze and greatlyoptimize real networks.

The rest of the paper is organized as follows. Section IIprovides the necessary modifications to the Markov model pro-posed in [12], while the employed traffic model is discussedin Section II-D. Section III proposes an analytical frameworkable to verify whether a network of 𝑁 contending stations isloaded. The novel proportional fairness criterion is presentedin Section IV, while Section V presents simulation results ofsome sample network scenarios. Finally, Section VI draws theconclusions.

II. THE NETWORK SCENARIO: OVERVIEW OF THE

MARKOVIAN MODEL CHARACTERIZING THE DCF

In [12], the authors derived a bi-dimensional Markov modelfor characterizing the behavior of the DCF in heterogeneousnetworks, where each station has its own traffic, which couldbe finite and characterized by the parameter 𝜆, expressingthe packet arrival rate. In order to deal with non-saturatedconditions, the traffic model is described by an exponentially



distributed packet inter-arrival process. In this paper we con-sider a more general network than [12]. Indeed, in the inves-tigated network, each station employs a specific bit rate, 𝑅(𝑠)

𝑑 ,a specific transmission packet rate, 𝜆𝑠, transmits packets withsize 𝑃𝐿(𝑠), and it employs a minimum contention windowwith size 𝑊

(𝑠)0 , which can differ from the one specified in

the IEEE 802.11 standard [1] (these modifications are at thevery basis of the proportional fairness criterion proposed inSection IV). A finite retry limit is considered in order to avoidinfinite number of retries when bad channel conditions inhibitthe station from successful transmission.

For the sake of greatly simplifying the evaluation of theexpected time slots required by the theoretical derivations thatfollow, we consider 𝑁𝑐 ≤ 𝑁 classes of channel occupancydurations1. First of all, given the payload lengths and thedata rates of the 𝑁 stations, the 𝑁𝑐 duration-classes arearranged in order of decreasing durations identified by theindex 𝑑 ∈ {1, ⋅ ⋅ ⋅ , 𝑁𝑐}, whereby 𝑑 = 1 identifies the slowestclass. Notice that in our setup a station is denoted fast ifit has a short channel occupancy. Furthermore, each stationis identified by an index 𝑠 ∈ {1, ⋅ ⋅ ⋅ , 𝑁}, and it belongsto a unique duration-class. In order to identify the classof a station 𝑠, we define 𝑁𝑐 subsets 𝑛(𝑑), each of themcontaining the indexes of the 𝐿𝑑 = ∣𝑛(𝑑)∣ stations within𝑛(𝑑), with 𝐿𝑑 ≤ 𝑁, ∀𝑑 and

∑𝑁𝑐

𝑑=1𝐿𝑑 = 𝑁 . As an example,𝑛(3) = {1, 5, 8} means that stations 1, 5, and 8 belong to thethird duration-class identified by 𝑑 = 3, and 𝐿𝑑 = 3.

A. Bi-dimensional Contention Markov Model

The modified bi-dimensional Markov model describing thecontention process of the 𝑠-th station2 in the network is shownin Fig. 1.

Let us elaborate. We consider an overall number of 𝑟different backoff stages, starting from the zero-th stage.The maximum Contention Window (CW) size is 𝑊𝑚𝑎𝑥 =2𝑚𝑊

(𝑠)0 , with 𝑚 ≤ 𝑟, whereas the notation 𝑊𝑖 =

min(2𝑚𝑊(𝑠)0 , 2𝑖𝑊

(𝑠)0 ) is used to identify the 𝑖-th contention

window size (𝑊 (𝑠)0 is the minimum contention window size

of the 𝑠-th station). Notice that after the 𝑚-th stage, thecontention window size is fixed to 𝑊𝑚𝑎𝑥 for the remain-ing (𝑟 − 𝑚) stages, after which the packet is dropped. Anadditional backoff stage, identified by (𝑃,−), with the samewindow size of the zero-th stage, is considered on top of thechain in order to account for the post-backoff stage enteredby the station after a successful packet transmission, or packetdrop [1]. Moreover, the state labelled (𝑃, 0) in Fig. 1, is usedfor emulating unloaded traffic conditions.

After the post-backoff stage, a station starts a new transmis-sion because a new packet is available in the queue, providedthat the channel is sensed idle for DIFS seconds. On the otherhand, a new zero-th stage backoff is employed if the channel issensed busy. Notice that the post-backoff stage is entered only

1This assumption relies on the observation that in actual networks somestations might transmit data frames presenting the same channel occupancy.As an instance, a station STA1 transmitting a packet of size 128 bytes at 1Mbps occupies the channel for the same time of a station STA2 transmittinga packet of size 256 bytes at 2 Mbps.

2In order to keep the notation concise, we omit the apex 𝑠 over theprobabilities involved in the model.

if the station has no longer packets to transmit after a packettransmission; otherwise, a zero-th stage is started. Moreover,if a new packet arrives during a post-backoff stage, the stationmoves into the zero-th stage, as depicted in Fig. 1. Indeed,backoff stages from 0 to 𝑟 assume that the station’s queuecontains at least a packet waiting for transmission.

A packet transmission is attempted only in the states la-belled (𝑖, 0), ∀𝑖 = 0, . . . , 𝑟, as well as in the state (𝑃, 0)only if there is a packet in the queue and the channelis sensed idle for DIFS seconds. In case of collision, ordue to the fact that transmission is unsuccessful because ofchannel errors, the backoff stage is incremented and the stationmoves in the state (𝑖 + 1, 𝑘), where 𝑘 = 0, . . . ,𝑊𝑖+1 − 1,with uniform probability 𝑃𝑒𝑞/𝑊𝑖+1, whereby 𝑃𝑒𝑞 , i.e., theprobability of equivalent failed transmission, is defined as𝑃𝑒𝑞 = 1 − (1 − 𝑃𝑒)(1 − 𝑃𝑐𝑜𝑙) = 𝑃𝑐𝑜𝑙 + 𝑃𝑒 − 𝑃𝑒 ⋅ 𝑃𝑐𝑜𝑙.Probabilities 𝑃𝑐𝑜𝑙 and 𝑃𝑒 are, respectively, the collision andthe packet error probabilities related to the 𝑠-th station.

The transition probabilities for the generic 𝑠-th station’sMarkov process in Fig. 1 could be separated as summarizedin what follows, depending on whether transitions start fromstandard backoff states or from post-backoff states.Backoff state transitions

𝑃𝑖,𝑘∣𝑖,𝑘+1 = 1, 𝑘 ∈ [0,𝑊(𝑠)𝑖 − 2], 𝑖 ∈ [0, 𝑟]

𝑃𝑃,𝑘∣𝑖,0 =(1−𝑃𝑒𝑞)(1−𝑞)

𝑊(𝑠)0

, 𝑘 ∈ [0,𝑊(𝑠)0 − 1], 𝑖 ∈ [0, 𝑟 − 1]

𝑃0,𝑘∣𝑖,0 =(1−𝑃𝑒𝑞)𝑞

𝑊(𝑠)0

, 𝑘 ∈ [0,𝑊(𝑠)0 − 1], 𝑖 ∈ [0, 𝑟 − 1]

𝑃𝑃,𝑘∣𝑟,0 = (1−𝑞)

𝑊(𝑠)0

, 𝑘 ∈ [0,𝑊(𝑠)0 − 1],

𝑃0,𝑘∣𝑟,0 = 𝑞

𝑊(𝑠)0

, 𝑘 ∈ [0,𝑊(𝑠)0 − 1].

(1)The meaning of the underlined probabilities is as follows.

The first equation in (1) states that, at the beginning ofeach slot time, the backoff time is decremented. The second(third) equation accounts for the fact that after a successfultransmission, the station goes in post-backoff because of anempty (non empty) queue. In both equations, 𝑞 is used toidentify the probability that the queue contains at least a packetwaiting for transmission after a time slot, and it will be betterdefined in Section II-D, where the employed traffic modelis described. The fourth equation deals with the situation inwhich the station has reached the retry limit and, after apacket transmission, the buffer of the station is empty. In thissituation, the station moves in the post-backoff stage with anempty queue. The last equation accounts for a scenario similarto the previous one with the difference that, after the packettransmission, the queue is not empty.Post-backoff state transitions

𝑃𝑃,𝑘∣𝑃,𝑘+1 = (1− 𝑞) 𝑘 ∈ [0,𝑊(𝑠)0 − 2]

𝑃0,𝑘∣𝑃,𝑘+1 = 𝑞 𝑘 ∈ [0,𝑊(𝑠)0 − 2]

𝑃𝑃,0∣𝑃,0 = (1− 𝑞)

𝑃𝑃,𝑘∣𝑃,0 =𝑞𝑃𝑖(1−𝑃𝑒𝑞)(1−𝑞)

𝑊(𝑠)0

𝑘 ∈ [0,𝑊(𝑠)0 − 1]

𝑃0,𝑘∣𝑃,0 = 𝑞(1−𝑃𝑖)+𝑞𝑃𝑖(1−𝑃𝑒𝑞)

𝑊(𝑠)0

𝑘 ∈ [1,𝑊(𝑠)0 − 1]

𝑃1,𝑘∣𝑃,0 =𝑞𝑃𝑖𝑃𝑒𝑞

𝑊(𝑠)1

𝑘 ∈ [1,𝑊(𝑠)1 − 1]

(2)

The meaning of the underlined probabilities is as follows. Thefirst equation states that the station remains in the post-backoffstage because the queue is empty, whereas the second equation



Fig. 1. Markov chain for the contention model of the generic 𝑠-th stationin general traffic conditions, based on the 2-way handshaking technique,considering the effects of channel induced errors, unloaded traffic conditions,and post-backoff.

accounts for a transition in the zero-th backoff stage becausea new packet arrives at the end of a backoff slot. The thirdequation models the situation in which there are no packetswaiting for transmission, and the station remains in the state(𝑃, 0) (idle state).

