1

Backoff Design for IEEE 802.11 DCF Networks:Fundamental Tradeoff and Design Criterion

Xinghua Sun, Member, IEEE and Lin Dai, Senior Member, IEEE

Abstract—Binary Exponential Backoff (BEB) is a key compo-nent of the IEEE 802.11 DCF protocol. It has been shown thatBEB can achieve the theoretical limit of throughput as long asthe initial backoff window size is properly selected. It, however,suffers from significant delay degradation when the networkbecomes saturated. It is thus of special interest for us to furtherdesign backoff schemes for IEEE 802.11 DCF networks whichcan achieve comparable throughput as BEB, but provide betterdelay performance.

This paper presents a systematic study on the effect of backoffschemes on throughput and delay performance of saturatedIEEE 802.11 DCF networks. In particular, a backoff schemeis defined as a sequence of backoff window sizes {Wi}. Theanalysis shows that a saturated IEEE 802.11 DCF networkhas a single steady-state operating point as long as {Wi} isa monotonic increasing sequence. The maximum throughput isfound to be independent of {Wi}, yet the growth rate of {Wi}determines a fundamental tradeoff between throughput and delayperformance. For illustration, Polynomial Backoff is proposed,and the effect of polynomial power x on the network performanceis characterized. It is demonstrated that Polynomial Backoff witha larger x is more robust against the fluctuation of the networksize, but in the meanwhile suffers from a larger second momentof access delay. Quadratic Backoff (QB), i.e., Polynomial Backoffwith x=2, stands out to be a favorable option as it strikes agood balance between throughput and delay performance. Thecomparative study between QB and BEB confirms that QB wellpreserves the robust nature of BEB, and achieves much betterqueueing performance than BEB.

Index Terms—IEEE 802.11 DCF networks, Binary Exponen-tial Backoff, Quadratic Backoff, Polynomial Backoff, maximumthroughput, access delay

I. INTRODUCTION

IEEE 802.11 wireless local area networks have gainedsignificant attention in both industry and academia. Fueledby the widespread popularity in commercial use, researchactivities have been intensified over the last few years, anda major focus has been put on the medium access control(MAC) layer with distributed coordination function (DCF).

As a key component of the IEEE 802.11 DCF protocol,Binary Exponential Backoff (BEB) plays a crucial role indetermining the whole network performance. With BEB, the

Manuscript received November 22, 2012; revised August 23, 2013 andNovember 22, 2013. The associate editor coordinating the review of this paperand approving it for publication was G. Bianchi.

This work was fully supported by the Research Grants Council (RGC) ofHong Kong under GRF Grant CityU 112810.

X. Sun is with the Key Laboratory of Wireless Communications, NanjingUniversity of Posts and Telecommunications, Nanjing, China. He was withCity University of Hong Kong (email: sxhgoahead@gmail.com).

L. Dai is with the Department of Electronic Engineering, City University ofHong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong, China (email:lindai@cityu.edu.hk).

backoff window size is doubled after each unsuccessful trans-mission. By doing so, the transmission probability of eachnode can be effectively reduced, thus alleviating the contentionin a distributed way. A widely adopted model of IEEE 802.11DCF networks was proposed by Bianchi in his landmark paper[1], where it was shown that BEB can achieve quite satisfyingthroughput performance if the initial backoff window size Wis carefully selected. A number of studies have been motivatedto further focus on the dynamic tuning of the backoff windowsize according to the estimated number of active nodes [2]–[5]or the transmission failure rate [6].

In the meanwhile, the so-called “capture phenomenon” ofBEB was also observed [7]–[9]. In particular, when the net-work becomes saturated, many nodes are pushed to deep stateswith extremely small retransmission probabilities such that thenode who once succeeds can dominate the channel for a longtime and produce a continuous stream of packets. In that case,the output process is no longer stationary, and nodes sufferfrom severe short-term unfairness [10]–[13]. A lot of efforthas been made to address this issue [14]–[24]. A commonbelief is that unfairness is caused by the fact that the backoffwindow size is reset to the initial value upon successfultransmission, which gives the fresh packets great advantagesto succeed in the contention. Based on this observation, manyschemes were proposed to slow down the decrement of thebackoff window size, including Exponential Decrease [16]–[19], Linear Decrease [20]–[22], Sliding Contention Window[23], and Gentle DCF [24] in which the window size is halvedafter several successful transmissions.

Distinct analytical models have been established in theaforementioned studies, which makes the performance eval-uation of key system parameters and various backoff schemesextremely difficult. In our recent work [25], a unified analyticalframework was proposed to study the stability, throughputand delay performance of homogeneous buffered IEEE 802.11DCF networks. In contrast to the classical Bianchi model [1],the behavior of each Head-of-Line (HOL) packet, includingbackoff, collision and successful transmission, is modeledas a discrete-time Markov renewal process, and two steady-state operating points are characterized by using the limitingprobability of successful transmission of HOL packets giventhat the network is in unsaturated or saturated conditions. Theanalysis shows that the maximum throughput λ̂max of IEEE802.11 DCF networks with Exponential Backoff is a functionof the holding time of HOL packets in successful transmissionand collision states, and BEB can achieve λ̂max as long as theinitial backoff window size W is linearly adjusted accordingto the network size n. Nevertheless, the second moment of

2

access delay may grow exponentially with the cutoff phase Kfor a small W . In a saturated network where nodes are pushedto deep states, BEB leads to a large second moment of accessdelay, from which the observed capture phenomenon and theshort-term unfairness stem.

The performance analysis of BEB stimulates a series ofquestions about the optimal backoff design criterion of IEEE802.11 DCF networks. For instance, it was shown in [25]that the maximum throughput λ̂max is independent of backoffparameters and solely determined by which access mechanism,i.e., the basic access mechanism or the request-to-send/clear-to-send (RTS/CTS) access mechanism, is adopted. Does itsuggest that all the backoff schemes achieve exactly the samemaximum throughput? If so, how to achieve it? Moreover,it seems that the growth rate of the backoff window sizecritically determines the network performance. With BEB, theexponential growth of backoff window size leads to a hugedifference of window sizes between a fresh packet and adeeply backlogged one, causing a large second moment ofaccess delay. Is it possible for us to find a “milder” backoffscheme which achieves good throughput performance as BEB,but with less severe delay jitter?

This paper is devoted to a comprehensive study on backoffdesign of IEEE 802.11 DCF networks. In particular, the unifiedanalytical framework proposed in [25] is further extended toincorporate a general backoff scheme which is defined as asequence of backoff window sizes {Wi}.1 It is shown thata saturated IEEE 802.11 DCF network has a single steady-state operating point, pA, as long as {Wi} is a monotonicincreasing sequence. pA is a function of {Wi}, indicating thatthe performance of a saturated IEEE 802.11 DCF network isclosely dependent on the backoff scheme.

The analysis verifies that the maximum throughput λ̂max

is independent of {Wi} and solely determined by the accessmechanism. To achieve λ̂max, however, the backoff parametersshould be carefully selected. According to whether the limitinggrowth rate of the backoff window size is larger than 1, thebackoff schemes can be categorized into two groups, i.e.,aggressive backoff with limi→∞

Wi+1

Wi> 1 and mild backoff

with limi→∞Wi+1

Wi= 1. The aggressive backoff schemes can

achieve the maximum throughput even as the network size ngoes to infinity. The second moment of access delay, however,may become unbounded if n exceeds a certain threshold. BEB,as a representative aggressive backoff scheme, suffers fromsuch a delay degradation when the network size n is large.

With the mild backoff schemes, the maximum throughputand a bounded second moment of access delay can be bothachieved for any finite network size. Moreover, it is foundthat the growth rate of the backoff window size can befurther steered to strike a balance between throughput anddelay performance. For demonstration, Polynomial Backoff2

is proposed, and the effect of polynomial power x on the

1Here i denotes the number of collisions that the packet has experienced,and Wi can be an arbitrary function of i. With BEB, for instance, Wi =W · 2i, i = 0, . . . ,K, where W is the initial backoff window size and K isthe cutoff phase (which is referred to as the maximum backoff stage in [1]).

2The backoff window size with Polynomial Backoff is Wi = W · (1 +i)x, i = 0, . . . ,K, where the polynomial power x is a non-negative integer.

network performance is characterized. Intuitively, a larger xindicates a faster growth rate of the backoff window size,with which the network is better capable of absorbing themounting contention. It, on the other hand, may lead to amore severe delay jitter due to a larger difference of backoffwindow sizes between a fresh packet and a deeply backloggedone. Quadratic Backoff (QB), i.e., Polynomial Backoff withx=2, stands out to be a favorable option as it provides a goodtradeoff between throughput and delay performance.

The comparative study between QB and BEB further showsthat both schemes achieve the same maximum throughput andare robust against the variation of network size n. Yet thequeueing performance can be significantly improved with QB,which is consistent with the observation in [26] and [27]. Theshort-term fairness performance of QB and BEB is also eval-uated, and compared with two representative slow-decrementbackoff schemes, Exponential Increase Exponential Decrease(EIED) [16] and Exponential Increase Linear Decrease (EILD)[20]. The comparison corroborates that the key to improvingfairness lies in the growth rate of the backoff window size.By choosing a mild growth rate, the proposed QB achievesthe best short-term fairness performance.

