1

A Unified Analysis of IEEE 802.11 DCFNetworks: Stability, Throughput and Delay

Lin Dai, Member, IEEE , and Xinghua Sun

Abstract—In this paper, a unified analytical framework is established to study the stability, throughput and delay performance ofhomogeneous buffered IEEE 802.11 networks with Distributed Coordination Function (DCF). Two steady-state operating points arecharacterized using the limiting probability of successful transmission of Head-of-Line (HOL) packets p given that the network is inunsaturated or saturated conditions.The analysis shows that a buffered IEEE 802.11 DCF network operates at the desired stable point p = pL if it is unsaturated. pLdoes not vary with backoff parameters, and a stable throughput can be always achieved at pL. If the network becomes saturated, incontrast, it operates at the undesired stable point p = pA, and a stable throughput can be achieved at pA if and only if the backoffparameters are properly selected. The stable regions of the backoff factor q and the initial backoff window size W are derived, andillustrated in cases of the basic access mechanism and the request-to-send/clear-to-send (RTS/CTS) mechanism. It is shown that thestable regions are significantly enlarged with the RTS/CTS mechanism, indicating that networks in the RTS/CTS mode are much morerobust. Nevertheless, the delay analysis further reveals that lower access delay is incurred in the basic access mode for unsaturatednetworks. If the network becomes saturated, the delay performance deteriorates regardless of which mode is chosen. Both the first andthe second moments of access delay at pA are sensitive to the backoff parameters, and shown to be effectively reduced by enlargingthe initial backoff window size W .

Index Terms—Stability, throughput, delay, IEEE 802.11 DCF networks, Binary Exponential Backoff.

F

1 INTRODUCTION

IEEE 802.11 Wireless Local Area Networks (WLANs)have gained significant attention in both industry

and academia [1]. Fueled by the widespread popularityof commercial WLANs, research activities have beenintensified over the last few years, and a major focus hasbeen put on the Medium Access Control (MAC) layerwith Distributed Coordination Function (DCF).

DCF is based on the Carrier Sense Multiple Access(CSMA) protocol with two access mechanisms includ-ing the basic access mechanism and the request-to-send/clear-to-send (RTS/CTS) mechanism. As a randomaccess protocol, DCF inherits the merits of minimumcoordination and distributed control, which, on the otherhand, also renders difficulty in modeling and perfor-mance evaluation. A widely adopted model of IEEE802.11 DCF networks was proposed by Bianchi in [2],where a two-dimensional Markov chain was establishedto characterize the backoff behavior of each single node.It is well supported by simulation results and shownto be a powerful, yet simple, analytical tool to evaluatethe throughput performance when the network becomessaturated, i.e., each node always has a packet to transmit.

Bianchi’s model has been refined by a series of follow-

Manuscript received May 16, 2011; revised January 9, 2012; accepted May3, 2012. This work was supported by the Research Grants Council (RGC) ofHong Kong under GRF Grant CityU 112810 and City University of HongKong under SRG Grant 7008049.L. Dai and X. Sun are with the Department of Electrical Engineering, CityUniversity of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong,China (email: lindai@cityu.edu.hk; xinghusun2@student.cityu.edu.hk).

up papers [3]–[15] to include more practical assumptionssuch as freezing of backoff counters [3], [5]–[7], finiteretransmission attempts [4] [8] and imperfect channelconditions [9]–[12]. A great deal of effort was also madeto extend the model to unsaturated networks [16]–[29].For instance, it was assumed in [16]–[19] that eachnode has a one-packet buffer and a new packet isgenerated with a certain probability after the previousone is successfully transmitted. A more general bufferednetwork was considered in [20]–[29], where each node isequipped with a buffer of finite [20] [21] or infinite size[22]–[29]. Most of them followed the analysis in [2] andused the conditional collision probability of each nodeas a key variable to characterize the network operatingpoint. No consensus, however, has been reached on thefixed-point equation of the conditional collision prob-ability in unsaturated scenarios. Moreover, an explicitexpression of the service time distribution is usuallytoo complicated to obtain based on these models. Ap-proximate methods are therefore adopted to simplify theanalysis. For example, in [20] and [28], the attempt ratein an unsaturated network is approximated by a scaledversion of the saturated attempt rate. Good accuracy wasdemonstrated via simulations when the traffic is light.

