exact formulations for the throughput of ieee 802.11 dcf in hoyt, rice, and nakagami-m fading...

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 5, MAY 2013 2261

Exact Formulations for the Throughput ofIEEE 802.11 DCF in Hoyt, Rice, and

Nakagami-m Fading ChannelsElvio J. Leonardo, Student Member, IEEE, and Michel D. Yacoub, Member, IEEE

Abstract—This paper investigates the throughput performanceof IEEE 802.11 Distributed Coordination Function (DCF) atthe Medium Access Control (MAC) layer with Hoyt, Rice andNakagami-m fading environments. The approach consideredincludes the signal capture model with incoherent addition ofinterfering signals and uniform attenuation for all terminals (orperfect power control). For the Hoyt and Rice fading channels,the results presented also cover the case of coherent additionof interfering signals. Furthermore, for the Nakagami-m fadingchannel, the case of unequal average power levels is also studied.For the sake of comparison, results for the carrier sense multipleaccess with collision avoidance (CSMA/CA) are also presented.The formulations derived here are exact and unprecedented inthe literature. Some of these formulations are given in infiniteseries form whereas several of them are presented in closed-form expressions. For the series-based ones, only a few termsare required in order to achieve high accuracy results.

Index Terms—Wireless communication, IEEE 802.11, DCF,MAC, CSMA, Hoyt fading, Rice fading, Nakagami-m fading,outage probability, capture.

I. INTRODUCTION

W IRELESS local area networks (WLANs) have beenexperiencing rapid development lately in part stim-

ulated by the deployment of systems compatible with theIEEE 802.11 standards [1]. They offer data communicationcapability between terminals within radio range while allowinga certain degree of mobility. These networks are organizedeither with or without a central node. In the cellular topology,a central node is responsible for controlling the access to thewireless medium and forwarding data to the intended users. Inad-hoc topologies, the central node is absent and all terminalsshare similar capabilities and responsibilities; the terminalscan communicate with each other either directly or by routingtheir data through intermediate nodes. In either case, the aimsare to provide connectivity among terminals and to efficientlyand fairly dispense the available bandwidth while employinglittle or no central coordination.

In order to serve terminals exhibiting bursty traffic be-havior, WLANs make use of packet radio techniques withrandom access to a transmission channel shared by multiple

Manuscript received June 14, 2012; revised October 24 and December 27,2012; accepted February 8, 2013. The associate editor coordinating the reviewof this paper and approving it for publication was Z. Han.

E. J. Leonardo is with the Department of Informatics, State University ofMaringa, Brazil (e-mail: [email protected]).

M. D. Yacoub is with the School of Electrical and Computer Engineering,State University of Campinas, Brazil (e-mail: [email protected]).

Digital Object Identifier 10.1109/TWC.2013.032513.120859

users. Specifically, variations of carrier sense multiple access(CSMA) protocols and, in particular, of CSMA with collisionavoidance (CSMA/CA), are generally used to access thewireless medium [2]–[8]. The capacity of the channel isthen influenced by the probability of packet collision andby the signal degradation due to mutual interference andsignal attenuation. In other words, it is influenced by themedium contention resolution algorithm and by the channelcharacteristics. Currently, it is fair to say that the use ofCSMA/CA is wide, and it has been growing continually, withvery recent adoption by some other IEEE standards [9], [10].

Intuitively, one might expect that original (wireline) CSMAsystems show better performance than wireless ones becauseof the more hostile channel characteristics found in the latter.However, this is not necessarily the case. For instance, in achannel model that takes into account the effects of fading,competing packets arriving at a common radio receiver an-tenna will not always destroy each other because they mayshow different and independent fading and attenuation levels[11], [12]. This leads to infer that wireless CSMA systems mayactually exhibit successful reception rate higher than that ofwireline ones. In fact, Arnbak and Blitterswijk have shown thisto happen with slotted Aloha over Rayleigh fading channels[13].

A practical and more sophisticated implementation of theCSMA/CA protocol is the IEEE 802.11. Such a system spec-ifies two operating modes: the Point Coordination Function(PCF) and the Distributed Coordination Function (DCF). Theformer is an access method planned to be implemented inan infrastructured network. The latter, which is similar toCSMA/CA, is the main focus of this work.

In this paper, we investigate the throughput performance ofIEEE 802.11 DCF with Hoyt, Rice and Nakagami-m fadingenvironments and capture effect. The performance of IEEE802.11 DCF with unsaturated traffic and non-ideal channelis presented in [14] and it is here extended to these fadingscenarios. The evolution of the performance analysis of IEEE802.11 DCF has been conducted in the following steps.Bianchi [15] presented analytical results and simulation forthe performance of IEEE 802.11 DCF assuming ideal channelconditions, finite number of terminals and saturated traffic. In[15], the MAC algorithm is modeled by a two dimensionalMarkov chain. Liaw et al. [16] added an idle state to theMarkov chain of Bianchi’s model, this way extending it tounsaturated traffic conditions, however keeping the channel

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2262 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 5, MAY 2013

ideal. Daneshgaran et al. [14] further extended the analysisto include non-ideal channel conditions and capture effect. In[14], the authors assume Rayleigh fading channel and they usesimulation to validate their results.

The IEEE 802.11 DCF is a commercially available, widelyused and practical system and it is interesting to see itevaluated with generic and potentially more realistic fadingmodels. In addition, for the sake of comparison, results forCSMA/CA in similar fading conditions are also presented.The work presented here introduces a number of exact closed-form as well as exact series-based expressions. To the best ofthe authors’ knowledge, unless otherwise cited, these resultsare novel contributions. As an intermediate outcome of thecalculation, novel results for the outage probability for theseenvironments are also introduced. Analysis of the outageprobability for fading channels has been an active investigationtopic for the last few years [17]–[22], in part because it may beused to estimate the performance of spectrum sharing systems.The work presented here offers contributions in this area andintroduces many original results.

The channel models considered in this work are amongthose commonly used to describe the short-term signal statis-tics of wireless communications links subject to fading [23]–[25]. The Rice model (also known as Nakagami-n) is usedwhen the random multipath signals are superimposed on anonfading dominant signal, for instance, when a line-of-sight(LOS) component is present. The Hoyt model (also known asNakagami-q) applies to the cases when no dominant signal ispresent and the in-phase and quadrature components of thereceived signal have non-identical powers or, otherwise, arecorrelated. Finally, the Nakagami-m model was inferred byNakagami [26] from experimental data. Its distribution canapproximate the Hoyt and Rice distributions, and it includesas special cases the one-sided Gaussian and the Rayleighdistributions.