The remainder of this paper is organized as follows. SectionII presents a detailed review of backoff algorithms proposedin the literature. Section III presents the analytical model andpreliminary analysis. Both throughput and delay performanceof a saturated IEEE 802.11 DCF network with a general back-off scheme is characterized in Section IV. Polynomial Backoffis proposed and analyzed in Section V, and a comparativestudy of QB and BEB is presented in Section VI. Finally,conclusions are summarized in Section VII.

II. OVERVIEW OF BACKOFF DESIGN

Backoff is a key component of random-access networks.It can be characterized as a sequence of transmission prob-abilities {qt}, where qt denotes the transmission probabilityof the HOL packet in each node’s buffer at time slot t,in a homogeneous random-access network. Early work hasshown that if qt is constant, the throughput of Aloha networkswould dramatically decrease as the network size increases[28], [29]. Intuitively, qt should be adaptively tuned to alleviatethe time-varying channel contention. The ultimate aim ofbackoff design is then to find how to properly set the sequenceof transmission probabilities {qt} to optimize the networkperformance.

A prevailing method in the literature is to adjust the trans-mission probability qt according to the number of collisionsthat the HOL packet has experienced by time slot t, i.e.,qt = Q(i), where Q(i) is an arbitrary monotonic decreasingfunction3 of the number of collisions i. The widely adoptedBEB [30] is a typical example, with the backoff functionQ(i) = 2−i. In addition to BEB, numerous backoff schemeshave been proposed and extensively studied based on differentbackoff functions such as exponential function [9], [27], [31],linear function [27], [32], µ-law function [32], step function

3Intuitively, to alleviate the channel contention, nodes should reduce theirtransmission probabilities as they experience more collisions.

3

[32] and polynomial function [26], [27]. It was shown in[32] that µ-law function and step function are preferableto the exponential one in terms of the delay performancewhen the network size is large. Similar observations weremade in [26], [27] that polynomial function could improvethe delay performance compared to BEB. Due to the lackof a unified analytical framework, nevertheless, it remainslargely unknown how to properly choose the backoff functionto optimize the network performance.

Note that there are also backoff schemes that do not fallinto the above category. For example, it was proposed in [33]that the transmission probability qt should be tuned accordingto the number of backlogged nodes nt in an Aloha network.Similar schemes could be found in CSMA networks [34],[35] and IEEE 802.11 DCF networks4 [2]–[5]. In [16]–[24],the initial backoff window size of a fresh HOL packet isdecremented according to the window sizes of its precedents,in which case the transmission probability of each HOL packetis determined by not only how many collisions it experiences,but when it enters the network.

In this paper, we limit our discussion to backoff schemeswhere the transmission probability qt can be fully charac-terized as a function of the number of collisions i that theHOL packet has experienced by time slot t. As IEEE 802.11DCF networks adopt the window mechanism, we focus onthe window-based counterpart, where the backoff scheme ischaracterized by a sequence of backoff window sizes {Wi}.We will analyze the effect of {Wi} on the throughput anddelay performance of a saturated IEEE 802.11 DCF network,and reveal the design criterion to achieve the optimal networkperformance.

III. PRELIMINARY ANALYSIS

Let us first briefly review the unified analytical frameworkproposed for IEEE 802.11 DCF networks in [25]. Specifically,we consider an n-node homogeneous IEEE 802.11 DCFnetwork with packet transmissions over a noiseless channel.Suppose that each node is equipped with a buffer of infinitesize and the maximum number of retransmission attemptsfor each HOL packet is infinite. Different from the widelyadopted Bianchi model [1] where the backoff process of eachnode was modeled as a two-dimensional Markov chain, in thefollowing subsection, a discrete-time Markov renewal processwill be established to characterize the complete behavior ofeach HOL packet including backoff, collision and successfultransmission.

A. State Characterization of HOL Packets

Let Xj denote the state of a tagged HOL packet at thej-th transition and Vj denotes the epoch at which the j-thtransition occurs. Fig. 1 shows the embedded Markov chainX = {Xj} of the discrete-time Markov renewal process(X,V) = {(Xj , Vj), j = 0, 1, . . . }.

4Note that a window-based backoff mechanism is adopted in IEEE 802.11DCF networks. Specifically, instead of attempting transmission with a certainprobability at each idle time slot, the node would choose a random numberfrom a backoff window, count down when the channel is idle, and transmitthe HOL packet if the counter is zero.

R0 R1 RK…...

1

11-p

pp

T

F0F1

1-p1 1

p

RK-1

FK

p

…...

1-p

FK-1

1

1-p

Fig. 1. Embedded Markov chain of the state transition process of anindividual HOL packet in IEEE 802.11 DCF networks.

The states of {Xj} can be divided into three categories: 1)waiting to request a transmission (State Ri, i = 0, . . . ,K),2) collision (State Fi, i = 0, . . . ,K) and 3) successfultransmission (State T). As Fig. 1 illustrates, a HOL packetmoves from State Ri to State T if the transmission requestis successful. Otherwise, it stays at State Fi until the end ofthe collision and then shifts to State Ri+1. Here i denotesthe number of collisions experienced by the HOL packet andis incremented until it reaches the cutoff phase K (which isreferred to as the maximum backoff stage in [1]). Note thatK can be any nonnegative integer from 0 to ∞.

In IEEE 802.11 DCF networks, each HOL packet canrequest a transmission only if it senses the channel idle. Letp represent the steady-state probability of successful trans-mission of HOL packets given that the channel is idle. Thestead-state probability distribution of the embedded Markovchain shown in Fig. 1 can be obtained as

πRi =

{(1− p)iπT i = 0, ...,K − 1(1−p)K

p πT i = K(1)

andπFi = πRi · (1− p), i = 0, . . . ,K. (2)

The interval between successive transitions, i.e., Vj+1−Vj ,is called the holding time in State Xj , which solely depends onState Xj . In IEEE 802.11 DCF networks, the holding time τTin State T and the holding time τF in State Fi, i = 0, . . . ,K,vary under different access mechanisms. A graphic illustrationof τT and τF in the basic and RTS/CTS access mechanismscan be found in Fig. 5 of [1] (corresponding to Ts and Tc,respectively, in unit of time slots), and the typical values areprovided in Table V of [1].

The mean holding time τRi in State Ri, i = 0, . . . ,K, onthe other hand, is determined by the backoff protocol. In IEEE802.11 DCF networks, when a HOL packet enters State Ri,it randomly selects a value from {0, . . . ,Wi − 1}, where Wi

is the backoff window size, i = 0, . . . ,K, and then countsdown at each idle time slot. It leaves State Ri and makes atransmission request when the channel is idle and the counteris zero. It is shown in [25] that the mean holding time τRi isgiven by

τRi =1

2α(Wi + 1), (3)

i = 0, . . . ,K, where α denotes the steady-state probability ofsensing the channel idle, which is given by

α =1

1 + τF − τF p− (τT − τF )p ln p. (4)

4

G′′D0

(1)=

K−1∑i=0

(1− p)iG′′Yi(1)+

(1− p)K

pG′′

YK(1)+2 (pτT+(1−p) τF ) ·

(K−1∑i=0

(1−p)iG′Yi(1)+

(1−p)K

pG′

YK(1)

)

+ 2K−1∑i=0

(τF+G′

Yi(1))·

K−1∑j=i+1

(1−p)j(pτT+(1− p)τF+G′

Yj(1))+(1−p)K

p

(pτT+(1− p)τF+G′

YK(1))

+ 2(1− p)K+1

p2(τF+G′

YK(1))·(pτT+(1− p)τF+G′

YK(1))+ τT (τT − 1)+

1− p

pτF (τF − 1) . (10)

Finally, the limiting state probabilities of the Markov re-newal process (X, V) are given by

π̃j =πj · τj∑i∈S πi · τi

, (5)

j ∈ S, where S is the state space of X. Specifically, theprobability of being in State T can be obtained as

π̃T = τT /

(τT +

1− p

pτF +

1+τF−τF p−(τT−τF )p ln p

2

·

(K−1∑i=0

(1− p)iWi +(1− p)K

pWK +

1

p

)), (6)

by substituting (1-4) into (5). Note that π̃T is also the servicerate of each node’s queue as each queue has a successful outputif and only if the HOL packet stays at State T.

B. First and Second Moments of Access Delay

The access delay performance can be also characterizedbased on the embedded Markov chain shown in Fig. 1. LetYi denote the holding time of a HOL packet in State Ri, andDi denote the time spent from the beginning of State Ri untilthe service completion, i = 0, . . . ,K. According to Fig. 1, wehave

Di =

{Yi + τT with probability pYi + τF +Di+1 with probability 1− p, (7)

i=0, . . . ,K−1, and

DK =

{YK + τT with probability pYK + τF +DK with probability 1− p, (8)

where τT and τF are holding time in State T and States Fi,i = 0, . . . ,K, respectively.