The main focus of the above studies is on the nu-merical calculation of throughput or delay for givennetwork configuration. Yet how to tune system param-eters to optimize the network performance is anotherinteresting problem, which attracts much attention aswell. For instance, it has been long observed that IEEE802.11 DCF networks may suffer from low throughput if

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the backoff parameters are improperly selected. Variousalgorithms were therefore developed to estimate thenumber of active nodes and adaptively adjust the initialbackoff window size [30]–[34]. On the other hand, severedelay jitter as well as short-term unfairness was foundto arise when an IEEE 802.11 DCF network becomessaturated. In that case, nodes are pushed to large phaseswith extremely small transmission probabilities such thatthe node who once succeeds can capture the channelfor a long time and produce a continuous stream ofpackets. To address this issue, a number of modified DCFprotocols have been proposed for IEEE 802.11 networks[35]–[39]. Most of these studies aimed at improvingthe fairness performance or enhancing the throughput.In many cases, however, the improvement is indeedobtained at the cost of delay degradation. It is, therefore,especially desirable to establish a coherent theory ofIEEE 802.11 DCF networks, based on which the effectof key system parameters can be evaluated within thesame framework.

In this paper, a unified analytical framework is es-tablished to study the stability, throughput and delayperformance of IEEE 802.11 DCF networks. Consider ann-node homogeneous buffered IEEE 802.11 DCF networkwhere each node is equipped with a buffer of infinitesize and an arrival rate of λ. In contrast to the classicmodel proposed in [2], the behavior of each Head-of-Line (HOL) packet, including backoff, collision andsuccessful transmission, is modeled as a discrete-timeMarkov renewal process. The corresponding steady-statedistribution is shown to be crucially determined by thelimiting probability of successful transmission of HOLpackets given that the channel is idle, p. Accordingto whether the network is unsaturated or saturated,distinct fixed-point equations of p are derived, and twosteady-state operating points, i.e., the desired stablepoint pL and the undesired stable point pA, are ob-tained as explicit functions of system parameters. Thenetwork throughput and delay performance at the bi-stable points pL and pA are further characterized.

Specifically, it is shown that a buffered IEEE 802.11DCF network operates at the desired stable point pLif it is unsaturated. pL does not vary with backoffparameters, and a stable network throughput λout=nλcan be always achieved at pL. Both the first and secondmoments of access delay at the desired stable point pLare shown to be insensitive to 1) the backoff factor q, 2)the cutoff phase K and 3) the number of nodes n. They,however, grow with the initial backoff window size W ,indicating that a small W is desirable to improve thedelay performance.

If the network becomes saturated, in contrast, it oper-ates at the undesired stable point pA which is a functionof 1) the backoff factor q, 2) the initial backoff windowsize W , 3) the cutoff phase K and 4) the number ofnodes n. It is shown that for saturated IEEE 802.11DCF networks, the network throughput λout may fallbelow the aggregate input rate λ=nλ if the backoff

parameters are not properly selected. The stable regionof backoff factor q is further characterized, within whicha stable throughput λout=λ can be always achieved atthe undesired stable point pA. The delay performanceat pA may also significantly deteriorate and becomesclosely dependent on backoff parameters. In this case,the key to reducing the access delay is to enlarge pA,which can be achieved by increasing the initial backoffwindow size W . With a small W , the second moment ofaccess delay exponentially grows with the cutoff phaseK, and eventually becomes infinite as K→∞.