In this paper, the channel is considered to be memoryless,i.e., failures to capture the channel and future attempts areuncorrelated. In addition, all packets are assumed to have fixedlength and to require p seconds to transmit, and that the 2-wayhandshaking mechanism is used for the packet transmissions.Finally, each packet is assumed to have a single destination.

This paper is organized as follows. Sections II describesthe framework used in this paper, while Section III applies itto the particular scenario assumed here. Section IV considersthe cases of incoherent signal addition at the receiver’s antennawith uniform attenuation for all terminals (or perfect powercontrol). Coherent signal addition and a model that includesspatial coverage, such as in a cell, are also explored. Numericalresults and conclusions are given in Sections V and VI,respectively.

II. FRAMEWORK

Consider a generic packet data communication system. Ifthe transmission of an arbitrary test packet is performed overa wireline channel, it is generally assumed that a successfulreception can only occur if no other transmission attempt ismade during the test packet reception, i.e., if there is no signaloverlap at the receiver’s end. However, in wireless systems the

radio receiver may be able to be captured by a test packet,even in the presence of n interfering packets, provided thepower ratio between the test signal and the joint interferingsignal exceeds a certain threshold during a given portion ofthe transmission period tw, 0 < tw < p, to lock the receiver[27], [28]. In such a case, the test packet is only destroyedif ws/wn ≤ z, during tw, with n > 0, where z is thecapture ratio, and ws and wn are the test packet power and thejoint interference power at the receiver’s antenna, respectively.Values for z and the capture window tw depend on, e.g., themodulation and the coding employed by the network. For atypical narrowband FM receiver, a z value of 6 dB is suggestedin [29]. The details about estimating the values of z and tw arebeyond the scope of this paper. The interested reader can findfurther information about the capture effect in the literature,including [30]–[33].

For the wireless channel, for which the signal assumes a ran-dom behavior, the capture phenomenon should, accordingly,be treated statistically. Let the random variable Z be definedas the signal-to-interference ratio (SIR) Z � Ws/Wn ≥ 0,where Ws ≥ 0 and Wn ≥ 0 are random variables representingthe desired signal power and the interference power at thereceiver’s antenna, respectively, with the latter one assumedto be an n-signal ensemble. If Ws and Wn are statisticallyindependent, the resulting probability density function (PDF)can be expressed as [34]

fZ(z) =

∫ ∞

0

yfWs(zy)fWn(y)dy (1)

where fWs(.) and fWn(.) are the PDFs of the desired signalpower and the interference power, respectively. The cumulativedistribution function (CDF) is then expressed as

FZ(z0) = Prob

{Ws

Wn≤ z0

}=

∫ z0

0

fZ(z)dz. (2)

If n is known and fixed, the resulting conditional captureprobability may be expressed as Pcp(z0|n) = 1 − FZ(z0).Of course, the statistics of Ws and Wn depend on the channelcharacteristics, and, in the current work, on the fading modelused.

In order to produce the unconditional capture probability,let γ be defined as the probability that a station starts atransmission in a randomly chosen time slot, with γ assumedto be constant across all time slots. Considering a scenario inwhich there are N stations apt to transmit, the unconditionalprobability of a test packet being able to capture the receiverin an arbitrary transmission period may be expressed by

Pcp(z0) =

N−1∑n=1

(N

n+ 1

)γn+1 (1−γ)N−n−1Pcp(z0|n). (3)

Hence, given the conditions described above, (3) yields theprobability of successful reception.

Furthermore, let S represent the normalized channelthroughput (or channel efficiency), defined as the fractionof time the channel is used to successfully transmit userinformation. It can be expressed as S = GPcp(z0), whereG is the offered traffic. In other words, S is evaluated as theportion of the offered traffic that is successfully received.

LEONARDO and YACOUB: EXACT FORMULATIONS FOR THE THROUGHPUT OF IEEE 802.11 DCF IN HOYT, RICE, AND NAKAGAMI-M FADING . . . 2263

It is interesting to note that the CDF in (2) also representsanother important measure of performance for wireless sys-tems: the outage probability. This is defined as the probabilitythat the SIR at the reference receiver falls below a certainspecified threshold required for successful reception. There-fore Pout(z0) = FZ(z0), where z0 is the successful receptionthreshold.

Another relevant aspect of our investigation is how theinterfering signal arises. In a wireless system, it typicallyresults from signals arriving at the receiver’s antenna frommultiple transmitters. Depending on how these random signalscombine during the observation interval, one of two scenariosmight occur [35]: coherent addition or incoherent addition.

Coherent addition occurs if the carrier frequencies areequal and if the random phase fluctuations are small duringthe capture time tw. For instance, coherent addition mighthappen when the deviation caused by the phase modulationis very small, and the observation interval is short comparedto the modulation rate. In other words, coherent additionshould be considered if the random phases of the individualinterferers barely vary during the capture period. Incoherentaddition occurs if the phases of the individual signals fluctuatesignificantly due to mutually independent modulation [13],[36].

Let the phasor x(t) = Re{r(t) ej[wct+θ(t)]} represent asignal reaching the receiver’s antenna, where r(t) and θ(t)are the random envelope and phase, respectively, and wc isthe carrier’s angular frequency. For the coherent addition of nsignals, the resulting phasor is [13] xn(t) =

∑ni=1 xi(t) where

the subscripts n and i represent the aggregate and individualvariables, respectively.

For the incoherent addition, the interference power wn

experienced during the observation interval is the sum ofthe individual signals’ powers wi, i.e., wn =

∑ni=1 wi =∑n

i=1 xi(t)x∗i (t), where x∗

i (.) is the complex conjugate ofphasor xi(.). Considering the current work, where the signalpower is a random variable, the PDF of the joint inter-ference power is therefore the convolution of the PDFs ofall contributing signal powers. If the individual componentsare independent and identically distributed (i.i.d.), then theinterference power is expressed as the n-fold convolution ofthe PDF of the individual signal power.

In the next section, the concepts introduced here are tailoredto the specific scenario of the current work.