Note that D0 is the service time of HOL packets (also theaccess delay). Let GD0(z) denote its probability generatingfunction. It is shown in [25] that

G′D0

(1)=τT+1−p

pτF+

K−1∑i=0

(1−p)iG′Yi(1)+

(1−p)K

pG′

YK(1),

(9)and G′′

D0(1) is given in (10), which is shown at the top of this

page, where G′Yi(1) and G′′

Yi(1) are given by

G′Yi(1) =

1

2α(Wi + 1) , (11)

andG′′

Yi(1) =

1

3α2W 2

i +1− α

α2Wi +

2− 3α

3α2, (12)

respectively, i=0, ...,K.Finally, the mean access delay E[D0] (in the unit of time

slots) is given by

E[D0]=G′D0

(1), (13)

and the second moment of access delay E[D20] is

E[D20] = G′′

D0(1) +G′

D0(1), (14)

which can be obtained by substituting (9-12) into (13) and(14), respectively.

The above analysis clearly indicates that the network per-formance critically depends on the steady-state probability ofsuccessful transmission of HOL packets given that the channelis idle, p. It is revealed in [25] that when the network isunsaturated, it operates at the desired stable point p = pLwhich is independent of backoff parameters and solely deter-mined by the aggregate input rate λ̂, the holding time τT inState T and the holding time τF in State Fi, i = 0, . . . ,K.The network throughput λ̂out at the desired stable point pL isalways equal to the aggregate input rate λ̂. In contrast, if all thenodes become saturated with non-empty queues, the networkoperating point will shift to the undesired stable point p = pA,at which the network performance is closely dependent onbackoff parameters such as the cutoff phase K and the backoffwindow size Wi, i = 0, . . . ,K. In the following section,we will characterize the effect of backoff parameters on thethroughput and delay performance of saturated networks.

IV. THROUGHPUT AND DELAY PERFORMANCE OFSATURATED NETWORKS

In this section, we consider a saturated network, in which allthe nodes are busy with non-empty queues, and the networkthroughput λ̂out falls below the aggregate input rate λ̂. In thatcase, all the HOL packets must be in State Ri, i = 0, . . . ,K,if the channel is idle. Moreover, for a given HOL packet, itstransmission request is successful if and only if the other n−1nodes are not requesting any transmission. The steady-stateprobability of successful transmission of HOL packets giventhat the channel is idle, p, can then be written as

p=

{∑Ki=0 π̃Ri(1−ri)∑K

i=0 π̃Ri

}n−1with a large n

≈ exp

{−n

K∑i=0

π̃Riri

},

(15)where ri is the conditional probability of a State-Ri HOLpacket making a transmission request given that the channelis idle. It is shown in [25] that

ri=2

1 +Wi, (16)

5

i = 0, . . . ,K. By substituting (1-3), (5) and (16) into (15), wehave

p=exp

{−n/

(α(τT · p+τF ·(1−p))+

1

2

(1+

K−1∑i=0

p(1−p)i·Wi

+(1−p)K ·WK

))}, (17)

where α is the probability of sensing the channel idle whichis given in (4). Intuitively, in a saturated network, there is asmall probability for each node to sense the channel idle orsuccessfully transmit due to a large number of transmissionrequests. With a small α and p, the first term of the denomi-nator in the right-hand side of (17) is much smaller than thesecond term, and can be ignored.5 (17) can then be written as

p = exp

{− 2n

1+∑K−1

i=0 p(1−p)iWi+(1−p)KWK

}. (18)

Note that (18) is consistent to Eq. (4) in [36].6 It was shownin Theorem 5.1 [36] that Eq. (4) has a unique fixed point ifthe sequence of mean backoff duration {bi} is non-decreasing.The following theorem states the existence and uniqueness ofthe root of the fixed-point equation (18).

Theorem 1. The fixed-point equation (18) has one single non-zero root pA if {Wi} is a monotonic increasing sequence.

Proof: See Appendix A.

Similar to [25], we refer to pA as the undesired stablepoint. It is clear from (18) that the undesired stable pointpA is determined by the sequence of backoff window sizes{Wi}, the network size n and the cutoff phase K. Intuitively,a smaller network size n leads to a better chance of successfultransmission, implying a larger pA. Corollary 1 summarizesthe monotonicity properties of pA with regard to the cutoffphase K and the network size n.

Corollary 1. If {Wi} is a monotonic increasing sequence,then 1) pA is a monotonic increasing function of the cutoffphase K;2) pA is a monotonic decreasing function of the network sizen.

Proof: See Appendix B.

According to Corollary 1, pA decreases as the network sizen increases. As n goes to infinity, it is clear from (18) that

limn→∞

pK<∞A = 0. (19)

The cutoff phase K determines the maximum backoff windowsize nodes can have. A larger cutoff phase K indicates thatnodes have more room to reduce their transmission probabili-ties if they experience collisions, and the network is then better

5Specifically, with a small p, α(τT ·p+τF ·(1−p)) ≈ τF1+τF

< 1, whichis much smaller than the second term of the denominator.

6Note that different notations were used in [36]. Specifically, in Eq. (4)of [36], γ and bi denote the probability of collision and the mean backoffduration for the i-th attempt, respectively, which can be written as γ = 1−p,bi=

1+Wi2

, for i≤K−1, and bi=1+WK

2, for i≥K by using our notations.

capable of absorbing the mounting contention as the networksize n grows. With n→∞, the huge contention brought byinfinite competing nodes cannot be alleviated if K is finite,thus dragging the probability of successful transmission downto zero. With an infinite cutoff phase K = ∞, on the otherhand, nodes can always back off to deeper states no matterhow large n is. Corollary 2 shows that with K = ∞, a positivepK=∞A > 0 as n → ∞ is possible if the limiting growth rate

of backoff window size is larger than 1.

Corollary 2. limn→∞ pK=∞A = 1− limi→∞

Wi

Wi+1.

Proof: See Appendix C.

With a monotonic increasing sequence of backoff windowsizes {Wi}, the limiting growth rate of backoff window sizeis

limi→∞

Wi+1

Wi≥ 1. (20)

Accordingly, we can divide the backoff schemes into twocategories:

1) Aggressive Backoff : limi→∞Wi+1

Wi> 1;

2) Mild Backoff : limi→∞Wi+1

Wi= 1.

It is clear from Corollary 2 that for the mild backoffschemes, limn→∞ pK=∞

A = 0. In this case, the growth rateof the backoff window size is not fast enough to catch upwith the ever-increasing contention as the network size ngoes to infinity, and thus the network will eventually collapsewith no packets getting through. For the aggressive backoffschemes, in contrast, a positive pK=∞

A > 0 can be achievedeven with an infinite number of nodes n. For instance, withExponential Backoff [31], i.e., Wi = W · q−i for someq ∈ (0, 1), the limiting growth rate of the backoff windowsize is limi→∞

Wi+1

Wi= 1/q > 1, and we have

limn→∞

pEB,K=∞A = 1− q > 0, (21)

according to Corollary 2. In this case, a non-zero networkthroughput can be achieved even as the network size ngrows without bound. In the following subsections, we willspecifically discuss the throughput and delay performance ofthe aggressive and mild backoff schemes.

A. Saturation Throughput

In a saturated network with non-empty queues, the through-put is usually referred to as saturation throughput λ̂s [1], whichis equal to the aggregate service rate nπ̃T . By combining (4),(6) and (17), the saturation throughput λ̂s can be written as

λ̂s =−τT pA ln pA

1+τF−τF pA−(τT−τF )pA ln pA. (22)

The following theorem presents the maximum throughputλ̂max=maxpA

λ̂s and the corresponding steady-state point p∗A.

Theorem 2. The maximum throughput λ̂max is given by

λ̂max =−W0

(− 1

e(1+1/τF )

)τF /τT − (1− τF /τT )W0

(− 1

e(1+1/τF )

) , (23)

6

E[D2,K=∞0,p=pA

] =3 + pA − 3αpA

6α2p2A+

(2

α+ (2− pA)τF

)(1− pA)τF

p2A+

(τT + 2

1− pApA

τF +1

αpA

)τT

+

(1− α

2α2+

1

2α2pA+

pAτT + (1− pA)τFαpA

) ∞∑i=0

(1− pA)iWi +

1

α

(τF +

1

2α

) ∞∑i=0

∞∑j=i+1

(1− pA)jWj

+

1

3α2

∞∑i=0

(1− pA)iW 2

i +1

2α2

∞∑i=0

Wi

∞∑j=i+1

(1− pA)jWj

. (28)

where W0 is the principal branch of the Lambert W function[37]. λ̂max is achieved when

pA = p∗A = −(1 + 1/τF )W0

(− 1

e(1+1/τF )

). (24)

Proof: See Appendix D.

We can see from Theorem 2 that the maximum throughputλ̂max is equal to that of IEEE 802.11 DCF networks withExponential Backoff which is given in Eq. (36) in [25]. Itis solely determined by τT and τF , i.e., the holding timeof HOL packets in the successful transmission and collisionstates, which indicates that the maximum throughput is thesame for all the backoff schemes. To achieve λ̂max, however,the backoff parameters should be carefully tuned such thatpA = p∗A.