The fourth equation deals with the situation in which thestation is in the idle state (𝑃, 0), and, at the end of a backoffslot, a new packet arrives in the queue. In this scenario, thepacket is successfully transmitted, the queue is empty, andthe station moves in another post-backoff stage. The term 𝑃𝑖identifies the probability that the channel is idle, and it isdefined as follows with respect to the 𝑠-th tagged station:

𝑃(𝑠)𝑖 =

𝑁∏𝑗=1,𝑗 ∕=𝑠

(1− 𝜏𝑗)

The fifth equation accounts for a scenario similar to theprevious one, except that the station queue is not empty afterthe immediate transmission of a packet or a situation of busychannel. The last equation models the scenario in which thestation goes from the idle state (𝑃, 0) to the first backoffstage because of a failure of the immediate transmission ofthe packet arrived in the head of the queue.

B. Throughput Evaluation

Next line of pursuit consists in finding the probability𝜏𝑠 that the 𝑠-th station starts a transmission in a randomlychosen time slot. Due to the lengthy algebra involved inthe derivations needed for solving the bidimensional Markovchain, the relation that defines 𝜏𝑠 has been derived in [40],

whereas for conciseness we show the final formula in (3)(shown at the bottom of this page), along with the other keyprobabilities needed in this paper. Given 𝜏𝑠 in (3), we canevaluate the aggregate throughput 𝑆 as follows:

𝑆 =

𝑁∑𝑠=1

𝑆𝑠 =

𝑁∑𝑠=1

1

𝑇𝑎𝑣𝑃 (𝑠)𝑠 ⋅ (1 − 𝑃 (𝑠)

𝑒 ) ⋅ 𝑃𝐿(𝑠) (4)

whereby 𝑇𝑎𝑣 is the expected time per slot, 𝑃𝐿(𝑠) is thepacket size of the 𝑠-th station, 𝑃 (𝑠)

𝑒 and 𝑃(𝑠)𝑠 are, respectively,

the packet error probability and the probability of successfulpacket transmission of the 𝑠-th station:

𝑃 (𝑠)𝑠 = 𝜏𝑠 ⋅

𝑁∏𝑗=1𝑗 ∕=𝑠

(1− 𝜏𝑗) (5)

The evaluation of the aggregate throughput in (4) requires theknowledge of the expected time per slot, 𝑇𝑎𝑣. Its evaluationis the focus of the next section.

C. Evaluation of the Expected Time per Slot

The expected time per slot, 𝑇𝑎𝑣, can be evaluated byweighting the times spent by a station in a particular statewith the probability of being in that state. First of all, weobserve that there are four different kinds of time slots, withfour different average durations:

∙ the idle slot, in which no station is transmitting over thechannel, with average duration 𝑇𝐼 ;

∙ the collision slot, in which more than one station isattempting to gain access to the channel, with averageduration 𝑇𝐶 ;

∙ the slot due to erroneous transmissions because of im-perfect channel conditions, with average duration 𝑇𝐸 ;

∙ the successful transmission slot, with average duration𝑇𝑆 .

The expected time per slot, 𝑇𝑎𝑣, can be evaluated by addingthe four expected slot durations:

𝑇𝑎𝑣 = 𝑇𝐼 + 𝑇𝐶 + 𝑇𝑆 + 𝑇𝐸. (6)

We will now evaluate 𝑇𝐼 , 𝑇𝐶 , 𝑇𝑆 , and 𝑇𝐸 .Upon identifying with 𝜎 an idle slot duration, and defining

with 𝑃𝑇𝑅 the probability that the channel is busy in a slotbecause at least one station is transmitting:

𝑃𝑇𝑅 = 1−𝑁∏𝑠=1

(1− 𝜏𝑠) (7)

the average idle slot duration can be evaluated as follows:

𝑇𝐼 = (1− 𝑃𝑇𝑅) ⋅ 𝜎 (8)

The average slot duration of a successful transmission, 𝑇𝑆 ,can be found upon averaging the probability 𝑃

(𝑠)𝑠 that only

the 𝑠-th tagged station is successfully transmitting over thechannel, times the duration 𝑇

(𝑠)𝑆 of a successful transmission

from the 𝑠-th station:

𝑇𝑆 =𝑁∑𝑠=1

𝑃 (𝑠)𝑠

(1− 𝑃 (𝑠)

𝑒

)⋅ 𝑇 (𝑠)

𝑆 (9)



𝜏𝑠 = 𝑞2(1−𝑃𝑟+1

𝑒𝑞 )

(1−𝑃𝑒𝑞)𝑊0

(1−(1−𝑞)𝑊0 )(1−𝑞)⋅ 𝑏𝑃,0

𝑏𝑃,0 ={𝑊0(𝑊0+1)

2(𝑋𝐵 +𝑋𝑃 )−

(𝑞𝑊0−(1−𝑞)[1−(1−𝑞)𝑊0 ]

𝑞2

)𝑋𝑃 + 𝑞𝑊0

𝑞𝑃𝑒𝑞𝐾+(1−𝑞)

[1−(1−𝑞)𝑊0 ](1−𝑞)

}−1

𝐾 = 12

[2𝑊0

1−(2𝑃𝑒𝑞)𝑚−1

1−2𝑃𝑒𝑞+

1−𝑃𝑚−1𝑒𝑞

1−𝑃𝑒𝑞

]+ (𝑊𝑚 + 1)𝑃𝑚−1

𝑒𝑞1−𝑃𝑟−𝑚+1

𝑒𝑞

1−𝑃𝑒𝑞

𝑋𝐵 = 𝑞𝑊0𝑞2+(1−𝑃𝑖)[1−(1−𝑞)𝑊0 ](1−𝑞)

𝑊0[1−(1−𝑞)𝑊0 ](1−𝑞)

𝑋𝑃 = 𝑞2

1−(1−𝑞)𝑊0

(3)

Notice that the term (1 − 𝑃(𝑠)𝑒 ) accounts for the probability

of packet transmission without channel induced errors.Analogously, the average duration of the slot due to erro-

neous transmissions can be evaluated as follows:

𝑇𝐸 =

𝑁∑𝑠=1

𝑃 (𝑠)𝑠 ⋅ 𝑃 (𝑠)

𝑒 ⋅ 𝑇 (𝑠)𝐸 (10)

Let us focus on the evaluation of the expected collision slot,𝑇𝐶 . There are 𝑁𝑐 different values of the collision probability𝑃

(𝑑)𝐶 , where 𝑑 identifies the tagged class. We assume that in a

collision of duration 𝑇(𝑑)𝐶 (class-𝑑 collisions), only the stations

belonging to the same class, or to higher classes (i.e., stationswhose channel occupancy is lower than the one of stationsbelonging to the tagged class indexed by 𝑑) might be involved.

In order to identify the collision probability 𝑃(𝑑)𝐶 , let

us first define the following three transmission probabilities(𝑃𝐶(𝑑)𝑇𝑅 , 𝑃

𝐻(𝑑)𝑇𝑅 , 𝑃𝐿(𝑑)

𝑇𝑅 ) with respect to a tagged class 𝑑. Prob-ability 𝑃

𝐿(𝑑)𝑇𝑅 represents the probability that at least another

station belonging to a lower class transmits, and it can beevaluated as

𝑃𝐿(𝑑)𝑇𝑅 = 1−

𝑑−1∏𝑖=1

∏𝑠∈𝑛(𝑖)

(1− 𝜏𝑠) (11)

Probability 𝑃𝐻(𝑑)𝑇𝑅 is the probability that at least one station

belonging to a higher class transmits, and it can be evaluatedas

𝑃𝐻(𝑑)𝑇𝑅 = 1−

𝑁𝑐∏𝑖=𝑑+1

∏𝑠∈𝑛(𝑖)

(1− 𝜏𝑠) (12)

Probability 𝑃𝐶(𝑑)𝑇𝑅 represents the probability that at least a

station in the same class 𝑑 transmits:

𝑃𝐶(𝑑)𝑇𝑅 = 1−

∏𝑠∈𝑛(𝑑)

(1− 𝜏𝑠) (13)

Therefore, the collision probability for a generic class 𝑑 takesinto account only collisions between at least one station ofclass 𝑑 and at least one station within the same class (internalcollisions) or belonging to higher class (external collisions).Hence, the total collision probability can be evaluated as:

𝑃(𝑑)𝐶 = 𝑃

𝐼(𝑑)𝐶 + 𝑃

𝐸(𝑑)𝐶 (14)

whereby

𝑃𝐼(𝑑)𝐶 = (1 − 𝑃

𝐻(𝑑)𝑇𝑅 ) ⋅ (1− 𝑃

𝐿(𝑑)𝑇𝑅 ) ⋅ (15)

⋅⎡⎣𝑃𝐶(𝑑)

𝑇𝑅 −∑𝑠∈𝑛(𝑑)

𝜏𝑠∏

𝑗∈𝑛(𝑑),𝑗 ∕= 𝑠

(1− 𝜏𝑗)

⎤⎦

represents the internal collisions between at least two stationswithin the same class 𝑑, while the remaining are silent, and

𝑃𝐸(𝑑)𝐶 = 𝑃

𝐶(𝑑)𝑇𝑅 ⋅ 𝑃𝐻(𝑑)

𝑇𝑅 ⋅ (1− 𝑃𝐿(𝑑)𝑇𝑅 ) (16)

concerns to the external collisions with at least one station ofclass higher than 𝑑.