III. SYSTEM MODEL

A. Medium Access Mechanism

The IEEE 802.11 DCF standard [1] is the communicationsystem considered in this work. It uses CSMA/CA as itsmedium contention resolution algorithm and, in essence, aterminal ready to transmit first senses the channel for aperiod of at least a Distributed Interframe Space (DIFS). Ifit senses the channel idle, it transmits its packet. Otherwise,it schedules the (re)transmission of the packet to a later timeaccording to some randomly distributed retransmission delay.After the retransmission delay has elapsed, the terminal repeatsthe procedure described above. A binary exponential backoffalgorithm is used to determine the retransmission delay. The

delay is uniformly chosen in the interval [0, CWi − 1],where CWi is the contention window size at the backoffstage i, i = 0, . . . , m. At the first transmission attempt(i = 0), the contention window size is set to its minimumvalue CW0 = CWmin. After each unsuccessful transmission,the backoff stage i is incremented up to the value of m andthe contention window is doubled up to its maximum valueCWm = CWmax = 2mCWmin.

Consider a setting with unsaturated traffic generated by Ncontending stations, non-ideal transmission channel, captureeffect, and that the number of packets generated in the networkfor new messages plus retransmissions follows the Poissondistribution, with mean generation rate of λ packets persecond. For such a scenario, Daneshgaran et al. [14] use a two-dimensional Markov process to model the protocol’s behaviorand produces a channel throughput given by

S=Pt Ps(1 − Pe)E{PL}

(1−Pt)σ+Pt(1−Ps)Tc+PtPs(1−Pe)Ts+PtPsPeTe, (4)

in which: (i) Pt is the probability that, in the considered timeslot, at least one of the contending stations is transmitting; (ii)Ps is the probability that a packet transmission is successful;(iii) Pe is the probability that errors due to the channel mayoccur on a transmitted packet; (iv) E{PL} is the averagepacket payload length; (v) σ is the duration of an emptytime slot; and (vi) Tc, Te and Ts are the average times achannel is sensed busy due to a collision, an error affected dataframe transmission and a successful data frame transmission,respectively. In order to calculate the channel throughput in(4), the analysis leads to the set of equations given below,which should be treated as a nonlinear system and should besolved numerically [14]:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Pt = 1− (1− γ)N (5)

Ps =Nγ(1− γ)N−1 + Pcp(z0)

Pt(6)

Pcol = 1− (1− γ)N−1 − Pcp(z0) (7)

Peq = (1− Pe)(1 − Pcol) = Pe + Pcol − Pe Pcol (8)

γ =2

(CW+1)+CWPeq1−(2Peq)m

1−2Peq+2(1−Peq)

1−qq

(9)

q = 1− e−λE{Sts} (10)

E{Sts} = (1−Pt)σ + Pt(1−Ps)Tc + PtPs(1−Pe)Ts

+PtPsPeTe, (11)

in which: (i) Pcol is the probability that collisions may occuron a transmitted packet; (ii) Peq is the probability of failedtransmission; (iii) q is the probability that there is at least onepacket to be transmitted in the buffer; and (iv) E{Sts} is theexpected time per slot.

In the current work, the Poisson model is used to describethe traffic generation process, i.e., the packet transmissionrequests from the upper layers. Accordingly, the resultinginter-arrival time is exponentially distributed. Although thePoisson model may not well represent bursty packet datatraffic, it offers an approach that is simple, tractable andcurrently widely used [37]–[40].

For the sake of comparison, the analysis of channel through-put for the CSMA/CA is also performed. In this case, a similar


analysis to the one presented above for the IEEE 802.11 DCFis assumed, with a difference that for the former a single-stage backoff algorithm is used to determine the retransmissiondelay, i.e., m = 0, with CWmin = CWmax

B. Channel Models

For the calculations presented in this subsection, as well asfor the remaining of this paper, let r represents the receivedsignal envelope, w = r2 the received signal power, and w itsaverage value.

1) Hoyt Fading Channel: The Hoyt fading model assumesthat the received signal is the result of the sum of a largenumber of multipath scattered waves, without the prevalenceof a single component (for instance, the LOS signal) [41].Let x and y be two independent Gaussian processes with zeromean and variances σ2

x and σ2y , respectively. The PDF of the

received signal envelope can be expressed as [41]

fR(r) =r

σxσye− r2

4

(1

σ2x+ 1

σ2y

)I0

[r2

4

(1

σ2y

− 1

σ2x

)](12)

where Iν(.) is the modified Bessel function of the first kindand ν-th order [42, Eq. 9.6.10].

The signal power PDF may be expressed as

fW (w) =

√h

we−hw

w I0

(H

w

w

)(13)

where w = σ2x +σ2

y , h � 14

(1√η+

√η)2

, H � 14

(1η − η

),

and η � σ2x/σ

2y is the power ratio between the in-phase and

quadrature signals. Knowing that η > 0, it can be seen that thePDF in (13) is symmetrical around η = 1 [43]. Therefore, asfar as the signal power distribution is concerned, consideringeither of the ranges η ≤ 1 or η ≥ 1 suffices. Also, it is easyto see that if η is set to unity, thus making σ2

x = σ2y , the Hoyt

distribution simplifies to the Rayleigh one.Considering the way the Hoyt phasor is produced, with its

in-phase and quadrature Gaussian components, it is easy tosee that the coherent addition of n uncorrelated Hoyt phasorsalso produces a Hoyt phasor with aggregate mean value wn =∑n

i=1 wi, in-phase signal power σ2X,n =

∑ni=1 σ

2X,i, and

quadrature signal power σ2Y,n =

∑ni=1 σ

2Y,i. If the n phasors

are i.i.d., the resulting phasor has wn = nwi, σ2X,n = nσ2

X,i,σ2Y,n = nσ2

Y,i, and ηn = ηi.2) Rice Fading Channel: The Rice fading model assumes

that the received signal is the result of a dominant component(such as a direct LOS signal) added to a large number ofmultipath scattered waves. Let x and y be two independentGaussian processes with zero mean and equal variances σ2.The in-phase and quadrature components of the signal enve-lope in a Rice fading channel can be expressed as x+ a andy, respectively, where the constant a represents the envelopeof the dominant signal (also, the mean value for the in-phasecomponent). The PDF of the received signal envelope can beexpressed as [23]

fR(r) =r

σ2e−

r2+a2

2σ2 I0

(arσ2

). (14)

The signal power PDF may be expressed as

fW (w) =κ+ 1

w eκe−(κ+1)w

w I0

(2

√κ(κ+ 1)

w

w

)(15)

where w = a2 + 2σ2 = 2σ2(κ + 1), and κ � a2/(2σ2) isthe power ratio between the dominant and scattered signals.It is easy to see that if κ is set to zero, thus eliminating thedominant component, the Rice distribution simplifies to theRayleigh one.