Recall that pA declines as the network size n increasesaccording to Corollary 1. For the mild backoff schemes,pA approaches zero as n goes to infinity, implying a zerothroughput. With the aggressive backoff schemes, in contrast,the maximum throughput λ̂max is achievable even with aninfinite number of nodes. For instance, with ExponentialBackoff, we can choose the retransmission factor q = 1− p∗A,such that limn→∞ pEB,K=∞

A = 1 − q = p∗A. The maximumthroughput λ̂max can then be achieved as the network sizen → ∞ according to Theorem 2. Corollary 3 presents thenecessary and sufficient condition to achieve λ̂max as n → ∞.

Corollary 3. λ̂max is achievable as n → ∞ if and only ifK = ∞ and limi→∞

Wi+1

Wi> 1.

Proof: See Appendix E.

B. Access Delay

The expressions of first and second moments of access delayhave been shown in Section III. At the undesired stable pointpA, the mean access delay E[D0,p=pA ] can be obtained as

E[D0,p=pA ]=τT+1−pApA

τF+1+τF−τF pA−(τT−τF )pA ln pA

2

·

(K−1∑i=0

(1−pA)iWi+

(1−pA)K

pAWK+

1

pA

), (25)

by combining (4), (9), (11) and (13). We can see from (25)and (6) that the mean access delay E[D0,p=pA

] is inverselyproportional to the saturation throughput λ̂s, i.e.,

E[D0,p=pA]=nτT /λ̂s. (26)

Theorem 2 shows that λ̂s is maximized when pA=p∗A; there-fore, E[D0,p=pA

] is minimized at

minpA

E[D0,p=pA ]=n(τT−τF

(1+1/W0

(− 1

e(1+1/τF )

))),

(27)

when pA = p∗A. It is clear from (27) that all the backoffschemes can achieve the same minimum mean access delay,which is solely determined by the network size n and theholding time in the successful transmission and collisionstates, τT and τF .

The second moment of access delay at the undesired stablepoint pA can be obtained by combining (9-12) and (14). Withan infinite cutoff phase K = ∞, an explicit expression of thesecond moment of access delay E[D2,K=∞

0,p=pA] can be obtained

as (28), which is shown at the top of the page. Theorem 3presents the convergence property of E[D2,K=∞

0,p=pA].

Theorem 3. If limi→∞Wi+1

Wi= 1, E[D2,K=∞

0,p=pA] < ∞ for any

finite n < ∞; otherwise, if limi→∞Wi+1

Wi> 1, there exists

n′< ∞ such that E[D2,K=∞

0,p=pA] = ∞ when n > n

′.

Proof: See Appendix F.

Theorem 3 indicates that for the mild backoff schemes,the second moment of access delay is always finite for anynetwork size n < ∞. For the aggressive backoff schemes,in contrast, the second moment of access delay may becomeunbounded if the network size n exceeds a certain thresholdn

′. With Exponential Backoff, for instance, it is shown in

[25] that its second moment of access delay is infinite ifn > n

′= −(1+q) ln(1−q2)

2 ·W .In saturated networks, HOL packets are normally pushed to

deep states due to intensive contention. With the aggressivebackoff schemes, the backoff window size Wi is constantlyenlarged as the number of collisions i grows. As a result, nodescan always effectively reduce their transmission probabilitiesby backing off to deeper states to alleviate the contention, suchthat a non-zero throughput is achievable even as the networksize n grows without bound. The cost is, nevertheless, a vastdifference of window sizes between fresh packets and deeplybacklogged ones. In that case, the node who once succeeds hasa much higher transmission probability than those with deeplybacklogged HOL packets. It can then capture the channel andproduce a continuous stream of packets while other nodes haveto wait for a long time. The access delay performance acrossthe network becomes extremely unbalanced, indicating a hugesecond moment of access delay. This irregular behavior haslong been observed in networks with Exponential Backoff,

7

and is referred to as the “capture phenomenon” [7]–[9]. HereTheorem 3 suggests that all the aggressive backoff schemessuffer from large delay jitter as well as severe short-termunfairness once the network grows to a certain size.

C. Tradeoff between Throughput and Delay

So far we have shown that the throughput and delayperformance of saturated IEEE 802.11 DCF networks criticallydepends on the sequence of backoff window sizes {Wi}. Ac-cording to whether the limiting growth rate of backoff windowsize exceeds one, the backoff schemes can be categorized intotwo groups: mild backoff and aggressive backoff. With theaggressive backoff schemes, the maximum throughput can beachieved even as the network size n goes to infinity. Thedelay performance, nevertheless, may significantly deterioratedue to an unbounded second moment of access delay whenthe network size becomes large. In contrast, the mild backoffschemes can reach a better balance: the maximum throughputand a bounded second moment of access delay can be bothachieved as long as the network size is finite.

For the mild backoff schemes, the growth rate of backoffwindow size can be further tuned to trade off between thethroughput and delay performance. For demonstration, letWi = W ·ω(i), where W is the initial backoff window size andω(i) is an arbitrary monotonic increasing function of i withω(0) = 1 and limi→∞

ω(i+1)ω(i) = 1. According to Theorem 2

and (26), the optimal initial backoff window size to achievethe maximum throughput λ̂max and the minimum mean accessdelay can be obtained from (18) as

Wm=− 2n

ln p∗A− 1∑K−1

i=0 p∗A(1−p∗A)i · ω(i)+(1−p∗A)

K · ω(K). (29)

(29) shows that the optimal initial backoff window size Wm

linearly increases with the network size n, and the slopeis determined by ω(i). With a higher growth rate of ω(i),Wm becomes less sensitive against n, implying that a smallerthroughput degradation is caused if the initial backoff windowsize is not updated with n in time.

In particular, suppose that the initial backoff window sizeW is set to be the optimal value Wn=n0

m when the networksize n is equal to n0. As n increases from n0, if the initialbackoff window size is not enlarged accordingly, the networkstead-state point will deviate from p∗A, causing a degradationof network throughput. With K=∞, the derivative of thesaturation throughput λ̂s with regard to the network size ncan be obtained from (18) and (22) as∣∣∣∣∣dλ̂s

dn

∣∣∣∣∣=∣∣∣∣∣ dλ̂s

dpA·dpAdn

∣∣∣∣∣= |−τT (1+τF )(1+ ln pA)+τT τF pA|(1+τF−τF pA−(τT−τF )pA ln pA)

2 ·

1

( n0

− ln p∗A−1

2)·∑∞

i=0(1−pA)i−1ω(i)((1+ln pA)(1−pA)−ipA ln pA)∑∞i=0 p∗

A(1−p∗A)iω(i) + 1

2pA

.

(30)

We can clearly see from (30) that the derivative decreases asthe growth rate of ω(i) increases, indicating that the networkthroughput becomes more robust against the variation of thenetwork size n.

On the other hand, the second moment of access delay withW=Wm and K=∞ can be explicitly written as

E[D2,K=∞0,p=p∗

A] = C +

4n2τ2T3λ̂2

max

·∑∞

i=0(1−p∗A)iω2(i)

(∑∞

i=0(1−p∗A)iω(i))

2 +2n2τ2Tλ̂2max

·∑∞

i=0

∑∞j=i+1(1−p∗A)

jω(i)ω(j)

(∑∞

i=0(1−p∗A)iω(i))

2 +2nτT

λ̂max

(τF−

τT p∗A ln p∗A

2λ̂max

)·∑∞

i=1 i(1−p∗A)iω(i)

1+∑∞

i=1(1−p∗A)iω(i)

, (31)

where C is a function of τT and τF , which is independentof ω(i). In contrast to (30), (31) indicates that E[D2,K=∞

0,p=p∗A]

sharply increases with the growth rate of ω(i). Intuitively,a higher growth rate of backoff window size leads to alarger difference of window sizes between fresh packets andbacklogged ones. The network is, on the other hand, also bettercapable of alleviating the contention and becomes more robust.In the next section, we will take the example of PolynomialBackoff to demonstrate how to steer the growth rate of backoffwindow size to strike a balance between the throughput anddelay performance.

V. POLYNOMIAL BACKOFF

In this section, we focus on Polynomial Backoff, where thebackoff window size Wi of a State-Ri HOL packet can bewritten as

Wi = W · (1 + i)x, (32)

i = 0, . . . ,K. The polynomial power x is a non-negativeinteger, which determines the growth rate of the backoffwindow size. It is clear from (32) that Polynomial Backoffbelongs to the group of mild backoff as limi→∞

Wi+1

Wi= 1.

In the following subsections, we will apply the analysis inSection IV to Polynomial Backoff, and illustrate the effect ofpolynomial power x on the network performance.

A. Undesired Stable Point pAIt has been shown in Section IV that a saturated IEEE

802.11 DCF network operates at the undesired stable pointpA. With Polynomial Backoff, the undesired stable point pAcan be obtained by combining (18) and (32). For instance,when x = 0 and K = ∞, it can be explicitly written as

pK=∞,x=0A = exp

{− 2n

1 +W

}. (33)

With x = 1 and K = ∞, we have

pK=∞,x=1A ≈ W

2nW0

(2n

W

). (34)

Both pK=∞,x=0A and pK=∞,x=1

A decline as the network sizen increases.