Our analysis sheds important light on the capability ofIEEE 802.11 DCF networks for quality of service (QoS)provisioning, and provides direct guidance on networkcontrol and optimization. For instance, the maximumthroughput of IEEE 802.11 DCF networks is derived inthis paper, and shown to be solely determined by theholding times of HOL packets in successful transmissionand collision states, τT and τF . The optimal backofffactor and the optimal initial backoff window size toachieve the maximum throughput are obtained as func-tions of τF and the number of nodes n. It is shown thatthanks to a drastic reduction of τF , a higher maximumthroughput is achieved in the RTS/CTS mode, which canbe approached with a wide range of backoff factor q orinitial backoff window size W .

In current IEEE 802.11 DCF networks, Binary Expo-nential Backoff (BEB) is adopted (i.e., the backoff fac-tor q is fixed to be 1/2), and small values of initialbackoff window size and cutoff phase are selected. Theanalysis shows that the default standard setting leadsto suboptimal performance, and may cause significantdegradation when the network size or the traffic levelincreases. The stable region of initial backoff windowsize W with BEB is characterized, and the optimal W toachieve the maximum throughput is obtained as a linearfunction of the number of nodes n. The delay analysisfurther reveals that the fundamental reason behind theobserved short-term unfairness of BEB is a high secondmoment of access delay in both the basic access and theRTS/CTS modes. To improve the delay performance atthe saturated point, a large initial backoff window sizeW should be adopted.

The remainder of this paper is organized as follows.Section 2 establishes the network model and presentsthe preliminary analysis. The stable regions are charac-terized in Section 3 and the delay analysis is presentedin Section 4. A special focus is given to BEB in Section5. Finally, conclusions are summarized in Section 6.

2 MODELING AND PRELIMINARY ANALYSIS

Consider an n-node IEEE 802.11 DCF network withpacket transmissions over a noiseless channel. A detaileddescription of the DCF protocol can be found in [2], andis omitted here. In this paper, we focus on a homoge-neous buffered network, where each node has an arrivalrate of λ and is equipped with a buffer of infinite size. We

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assume that each HOL packet independently follows anidentical transition process which will be characterizedin the following subsection. Note that this assumption isalso widely adopted in previous studies [2]-[29].

2.1 State Characterization of HOL PacketsA discrete-time Markov renewal process (X,V) ={(Xj , Vj), j = 0, 1, . . . } is established in this subsection tomodel the behavior of each HOL packet. Xj denotes thestate of a tagged HOL packet at the j-th transition andVj denotes the epoch at which the j-th transition occurs.Fig. 1 shows the embedded Markov chain X = {Xj}.

R0R1

…...

11-pt

pt

T

F1

1

…...FK

pt

ptpt

RK

F0

11-pt 1

FK-1

1-pt

RK-1

1-pt 1

Fig. 1. Embedded Markov chain {Xj} of the state transi-tion process of an individual HOL packet in IEEE 802.11DCF networks.

The states of {Xj} can be divided into three categories:1) waiting to request (State Ri, i = 0, . . . ,K), 2) collision(State Fi, i = 0, . . . ,K) and 3) successful transmission(State T). As Fig. 1 illustrates, a HOL packet movesfrom State Ri to State T if the request of transmission issuccessful. 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 packetand is incremented until it reaches the cutoff phase K(which is referred to as the maximum backoff stage in [2]).