Finally, the expected duration of a collision slot is:

𝑇𝐶 =

𝑁𝑐∑𝑑=1

𝑃(𝑑)𝐶 ⋅ 𝑇 (𝑑)

𝐶 (17)

Constant time durations 𝑇 (𝑠)𝑆 , 𝑇 (𝑠)

𝐸 and 𝑇(𝑑)𝐶 are defined in a

manner similar to [22] with the slight difference that the firsttwo durations are associated to a generic station 𝑠, while thelatter is associated to each duration class, which depends onthe combination of both payload length and data rate of thestation of class 𝑑.

D. Traffic Model

The employed traffic model for each station assumes aPoisson distributed packet arrival process, whereby the inter-arrival times among packets are exponentially distributed withmean 1/𝜆𝑡, where 𝑡 identifies the 𝑡-th station. In order togreatly simplify the analysis, we consider small queue, asproposed in [12], even though the proposed analysis maybe easily extended to queues with any length. The traffic ofeach station is accounted for within the Markov model byemploying a probability3, 𝑞, that accounts for the scenariowhereby at least one packet is available in the queue at theend of a slot. In our setting, each station is characterized byits own traffic, and the probability 𝑞(𝑡) of the 𝑡-th station canbe evaluated by averaging over the four types of time slots,namely idle, success, collision, and channel error time slot.Upon noticing that, with the underlined packet model, theprobability of having at least one packet arrival during time𝑇 is equal to 1− 𝑒−𝜆𝑡⋅𝑇 , 𝑞(𝑡) can be evaluated as:

𝑞(𝑡) = (1− 𝑃𝑇𝑅) ⋅ (1 − 𝑒−𝜆𝑡⋅𝜎)+

+∑𝑁

𝑠=1 𝑃(𝑠)𝑠

(1− 𝑃

(𝑠)𝑒

)⋅ (1− 𝑒−𝜆𝑡⋅𝑇 (𝑠)

𝑆 )+

+∑𝑁

𝑠=1 𝑃(𝑠)𝑠 ⋅ 𝑃 (𝑠)

𝑒 ⋅ (1− 𝑒−𝜆𝑡⋅𝑇 (𝑠)𝐸 )+

+∑𝑁𝑐

𝑑=1 𝑃(𝑑)𝐶 ⋅ (1− 𝑒−𝜆𝑡⋅𝑇 (𝑑)

𝐶 )(18)

whereby the probabilities 𝑃𝑇𝑅, 𝑃 (𝑠)𝑠 , and 𝑃

(𝑑)𝐶 are, respec-

tively, as defined in (7), (5), and (14), whereas 𝑃(𝑠)𝑒 is the

packet error rate of the 𝑠-th station.

3A superscript (𝑡) is used for discerning the probability 𝑞 among thestations.



III. EVALUATING THE NETWORK LOADING CONDITIONS

In a previous paper [14], we proved that the behaviour ofthe aggregate throughput in a network of 𝑁 homogeneous4

contending stations is a linear function of the packet arrivalrate 𝜆 with a slope depending on both the number of contend-ing stations and the average payload length. We also derivedthe interval of validity of the proposed model by showing thepresence of a critical 𝜆, above which all the stations beginoperating in saturated traffic conditions.

This kind of behaviour, with appropriate generalizations,is also observed when multirate and variable loaded stationsare present in the network. We have to identify a set ofconditions for a network to be considered as loaded. We noticein passing that this framework is not generally consideredin the literature, since most papers assume saturated trafficconditions. A key observation from the analysis developed inthis section is that in an unloaded network there is no need toguarantee fairness.

Under the traffic model described in section II-D, we defineunloaded a network in which each contending station has apacket rate 𝜆𝑡 less than or equal to its packet service rate�̃�(𝑡)𝑆 :

𝜆𝑡 ≤ �̃�(𝑡)𝑆 , ∀ 𝑡 ∈ {1, . . . , 𝑁} (19)

The reason is simple: this condition ensures that the averagepacket inter-arrival time is greater than or equal to the averageservice time of the 𝑡-th station (stability condition [43]). Insuch a scenario, the probabilities of collisions among stationsare very low, and each contending station is able, on theaverage, to gain the access to the channel as soon as a newpacket arrives in its queue. Notice that �̃�(𝑡)

𝑆 only depends onthe packet rates 𝜆𝑖 of the other 𝑁 − 1 stations other than thetagged one.

The evaluation of the packet service rate �̃�(𝑡)𝑆 in a multirate

and heterogeneous network is quite difficult [12] since packetarrivals may occur during the stage of post-backoff, as well asduring the usual backoff stages accomplished by each stationbefore gaining the channel for transmission. Since we areinterested in a threshold which differentiates the unsaturatedfrom the saturated loading conditions of the stations, we canemploy an upper bound defined by the saturation servicerate, identified as 𝜇

(𝑡)𝑆 , in place of the actual service rate

�̃�(𝑡)𝑆 . The advantage relies on the observation that such a

bound is always evaluated considering a post-backoff stage.Indeed, after a packet transmission, a new packet is alwaysavailable in the queue assuming saturated traffic; therefore,the service time starts from a post-backoff phase wherebythe contention window is 𝑊

(𝑠)0 . We notice that the saturation

service time always includes the post-backoff stage, thusits duration is longer than the actual service time evaluatedwithout considering the post-backoff time.

Hence, in the remaining part of this section, we evaluate thesaturation service rate 𝜇

(𝑡)𝑆 = 1/𝑇

(𝑡)𝑠𝑒𝑟𝑣, i.e., the rate at which

packets are taken from the queue of the 𝑡-th station undersaturated conditions.

4By homogeneous we simply mean that the network is characterized by 𝑁stations transmitting with the same bit rate (no multirate hypothesis) and thesame load.

0 100 200 300 400 500 600 700 8000

1

2

3

4

5

6

7

8

9

10

λ1 [pkt/s]

S1−1 MbpsS2−11 Mbps, λ

2=100pkt/s

S3−11 Mbps, λ3=500pkt/s

λ2 / μ(2)

s

λ3 / μ(3)

s

λ1 / μ(1)

s

Saturation Point: 89.19 pkt/s

Fig. 2. Behaviour of the ratio 𝜆𝑡/𝜇(𝑡)𝑆 = 𝜆𝑡 ⋅ 𝑇 (𝑡)

𝑠𝑒𝑟𝑣 in (19) in a networkof three contending stations (labelled S1, S2 and S3) as a function of thepacket rate 𝜆1 of the slowest station S1 transmitting at 1 Mbps. The othertwo stations transmit at 11 Mbps with constant packet rates, respectively equalto 100pkt/s and 500pkt/s. The packet size PL is equal to 1028 bytes for thethree contending stations.

Upon considering the tagged station identified by the index𝑡 ∈ {1, ⋅ ⋅ ⋅ , 𝑁}, the saturation service time 𝑇

(𝑡)𝑠𝑒𝑟𝑣 can be

defined as follows [28]:

𝑇(𝑡)𝑠𝑒𝑟𝑣 =

{∑𝑟𝑖=0(𝑃

(𝑡)𝑒𝑞 )𝑖

(𝑖𝑇𝐶 +

∑𝑖𝑗=0 𝑊

(𝑡)𝑗 ⋅ 𝑇 (𝑡)

𝑏𝑜 + 𝑇(𝑡)𝑆

)+

(𝑃 (𝑡)𝑒𝑞 )𝑟+1

((𝑟 + 1)𝑇𝐶 +

𝑟∑𝑗=0

𝑊(𝑡)𝑗 ⋅ 𝑇 (𝑡)

𝑏𝑜

)︸︷︷︸

𝐷𝑅𝑂𝑃

⎫⎬⎭

/∑𝑟+1

𝑗=0(𝑃(𝑡)𝑒𝑞 )𝑗

(20)The first term in the summation represents the average timethat a station spends through the backoff stages from 0 to 𝑟before transmitting a packet, i.e., the so called MAC accesstime. We notice that for the 𝑖-th stage, 𝑖 collisions of averageduration 𝑇𝐶 , as well as 𝑖 backoff stages from 0 to 𝑖 (each ofthem with an average number 𝑊

(𝑡)

𝑗 of slot of duration 𝑇(𝑡)𝑏𝑜 )

occurred, after which the packet is successfully transmittedwith duration 𝑇

(𝑡)𝑠 . The second term of the summation takes

into account the average duration of a packet drop thatoccurs after (𝑟 + 1) collisions and backoff stages. The wholesummation is scaled by a normalization factor that takes intoaccount the probability set over which the service time isevaluated.

The average number of slots for the 𝑖 backoff stages, isdefined as

𝑊(𝑡)

𝑗 = (2min(𝑗,𝑚) ⋅𝑊 (𝑡)0 − 1)/2.