Fig. 2 illustrates the undesired stable point pA of a 50-node IEEE 802.11 DCF network with Polynomial Backoff.It can be clearly observed from Fig. 2 that pA increases asthe network size n decreases or the cutoff phase K increases,which corroborates Corollary 1. Moreover, pA is improvedwith a larger polynomial power x owing to a faster growthrate of backoff window size.

8

10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n

K Ap 0 8x

(a)

0 8x

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

K

Ap

(b)

Fig. 2. Undesired stable point pA of saturated IEEE 802.11 DCF networks with Polynomial Backoff. W = 32. (a) pK=∞A versus network size n. K = ∞;

(b) pA versus cutoff phase K. n = 50.

2 4 8 16 32 64 128 256 512 1024 2048 4096 81920.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

=0.9max

Basic

s

W

denotesm

W

0 8x !

(a)

2 4 8 16 32 64 128 256 512 1024 2048 4096 81920.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

=0.97max

RTS

s

W

denotesm

W

0 8x !

(b)

Fig. 3. Saturation throughput λ̂s versus initial backoff window size W in IEEE 802.11 DCF networks with Polynomial Backoff. n = 50 and K = ∞. (a)Basic access mechanism; (b) RTS/CTS access mechanism.

B. Saturation Throughput

With Polynomial Backoff, the saturation throughput λ̂s canbe obtained by combining (18), (22) and (32). The curvesof λ̂s versus the initial backoff window size W in the basicand RTS/CTS access mechanisms are presented in Fig. 3a andFig. 3b, respectively. Here we use the typical values of τT andτF provided in Table V of [1] to demonstrate the numericalresults. That is, τBasic

T = 180 time slots and τBasicF = 175

time slots for the basic access mechanism, and τRTST = 192

time slots and τRTSF = 9 time slots for the RTS/CTS access

mechanism. We can clearly see from Fig. 3 that the maximumthroughput λ̂max is independent of the polynomial power x,and solely determined by τT and τF . The optimal initialbackoff window size to achieve λ̂max can be obtained bycombining (29) and (32) as

Wm=− 2n

ln p∗A− 1∑K−1

i=0 p∗A(1−p∗A)i·(1+i)x+(1−p∗A)

K ·(1+K)x, (35)

where p∗A is given by (24). We can see from (35) that theoptimal initial backoff window size Wm decreases as thenetwork size n decreases or the polynomial power x increases.

Section IV-C also shows that the network becomes morerobust with a larger growth rate of the backoff window size.As Fig. 4 illustrates, the saturation throughput λ̂s starts todecline from λ̂max as the network size n increases from 10,if the initial backoff window size W is fixed to be W =WK=∞,n=10

m . The throughput degradation is, nevertheless,much less significant with a larger polynomial power x. Withthe RTS/CTS access mechanism, the throughput barely varieswith n when x ≥ 2. We can conclude that the network isincreasingly robust against the variation of the network size nas the polynomial power x grows.

C. Access Delay

With Polynomial Backoff, the mean access delay at the un-desired stable point E[D0,p=pA ] can be obtained by combining

9

2 4 8 16 32 64 128 256 512 1024 2048 4096 8192

W

0 8x !

0,

[]

A

K

pp

ED

!

1

2

3

4

5denotes

mW

x104

(a)

4 8 16 32 64 128 256 512 1024 2048 4096 8192

1

2

3

4

5

2

W

0,

[]

A

K

pp

ED

!

0 8x !

denotesm

W

x104

(b)

Fig. 5. Mean access delay E[DK=∞0,p=pA

] versus initial backoff window size W in saturated IEEE 802.11 DCF networks with Polynomial Backoff. n = 50and K = ∞. (a) Basic access mechanism; (b) RTS/CTS access mechanism.

=0.9max

10 20 30 40 50 60 70 80 90 1000.55

0.6

0.65

0.7

0.75

0.8

0.85

0.95

n

0 8xBasic

=0.97max

RTS

RTS/CTS

Basic

S

Fig. 4. Saturation throughput λ̂s versus network size n in IEEE 802.11 DCFnetworks with Polynomial Backoff. K = ∞ and W = WK=∞,n=10

m .

(18), (25) and (32). As we can see from Fig. 5, the minimummean access delay is independent of the polynomial power x,and is achieved when the initial backoff window size W istuned to be Wm according to (35).

The second moment of access delay at the undesired stablepoint E[D2

0,p=pA] can be obtained by combining (9-14), (18)

and (32). Fig. 6 presents the curves of E[D2,K=∞0,p=pA

] versusthe initial backoff window size W under various values ofpolynomial power x. We can see from Fig. 6 that similar tothe mean access delay, E[D2,K=∞

0,p=pA] is also sensitive to the

initial backoff window size W when x is small. It is minimizedwhen W is carefully tuned, and the minimum second momentof access delay increases with the polynomial power x.

As we have pointed out in Section IV-C, a faster growth rateof backoff window size leads to a larger difference of windowsizes between fresh packets and backlogged ones, and thusincurs a higher second moment of access delay. This can beclearly observed from Fig. 7, where the second moment ofaccess delay E[D2,K=∞

0,p=p∗A] with W = Wm is plotted against

2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384

108

109

1010

1011

1012

2,

0,

[]

A

K pp

ED

W

0 8x

RTS/CTS

Basic

Fig. 6. Second moment of access delay E[D2,K=∞0,p=pA

] versus initial backoffwindow size W in saturated IEEE 802.11 DCF networks with PolynomialBackoff. n = 50 and K = ∞.

0 1 2 3 4 5 6 7 810

8

109

1010

1011

1012

*

2,

0,

[]

A

K pp

ED

x

RTS/CTS

Basic

Fig. 7. Second moment of access delay E[D2,K=∞0,p=p∗

A] versus polynomial

power x in saturated IEEE 802.11 DCF networks with Polynomial Backoff.K = ∞, W = Wm and n = 50.

10

the polynomial power x.So far we have shown that Polynomial Backoff with various

polynomial powers can achieve the same maximum throughputand the same minimum mean access delay when the initialbackoff window size W is carefully tuned. The polynomialpower x determines the growth rate of the backoff windowsize, and also the tradeoff between the throughput and delayperformance. With a larger x, the network becomes morerobust, while the delay performance deteriorates. To balancebetween throughput and delay, Quadratic Backoff (QB), i.e.,Polynomial Backoff with x = 2, stands out as a favorableoption. In the following section, we will further compare theperformance of QB with the default backoff scheme of IEEE802.11 DCF networks, Binary Exponential Backoff (BEB).

VI. PERFORMANCE COMPARISON OF QUADRATICBACKOFF AND BINARY EXPONENTIAL BACKOFF

With BEB, the backoff window size Wi of a State-Ri HOLpacket is given by

Wi = W · 2i, (36)

i = 0, . . . ,K. BEB belongs to the group of aggressive backoff,as limi→∞

Wi+1

Wi=2>1. According to Theorem 3, the second

moment of access delay of BEB diverges if the network sizen is too large. With QB, the backoff window size is given by

Wi = W · (1 + i)2, (37)

i = 0, . . . ,K. As a mild backoff scheme, QB can alwaysachieve a bounded second moment of access delay.

The backoff window sizes with both QB and BEB arepresented in Fig. 8. As we can see from Fig. 8, when i issmall, the backoff window size with QB grows faster thanthat with BEB. A smaller increasing rate of Wi, however,is observed with QB when i is large. Intuitively, when thenetwork becomes saturated, a fast increase of the backoffwindow size in the first few states, i.e., with a small i, ishelpful for effective alleviation of the channel contention.After nodes are pushed to deep states, a small backoff windowsize would be desirable to reduce the delay jitter. In thefollowing subsections, we will compare the throughput anddelay performance of QB with that of BEB.

A. Undesired Stable Point pAThe undesired stable points of BEB and QB can be obtained

by combining (18) and (36-37), respectively. With an infinitecutoff phase K=∞, the undesired stable point of BEB can beexplicitly written as

pBEB,K=∞A ≈ 2n/W

W0 (2n/W · exp (4n/W )). (38)

Similarly, pQB,K=∞A can be numerically obtained by solving

p = exp

{− 2n

1 +W (2− p)/p2

}. (39)

(38-39) is verified by the simulation results presented inFig. 9. In this section, all the simulations are conducted usingthe ns-2 simulator, and the values of system parameters are

0 1 2 3 4 5 6 70

20

40

60

80

100

120

140

i

/iWW

QB

BEB

Fig. 8. Backoff window sizes of BEB and QB.

in accordance with [1] (which were summarized in Table IIof [1]). As we can see from Fig. 9a, with an infinite cutoffphase K = ∞, pQB,K=∞

A is comparable to pBEB,K=∞A if

the initial backoff window size W is not too small. With afinite K < ∞, since the backoff window sizes with QB in thefirst few states are larger than those with BEB, pQB

A convergesfaster to pQB,K=∞

A than pBEBA does, as shown in Fig. 9b. A

perfect match between the theoretical and simulation resultscan be observed from Fig. 9.