Similarly to the Hoyt case described above, the coherentaddition of n uncorrelated Rice phasors produces a Ricephasor with aggregate mean value wn =

∑ni=1 wi, dominant

signal power a2n =∑n

i=1 a2i , and scattered signal power

σ2n =

∑ni=1 σ

2i . If the n phasors are i.i.d., the resulting phasor

has wn = nwi, a2n = na2i , σ2n = nσ2

i , and κn = κi.3) Nakagami-m Fading Channel: In a Nakagami fading

channel, PDF of the signal envelope is given by [26]

fR(r) =2r2m−1

Γ(m)

(mr2

)m

e−mr2

r2 (16)

where r2 � E{r2} is the mean square value, m ≥ 0 is afading parameter, and Γ(.) is the gamma function [42, Eq.6.1.1]. For m = 1/2, the Nakagami distribution reduces tothe one-sided Gaussian PDF; for m = 1, it reduces to theRayleigh PDF while m → ∞ corresponds to a non-fadingsituation. The signal power PDF may be expressed as

fW (w) =wm−1

Γ(m)

(mw

)m

e−mww . (17)

For the Nakagami-m channel, treatment of coherent addi-tion of signals is still under investigation by the authors. Themain problem is that the exact analysis leads to very intricateformulation and the idea to use some approximation has notyet produced interesting results.

IV. ANALYTICAL RESULTS

For the remaining of this paper and wherever applicable, thesubscripts s, i and n are used to represent the desired signalvariables, the interference signal’s individual component vari-ables, and the joint interference signal variables, respectively.Also, for compactness, let z0 be defined as z0 � z0/(ws/wn)where the ratio ws/wn is commonly denoted as the averageSIR.

A. Perfect Power Control

1) Hoyt Fading Channel: For the calculations presented inthis subsection, let (13) represent the desired signal powerPDF as well as, with different parameters, the signal powerPDF of an individual component of the interference signal.

If coherent addition of phasors is assumed, then (13) shouldbe used to represent the joint interference signal power PDF.With (1) and (2), changing the integration order, using [42, Eq.9.6.10], integrating over z (see [44, Eq. 3.351.1]), and using[45], the integral solves to

FZ(z0) =1− v1√hshn

∞∑i=0

(2i)!

(i!)2

(Hs

2hs

)2i 2i∑j=0

(1− v1)j

× 2F1

[j + 1

2,j

2+ 1, 1,

(Hn

hnv1

)2] (18)


TABLE IRELATION BETWEEN NUMBER OF TERMS AND ACCURACY IN THE

INFINITE SUMMATION OF (24).

Parameters smallest J for accuracy ofn ηs ηn z0 3-decimal-place 6-decimal-place1 0.01 0.01 0.1 27 59

1.0 117 2745.0 167 423

0.1 1.0 115 2655.0 169 421

0.99 1.0 104 2175.0 179 415

0.1 0.01 1.0 14 315.0 18 44

0.99 1.0 13 265.0 19 44

0.99 0.01 1.0 1 35.0 1 3

0.99 1.0 1 35.0 2 3

10 0.01 0.01 1.0 67 1105.0 186 337

0.99 1.0 57 865.0 181 293

0.99 0.01 1.0 1 35.0 1 3

0.99 1.0 1 35.0 1 3

50 0.01 0.01 5.0 170 2430.99 5.0 163 217

0.99 0.01 5.0 1 30.99 5.0 1 3

where 2F1(.) is the Gauss hypergeometric function [42, Eq.15.1.1], and

v1 =hn

hn + hsz0. (19)

Changing the summation order, using [42, Eq. 6.1.18] and[45], the signal-to-interference CDF may be expressed as

FZ(z0)=1− v1√πhshn

∞∑j=0

(1−v1)j2F1

[j+1

2,j

2+1,1,

(Hn

hnv1

)2]

×(Hs

hs

)2cj Γ (cj+

12

)Γ (cj+1)

2F1

[1, cj+

1

2, cj+1,

(Hs

hs

)2](20)

where cj=�j/2�, �.� is the ceiling function (defined as �x�=k, with x∈R and k being the smallest integer such that x≤k).

Although (20) includes an infinite summation, the evalua-tion of the CDF converges rapidly for cases of interest. In orderto estimate the error if the summation in (20) is truncated,all summands whose absolute value is larger than 10−20 arecalculated, and those smaller than this value are discardedbecause it was observed that they do not affect the desiredaccuracy. The error of the truncated summation may now beestimated and, of course, these results vary depending on thegiven set of parameters used. Let J be defined as the numberof terms in a truncated summation, i.e., 0 ≤ j < J . The entriesfor n = 1 in Table I give the value of J necessary to obtaina three-decimal-place accuracy (error < 0.0005) and a six-decimal-place accuracy (error < 0.0000005) for the infinitesummation of (20).

For the incoherent addition, assume that the interferencesignal is composed of n i.i.d. variables. As a result, the jointinterference signal power PDF is the n-fold convolution of the

individual signal power PDF which, on its turn, is expressedby (13). This calculation gives

fWn(wn)=

√πn

n+12 h

n2n

Γ(n2 )wn

(wn

2Hnwn

)n−12

e−nhnwnwn In−1

2

(nHn

wn

wn

)(21)

where wn = nwi, ηn = ηi, hn = hi and Hn = Hi. It can beseen that the n-fold convolution used to obtain (20) producedthe η-μ distribution for μ = n/2 [43].

With (1) and (2), changing the integration order, using [42,Eq. 9.6.10], integrating over z (see [44, Eq. 3.351.1]), andusing [45], the integral solves to

FZ(z0) = 1− vnnΓ(n)

√hshn

n

∞∑i=0

(2i)!

(i!)2

(Hs

2hs

)2i 2i∑j=0

Γ(j+n)

j!

× (1−vn)j 2F1

[j+n

2,j+n+1

2,n+1

2,

(Hn

hnvn

)2](22)

where

vn =nhn

nhn + hsz0. (23)

Changing the summation order, using [42, Eq. 6.1.18] and[45], the signal-to-interference CDF may be expressed as

FZ(z0)= 1− vnn√πhshn

n

∞∑j=0

(1−vn)j

(j+n)B(j+1,n)

(Hs

hs

)2cj

× Γ(cj+

12

)Γ(cj+1)

2F1

[j+n

2,j+n+1

2,n+1

2,

(Hn

hnvn

)2]

× 2F1

[1, cj+

1

2, cj+1,

(Hs

hs

)2](24)

where B(.) is the beta function [42, Eq. 6.2.2], and cj asalready defined.