Each slot has average duration 𝑇(𝑡)𝑏𝑜 , which is substantially

evaluated as 𝑇𝑎𝑣 in (6) except that the tagged station (𝑡) isnot considered because it is either idle, or in a backoff state.

Let us discuss two sample scenarios in order to derivea variety of observations that are at the very basis of thefairness problem developed in the next section. The networkparameters used in the investigated IEEE802.11b MAC layer



are reported in Table I [1]. The first investigated scenarioconsiders a network of 3 contending stations. Two stations,namely S2 and S3, transmit packets with constant rates𝜆 = 100 pkt/s and 𝜆 = 500 pkt/s, respectively. The bit rateof the two stations S2 and S3 is 11 Mbps. The third station,S1, has a bit rate equal to 1 Mbps and a packet rate 𝜆1 thatis varied in the range [0, 2000] pkt/s in order to investigate itseffects on the network load. The behaviour of the three ratios𝜆𝑡/𝜇

(𝑡)𝑆 is shown in Fig. 2. Some observations are in order.

First of all, notice that as far as S1 increases its packet rate,the curves 𝜆2 ⋅ 𝑇 (2)

𝑠𝑒𝑟𝑣 and 𝜆3 ⋅ 𝑇 (3)𝑠𝑒𝑟𝑣 tend to increase because

of the increasing values of the service times 𝑇(2)𝑠𝑒𝑟𝑣 and 𝑇

(3)𝑠𝑒𝑟𝑣

experienced by S2 and S3. Notice that, as 𝜆1 increases, theslowest station S1 tends to transmit more often. The stationS3 goes in loaded condition when 𝜆1 ≈ 20pkt/s, while S2can be considered loaded for 𝜆1 ≈ 200 pkt/s. When a stationbecomes loaded, the incoming packets tend to be stored in thestation queue waiting for transmission since the service rate,i.e., the number of packets that on average are serviced bythe MAC, is below the rate by which the packets arrive in thestation queue.

The per-station throughput achieved by the three stations inthe investigated scenario is shown in Fig. 3. The throughputgained by the two fastest stations, S2 and S3, tends to decreasebecause of the anomaly problem in the multirate scenario: theslowest station tends to occupy the channel longer and longeras far as its packet rate 𝜆1 increases. In the same figure, weshow two tick curves. The horizontal line L2 corresponds tothe saturation throughput of S1, while the straight line L1 isthe tangent to the throughput curve passing through the origin.For very small values of 𝜆1, the throughput of the station S1grows linearly with 𝜆1. Packets are mainly transmitted as soonas they arrive at the MAC layer, and the station throughput isapproximately equal to 𝜆1 ⋅𝑃𝐿(1). However, when the stationapproaches the transition point 𝜆∗

1 = 89.19 pkt/s derived withthe proposed framework (this is the value of 𝜆1 correspondingto the relation 𝜆1/𝜇

(1)𝑆 = 1), the throughput curve tends to

reach the asymptote L2, which corresponds to the saturationthroughput of S1. Notice that the curve L2 approximatelycorresponds to 0.73 Mbps, which is 𝜆∗

1 ⋅𝑃𝐿(1) = 89.19⋅1028⋅8bps.

In the second scenario, the stations S1 and S2 are interestedby a constant packet rate noticed in the label of Fig. 4 (S1 has apacket rate of 10 pkt/s, while S2 100 pkt/s), whereas S3, one ofthe 11 Mbps stations, has an increasing packet rate in the range[0, 2000]pkt/s. We notice in passing that S1 is operating belowthe critical value 89.19 pkt/s obtained in the first scenario.These two stations do not get loaded by the increasing packetrate of the station S3 since the curves 𝜆1/𝜇

(1)𝑆 and 𝜆2/𝜇

(2)𝑆

are strictly less than one. As a consequence, the per-stationthroughput of both S1 and S2 is approximately constant acrossthe range of values of the packet rate 𝜆3 as noticed in Fig. 5.On the other hand, the throughput achieved by the station S3tends to saturate as soon as 𝜆3 reaches the value 𝜆∗

3 = 533pkt/s noticed in Fig. 4.

In the light of the previous two sample scenario, let ussummarize the main ingredients of the results proposed in thissection. As observed in the two previous sample scenarios,

0 100 200 300 400 500 600 700 8000

0.5

1

1.5

2

2.5

3

3.5

4x 10

6

Thr

ough

put−

[bps

]

λ1 [pkt/s]

S1−1 MbpsS2−11 Mbps, λ

2=100pkt/s



L1

L2, Throughput=0.73 Mbps

Fig. 3. Behaviour of the per-station throughput in a network of threecontending stations (labelled S1, S2 and S3) as a function of the packetrate 𝜆1 of the slowest station S1 transmitting at 1 Mbps. The other twostations transmit at 11 Mbps with constant packet rates, respectively equalto 100pkt/s and 500pkt/s. The packet size PL is equal to 1028 bytes for thethree contending stations.

this method allows to identify whether the network is loadedby establishing the thresholds of each contending station inthe network. This issue has been overlooked in the literature,where the saturation assumption is widely adopted. Moreover,this issue is at the very basis of any throughput optimizationstrategy since an unloaded network does not need to beoptimized.

We say that the network is loaded when every station has atraffic above its proper threshold. On the other hand, it shouldbe noticed that a network can be unloaded even if a subset ofthe stations is loaded. This was the case of the second scenariodescribed above, where, despite the fact that the station S3was interested by an increasing traffic load 𝜆3, the stations S1and S2 did not experience any considerable performance loss(see Fig. 5) because their traffics were below the respectivethresholds.

IV. THE PROPORTIONAL FAIRNESS THROUGHPUT

ALLOCATION ALGORITHM

This section presents the novel resource allocation criterion,which aims at improving fairness among the 𝑁 contendingstations. In order to face the fairness problem in the mostgeneral scenario, i.e., multirate DCF and general stationloading conditions, we propose a novel Proportional FairnessCriterion (PFC) by starting from the PFC defined by Kellyin [41], and employed in [25] in connection to proportionalfairness throughput allocation in multirate and saturated DCFoperations.

Let us briefly mention the rationales at the very basis of theclassical PFC by Kelly. A proportional fairness optimizationcriterion allocates to each station a throughput proportionalto the station transmission rate. Resorting to the notationproposed in [41], a throughput allocation vector 𝑥 = {𝑥𝑠; 𝑠 =1, ⋅ ⋅ ⋅ , 𝑁} is proportional fair if the following condition



0 200 400 600 800 1000 1200 1400 1600 1800 20000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

λ3 [pkt/s]


S2−11Mbps, λ2=100pkt/s

S3−11 Mbps

λ3 / μ(3)

s

λ2 / μ(2)

s λ1 / μ(1)

s


Fig. 4. Behaviour of the ratio 𝜆𝑡/𝜇(𝑡)𝑆 = 𝜆𝑡 ⋅ 𝑇 (𝑡)

𝑠𝑒𝑟𝑣 in (19) in a networkof three contending stations (labelled S1, S2 and S3) as a function of thepacket rate 𝜆3 of the fastest station S3 transmitting at 11 Mbps. The stationS1 transmits with constant packet rate 10pkt/s at 1 Mbps, whereas the stationS2 transmits with constant packet rate 100pkt/s at 11 Mbps. The packet sizePL is equal to 1028 bytes for the three contending stations.

500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3

3.5

4x 10

6

Thr

ough

put−

[bps

]

λ3 [pkt/s]

S1−1 Mbps, λ1=10 pkt/s

S2 −11 Mbps, λ2=100 pkt/s

S3 − 11 Mbps

Fig. 5. Behaviour of the per-station throughput in a network of threecontending stations (labelled S1, S2 and S3) as a function of the packetrate 𝜆3 of the fastest station S3 transmitting at 11 Mbps. The other twostations transmit with constant packet rates, respectively equal to 10pkt/s and100pkt/s. The packet size PL is equal to 1028 bytes for the three contendingstations.

holds:𝑁∑𝑠=1

𝑦∗𝑠 − 𝑥𝑠𝑥𝑠

≤ 0 (21)

for any other feasible throughput allocation vector 𝑦∗. ThePFC maximization problem satisfying (21) can be formalizedas follows:

max𝑁∑𝑠=1

log(𝑥𝑠) (22)

over 𝑥𝑠 ∈ [0, 𝑥𝑠,𝑚], 𝑠 = 1, . . . , 𝑁

whereby 𝑥𝑠,𝑚 is the maximum throughput of the 𝑠-th station.

Due to the strict concavity of the logarithmic function andbecause of the compactness of the feasible region 𝑥𝑠 ∈[0, 𝑥𝑠,𝑚], 𝑠 = 1, . . . , 𝑁 , there exists a unique solution to theoptimization problem (22). This implies that a local maximumis also global.

It is known that one of the main drawbacks of the basicDCF operating in a multirate scenario relies on the fact thatit behaves in such a way as to guarantee equal long-termchannel access probability to the various contending stations[25], [30] regardless of both transmission rates and packetrates, thus causing the performance anomaly problem. Inorder to solve this problem, various optimization algorithmshave been proposed in the literature (see, for instance, [25]-[37]). These contributions allowed to highlight the behaviourof the DCF as well as various drawbacks when operatingin a multirate scenario. For instance, it is known that theaggregate throughput of multirate IEEE802.11-like networksis maximized when only the high rate stations transmit, whilethe low rate stations are kept silent. Of course, this result isnot desirable from a fairness point of view, even though itmitigates the DCF performance anomaly noticed in [2].