B. Saturation Throughput

The saturation throughput of BEB and QB can be obtainedby combining (18), (22) and (36-37). It can be clearly seenfrom Fig. 10 that both QB and BEB can achieve the samemaximum throughput λ̂max when the initial backoff windowsize W is properly tuned. The optimal initial backoff windowsizes of BEB and QB to achieve the maximum throughputλ̂max can be obtained as

WBEB,K=∞m =

−2n(21+τFτF

W0

(− τF

e(1+τF )

)+1

)1+τFτF

W0

(− τF

e(1+τF )

)ln(− 1+τF

τFW0

(− τF

e(1+τF )

))(40)

and

WQB,K=∞m =

−2n(− 1+τF

τFW0

(− τF

e(1+τF )

))2(2+

1+τFτF

W0

(− τF

e(1+τF )

))ln(− 1+τF

τFW0

(− τF

e(1+τF )

))(41)

by combining (24), (29) and (36-37), respectively.We can see from (40-41) that to achieve the maximum

throughput λ̂max, the initial backoff window size Wm shouldbe carefully tuned according to the network size n. Otherwise,the saturation throughput may decline as n grows. Fig. 11presents the throughput performance of QB and BEB with Wfixed to be WK=∞,n=10

m . A small throughput degradation canbe observed in both cases, indicating that both QB and BEBare robust against the variation of network size.

The above results are obtained when the cutoff phaseK=∞. With a finite cutoff phase K<∞, the saturation

11

2 4 8 16 32 64 128 256 512 1024 2048 40960.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1K A

p

QB

BEB

Analysis

Simulation

Analysis

Simulation

W

(a)

0 2 4 6 8 10 12 14 160

0.1

0.2

0.3

0.4

0.5

0.6

Ap

K

QB

BEB

Analysis

Simulation

Analysis

Simulation

(b)

Fig. 9. Undesired stable point pA in saturated IEEE 802.11 DCF networks with BEB and QB. (a) pK=∞A versus initial backoff window size W . n = 50

and K = ∞; (b) pA versus cutoff phase K. W = 32 and n = 50.

2 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.1

=0.9

=0.97

max

Basicmax

RTS

W

RTS/CTS

Basic

ˆ s

BEB

Analysis

Simulation

QB

Analysis

Simulation

,

QB

m RTSW

,

QB

m BasicW

,

BEB

m BasicW,

BEB

m RTSW

Fig. 10. Saturation throughput λ̂s versus initial backoff window size W inIEEE 802.11 DCF networks with BEB and QB. n = 50 and K = ∞.

throughput may be significantly lower than that with K=∞due to a smaller pA. It has been shown in Fig. 9b thatpQBA converges to pQB,K=∞

A at a faster rate. Fig. 12 furtherillustrates that the saturation throughput of QB also convergesfaster to the limiting value than that of BEB. In particular,with QB, K=4 is good enough to approach the limitingthroughput with K=∞ for both the basic and RTS/CTS accessmechanisms, which is much smaller than that with BEB,i.e., K=16 for the basic access mechanism and K=6 forthe RTS/CTS access mechanism. Fig. 12 validates that withenlarged backoff window sizes in the first few states, QB canbetter alleviate the channel contention, resulting in a largerthroughput. Nodes with BEB, in contrast, have to back offto much deeper states to achieve a comparable throughput. Alarger cutoff phase K is therefore required to approach thelimiting throughput.

10 20 30 40 50 60 70 80 90 1000.7

0.75

0.8

0.85

0.95

=0.97max

RTS

=0.9max

Basic

RTS/CTS

Basic

n

QB

BEB

AnalysisSimulation

Analysis

Simulation

S

Fig. 11. Saturation throughput λ̂s versus network size n in IEEE 802.11DCF networks with BEB and QB. K = ∞ and W = WK=∞,n=10

m .

C. Access Delay

The mean access delay at the undesired stable pointE[D0,p=pA

] with BEB and QB can be obtained by combining(18), (25) and (36-37). Both BEB and QB achieve the sameminimum mean access delay if their initial backoff windowsizes are tuned to be WBEB,K=∞

m and WQB,K=∞m according

to (40) and (41), respectively. Fig. 13 shows that both schemeshave similar mean access delay performance as the networksize n increases.

The second moments of access delay at the undesired stablepoint E[D2

0,p=pA] with BEB and QB can be obtained by

combining (9-14), (18) and (36-37). With K = ∞, they can

12

be explicitly written as

E[D2,BEB,K=∞0,p=pA

]=3+pA−3αpA

6α2p2A+

(2

α+(2−pA)τF

)(1−pA)τF

p2A

+

(τT+2

1−pApA

τF+1

αpA

)τT+

(τT+

1−pApA

τF+2(1−pA)

2pA−1

·(τF+

1

2α

)−1

2+1+pA2αpA

)· W

α (2pA−1)+

(1

3+

1−pA2pA−1

)·

∞∑i=0

(4−4pA)i · W

2

α2, (42)

and

E[D2,QB,K=∞0,p=pA

]=3+pA−3αpA

6α2p2A+

(2

α+(2−pA)τF

)(1−pA)τF

p2A

+

(τT+2

1−pApA

τF+1

αpA

)τT+

(3

2pA(2−pA)τT+(1−pA)

·(8−3pA)τF+α(1+τT )(pA−2)pA+p2A−7pA+8

2α

)· W

αp4A

+ (p4A − 26p3A + 135p2A − 228pA + 120) · W 2

6α2p6A, (43)

respectively, where α is given by (4). As Theorem 3 indicates,the second moment of access delay for mild backoff is alwaysfinite. With aggressive backoff, however, it diverges if thenetwork size n is too large. Here we can clearly see from(42) that E[D2,BEB,K=∞

0,p=pA] becomes infinite if

pBEBA <3/4, (44)

which is equivalent to

n >3

4

(ln

4

3

)W, (45)

according to (38).The second moments of access delay E[D2

0,p=pA] with BEB

and QB with W=WK=∞,n=10m are illustrated in Fig. 13.

WK=∞,n=10m with BEB in the basic access and the RTS/CTS

access modes can be obtained from (40) as 173 and 26,respectively. According to (45), the second moment of accessdelay with BEB E[D2,BEB,K=∞

0,p=pA] is infinite if the network

size n>37 in the basic access mode, and n>5 in the RTS/CTSaccess mode. As Fig. 13 shows, E[D2,BEB,K=∞

0,p=pA] sharply

grows as the network size n increases, and eventually becomesunbounded. In contrast, a finite second moment of access delaycan be always achieved with QB.

For a 50-node IEEE 802.11 network with BEB, (45) in-dicates that the initial backoff window size W should belarger than 232. Otherwise, the second moment of access delayE[D2,BEB

0,p=pA] will sharply grow with the cutoff phase K, and

become infinite as K→∞. This fact can be clearly observedfrom Fig. 14. Also note that a large cutoff phase K is requiredfor BEB to achieve the limiting throughput with K=∞. WithW=32, for instance, Fig. 12 shows that K should be at least16 for BEB to approach the limiting throughput when the basicaccess mechanism is adopted. In contrast, K≥4 is sufficient

=0.9

max

ˆRTS=0.97

max

ˆBasic

0 2 4 6 8 10 12 14 160.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

QB

BEB

AnalysisSimulation

Analysis Simulation

ˆ s

K

RTS/CTS

Basic

Fig. 12. Saturation throughput λ̂s versus cutoff phase K in IEEE 802.11DCF networks with BEB and QB. W = 32 and n = 50.

1 10 20 30 40 50 60 70 80 90 100

103

104

105

106

107

108

109

n

SimulationAnalysis

BEBBasic

AnalysisRTS/CTS

Analysis

QBBasic

AnalysisRTS/CTS

Simulation

Simulation

Simulation

2

0,[ ]

Ap pE D

0 ,[ ]

Ap pE D

Fig. 13. Mean access delay E[D0,p=pA ] (in unit of time slots) and secondmoment of access delay E[D2

0,p=pA] versus network size n in saturated IEEE

802.11 DCF networks with BEB and QB. K = ∞ and W = WK=∞,n=10m .

for QB, and the corresponding second moment of access delayis much smaller than that with BEB.

Recall that a huge second moment of access delay indicatesthat the access delay performance across the network is fairlydisproportionate: some node may produce a continuous streamof packets, while others have to wait for a long time. It isthe inherent reason behind the capture phenomenon and thepoor delay performance of BEB. The above results clearlyshow that QB can dramatically reduce the second momentof access delay, and therefore greatly improve the queueingperformance.