Although (24) includes an infinite summation, the evalua-tion of the CDF converges rapidly for cases of interest. Let Jbe the number of terms in a truncated summation (as definedearlier). Table I gives the value of J necessary to obtaina three-decimal-place accuracy (error < 0.0005) and a six-decimal-place accuracy (error < 0.0000005) for the infinitesummation of (24).

2) Rice Fading Channel: For the calculations presented inthis subsection, let (15) represent the desired signal powerPDF as well as, with different parameters, the signal powerPDF of an individual component of the interference signal.

If coherent addition of phasors is assumed, then (15) shouldbe used to represent the joint interference signal power PDF.With (1) and (2), changing the integration order, using [42, Eq.9.6.10], integrating over z (see [44, Eq. 3.351.1]), and using[45], the integral solves to

FZ(z0)=1− u1

eκs+κn

∞∑i=0

κis

i!

i∑j=0

(1−u1)j1F1(j+1, 1, κnu1) (25)

where 1F1(.) is the Kummer confluent hypergeometric func-tion [42, Eq. 13.1.2], and

u1 =(κn + 1)

(κn + 1) + (κs + 1) z0. (26)


TABLE IIRELATION BETWEEN NUMBER OF TERMS AND ACCURACY IN THE

INFINITE SUMMATION OF (31).

Parameters smallest J for accuracy ofn κs κn z0 3-decimal-place 6-decimal-place1 0.01 0.01 0.1 2 2

1.0 2 35.0 2 3

1.0 1.0 2 35.0 2 3

10.0 1.0 2 35.0 2 3

1.0 0.01 1.0 5 85.0 5 8

10.0 1.0 5 85.0 5 9

10.0 0.01 1.0 18 265.0 18 26

10.0 1.0 19 255.0 17 26

10 0.01 0.01 1.0 2 35.0 2 3

10.0 1.0 2 35.0 1 3

10.0 0.01 1.0 19 255.0 12 25

10.0 1.0 18 235.0 0 15

50 0.01 0.01 5.0 1 310.0 5.0 1 3

10.0 0.01 5.0 0 1810.0 5.0 0 0

Changing the summation order, using [44, Eqs. 1.211.1 and8.352.2], and the signal-to-interference CDF may be expressedas

FZ(z0) =1− u1

eκn

∞∑j=0

(1− u1)j [1−Q(j, κs)]

× 1F1(j + 1, 1, κnu1)

(27)

where Q(.) is the regularized incomplete gamma function de-fined as Q(a, b) = Γ(a, b)/Γ(a), and Γ(., .) is the incompletegamma function [42, Eq. 6.5.3].

If both κs and κn are set to zero in (27), the CDFsimplifies to the Rayleigh channel model. If this result isfurther simplified by assuming that ws = wi = wn/n (hencemaking z0 = nz0), then the same expression presented in [13,Eq. 20.b] is produced.

Although (27) includes an infinite summation, the eval-uation of the CDF converges rapidly for cases of interest.Let J be the number of terms in a truncated summation (asdefined earlier). The entries for n = 1 in Table II give thevalue of J necessary to obtain a three-decimal-place accuracy(error < 0.0005) and a six-decimal-place accuracy (error< 0.0000005) for the infinite summation of (27).

For the incoherent addition, assume that the interferencesignal is composed of n i.i.d. variables. As a result, the jointinterference signal power PDF is the n-fold convolution of theindividual signal power PDF which, on its turn, is expressed

by (15). This calculation gives

fWn(wn) =n

enκn

(κn+1

wn

)n+12(wn

κn

)n−12

e−n(κn+1)wnwn

× In−1

(2n

√κn(κn + 1)

wn

wn

) (28)

where wn = nwi, and κn = κi. It can be seen that the n-foldconvolution used to obtain (28) produced the κ-μ distributionfor integer values of μ = n [43].

With (1) and (2), changing the integration order, using [42,Eq. 9.6.10], integrating over z (see [44, Eq. 3.351.1]), andusing [45], the integral solves to

FZ(z0) =1− unn

eκs+nκn

∞∑i=0

κis

i!

i∑j=0

Γ(j + n)

Γ(n)j!(1 − un)

j

× 1F1(j + n, n, nκnun)

(29)

where

un =n(κn + 1)

n(κn + 1) + (κs + 1) z0, (30)

Changing the summation order, using [44, Eqs. 1.211.1 and8.352.2], and the signal-to-interference CDF may be expressedas

FZ(z0) =1− unn

enκn

∞∑j=0

(1− un)j [1−Q(j, κs)]

(j + n)B(j + 1, n)

× 1F1(j + n, n, nκnun).

(31)

If both κs and κn are set to zero in (31), the CDF simplifies tothe Rayleigh channel model. If this result is further simplifiedby assuming that ws = wi = wn/n (hence making z0 = nz0),then the same expression presented in [13, Eq. 20.a] is found.

Although (31) includes an infinite summation, the evalua-tion of the CDF converges rapidly for cases of interest. Let Jbe the number of terms in a truncated summation (as definedearlier). Table II gives the value of J necessary to obtaina three-decimal-place accuracy (error < 0.0005) and a six-decimal-place accuracy (error < 0.0000005) for the infinitesummation of (31).

3) Nakagami-m Fading Channel: For the calculations pre-sented in this subsection, let (17) represent the desired signalpower PDF as well as, with different parameters, the signalpower PDF of an individual component of the interferencesignal.

For the incoherent addition, assume that the interferencesignal is composed of n i.i.d. variables. As a result, the jointinterference signal power PDF is the n-fold convolution of theindividual signal power PDF which, on its turn, is expressedby (17). This calculation yields

fWn(wn) =wmn−1

n

Γ(mn)

(mn

wn

)mn

exp

(−mnwn

wn

)(32)

where mi and mn = nmi are the individual and the jointfading parameters, respectively, and wn = nwi is the jointmean power. It can be seen from (17) and (32) that bothsignal power and interference power are described by the samedistribution, except that they have distinct parameters.