To the best of our knowledge, the solutions proposed so farin the literature refer to homogeneous networks in that all thecontending stations operate with the same traffic, as exempli-fied by the station packet rate 𝜆𝑠, and ideal channel conditions.Furthermore, almost all the works focus on saturated trafficconditions. As already mentioned before, in a practical settingthe contending stations have their own traffic and are affectedby different channel conditions. The key observation here isthat a fair throughput allocation should account for the stationpacket rate, as well as for the specific channel conditionsexperienced by each contending station.

Given the aforementioned preliminaries, in our proposedmodel, the traffic of each station is characterized by thepacket arrival rate 𝜆𝑠, ∀𝑠 = 1, . . . , 𝑁 , which depends mainlyon the application layer. Let 𝜆𝑚𝑎𝑥 be the maximum packetrate among 𝜆1, . . . , 𝜆𝑁 . Furthermore, let us set the followingbounds on the stations packet arrivals 𝜆𝑠, ∀𝑠 = 1, . . . , 𝑁 :

𝜆∗𝑠 =

{𝜆𝑠, if 𝜆𝑠 ⋅ 𝑃𝐿(𝑠) ⋅ 8 ≤ 𝑅

(𝑠)𝑑

𝑅(𝑠)𝑑

8⋅𝑃𝐿(𝑠) , if 𝜆𝑠 ⋅ 𝑃𝐿(𝑠) ⋅ 8 > 𝑅(𝑠)𝑑

(23)

and let 𝑔𝑠 be 𝜆∗𝑠

𝜆∗𝑚𝑎𝑥

⋅(1− 𝑃

(𝑠)𝑒

), where 𝑃 (𝑠)

𝑒 is the packet errorrate of the s-th contending station, and 𝜆∗

𝑚𝑎𝑥 = max𝑠 𝜆∗𝑠 .

With this setup, we weight the throughput 𝑆𝑠 of the 𝑠-th station by the ratio 𝜆∗

𝑠

𝜆∗𝑚𝑎𝑥

in order to allocate networkresources proportionally to the real need of the 𝑠-th station totransmit (as exemplified by 𝜆∗

𝑠) normalized by the maximumpacket rate 𝜆∗

𝑚𝑎𝑥 in the network (the station with maximum𝜆∗𝑠 gets unitary weight). As pointed out subsequently, the

bounds (23) are meant to provide the stations a resourceallocation that can be truly managed compared to actualstation bit rates at the physical layer. Moreover, the overallweight 𝑔𝑠 =

𝜆∗𝑠

𝜆∗𝑚𝑎𝑥

⋅(1− 𝑃

(𝑠)𝑒

)accounts for the real channel

conditions experienced by the 𝑠-th station in such a way asto provide more resources to the stations experiencing goodchannel conditions compared to the ones in the network whosechannel quality corrupts their transmitted packets. We noticein passing that the overall throughput of the network may



be seriously degraded by the stations transmitting with badchannel conditions.

Taking into account the aforementioned rationales, we pro-pose the following optimization problem

max 𝑈 = 𝑈(𝑆1, ⋅ ⋅ ⋅ , 𝑆𝑁) =∑𝑁

𝑠=1 𝑔𝑠 ⋅ log(𝑆𝑠)over 𝑆𝑠 ∈ [0, 𝑆𝑠,𝑚], 𝑠 = 1, . . . , 𝑁

(24)

whereby 𝑆𝑠 is the throughput of the 𝑠-th station, and 𝑆𝑠,𝑚 isits maximum value. In our scenario, the individual through-puts, 𝑆𝑠, are interlaced because of the interdependence ofthe probabilities involved in the transmission probabilities𝜏𝑠, ∀𝑠 = 1, . . . , 𝑁 . For this reason, we reformulate themaximization problem in order to find the 𝑁 optimal values of𝜏𝑠 for which the cost function 𝑈 in (24) gets maximized. Theoptimal values 𝜏∗𝑠 are then used to set the network parametersof each station.

In the following, the optimization criterion summarized in(24) will be denoted as Modified Load Proportional Fairness(MLPF) criterion. For comparisons, the optimization criterionsummarized in (24) will be denoted as Load ProportionalFairness (LPF) criterion when (23) is not imposed on thepacket rates (i.e., 𝜆∗

𝑠 = 𝜆𝑠). The MLPF (LPF) optimizationproblem in (24) is solved by first numerically obtaining theoptimal values 𝜏∗𝑠 , ∀𝑠, and then choosing the value of theminimum contention window sizes 𝑊

(𝑠)0 by equating the

optimizing 𝜏∗𝑠 to (3) independently for any 𝑠 ∈ {1, . . . , 𝑁}.Let us derive some observations on the proposed throughput

allocation algorithm by contrasting it to the classical PFalgorithm in (21). Consider two contending stations withpacket rates 𝜆1 = 50 pkt/s and 𝜆2 = 100 pkt/s, respectively.Employing the classical PF method, a throughput allocationis proportionally fair if a reduction of 𝑥% of the throughputallocated to one station is counterbalanced by an increaseof more than 𝑥% of the throughputs allocated to the othercontending stations.

In our setup, the ratio 𝜆1/𝜆2 can be interpreted as thefrequency by which the first station tries to get access to thechannel relative to the other station. Therefore, a throughputallocation is proportionally fair if, for instance, a reduction of20% of the throughput allocated to the first station, which hasa relative frequency of 1/2, is counterbalanced by an increaseof more than 40% of the throughput allocated to the secondstation. In a scenario with multiple contending stations, therelative frequency is evaluated with respect to the station withthe highest packet rate in the network, that gets unitary relativefrequency.

Based on extensive analysis, we found that the LPF op-timization problem (24) without packet rate saturation (23)sometimes yields throughput allocations that cannot be actu-ally managed by the stations. As a reference example, assumethat, due to the specific channel conditions experienced, thefirst station has a bit rate equal to 1 Mbps and needs totransmits 200 pkt/s. Given a packet size of 1024 bytes,that is 8192 bits, the first station would need to transmit8192× 200 bps ≈ 1.64Mbps far above the maximum bit ratechosen at the physical layer. In this scenario, such a stationcould not transmit data over the channel with a throughputgreater than 1Mbps. The same applies to the other contendingstations in the network experiencing similar conditions. That’s

TABLE ITYPICAL NETWORK PARAMETERS

MAC header 28 bytes Propag. delay 𝜏𝑝 1 𝜇𝑠PLCP Preamble 144 bit PLCP Header 48 bit

PHY header 24 bytes Slot time 20 𝜇𝑠PLCP rate 1Mbps W0 32

No. back-off stages, m 5 W𝑚𝑎𝑥 1024Payload size 1028 bytes SIFS 10 𝜇𝑠

ACK 14 bytes DIFS 50 𝜇𝑠ACK timeout 364𝜇𝑠 EIFS 364 𝜇𝑠

the reason for considering the MLPF optimization criterionwith bounds on the stations packet rates.

Let us derive a simplified closed-form formula for theoptimization problem stated in (24). Due to the compactnessof the feasible region 𝑆𝑠 ∈ [0, 𝑆𝑠,𝑚], ∀𝑠, the maximum of𝑈(𝑆1, ⋅ ⋅ ⋅ , 𝑆𝑁 ) can be found among the solutions of ∇𝑈 =(∂𝑈∂𝜏1

, ⋅ ⋅ ⋅ , ∂𝑈∂𝜏𝑁)= 0. After some algebra (the derivations are

reported in the Appendix), the optimal solutions 𝜏𝑗 can befound by the following set of equations:

𝑔𝑗𝜏𝑗− 1

1− 𝜏𝑗(𝐶𝑔−𝑔𝑗)− 𝐶𝑔

𝑇𝑎𝑣

∂𝑇𝑎𝑣∂𝜏𝑗

= 0, ∀𝑗 = 1, . . . , 𝑁 (25)

where 𝐶𝑔 =∑𝑁

𝑘=1 𝑔𝑘, and 𝑇𝑎𝑣 is a function of 𝜏1, ⋅ ⋅ ⋅ , 𝜏𝑁 asnoticed in (6).

Due to the presence of 𝑇𝑎𝑣, a closed form of the maximumof 𝑈(𝑆1, ⋅ ⋅ ⋅ , 𝑆𝑁 ) cannot be found. Notice that it is quitedifficult to derive the contribution of the partial derivative of𝑇𝑎𝑣 on 𝜏𝑗 , especially when 𝑁 ≫ 1, because of the largenumber of network parameters belonging to different stations.The definition of 𝑇𝑎𝑣 in (6) is composed by four differentterms, which include the whole set of 𝜏𝑠, ∀𝑠. In order toovercome this problem, we derived approximated expressionsof 𝑇𝑎𝑣 and its derivatives, as detailed in the Appendix. Aftersome algebra, (25) yields a set of relationships between the𝜏𝑗 of the contending stations and the 𝜏𝑠 of a reference station.Taking the station indexed by 𝑗 = 1 as reference station, wefirst obtain the approximated value 𝜏∗1 for the first station (Equ.(34) in the Appendix) and derive 𝜏∗𝑠 , ∀𝑠 ∈ {2, . . . , 𝑁}, as afunction of 𝜏∗1 (Equ. (28) in the Appendix). Finally, given theoptimal values 𝜏∗𝑠 , ∀𝑠 ∈ {1, . . . , 𝑁}, we obtain the value ofthe minimum contention window size, 𝑊 (𝑠)

0 , by equating theoptimizing 𝜏∗𝑠 to (3) independently for any 𝑠 ∈ {1, . . . , 𝑁}. Asshown in Section V, these closed-form approximated solutionsyielded optimal allocations very close to the ones obtained bysolving the optimization problem (24).