In IEEE 802.11 DCF networks, a HOL packet can re-quest a transmission only if it senses the channel idle. Letpt represent the probability of successful transmission ofHOL packets at time slot t given that the channel is idleat t−1. It can be easily shown that the Markov chain inFig. 1 is uniformly strongly ergodic if and only if thelimit

limt→∞

pt = p (1)

exists [40]. The steady-state probability distribution ofthe embedded Markov chain can be further obtained as

πRi =

{(1− p)iπT i = 0, ...,K − 1(1−p)K

p πT i = K(2)

andπFi

= πRi· (1− p), i = 0, . . . ,K. (3)

The interval between successive transitions, i.e., Vj+1−Vj , is called the holding time in State Xj , which solelydepends on State Xj , j = 0, 1, . . . . In IEEE 802.11 DCFnetworks, the holding time τT in State T and the holdingtime τF in State Fi, i = 0, . . . ,K, vary under differentaccess mechanisms. A graphic illustration of τT and τF

in the basic access and RTS/CTS modes can be found inFig. 5 of [2] (corresponding to Ts and Tc, respectively, inunit of time slots), and the typical values are providedin Table V of [2].

The mean holding time τRi in State Ri, i = 0, . . . ,K, onthe other hand, is determined by the backoff protocol. InIEEE 802.11 DCF networks, when a HOL packet entersState Ri, it randomly selects a value from {0, . . . ,Wi−1},where Wi is the backoff window size, i = 0, . . . ,K, andthen counts down at each idle time slot. It leaves StateRi and makes a transmission request when the channelis idle and the counter is zero. Let Gi

t denote the stateof the backoff counter of a State-Ri HOL packet at timeslot t, i = 0, . . . ,K. The transition process of {Gi

t} can bedescribed by the Markov chain shown in Fig. 2, whereαt represents the probability of sensing the channel idleat time slot t.

…...t

1

t

iW

1

t

iW1

t

iW

1

t

iW

t t t

t t tB0 1

iWBB1 B2

t

Fig. 2. State transition diagram of a State-Ri HOL packetin IEEE 802.11 DCF networks, i = 0, . . . ,K.

Similarly, the Markov chain in Fig. 2 is uniformlystrongly ergodic if and only if the limit

limt→∞

αt = α (4)

exists [40]. The mean holding time τRi can be thenobtained as

τRi =1

α· 1 +Wi

2, (5)

i = 0, . . . ,K. Appendix A presents the detailed deriva-tion of (5). Appendix B further shows that

α =1

1 + τF − τF p− (τT − τF )p ln p. (6)

By substituting (6) into (5), the mean holding time τRi

can be written as

τRi =1

2(1 +Wi) (1 + τF − τF p− (τT − τF )p ln p) . (7)

Finally, the limiting state probabilities of the Markovrenewal process (X, V) are given by [41]

πj =πj · τj∑i∈S πi · τi

, (8)

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

πT=1/

(1+

τFτT

·1−p

p+

(1

τT+τFτT

·(1−p)−(1−τF

τT

)·p ln p

)

·

(K−1∑i=0

(1−p)i·1+Wi

2+(1−p)K

p·1+WK

2

))(9)

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by substituting (2-3) and (7) into (8). Note that πT is alsothe service rate of each node’s queue as each queue hasa successful output if and only if the HOL packet staysat State T. The offered load of each node’s queue, ρ, isthen given by

ρ = λ/πT , (10)

where λ is the input rate of each node.

2.2 Limiting Probability of Success p

The analysis in Section 2.1 indicates that the steady-stateperformance of IEEE 802.11 DCF networks is cruciallydetermined by p, the limiting probability of successfultransmission of HOL packets given that the channel isidle. In this subsection, the steady-state operating pointsof IEEE 802.11 DCF networks will be characterized basedon the fixed-point equations of p.

2.2.1 Steady-state Point of Unsaturated NetworksLet us first consider an unsaturated network. Each nodemust be in one of the following four states:S1: the queue is empty;S2: the HOL packet is in State Ri = 0, . . . ,K;S3: the HOL packet is in State T;S4: the HOL packet is in State Fi = 0, . . . ,K.

For a given HOL packet, its transmission request issuccessful if and only if the other n−1 nodes are either instate S1, or, in state S2 and not requesting any transmis-sion. The limiting probability of successful transmissionof HOL packets given that the channel is idle, p, is thengiven by

p = (Pr{node is in S1|channel is idle}+ Pr{node is in S2 with no request|channel is idle})n−1.