Using the appropriate expressions in (1), and after somemanipulation, the signal-to-interference PDF is found as

fZ(z) =1

B(ms,mn)

wn

ws

ms

mnbms−1 (1− b)mn+1 (33)

where

b =wnmsz

wnmsz + wsmn. (34)

The corresponding CDF may be expressed as [46], [47]

FZ(z0) = Ibn(ms,mn) = 1− I1−bn(mn,ms) (35)

where Ix(.) is the regularized incomplete beta function [42,Eq. 6.6.2], and

bn =msz0

msz0 +mn. (36)

B. Spatial Coverage

The analysis presented above for the incoherent additionof interferers assumes that the components of the interferencesignal have identical mean power wi , i = 1, . . . , n. Thisrestriction limits the results to systems where perfect powercontrol is employed or to terminals placed at a fixed distancefrom the receiving antenna (i.e., on a circular ring) and in anenvironment without any shadowing effects. Let us now extendthe model presented above to include the case of packetsarriving with different mean powers, e.g., from terminalswith a given spatial distribution across the cell radius andat different transmission distances to the receiver’s antenna.Therefore, the statistical behavior of the packet mean powerneeds to be specified and taken into account.

The mean power of a packet received from a terminal at adistance d is of the general form [23]

w = Λ d−α (37)

where α gives the channel attenuation with the distance, andΛ is a value that depends on, e.g., the transmit power and theheight and gain of the antennas. Typical values of the exponentα are α = 2 in free space and α = 4 in urban land mobilecellular systems. Using a similar approach to that presentedin [13], let ρ � dΛ−1/α be defined and used to rewrite (37)as

w = ρ−α. (38)

Let the offered traffic density function G(ρ) be defined asthe number of packets offered per transmission period per unitof area at distance ρ. The total offered traffic can be calculatedby Gt = 2π

∫∞0 ρG(ρ) dρ.

The spatial CDF of the offered traffic as a function of thedistance ρ can be expressed as

FG(ρ) = Prob{packet generated within distance ρ}=

2π

Gt

∫ ρ

0

uG(u) du(39)

and the corresponding PDF is

fG(ρ) =2π

GtρG(ρ). (40)

The PDF of the received packet mean power fW (w) =

fG(ρ)∣∣∣ dρdw ∣∣∣ is calculated using (38) and (40) and can be

expressed as

fW (w) =2π

αGt w1+2/α

G(w−1/α). (41)

The capture probability for spatial coverage is now con-sidered. Given an arbitrary spatial traffic density G(ρ), (41)can be used to calculate the PDF of the test packet meanpower fW s

(ws). The PDF of the mean interference power ofn packets fWn

(wn) is calculated by convolving (41) n times.With these results and assuming that the signal power and theinterference power are statistically independent, the CDF ofthe signal-to-interference ratio can be calculated as

FZ(z0)=

∫ z0

0

dz

∫ ∞

0

dwn

∫ ∞

0

fZ(z)fWn(wn)fW s

(ws) dws (42)

where fZ(.) is given by (1). With (3) and (42), the captureprobability is then calculated. As an example, let us use thequasi-constant traffic density given in [13], expressed as

G(ρ) =Gt

π

√ξ exp

(−π

4ξρ4

), ξ > 0, ρ ≥ 0. (43)

It is possible to see that the traffic density is roughly constantinside the cell of radius ρ = 1, falling rapidly beyond the cellboundary. If we select α = 4, it can be seen that

fW s(ws) =

√ξ

2w3/2s

exp

(− πξ

4ws

)(44)

and

fWn(wn) =

n√ξ

2w3/2n

exp

(−n2πξ

4wn

). (45)

Using these results in (42), and considering the Nakagami-m channel, where fZ(.) is given by (33), the signal-to-interference CDF is given by

FZ(z0)=sec(msπ)

[cms0 2F1(ms,ms+mn,ms+1, c0)

msB(ms,mn)

− 2√c0

Γ(mn + 1

2

)Γ(ms)Γ(mn)Γ

(−ms +32

)× 3F2

(1

2, 1,mn +

1

2,3

2,−ms +

3

2, c0

)] (46)

where 3F2(.) is a generalized hypergeometric function [44,Eq. 9.14.1], and

c0 = n2z0ms

mn. (47)

Note that the parameter ξ does not influence the results in (46).This is due to the fact that it appears as a multiplicative factorof the mean power (Eqs. 44 and 45), and in (42) the averageis performed over the ratio of these powers. In physical terms,this parameter simply indicates more or less concentration oftraffic within the center of the cell, which impacts equally onthe effect of the desired signal as well as of the interferenceones.


TABLE IIITYPICAL NETWORK PARAMETERS.

MAC header 34 bytesPHY header 24 bytesACK frame 14 bytes + PHY header

packet payload 1020 byteschannel bit rate 1 Mbps

DIFS 50 μs

SIFS 10 μs

ACK timeout 300 μs

σ 20 μs

m 5CWmin 8

10−4

10−3

10−2

10−1

100

0 5 10 15 20 25 30

Pout

1/z0 [dB]

��

��

��1 �2

�3 �4�5

�6

curve 1 2 3 4 5 6

η 1 1 0.1 0.1 0.01 0.01

n 50 1 50 1 50 1

Fig. 1. Outage probability for incoherent Hoyt channel with η = ηs = ηn.The dashed lines correspond to the Rayleigh channel (η = 1).

V. NUMERICAL RESULTS

This section presents numerical results assuming, whereverapplicable and unless otherwise indicated, the networks pa-rameters listed in Table III. These parameters belong to theIEEE 802.11b protocol. However, the mathematical modelsused here hold for any wireless protocol with similar MACfunctionality.

A. Outage Probability

The outage probabilities for the Hoyt, Rice and Nakagami-m channels, considering incoherent addition of interfering sig-nals and perfect power control, are pictured in Figs. 1, 2 and 3,respectively. In all cases the outage probability decreases withhigher values of average SIR, as expected. In addition, usingthe results for the Rayleigh channel as a reference, the figuresallow a comparison between these channels with regards to theoutage probabilities. Considering a line of decreasing fadingintensity, the Rice channel spans from Rayleigh fading (κ = 0)to a no-fading situation (κ → ∞); the Hoyt channel rangesfrom one-sided Gaussian fading (η = 0) to Rayleigh fading(η = 1); and the Nakagami-m channel varies from one-sided Gaussian fading (m = 1/2) to a no-fading situation(m → ∞), with Rayleigh fading (m = 1) in-between. Thefigures show that higher fading intensity values translate intohigher outage probabilities. Also, considering the average SIRvalue constant, the number of interfering signals n has only aminor influence on the outage probability.

10−4

10−3

10−2

10−1

100

0 5 10 15 20 25 30

Pout

1/z0 [dB]

��

�

��

��

�

�1 �2�3

�4�5

�6�7

�8

curve 1 2 3 4 5 6 7 8

κ 0 0 1 2 5 5 10 10

n 50 1 1 1 1 50 1 50

Fig. 2. Outage probability for incoherent Rice channel with κ = κs = κn.The dashed lines correspond to the Rayleigh channel (κ = 0).