Hereafter the optimization problem solved employing theapproximated solutions will be identified by LPF-approx andMLPF-approx depending on whether the bound (23) on thestations packet rate is adopted.

V. SIMULATION RESULTS

This section presents simulation results obtained for a va-riety of network scenarios optimized with the fairness criteriaproposed in the previous section.

We have developed a C++ simulator modelling both theDCF protocol details in 802.11b and the backoff proceduresof a specific number of independent transmitting stations.



0 100 200 250 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

350

400

λ1 [pkt/s]

1 / T

SE

RV

ST − 11 Mbps, λ2=λ

3 =500 pkt/s

ST − 1 Mbps, λ1

SCENARIO ASCENARIO B

Fig. 6. Behaviour of the critical packet rates (𝜇(𝑡)𝑆 = 1/𝑇 𝑡

𝑠𝑒𝑟𝑣 ) in a networkof three contending stations as a function of the packet rate 𝜆1 of the sloweststation transmitting at 1 Mbps. The other two stations transmit with a constantpacket rate equal to 500pkt/s at 11 Mbps. Notice that curves 𝜇

(𝑡)𝑆 related to

the stations at 11 Mbps are superimposed since they both employ the samenetwork parameters.

The simulator considers an Infrastructure BSS (Basic ServiceSet) with an Access Point (AP) and a certain number offix stations which communicate only with the AP. Traffic isgenerated following the exponential distribution for the packetinterarrival times. Moreover, the MAC layer is managed by astate machine which follows the main directives specified inthe standard [1], namely waiting times (DIFS, SIFS, EIFS),post-backoff, backoff, basic and RTS/CTS access modes. Thetypical MAC layer parameters for IEEE802.11b reported inTable I [1] have been used for performance validation.

The first investigated scenario, namely scenario A, consid-ers a network with 3 contending stations and ideal channelconditions to have a fair comparison with other techniquesproposed in the literature. Two stations transmit packets withrate 𝜆 = 500 pkt/s at 11 Mbps. The payload size, assumedto be common to all the stations, is 𝑃𝐿 = 1028 bytes. Thethird station has a bit rate equal to 1Mbps and a packet rate𝜆 = 1000 pkt/s in order to simulate saturated traffic.

From (19), it is straightforward to notice that the scenario Arefers to a loaded network. Fig. 6 shows the service rates 𝜇(𝑡)

𝑆

of the three contending stations as a function of the packetrate 𝜆1 of the station transmitting at 1 Mbps. The operatingpoint of the considered scenario A is highlighted in Fig. 6.Notice that, since the service rates 𝜇

(𝑡)𝑆 of the three stations

are below the respective packet rates 𝜆𝑡, the network is loaded.Moreover, the service rates of the three stations tend to thesame values because of the rate anomaly problem: the stationtransmitting at 1 Mbps reduces the service rates of the otherstations.

The simulated normalized throughput achieved by eachstation in this scenario is depicted in the left subplot ofFig. 7 for the following four setups. The three bars labelled1-DCF represent the normalized throughput achieved by thethree stations with a classical DCF. The second set of bars,labelled 2-PF, identifies the simulated normalized throughput

TABLE IIAGGREGATE THROUGHPUT

Scenarios 1-DCF 2-PF 3-LPF 4-LPF 5- 6-MLPFapprox MLPF approx

AJain’s Ind. 0.451 0.872 0.715 0.755 0.874 0.869S [Mbps] 1.85 3.60 3.06 3.17 5.01 5.0

BJain’s Ind. 0.474 0.881 0.987 0.981 0.874 0.869S [Mbps] 1.99 3.63 4.54 4.58 5.0 5.0

achieved by the DCF optimized with the PF criterion [25],[41], whereby the actual packet rates of the stations are notconsidered. The third and fourth sets of bars, labelled 3-LPF and 4-LPF approx, represent the normalized throughputachieved by the three stations when the allocation problem(24) and the approximated solutions obtained using (34) and(28) in the Appendix are respectively employed. Finally, thelast two sets of bars, labelled 5-MLPF and 6-MLPF approx,represent the simulated normalized throughput achieved by thecontending stations when the CW sizes are optimized with themodified fairness criterion in (24) and using the approximatedsolutions obtained using (34) and (28) in the Appendix withstation packet rate saturation, respectively. Notice that thethroughput allocations guaranteed by LPF and MLPF, andtheir approximated solutions, improve over the classical DCF.When the station packet rate is considered in the optimizationframework, a higher throughput is allocated to the first stationpresenting the maximum value of 𝜆 among the considered sta-tions. However, the highest aggregate throughput is achievedwhen the allocation is accomplished with the optimizationframework 4-MLPF. The reason for this behaviour lies in thefollowing observation: the first station requires a traffic equalto 8.22 Mbps = 103 pkt/s ⋅ 1028 bytes/pkt ⋅ 8 bits/pkt, whichis far above the maximum traffic (1 Mbps) that it would beable to deal with in the best scenario. In this respect, theMLPF criterion results in better throughput allocations sinceit accounts for the real traffic that the contending station wouldtheoretically be able to manage in the specific scenario at hand.

Similar considerations can be drawn from the results shownin the right subplot of Fig. 7 (related to scenario B), wherebyin the simulated scenario the two fastest stations are also char-acterized by a packet rate greater than the one of the sloweststation. Once again, the transmission channel in scenario Bis ideal in order to guarantee a fair comparison with othertechniques in the literature.

Notice that the optimization framework 3-LPF is able toguarantee improved aggregate throughput with respect to boththe non-optimized DCF and the classical PF algorithms. Theoperating point of the scenario B is highlighted in Fig. 6; basedon the considerations above, this is a loaded network.

The aggregate throughputs achieved in the two investigatedscenarios are shown in Table II where we also show the fair-ness Jain’s index [42] evaluated on the normalized throughputsnoted in the subplots of Fig. 7. It is worth noticing thatthe proposed MLPF throughput allocation criterion is ableto guarantee improved aggregate throughput relative to boththe classical DCF and the PF algorithm, with fairness levelson the same order of the ones guaranteed by the classicalPF algorithm. Finally, we point out the effectiveness of theapproximated solutions LPF and MLPF approx derived in



1 2 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7N

orm

aliz

ed T

hrou

ghpu

t

1 Mbps, λ=100011 Mbps, λ=50011 Mbps, λ=500

1 2 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Nor

mal

ized

Thr

ough

put

1 Mbps, λ=25011 Mbps, λ=50011 Mbps, λ=500

Scenario A1 − DCF2 − PF3 − LPF4 − LPF approx5 − MLPF6 − MLPF approx

Scenario B1 − DCF2 − PF3 − LPF4 − LPF approx5 − MLPF6 − MLPF approx

Fig. 7. Simulated normalized throughput achieved by three contendingstations upon employing 1) a classical DCF; 2) DCF with PF allocation;3) DCF optimized as noted in (24); 4) DCF optimized with the LPFapproximated criterion; 5) DCF optimized with the MLPF criterion; and 6)DCF optimized with the MLPF approximated criterion. Left and right plotsrefer to scenarios A and B, respectively.

the Appendix in guaranteeing fairness and throughput valuescomparable with the ones of the LPF and MLPF in (24)without the need of executing any optimization problem, butsimply relying on closed-form formulas.

Let us discuss another scenario in order to compare the pro-posed optimization technique with the one presented in [44](see also [45]), whereby an IEEE 802.11b network is con-sidered. Once again, the contending stations experience idealchannel conditions. In [44] the authors proposed a techniqueto maximize the total aggregate throughput in such a waythat the channel occupancy times among the stations trans-mitting at different rates are kept at a given time fairnessindex ratio. The scenario investigated in [44] is saturatedin that the saturation hypothesis holds for every contendingstation in the network and the proposed analysis does notaccount for unsaturated networks. Four stations, each witha different bit rate in the set {1,2,5.5,11} Mbps, transmitusing different contention windows and packet sizes, whichare the solutions of the optimization problem. For an im-posed fairness index ratio equal to 0.8, the optimal solution{𝑊 (1)

0 = 386.41;𝑊(2)0 = 229.5;𝑊

(3)0 = 64.948;𝑊

(4)0 =

35.24;𝑃𝐿(1) = 912.68;𝑃𝐿(2) = 1608.4;𝑃𝐿(3) =2100.8;𝑃𝐿(4) = 2304}, yields an aggregate throughput equalto 4.1958 Mbps. For comparisons, we used the same 4-stationsscenario with the packet sizes 𝑃𝐿(1) = 912.68;𝑃𝐿(2) =1608.4;𝑃𝐿(3) = 2100.8;𝑃𝐿(4) = 2304 bytes since in our al-gorithm the packet size is not the objective of the optimization.We recall that our algorithm does account for variable-loadedstations, but for fair comparisons in this scenario we consid-ered saturated stations. The optimal set of minimum contentionwindows obtained as a result of our MLPF optimization algo-rithm is {𝑊 (1)

0 = 327;𝑊(2)0 = 253;𝑊

(3)0 = 60;𝑊

(4)0 = 16}

leading to a fairness index of 0.78 and a total aggregatethroughput of 5.10 Mbps. We notice that our algorithm is ableto increase the throughput by more than 1Mbps compared to

10 50 100 200 500 1000 2000 30000

0.2

0.4

0.6

DC

F

λ1 [pkt/s]

10 50 100 200 500 1000 2000 30000

0.2

0.4

0.6

LPF

λ1 [pkt/s]

10 50 100 200 500 1000 2000 30000

0.2

0.4

0.6

MLP

F

λ1 [pkt/s]

1 Mbps, λ1

11 Mbps, λ=50011 Mbps, λ=500

Fig. 8. Simulated normalized throughput achieved by three contendingstations as a function of the packet rate of the slowest station in DCF, LPFand MLPF modes.

the scenario investigated in [44] against a slight fairness indexreduction of 0.02.