(11)

If the channel is idle, each node must be in either state S1

or S2. The probabilities that a node is in state Sj , j=1, 2,are given by

Pr{node is in S1} = 1− ρ, (12)

Pr{node is in S2} = ρK∑i=0

πRi , (13)

where the offered load of each node’s queue ρ < 1. Wethen have

Pr{node is in S1|channel is idle}=1−ρ

1−ρ+ρ∑K

i=0 πRi

, (14)

and

Pr{node is in S2 with no request|channel is idle}

=ρ∑K

i=0 πRi(1− ri)

1− ρ+ ρ∑K

i=0 πRi

, (15)

where ri is the conditional probability of a State-Ri HOLpacket making a transmission request given that thechannel is idle, which can be obtained as

ri=2

1 +Wi, (16)

i = 0, . . . ,K. Detailed derivation of (16) can be foundin Appendix A. By substituting (14-15) into (11), thelimiting probability of success p can be written as

p =

{1− ρ+ ρ

∑Ki=0 πRi(1− ri)

1− ρ+ ρ∑K

i=0 πRi

}n−1

. (17)

With a large number of nodes n, we have

p ≈ exp

{−nρ

K∑i=0

πRiri

}= exp

{−λ

K∑i=0

πRi

πTri

}, (18)

according to (10), where λ = nλ is the aggregate inputrate.

Finally, by combining (2-3), (7-8) and (16), (18) can bewritten as

p=exp

{λτF /τT

1−(1−τF /τT )λ

}· exp

{− λ(1+τF )/τT

1−(1−τF /τT )λ·1p

}.

(19)(19) has two non-zero roots:

pL=exp

{W0

(− λ(1+τF )/τT

1−(1−τF /τT )λ· exp

{− λτF /τT

1−(1−τF /τT )λ

})

+λτF /τT

1−(1−τF /τT )λ

}(20)

and

pS=exp

{W−1

(− λ(1+τF )/τT

1−(1−τF /τT )λ· exp

{− λτF /τT

1−(1−τF /τT )λ

})

+λτF /τT

1−(1−τF /τT )λ

}, (21)

if the aggregate input rate λ is no larger than

λmax =−W0

(− 1

e(1+1/τF )

)τF /τT − (1− τF /τT )W0

(− 1

e(1+1/τF )

) . (22)

W0(·) and W−1(·) in (20-22) are two branches of theLambert W function 1 [42], and we have pS ≤ pL.It is interesting to note that only the larger root pLis a steady-state operating point. It is determined bythe aggregate input rate λ and the holding times insuccessful transmission and collision states, τT and τF ,and does not vary with the backoff parameters. A stablethroughput λout = λ can be always achieved at pL asthe offered load ρ of each node’s queue is lower than 1if the network is unsaturated.

2.2.2 Steady-state Point of Saturated NetworksAs the aggregate input rate λ increases, the networkwill eventually become saturated, i.e., all the nodes arebusy with non-empty queues. In this case, given that thechannel is idle, each node must be in state S2. Similar to

1. The defining equation for the Lambert W function W(z) is z =W(z)eW(z) for any complex number z.

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(11), the limiting probability of success p in a saturatednetwork can be written as

p = (Pr{node is in S2 with no request|channel is idle})n−1,(23)

which is given by

p=

{∑Ki=0 πRi(1−ri)∑K

i=0 πRi

}n−1with a large n

≈ exp

{−n

K∑i=0

πRiri

}.

(24)By substituting (2-3), (5), (8) and (16) into (24), we have

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

))}, (25)

where α is the limiting probability of sensing the channelidle which is given in (6).