10−4

10−3

10−2

10−1

100

0 5 10 15 20 25 30

Pout

1/z0 [dB]

��

�

�

�1 �2�3 �4�5

�6

curve 1 2 3 4 5 6

m 0.5 0.5 1 1 2 2

n 50 1 50 1 1 50

Fig. 3. Outage probability for incoherent Nakagami-m channel with m =ms = mi. The dashed lines correspond to the Rayleigh channel (m = 1).

B. Channel Throughput: Perfect Power Control

With the results obtained in the previous sections, using(3), (4), (31), (24) and (35), it is now possible to calculatethe IEEE 802.11 DCF channel throughput for the Hoyt, Riceand Nakagami-m channels, assuming incoherent addition ofinterfering signals and perfect power control. Let τ be theworst case propagation delay and τ = τ/p its normalizedversion. For the Hoyt channel, the throughput for variousvalues of η = ηs = ηn, z0, τ and Pe is shown in Fig. 4. Forthe Rice channel, Fig. 5 depicts the throughput for variousvalues of κ = κs = κn, z0, τ and Pe. For the Nakagami-mchannel, considering various values of m = ms = mi, z0, τand Pe, the throughput is presented in Fig. 6.

The figures indicate that the behavior of the throughput Sas a function of the packet rate λ consists, basically, of tworegions: a linear growth region where S is a linear function ofλ, and a saturation region where S remains almost constant.The transition between the two regions may be characterizedby a peak in the throughput, in particular if the number ofnodes N is high.

In addition, the throughput graphs clearly indicate theimportant role the normalized worst case propagation delayτ plays to determine the channel throughput. Also, highernormalized capture threshold z0 values mean lower channel


0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

S

λ [pkt/s]

��

��

��

�1 �2 �3 �4 �5�6 �7�8�9 �10 �11�12 �13 �14

curve 1 2 3† 4 5 6 7 8‡ 9 10 11‡ 12 13 14‡N 4 4 4 10 10 4 4 4 10 10 10 20 20 20

η 1 0.1 0.1 1 0.1 0.1 1 0.1 0.1 1 0.1 0.1 1 0.1

z0 [dB] 6 6 6 6 6 20 20 20 20 20 20 20 20 20

Note: τ = 0.01 and Pe = 0 except †: τ = 2, and ‡: Pe = 0.1.

Fig. 4. Throughput S for IEEE 802.11 DCF with 2-way handshake,incoherent Hoyt channel, perfect power control and η = ηs = ηi = ηn.The dashed lines correspond to the Rayleigh channel (η = 1)

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

S

λ [pkt/s]

��

��

��

�1 �2 �3 �4 �5�6 �7

�8�9 �10�11 �12 �13 �14

curve 1 2 3† 4 5 6 7 8‡ 9 10 11‡ 12 13 14‡N 4 10 4 4 10 4 4 4 10 10 10 20 20 20

κ 10 10 10 0 0 0 10 10 0 10 10 0 10 10

z0 [dB] 6 6 6 6 6 20 20 20 20 20 20 20 20 20


Fig. 5. Throughput S for IEEE 802.11 DCF with 2-way handshake,incoherent Rice channel, perfect power control and κ = κs = κi = κn.The dashed lines correspond to the Rayleigh channel (κ = 0).

throughput, which is expected since it indicates a diminishedability of receiver detection of the intended signal among theinterfering signals. Examining same-z0 data, compared to theRayleigh channel results, higher values of κ in Rice channelstend to produce slightly lower throughput. The reason behindit is that higher κ implies a more deterministic scenario,approaching that of the wireline case. On the other hand, lowervalues of η in Hoyt channels tend to produce slightly higherthroughput when compared to Rayleigh channel results. Thereason for this is that lower η implies a more random scenariowhich may increase the chance of capture. For the Nakagami-m channel, depending on the value of m, throughput resultscan be higher or lower than those seen for the Rayleighchannel. In all cases, higher fading intensity values tend tohigher throughput, although the differences observed are rathermarginal. Also, in all cases, higher probability of channel errorPe translates into lower throughput.

It is noteworthy that the Nakagami-m distribution approxi-mates Hoyt and Rice and includes Rayleigh. The Hoyt channelis approximated if the parameter m is such that 1/2 < m < 1,

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

S

λ [pkt/s]

��

��

�1 �2 �3 �4 �5 �6�7 �8 �9 �10�11 �12 �13�14 �15 �16

curve 1 2 3 4† 5 6 7 8 9 10‡ 11 12 13‡ 14 15 16‡N 4 4 4 4 10 10 4 4 4 4 10 10 10 20 20 20

m 5 1 0.5 0.5 1 0.5 0.5 1 5 0.5 0.5 1 0.5 0.5 1 0.5

z0 [dB] 6 6 6 6 6 6 20 20 20 20 20 20 20 20 20 20


Fig. 6. Throughput S for IEEE 802.11 DCF with 2-way handshake,incoherent Nakagami-m channel, perfect power control and m = ms =mi = mn/n. The dashed lines correspond to the Rayleigh channel (m = 1).

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

S

λ [pkt/s]

��

Nakagami-m, m = 5

Rice, κ = 8.47

Rayleigh

Nakagami-m, m = 0.51

Hoyt, η = 0.01

Fig. 7. Throughput S for IEEE 802.11 DCF with 2-way handshake,incoherent Hoyt, Rice, Nakagami-m and Rayleigh channels with perfectpower control, η = ηs = ηi = ηn, κ = κs = κi = κn andm = ms = mi = mn/n, τ = 0.01, N = 4, z0 = 6 dB and Pe = 0.

with [26]

η =1−

√1m − 1

1 +√

1m − 1

. (48)

On the other hand, the Rice channel is approximated if m > 1,with [26]

κ =

√m2 −m

m−√m2 −m

. (49)

Of course, m = 1 produces the Rayleigh channel. Fig. 7illustrates these situations by showing the throughput for theHoyt, Rice and Rayleigh channels and equivalent curves forthe matching Nakagami-m channel. It can be seen that theoriginal and the matching curves are indistinguishable. Theseresults show that the formulation obtained for the spatialcoverage for the Nakagami-m case can be directly appliedto the Hoyt (1/2 < m < 1) and Rice (m > 1) cases, forwhich the exact formulations are mathematically intractable.