For the sake of investigating the behaviour of the proposedallocation criteria in ideal channel conditions as a functionof the packet rate of the slowest station, we simulated thethroughput allocated to a network composed by three stations,whereby the slowest station, transmitting at 1Mbps, presentsan increasing packet rate in the range 10−3000 pkt/s. Theother two stations transmit packets at the constant rate 𝜆 =500 pkt/s at 11 Mbps. The simulated throughput of the threecontending stations is shown in the three subplot of Fig. 8 forthe unoptimized DCF, as well as for the two criteria LPF andMLPF. Some considerations are in order. Let us focus on thethroughput of the DCF (uppermost subplot in Fig. 8). As far asthe packet rate of the slowest station increases, the throughputallocated to the fastest stations decreases quite fast because ofthe performance anomaly of the DCF [2]. The three stationsreach the same throughput when the slowest station presentsa packet rate equal to 500 pkt/s, corresponding to the one ofthe other two stations. From 𝜆 = 500 pkt/s all the way up to3000 pkt/s, the throughput of the three stations do not changeanymore, since all the stations have a throughput imposed bythe slowest station in the network. Let us focus on the resultsshown in the other two subplots of Fig. 8, labelled LPF andMLPF, respectively. A quick comparison among these threesubplots in Fig. 8 reveals that the MLPF allocation criterionguarantees improved aggregate throughput for a wide range ofpacket rates of the slowest station, greatly mitigating the rateanomaly problem of the classical DCF operating in a multiratesetting. In terms of aggregate throughput, the best solutionis achieved with the MLPF criterion, which avoids that theslowest station receives a resource allocation that would notbe able to employ due to its reduced bit rate (1 Mbps).

The last set of simulations is meant to provide insightson the network optimization when some of the contendingstations in the network experience bad channel quality. Topresent the resource allocation guaranteed by the optimization



problem, we considered a network scenario whereby 5 satu-rated stations (3 of them transmit without packet error rates,i.e., 𝑃𝑒 = 0, with a bit rate 1Mbps, whereas the other twostations transmit at 11Mbps with different values of packeterror probability, 𝑃𝑒) transmit packets of size 1000 bytes.The packet error probability 𝑃𝑒 for the two fastest stations isidentified by 𝑃 11𝑀𝑏𝑝𝑠

𝑒 in the first column of the table below.For each 𝑃 11𝑀𝑏𝑝𝑠

𝑒 , we run the MLPF algorithm based on (24)considering the two cases in which the weights 𝑔𝑠 are set to𝜆∗𝑠

𝜆∗𝑚𝑎𝑥

⋅(1− 𝑃

(𝑠)𝑒

)(case 1) and 𝑔𝑠 =

𝜆∗𝑠

𝜆∗𝑚𝑎𝑥

(case 2).The optimization problem provided the optimal 𝜏𝑠 used

for obtaining the optimal minimum contention windowsidentified in the second and third column for the two differentbit rates. The aggregate throughput is shown in the fifthcolumn, while the last column contains the Jain’s fairnessindex obtained in each optimized scenario. In each entry ofthe table, the left value refers to the optimization for the case1, while the right values for case 2.

𝑃 11𝑀𝑏𝑝𝑠𝑒 CW- CW- S Jain’s

1Mbps 11Mbps [Mbps] Index0.077 252, 263 42, 40 2.32, 2.40 0.90, 0.920.55 176, 263 10, 7 1.15, 1.42 0.68, 0.780.98 76, 263 16, 1 0.8536, 0.53 0.60, 0.611 70, 263 798, 1 0.8631, 0.49 0.60, 0.60

Some observations are in order. Consider case 2; in theattempt to guarantee fairness among the contending stations,the algorithm tends to reduce the contention windows of thefastest stations for increasing packet error rates. By doingso, however, the aggregate throughput decreases reaching avalue around 0.5 Mbps because the fastest stations experienceincreasingly bad channel conditions corrupting almost all thetransmitted packets.

In case 1 the resource allocation accounts for this situation.Therefore, for low values of 𝑃 11𝑀𝑏𝑝𝑠

𝑒 , i.e., for good channelconditions, the assigned contention windows are almost equalto the ones allocated in case 2. However, for increasinglybad channel conditions, the optimization algorithm preventsthe fastest stations from transmitting (the assigned contentionwindows increase) thus improving the aggregate throughput,which reaches the value 0.86 Mbps. We notice in passing thatthe algorithm avoids wasting network resources by allocatingtransmission slots to stations that experience bad channelconditions corrupting the transmitted packets. Finally, noticethat under bad channel conditions, in case 1 the algorithmguarantees fairness values (0.6) equal to the ones for case 2,but with improved aggregate network throughput.

VI. CONCLUSIONS

Focusing on multirate IEEE 802.11 Wireless LAN employ-ing the mandatory Distributed Coordination Function (DCF)option, this paper established the conditions under which a net-work constituted by a certain number of stations transmittingwith their own bit rates and packet rates can be consideredloaded. It then proposed a modified proportional fairnesscriterion suitable for mitigating the rate anomaly problem ofmultirate loaded IEEE 802.11 Wireless LANs.

Simulation results were presented for some sample scenar-ios showing that the proposed throughput allocation was able

to greatly increase the aggregate throughput of the DCF, whileensuring fairness levels among the stations of the same orderof the ones available with the classical proportional fairnesscriterion.

APPENDIX

The objective of this section is to derive a closed-formapproximated solution of the maximization problem in (24).

Upon substituting (5) and (6) in (4), and deriving withrespect to 𝜏𝑗 , we obtain:

∂

∂𝜏𝑗

𝑁∑𝑖=1

𝑔𝑖 ⋅ log

⎡⎢⎣ 1

𝑇𝑎𝑣𝜏𝑖

𝑁∏𝑘=1𝑘 ∕=𝑖

(1− 𝜏𝑘)(1 − 𝑃 (𝑖)𝑒 )𝑃𝐿(𝑖)

⎤⎥⎦ =

∂

∂𝜏𝑗

𝑁∑𝑖=1

𝑔𝑖 ⋅

⎡⎢⎣log 𝜏𝑖 + 𝑁∑

𝑘=1𝑘 ∕=𝑖

log(1− 𝜏𝑘) + log(1− 𝑃 (𝑖)𝑒 )+

+ log(𝑃𝐿(𝑖))− log(𝑇𝑎𝑣)]

=

∂

∂𝜏𝑗

𝑁∑𝑖=1

𝑔𝑖 ⋅

⎡⎢⎣log 𝜏𝑖 + 𝑁∑

𝑘=1𝑘 ∕=𝑖

log(1− 𝜏𝑘)− log(𝑇𝑎𝑣)

⎤⎥⎦ (26)

whereby the last relation stems from the independence of both𝑃

(𝑖)𝑒 and 𝑃𝐿(𝑖) on 𝜏𝑗 . Exchanging the derivative with the

summation yields:

𝑔𝑗1

𝜏𝑗− 1

1− 𝜏𝑗

𝑁∑𝑘=1,𝑘 ∕=𝑗

𝑔𝑘− 𝐶𝑔𝑇𝑎𝑣


, ∀𝑗 = 1, . . . , 𝑁 (27)

whereby 𝐶𝑔 =∑𝑁

𝑖=1 𝑔𝑖. By equating (27) to zero, we get

𝑔𝑗𝜏𝑗

− 1

1− 𝜏𝑗(𝐶𝑔 − 𝑔𝑗)− 𝐶𝑔

𝑇𝑎𝑣


= 0, ∀𝑗 = 1, . . . , 𝑁

After some algebra, the previous equation can be rewritten asfollows:

𝑇𝑎𝑣𝐶𝑔

=𝜏𝑗(1 − 𝜏𝑗)

𝑔𝑗(1 − 𝜏𝑗)− (𝐶𝑔 − 𝑔𝑗)𝜏𝑗


Upon noting that the ratio 𝑇𝑎𝑣

𝐶𝑔does not depend on the specific

𝑗-th station considered, we can find a relation between thetransmission probabilities 𝜏𝑗 and 𝜏𝑟 of two generic stationsin the network by equating the right sides of the previousequation corresponding to the two stations indexed by 𝑗 and𝑟:

𝜏𝑗(1− 𝜏𝑗)

𝑔𝑗(1− 𝜏𝑗)− (𝐶𝑔 − 𝑔𝑗)𝜏𝑗


=𝜏𝑟(1− 𝜏𝑟)

𝑔𝑟(1− 𝜏𝑟)− (𝐶𝑔 − 𝑔𝑟)𝜏𝑟

∂𝑇𝑎𝑣∂𝜏𝑟

After some algebra, this equation simplifies to:

𝑔𝑟𝜏𝑗 − 𝐶𝑔𝜏𝑟𝜏𝑗 − 𝑔𝑟𝜏𝑗2 − 𝐶𝑔𝜏𝑟𝜏𝑗

2

𝑔𝑗𝜏𝑟 − 𝐶𝑔𝜏𝑗𝜏𝑟 − 𝑔𝑗𝜏𝑟2 − 𝐶𝑔𝜏𝑗𝜏𝑟2=

∂𝑇𝑎𝑣

∂𝜏𝑟∂𝑇𝑎𝑣

∂𝜏𝑗

Upon neglecting5 the terms 𝜏𝑝𝑟(𝑗) of order 𝑝 greater than 1, weobtain the following equation:

𝑔𝑟𝜏𝑗𝑔𝑗𝜏𝑟

=𝑇

(𝑟)𝑠 − 𝜎

𝑇(𝑗)𝑠 − 𝜎

5This approximation is justified by noting that 𝜏𝑗 ≪ 1,∀𝑗.