D. Short-term Unfairness

Note that the capture phenomenon also indicates seriousshort-term unfairness: the node who captures the channel hasa much higher throughput than those with deeply backloggedHOL packets in a certain time period.7 To better understand

7 In the long run, nodes would capture the channel in turn and achievesimilar throughput performance.

13

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1610

8

109

1010

1011

K

2 0,

[]

Ap

pED

W=32

AnalysisSimulation

W=128

AnalysisSimulation

QB

BEB

(a)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1610

8

109

1010

1011

K

W=32

AnalysisSimulation

W=128

AnalysisSimulation

QB

BEB

2 0,

[]

Ap

pED

(b)

Fig. 14. Second moment of access delay E[D20,p=pA

] versus cutoff phase K in saturated IEEE 802.11 DCF networks with BEB and QB. n = 50. (a)Basic access mechanism; (b) RTS/CTS access mechanism.

t (second)

Ft

QB

BEB

EIED

EILD

EILD

(d=32)

(d=64)

Fig. 15. Jain’s Fairness Index Ft versus time t in saturated IEEE 802.11 DCFnetworks with BEB, QB, EIED and EILD. W = 32, K = 6 and n = 50.With EIED, the backoff window size is doubled upon collision and halvedupon success. With EILD, the backoff window size is doubled upon collisionand linearly decreased by some value d upon success.

the short-term fairness performance, let us introduce Jain’sFairness Index [38]:

Ft =

(∑ni=0 λ

it

)2n∑n

i=0(λit)

2, (46)

where λit is the throughput of node i measured in the time

interval (0, t). The index characterizes the difference of thethroughput performance of nodes. It is clear from (46) thata higher Ft indicates better short-term fairness. It approachesunity when each node has an equal throughput. Fig. 15 showsthe fairness indexes of BEB and QB. It can be clearly observedthat the fairness index of QB converges much faster to unitythan that of BEB, which validates that QB can effectivelymitigate the capture phenomenon and significantly improvethe short-term fairness performance.

As mentioned in Section I, it is widely believed that the

short-term unfairness of BEB is rooted in the setting of theinitial backoff window size: by setting the window size offresh HOL packets to a small value, they have a much highertransmission probability than those deeply backlogged ones.Various schemes were then proposed to decrement the backoffwindow sizes of fresh HOL packets from their precedingones. Fig. 15 illustrates the short-term fairness performance oftwo representative schemes: Exponential Increase ExponentialDecrease (EIED) [16] and Exponential Increase Linear De-crease (EILD) [20]. On the contrary to the common belief, thefairness indexes of both schemes are found to be significantlylower than that of BEB.

The reason lies in the difference of backoff window sizes ofnodes. With a slow decrement of the window size upon suc-cessful transmission, nodes that ever have deeply backloggedHOL packets would stay with a large backoff window sizefor a long time. They can hardly access the channel due toa small transmission probability, and thus have much worsethroughput performance. As the window size decreases moreslowly, it takes more time for them to return to a small backoffwindow size. The channel is then captured by other nodes forlonger time, leading to more severe short-term unfairness. Aswe can see from Fig. 15, compared to EIED where the windowsize is exponentially decreased upon successful transmission,the fairness performance of EILD is much worse due to alinear decrement. Although the fairness index of EILD can beimproved by choosing a larger decrement d of the windowsize, it is far below the other backoff schemes.

Similar to the second moment of access delay, unfairnessoriginates from a large difference of backoff window sizes ofnodes. By decrementing the backoff window sizes of freshHOL packets from their preceding ones, the window sizedifference of packets from the same node is reduced, but thedifference among nodes is enlarged. That is why the short-term unfairness is worsened rather than improved by usingEILD/EIED. Here we can conclude that the key to improvingfairness lies in the growth rate of backoff window size. Witha slower growth rate, the difference of backoff window size

14

2 4 8 16 32 64 128 256 512 1024 2048 4096 8192163840.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1

W

ˆ s =0.9max

Basic

BEB

Simulation

QB

Analysis

,

QB

m BasicW

,

BEB

m BasicW

QB

BEB

EIED

EILD

(a)

1 5 10 15 20 25 30 35 40 45 50

103

104

105

i

Me

an

Acce

ss D

ela

y o

f N

od

e i

QB

BEB

EIED

EILD

(b)

Fig. 16. Network throughput and mean access delay with BEB, QB, EIED and EILD in saturated IEEE 802.11 DCF networks with the basic accessmechanism. n = 50. K = 6. d = 64 for EILD. (a) Saturation throughput λ̂s versus initial backoff window size W . (b) Mean access delay (in unit of timeslots) of each node with W = Wm. The x-axis i denotes the index of a node.

between nodes with fresh packets and those with deeplybacklogged ones becomes smaller, and thus better fairnessperformance can be achieved by QB.

For the sake of comparison, the throughput and delayperformance of EIED and EILD is also presented in Fig.16. It can be observed from Fig. 16a that EIED and EILDachieve the same maximum network throughput as BEB andQB, though with different initial backoff window sizes, i.e.,Wm = 1024 for EIED and Wm = 64 for EILD. Fig. 16billustrates the mean access delay performance of each nodewhere the initial backoff window size W is set to be theoptimal value Wm according to Fig. 16a. It can be clearlyseen from Fig. 16b that although the minimum mean accessdelay across the network, which is inversely proportional tothe maximum network throughput, is equal for all the backoffschemes, the delay performance with EILD significantly variesfrom node to node. As we have observed from Fig. 15, withEILD, only a small subset of nodes could capture the channeland have superior performance. The rest of them still sufferfrom a long delay as they can rarely access the channel,thus leading to serious unfairness. Each node with EIED,on the other hand, achieves comparable mean access delayperformance as that with QB and BEB. Fig. 17 further showsthe second moment of access delay with EIED. We can seefrom Fig. 17 that although it is lower than that of BEB whenthe initial backoff window size is small, i.e., W = 32, thegap is diminished as W increases. Similar to BEB, the secondmoment of access delay with EIED increases with the cutoffphase K unboundedly, and becomes much higher than thatwith QB when K is large.

VII. CONCLUSION

In this paper, the effect of backoff schemes on saturatedIEEE 802.11 DCF networks is characterized based on a uni-fied analytical framework. According to whether the limitinggrowth rate of the backoff window size is larger than one, thebackoff schemes are categorized into two groups: aggressive

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16108

109

1010

1011

2 0,

[]

Ap

pED

K

W=32

Analysis

Simulation

W=128

QB

EIED

BEB Analysis

Simulation

Simulation

Fig. 17. Second moment of access delay E[D20,p=pA

] with BEB, QB andEIED versus cutoff phase K in saturated IEEE 802.11 DCF networks withthe basic access mechanism. n = 50.

backoff and mild backoff. It is shown that both groups have thesame maximum throughput λ̂max which is solely determinedby the access mechanism, i.e., the basic access or the RTS/CTSaccess. Yet with the aggressive backoff schemes, an infinitesecond moment of access delay may be incurred if the networksize exceeds a certain threshold. The default backoff schemeof IEEE 802.11 DCF networks, BEB, belongs to the groupof aggressive backoff, and therefore suffers from such a delaydegradation when the network size is large.

To further demonstrate how to trade off between throughputand delay by tuning the growth rate of the backoff windowsize for mild backoff schemes, the performance of PolynomialBackoff with various polynomial powers is evaluated, andQB is selected as a competitive candidate for IEEE 802.11DCF networks. The comparative study of QB and BEB showsthat both schemes achieve the same maximum throughput andare robust against the variation of network size. With BEB,

15

however, the second moment of access delay rapidly grows asthe cutoff phase K increases, indicating deteriorating queueingperformance as well as severe unfairness. With QB, in contrast,the second moment of access delay is insensitive to the valueof K. Much better queueing and fairness performance istherefore achieved, which makes QB a favorable option forIEEE 802.11 DCF networks.

APPENDIX APROOF OF THEOREM 1

Proof: Denote

gK(p) =

K−1∑i=0

p(1−p)iWi+(1−p)KWK . (47)

(18) can be written as

p = exp

{− 2n

1 + gK(p)

}. (48)

Let us first prove the monotonicity of gK(p) with regard to p.

Lemma 1. gK(p) is a monotonic non-increasing function ofp ∈ (0, 1) if {Wi} is a monotonic increasing sequence.

Proof: Define {W̃i} as an infinite sequence with W̃i =Wi for 0 ≤ i ≤ K−1 and W̃i = WK for i ≥ K. gK(p) canthen be written as

gK(p) =∞∑i=0

p(1−p)iW̃i = EX [W̃X ], (49)

where X is a geometric random variable with parameter p,and EX [·] denotes expectation with respect to X .

Suppose that 0<p1<p2<1. Let X1 and X2 denote geo-metric random variables with parameters p1 and p2, respec-tively. Clearly we have X1 ≥st X2 [39]. Further note that{W̃i} is a monotonic non-decreasing sequence. As a result,W̃X1 ≥st W̃X2 , and we can then conclude from (49) thatg(p1) ≥ g(p2).

According to Lemma 1, exp{− 2n

1+gK(p)

}− p is a mono-

tonic decreasing function of p ∈ (0, 1). Moreover,

limp→0

exp{− 2n

1+gK(p)

}− p = exp

{− 2n

1+WK

}> 0, (50)

and

limp→1

exp{− 2n

1+gK(p)

}− p = exp

{− 2n

1+W0

}− 1 < 0. (51)

Therefore, (18) has one single non-zero root.

APPENDIX BPROOF OF COROLLARY 1

The proof is divided into two parts. In the first part, we willprove the monotonicity of pA with regard to K; in the secondpart, we will prove the monotonicity of pA with regard to n.