It is difficult to derive the exact root of (25). Neverthe-less, it will be shown in Fig. 4 that the idle probability αis close to zero when the network becomes saturated. Byignoring the first term of the denominator in the right-hand side of (25), (25) can be written as

p = exp

{− 2n

1+∑K−1

i=0 p(1−p)i·Wi+(1−p)K ·WK

}. (26)

In IEEE 802.11 DCF networks, the backoff window sizeWi is set as

Wi = W · q−i, (27)

i = 0, . . . ,K. W is called the initial (or minimum) backoffwindow size and q is the backoff factor. By combining(26) and (27), we have

p = exp

− 2n

1+W

(qp

q+p−1−(

qpq+p−1−1

)·(

1−pq

)K) .

(28)Recall that in [2], the fixed-point equation of the collisionprobability (i.e., corresponding to 1 − p in this paper)was derived for saturated IEEE 802.11 DCF networkswith Binary Exponential Backoff (q=1/2). It can be easilyshown that (28) is consistent with Eq. (9) in [2] if q isequal to 1/2.

In contrast to (19), (28) has a single non-zero root pAfor any cutoff phase K = 0, . . . ,∞. Moreover, the non-zero root pA is closely dependent on backoff parameterssuch as the cutoff phase K, the backoff factor q and theinitial backoff window size W . With K = ∞, pA can beexplicitly written as

pK=∞A

for large W≈ 2n(1− q)/(Wq)

W0 (2n(1− q)/(Wq) · exp (2n/W/q)).

(29)

2.3 Desired Stable Point versus Undesired StablePointSo far we have demonstrated that for any aggregateinput rate λ ≤ λmax, an IEEE 802.11 DCF networkoperates at the steady-state point pL if the network isunsaturated. As the aggregate input rate λ increases,however, the network may become saturated and shiftto another steady-state point pA. As IEEE 802.11 DCFnetworks possess the same bi-stable property as Alohanetworks [43], we follow the terminology in [43] andrefer to pL and pA as the desired stable point and theundesired stable point, respectively.

2.3.1 Desired Stable Point pL(20) indicates that the desired stable point pL does notvary with backoff parameters, and is crucially deter-mined by the holding times in successful transmissionand collision states, τT and τF . Typical values of τT andτF in the basic access and RTS/CTS modes have beensummarized in Table V of [2]. In particular, with thebasic access mechanism, nodes are unaware of whethertheir transmissions are successful or not until the endof the packet frame, and hence τBasic

T and τBasicF are

approximately equal to each other. With the RTS/CTSmechanism, collisions occur only on the RTS frames thatare much shorter than the payload frames [2]. The hold-ing time in collision states τRTS

F is therefore drasticallyreduced and becomes much smaller than τRTS

T . Fig. 3aillustrates how the desired stable point pL varies withthe aggregate input rate λ in the basic access mode (i.e.,τBasicT =180 time slots and τBasic

F =175 time slots) and theRTS/CTS mode (i.e., τRTS

T =192 time slots and τRTSF =9

time slots). It can be clearly seen from Fig. 3a that pLdeclines as λ increases in both modes, and for givenaggregate input rate λ, a higher pL is always obtainedin the RTS/CTS mode thanks to the reduction of theholding time in collision states.

2.3.2 Undesired Stable Point pAIn contrast, the undesired stable point pA is a function of1) the cutoff phase K, 2) the backoff factor q, 3) the initialbackoff window size W and 4) the number of nodesn. For instance, (29) indicates that pK=∞

A monotonicallydecreases as the backoff factor q or the ratio of thenumber of nodes n and the initial backoff window sizeW increases, which can be clearly observed from Fig.3b. pA could be much lower than pL if the backoffparameters are not properly selected, implying that boththroughput and delay performance may significantlydeteriorate when the network becomes saturated.