This equation can be employed to relate the transmission prob-abilities of any pair of stations in the network. Consideringthe station indexed 1 as reference station, the transmissionprobability of the 𝑗-th station, ∀𝑗 ∈ {2, 3, . . . , 𝑁}, can berelated to 𝜏1 of the first station as follows:

𝜏𝑗 =𝑇

(1)𝑠 − 𝜎


⋅ 𝑔𝑗𝑔1𝜏1 = 𝛾𝑗𝜏1; 𝛾𝑗 =

𝑇(1)𝑠 − 𝜎


⋅ 𝑔𝑗𝑔1

(28)

The problem now turns to be the evaluation of 𝜏1 given that allthe other 𝜏𝑗 can be easily related to 𝜏1. Unfortunately, givento the many variables involved in the optimization problem,it is not easy to find 𝜏1. However, we can resort to some keyapproximations to find a closed-form expression for 𝜏1.

To proceed further, we consider the following hypotheses.

1) We assume that the durations of the three time slots𝑇

(𝑠)𝑐 , 𝑇

(𝑠)𝑠 , and 𝑇

(𝑠)𝑒 are equal to each other for the

same tagged station indexed by 𝑠, and call 𝑇 (𝑠) sucha duration. This hypothesis stems from the definitionsof 𝑇 (𝑠)

𝑐 , 𝑇 (𝑠)𝑠 , and 𝑇

(𝑠)𝑒 upon noting that such definitions

differ for a very small value compared to the durationof the station’s data frame.

2) When the system is optimized and the number ofstations in the network is 𝑁 ≫ 1, which is a com-mon assumption in practical network scenarios, theoptimum set of transmission probabilities 𝜏∗ is suchthat 𝜏𝑠 ≪ 1, ∀𝑠 = 1, ⋅ ⋅ ⋅ , 𝑁 . Therefore, a reasonableapproximation is to take into account only terms 𝜏𝑝𝑠 oforder 𝑝 lower than or equal to 2.

Using these assumptions and recalling (28), first we canexpress6 𝑇𝑎𝑣 as a function of 𝜏1:

𝑇𝑎𝑣 ≈ 𝜎 + (𝐺3 −𝐺1𝜎)𝜏1 + (𝐺2𝜎 −𝐺4)𝜏21 + 𝑜(𝜏21 )

(29)whereby the constants 𝐺𝑖 are defined as

𝐺1 =∑𝑁

𝑠=1 𝛾𝑠𝐺2 =

∑𝑁𝑠=1 𝛾𝑠

∑𝑁𝑘=𝑠+1 𝛾𝑘

𝐺3 =∑𝑁

𝑑=1 𝛾𝑑 ⋅ 𝑇 (𝑑)

𝐺4 =∑𝑁

𝑑=1 𝛾𝑑∑𝑑−1

𝑠=1 𝛾𝑠 ⋅ 𝑇 (𝑑)

(30)

Let us recall the optimization problem:

𝑔𝑗𝜏𝑗− 1

1− 𝜏𝑗(𝐶𝑔−𝑔𝑗)− 𝐶𝑔

𝑇𝑎𝑣


= 0, ∀𝑗 = 1, . . . , 𝑁 (31)

We need an approximation of ∂𝑇𝑎𝑣

∂𝜏𝑗to solve (31) for 𝜏𝑗 ,

independently from the other 𝜏𝑘, 𝑘 ∕= 𝑗.Under the two hypotheses above, (6) can be simplified as

follows:

𝑇𝑎𝑣 ≈ 𝜎 −∑𝑁𝑠=1 𝜏𝑠 ⋅ 𝜎+

+∑𝑁

𝑑=1 𝜏𝑑 ⋅ 𝑇 (𝑑) −∑𝑁𝑑=1 𝜏𝑑

∑𝑑−1𝑠=1 𝜏𝑠 ⋅ 𝑇 (𝑑)

(32)where we resorted to the following first order approximationfor 𝜏𝑠 ≪ 1, ∀𝑠:

𝑁∏𝑠=1

(1− 𝜏𝑠) ≈ 1−𝑁∑𝑠=1

𝜏𝑠 + 𝑜(𝜏𝑠)

6The interested reader may refer to [40] for the analytical derivations.

From (32), ∂𝑇𝑎𝑣

∂𝜏𝑗becomes:

∂

∂𝜏𝑗𝑇𝑎𝑣 ≈ −𝜎 + 𝑇 (𝑑) + 𝑜(𝜏𝑗) = 𝑇 (𝑑) − 𝜎 (33)

where, once again, we neglected terms of 𝜏𝑠 of order greaterthan or equal to 2.

By virtue of (28), we can rewrite the optimization problemwith respect to the station 𝑗 = 1 as follows:

𝑔1𝜏1

− 1

1− 𝜏1(𝐶𝑔 − 𝑔1)− 𝐶𝑔

𝑇𝑎𝑣[𝑇 (1) − 𝜎] = 0

After some algebra, we obtain the solution 𝜏1 = 𝜏∗1 for thestation 1 as the positive root among the roots:

𝜏∗1 ≈ −𝑍1 ±√𝑍21 − 4𝑍2𝑍0

2𝑍2(34)

whereby 𝑍0 = 𝑔1𝜎, 𝑍1 = 𝑔1(𝐺3 − 𝜎𝐺1) − 𝐶𝑔𝜎 − 𝐶𝑔Δ(1),

and 𝑍2 = 𝑔1(𝐺2𝜎 −𝐺4)− 𝐶𝑔(𝐺3 − 𝜎𝐺1) + 𝐶𝑔Δ(1).

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Massimiliano Laddomada (M’03-SM’ 08) is anassistant professor of Electrical Engineering at TexasA&M University, Texarkana, and an adjunct profes-sor at California State University, Los Angeles. Mostrecently (2003-2008) he was visiting assistant pro-fessor at Polytechnic University of Turin. Previously,he was a senior engineer at Technoconcepts, Inc.,Los Angeles. In the past nine years he worked fora number of projects funded by telecommunicationscompanies, as well as by the European Commission.

He holds a Ph.D. degree (2003) in Communica-tions and Electronics Engineering and a Master degree (1999) in ElectricalEngineering from Polytechnic University of Turin.

His research is mainly in wireless communications, especially modulationand coding, including turbo codes and, more recently, networks coding.A senior member of IEEE, currently he is serving as a member of theeditorial boards of IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I:REGULAR PAPERS, INTERNATIONAL JOURNAL OF DIGITAL MULTIMEDIA

BROADCASTING and IEEE COMMUNICATIONS SURVEYS AND TUTORIALS.

Fabio Mesiti was born in Torino, Italy in 1979.He received the Degree in Telecommunication En-gineering, in 2005, and the PhD in Electronics andCommunications Engineering in 2010 from Politec-nico di Torino. His research interests include mainlycross-layer modeling, optimization and performanceanalysis of wireless communications systems and,more recently, channel coding for quantum commu-nication systems. He is a student member of IEEE.

Marina Mondin was born in Torino, Italy. Shereceived the Degree in Electronic Engineering in1986, and the PhD in Electronic Engineering in1990, both from Politecnico di Torino, Italy. Shewas a recipient of the 1987 ”De Castro” scholarship,and she spent the year 1987-88 as visiting scholarin the Department of Electrical Engineering at theUniversity of California, Los Angeles.

Since 1990 she has been with DELEN, Politec-nico di Torino, where she is an Associate Professor.Her current interests are in the area of modulation

and coding, simulation of communication systems. She is a member of IEEE.

Fred Daneshgaran received the B.S. degree inElectrical and Mechanical Engineering from Cal-ifornia State University, Los Angeles (CSLA) in1984, the M.S. degree in Electrical Engineering fromCSLA in 1985, and the Ph.D. degree in Electri-cal Engineering from University of California, LosAngeles (UCLA), in 1992. Since 1997 he is a fullprofessor of the ECE department at CSLA.


On the throughput performance of multirate ieee 802 11 networks with variable loaded stations...

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