1) If 0 ≤ K1 < K2, then pK1

A < pK2

A .

Proof: As pA is a function of gK(pA) which is shown in(48), let us first prove the monotonicity of gK(p) with regardto K.

Lemma 2. gK(p) is a monotonic increasing function of K ≥ 0if {Wi} is a monotonic increasing sequence.

Proof: Suppose that 0≤K1<K2. Define {W̃Kj

i }, j=1, 2,as two infinite sequences with W̃

Kj

i =Wi for 0 ≤ i ≤ Kj−1

and W̃Kj

i =WKj for i ≥ Kj . We have

W̃K1i = W̃K2

i if i ≤ K1, and W̃K1i < W̃K2

i if i > K1.(52)

Further define X as a geometric random variable with param-eter p. gKj (p), j=1, 2, can then be written as

gKj (p) =∞∑i=0

p(1− p)i · W̃Kj

i = EX

[W̃

Kj

X

]. (53)

By combining (52) and (53), we can easily see that gK1(p) <gK2(p).

According to (48), we have

− ln pK1

A ·(1+gK1

(pK1

A

))=− ln pK2

A ·(1+gK2

(pK2

A

))=2n.

(54)

According to Lemma 2,

− ln pK1

A ·(1 + gK1

(pK1

A

))< − ln pK1

A ·(1 + gK2

(pK1

A

)),

(55)

if 0 ≤ K1 < K2. By combining (54) and (55), we can obtainthat

− ln pK2

A ·(1 + gK2

(pK2

A

))< − ln pK1

A ·(1 + gK2

(pK1

A

)).

(56)

Lemma 1 shows that gK(p) is a monotonic non-increasingfunction of p ∈ (0, 1). As a result, − ln p · (1 + gK2(p)) is amonotonic decreasing function of p ∈ (0, 1), and we can thenconclude from (56) that pK1

A < pK2

A .

2) If 0 < n1 < n2, then pn1

A > pn2

A .

Proof: With 0 < n1 < n2, we have

2n1=− ln pn1

A · (1+gK (pn1

A ))<2n2=− ln pn2

A · (1+gK (pn2

A )) ,(57)

according to (48). As − ln p · (1 + gK(p)) is a monotonicdecreasing function of p ∈ (0, 1), we can then conclude from(57) that pn1

A > pn2

A .

APPENDIX CPROOF OF COROLLARY 2

Proof: It is clear from (18) that∑∞i=0 p

K=∞A

(1− pK=∞

A

)iWi < ∞ for any finite n < ∞.

We then have

limn→∞

pK=∞A ≥ 1− lim

i→∞Wi

Wi+1. (58)

Now suppose that limn→∞ pK=∞A >1− limi→∞

Wi

Wi+1.

As limi→∞Wi

Wi+1≤ 1 according to (20), we

have limn→∞ pK=∞A > 0. On the other hand,

if limn→∞ pK=∞A >1− limi→∞

Wi

Wi+1, the series∑∞

i=0 pK=∞A

(1− pK=∞

A

)iWi would converge as n → ∞.

16

We can then obtain from (18) that limn→∞ pK=∞A = 0,

which results in a contradiction. Therefore, the postulatelimn→∞ pK=∞

A >1− limi→∞Wi

Wi+1is not true, and we can

conclude from (58) that limn→∞ pK=∞A = 1− limi→∞

Wi

Wi+1.

APPENDIX DPROOF OF THEOREM 2

Proof: According to (22), the derivative of λ̂s is given by

dλ̂s

dpA=

−τT (1 + τF )(1 + ln pA) + τT τF pA

(1+τF−τF pA−(τT−τF )pA ln pA)2

=f(pA)

(1+τF−τF pA−(τT−τF )pA ln pA)2 , (59)

where f(pA) = −τT (1 + τF )(1 + ln pA) + τT τF pA. It can beeasily obtained that p∗A = −(1 + 1/τF )W0

(− 1

e(1+1/τF )

)is

the root of f(pA) = 0. Moreover, the derivative of f(pA) isgiven by

df(pA)

dpA=τT τF − τT (1+τF )/pA=τT τF (1−1/pA)−τT /pA<0

(60)

for pA ∈ (0, 1). Therefore, f(pA) > 0 if pA ∈ (0, p∗A),and f(pA) < 0 if pA ∈ (p∗A, 1). We can then concludefrom (59) that λ̂s is monotonically increasing with pA ifpA ∈ (0, p∗A), and decreasing with pA if pA ∈ (p∗A, 1). Itreaches the maximum value with pA = p∗A. (23) can beobtained by substituting (24) into (22).

APPENDIX EPROOF OF COROLLARY 3

Proof: 1) If : if K = ∞ and limi→∞Wi+1

Wi> 1, we

have limn→∞ pK=∞A = 1 − limi→∞

Wi

Wi+1> 0, according to

Corollary 2. We can choose limi→∞Wi+1

Wi= 1

1−p∗A

such that

limn→∞ pK=∞A = p∗A, and λ̂max can be achieved according

to Theorem 2.

2) Only if : if K < ∞, we have limn→∞ pK<∞A = 0 accord-

ing to (19). If limi→∞Wi+1

Wi= 1, we have limn→∞ pK=∞

A =0 according to Corollary 2. In both cases, the saturationthroughput λ̂s approaches 0 as n→∞ according to (22).

APPENDIX FPROOF OF THEOREM 3

Proof: 1) Let us first consider the case whenlimi→∞

Wi+1

Wi= 1.

Lemma 3. If limi→∞Wi+1

Wi= 1,

∑∞i=0(1−pA)

iW 2i < ∞ and∑∞

i=0 Wi

(∑∞j=i+1(1− pA)

jWj

)< ∞ for any finite n < ∞.

Proof: For the series∑∞

i=0(1− pA)iW 2

i , we have

limi→∞

(1− pA)i+1W 2

i+1

(1− pA)iW 2i

= (1−pA) limi→∞

(Wi+1

Wi

)2

= 1−pA.

(61)

For any finite n, pA ≥ exp{−2n} > 0 according to(18). Therefore, limi→∞

(1−pA)i+1W 2i+1

(1−pA)iW 2i

< 1, and we have∑∞i=0(1− pA)

iW 2i < ∞.

Similarly, for the series∑∞

i=0 Wi

(∑∞j=i+1(1− pA)

jWj

),

because

limi→∞

Wi+1

(∑∞j=i+2(1− pA)

jWj

)Wi

(∑∞j=i+1(1− pA)jWj

) = (1− pA)

· limi→∞

∑∞j=i+1(1− pA)

jWj+1∑∞j=i+1(1− pA)jWj

= 1− pA < 1, (62)

we have∑∞

i=0 Wi

(∑∞j=i+1(1− pA)

jWj

)< ∞.

If both∑∞

i=0 Wi

(∑∞j=i+1(1− pA)

jWj

)and∑∞

i=0(1 − pA)iW 2

i converge, then∑∞

i=0(1 − pA)iWi

and∑∞

i=0

(∑∞j=i+1(1− pA)

jWj

)converge. As a result,

the second moment of access delay E[D2,K=∞0,p=pA

] convergesaccording to (28).

2) When limi→∞Wi+1

Wi> 1, we have

limn→∞

pA=1− limi→∞

Wi

Wi+1<1− lim

i→∞

(Wi

Wi+1

)2

<1= limn→0

pA.

(63)Corollary 1 shows that pA is a monotonic decreasing functionof n. Therefore, there must exist n

′< ∞, such that when

n > n′, pA < 1− limi→∞

(Wi

Wi+1

)2. In this case,

limi→∞

(1− pA)

(Wi+1

Wi

)2

> 1. (64)

We can then conclude that the series∑∞

i=0(1 − pA)iW 2

i

diverges, leading to the divergence of E[D2,K=∞0,p=pA

] whenn > n

′.

ACKNOWLEDGMENT

The authors would like to thank an anonymous reviewer forthe insightful comments on the proof of Corollary 2.

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Xinghua Sun (M’13) received the BS degree fromNanjing University of Posts and Telecommunica-tions (NUPT) in 2008 and the PhD degree from CityUniversity of Hong Kong (CityU) in 2013. FromMarch 2010 to July 2010, he was a visiting studentwith INRIA, France. From March 2013 to August2013, he was a postdoctoral fellow at CityU. He isnow an assistant professor in NUPT. His researchinterests are in the area of wireless communicationsand networking.

Lin Dai (S’00-M’03-SM’13) received the B.S. de-gree from Huazhong University of Science andTechnology, Wuhan, China, in 1998, and the M.S.and Ph.D. degrees from Tsinghua University, Bei-jing, China, in 2003, all in electronic engineer-ing. She was a postdoctoral fellow at the HongKong University of Science and Technology andUniversity of Delaware. Since 2007, she has beenwith City University of Hong Kong, where she isan associate professor. She has broad interest incommunications and networking theory, with special

interest in wireless communications. She received the Best Paper Award atIEEE Wireless Communications and Networking Conference (WCNC) 2007and the IEEE Marconi Prize Paper Award in 2009.