C. Channel Throughput: Spatial Coverage

With the results obtained in Section IV-B, using (3), (4)and (46), the throughput can be calculated for the Nakagami-m channel model. Fig. 8 presents the throughput for various


0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

S

λ [pkt/s]

��

��

�1 �2 �3 �4 �5 �6 �7�8 �9 �10 �11�12 �13 �14

curve 1 2 3 4† 5 6 7 8 9 10† 11 12 13 14

N 4 4 4 4 4 4 4 4 4 4 10 10 10 10

m 5 1 0.5 0.5 0.5 1 5 5 0.5 0.5 1 0.5 1 5

z0 [dB] 6 6 6 6 20 20 20 6 6 6 6 20 20 20

Pe 0 0 0 0 0 0 0 0.1 0.1 0.1 0.1 0.1 0.1 0.1

Note: 802.11 except †: CSMA/CA.

Fig. 8. Throughput S for IEEE 802.11 DCF with 2-way handshake, incoher-ent Nakagami-m channel with spatial coverage, m = ms = mi = mn/nand τ = 0.01. The dashed lines correspond to the Rayleigh channel (m = 1).The dotted lines correspond to CSMA/CA.

values of m = ms = mi, z0 and Pe. In addition, itshows how the IEEE 802.11 DCF compares against a simplerversion of CSMA/CA. In this case, CSMA/CA is assumed toimplement a single-stage backoff algorithm, i.e., m = 0, withCWmin = CWmax = 256. Similarly to the perfect powercontrol scenario presented above, it can be seen that lowervalues of z0 or Pe translates into higher throughput figures.However, it seems that the influence of the fading parameter mis somewhat weaker when compared to the results presentedearlier. On the other hand, the results indicate that z0 appearsto play a more significant role in determining the channelperformance. Also, the results, presented for N = 4, show thatthe IEEE 802.11 DCF outperforms the CSMA/CA althoughthe difference seems narrower at the both ends of the curves,i.e., with low and high packet rates. Also, the transitionbetween the linear and the saturation regions for CSMA/CAis not as sharp as the one observed for the IEEE 802.11 DCF.

Fig. 9 presents the throughput for various values of m =ms = mi and average packet payload length E{PL}. Asexpected, the throughput is lower for shorter payload packets,although the relation between packet payload length andthroughput is not a one-to-one. For instance, a reduction of49% in the average packet payload length (from 1020 to 520bytes) causes the saturated throughput to drop only about 8%,and an 87% reduction (from 1020 to 128 bytes) causes thesaturated throughput to drop about 37%.

As already mentioned, it is important to note that sincethe Nakagami-m distribution approximates Hoyt and Rice, theformer may be used to estimate the spatial coverage when thedesired channel model is either one of the latter.

VI. CONCLUSIONS

This paper investigates the throughput performance of IEEE802.11 DCF in a packet radio network and Hoyt, Rice andNakagami-m fading environments. Analytical and numericalresults are presented considering the signal capture model withcoherent and incoherent addition of interfering signals. Theapproach used here includes the signal capture model withuniform attenuation for all terminals (or perfect power control)and unequal average power levels (or spatial coverage). The

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

S

λ [pkt/s]

��

��

�1 �2 �3�4 �5 �6

�7 �8 �9

curve 1 2 3 4† 5† 6† 7‡ 8‡ 9‡m 5 1 0.5 5 1 0.5 5 1 0.5

Note: E{PL} = 1020 bytes except †: E{PL} = 520 bytes,

and ‡: E{PL} = 128 bytes.

Fig. 9. Throughput S for IEEE 802.11 DCF with 2-way handshake, incoher-ent Nakagami-m channel with spatial coverage, m = ms = mi = mn/n,τ = 0.01, N = 10, z0 = 6 dB and Pe = 0. The dashed lines correspond tothe Rayleigh channel (m = 1).

results indicate that higher fading intensity, lower capturethreshold or lower propagation delay contributes to higherchannel throughput. Also, since the fading intensity valuesof the Nakagami-m channel model approximate those foundin the Hoyt and Rice channels, the results presented for theformer channel offer a good indication of what range the latterchannels may exhibit. It is certainly of interest to extend theseresults to more general fading scenarios, such as those of κ-μ,η-μ and α-μ [48].

The IEEE 802.11 DCF algorithm and its correspondingMarkov model as used here, and that has been extractedfrom [14], has already been fully validated in [14] by meansof simulation. Apart from the new channel model statisticsproposed here, which have no effect on the traffic model of[14], the formulations of [14] are kept intact. The new fadingstatistics added to the model of [14] have been fully validatedby means of numerical integration. Therefore, the full modelis validated.

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Elvio J. Leonardo was born in Maringa, Brazil, in1961. He received the B.S. and M.Sc. degrees inElectrical Engineering from the School of Electricaland Computer Engineering of the State Universityof Campinas, Brazil, in 1984 and 1992, respectively.From 1985 to 1992, Mr. Leonardo worked at as anEngineer at the Research and Development Centerof Telebras, the then holding of the Brazilian state-owned telephone operators. In 1992 he moved toAustralia, where he first worked as a ResearchAssistant at the Sydney University, joining later the

Motorola Australia Software Centre as a Software Engineer. In 1998 hemoved to Libertyville, USA, still working for Motorola, where he helpedthe development of cellular phones. In 2002 he moved back to Brazil, wherehe started his academic career. Currently he is an Assistant Professor at theState University of Maringa. His general research interests include embeddedsystems and wireless communications.

Michel Daoud Yacoub was born in Brazil in 1955.He received the B.S.E.E. and the M.Sc. degrees fromthe School of Electrical and Computer Engineeringof the State University of Campinas, UNICAMP,Brazil, in 1978 and 1983, respectively, and the Ph.D.degree from the University of Essex, U.K., in 1988.From 1978 to 1985, he worked as a Research Spe-cialist at the Research and Development Center ofTelebras, Brazil, in the development of the Tropicodigital exchange family. He joined the School ofElectrical and Computer Engineering, UNICAMP,

in 1989, where he is presently a Full Professor. He consults for severaloperating companies and industries in the wireless communications area.He is the author of Foundations of Mobile Radio Engineering (Boca Raton,FL: CRC, 1993), Wireless Technology: Protocols, Standards, and Techniques(Boca Raton, FL: CRC, 2001), and the co-author of Telecommunications:Principles and Trends (So Paulo, Brasil: Erica, 1997, in Portuguese). He holdstwo patents. His general research interests include wireless communications.

exact formulations for the throughput of ieee 802.11 dcf in hoyt, rice, and nakagami-m fading...

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