The above analysis is verified by the simulation resultspresented in Fig. 4. In this paper, all the simulationsare conducted using the ns-2 simulator, and the valuesof system parameters are in accordance with [2] (whichwere summarized in Table II of [2]). In the simulations,each node is assumed to have Bernoulli arrivals withrate λ, i.e., each node has probability λ to generate a

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

max

ˆBasic

x x

max

ˆRTS

=0.97=0.9

Basic

RTS/CTS

pL

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

5n

W

0.5n

W

0.05n

W

K Ap

!

(b)

Fig. 3. Stable points of IEEE 802.11 DCF networks. (a) Desired stable point pL versus aggregate input rate λ. (b)Undesired stable point pK=∞

A versus backoff factor q.

new packet every τT time slots. The new arrival packetis attached to the end of the waiting queue, and thebuffer size is assumed to be infinite. It can be clearlyseen from Fig. 4 that with a low aggregate input rateλ = 0.2, the network operates at the desired stable pointpL, which does not vary with the backoff factor q. Whilethe aggregate input rate λ increases to 0.8, the networkbecomes saturated at the undesired stable point pA thatdeclines as q increases. It is also shown in Fig. 4 that thelimiting probability of sensing the channel idle α is closeto zero when the network operates at pA. It verifies that(26) can serve as a good approximation of (25).

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

p=pL

p=pA

Probability of Sensing

the channel Idle

Eq. (6)

Simulation

Probability of Success p

Eq. (20)

Simulation

Eq. (29)

ˆ 0.8 !

ˆ 0.2 !

Fig. 4. Desired and undesired stable points of IEEE802.11 DCF networks with the basic access mechanism.n = 50, K = ∞ and W = 16.

Note that the network exhibits distinct performancesat different stable points. A stable throughput λout = λcan be always achieved when it operates at the desiredstable point pL. In contrast, the undesired stable pointpA depends on backoff parameters, indicating that thenetwork throughput λout may fall below the aggregateinput rate λ unless the backoff parameters are care-fully selected. In the next section, we will focus on thethroughput performance when the network operates atthe undesired stable point pA. We are particularly inter-ested in how to properly choose the backoff parametersto stabilize the network at pA, and how to achieve themaximum stable throughput.

3 STABLE REGION AND MAXIMUM STABLETHROUGHPUT

Section 2 shows that an IEEE 802.11 DCF network op-erates at the undesired stable point pA if it is saturated.According to (24), the aggregate service rate at pA canbe obtained as

nπT =−τT pA ln pA

1 + τF − τF pA − (τT − τF )pA ln pA. (30)

To achieve a stable throughput λout = λ, the aggregateservice rate should be no smaller than the aggregateinput rate λ:

nπT ≥ λ. (31)

By combining (30-31) and (19), we can conclude that astable throughput can be achieved at pA if and only if

pS ≤ pA ≤ pL, (32)

where pL and pS are given in (20-21).

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Note that the undesired stable point pA is a function ofbackoff parameters. In this section, we will demonstratehow to stabilize the network at pA by properly choosingthe backoff factor q. Specifically, define Sq

A as the stableregion of backoff factor q, within which a stable through-put λout = λ is achieved at the undesired stable point pA.According to (32), Sq

A can be written as

SqA = {q|pS ≤ pA ≤ pL}. (33)

According to (20-21), pL and pS are monotonic decreas-ing and increasing functions of the aggregate inputrate λ, respectively. As a result, Sq

A will shrink as λincreases, and eventually becomes a single point whenλ reaches the maximum stable throughput λmax Sq

A.

With λ>λmax SqA

, SqA is an empty set, implying that the

network cannot be stabilized. We will elaborate on theabove results in the following subsections.

3.1 Stable Region of Backoff Factor q with CutoffPhase K=∞Let us first assume an infinite cutoff phase K=∞. Theundesired stable point pK=∞

A has been given in (29),and the corresponding stable region can be obtained bycombining (33) and (29) as

Sq,K=∞A =

[1− pL

1 + W2npL ln pL

,1− pS

1 + W2npS ln pS

]. (34)

According to (34), the stable region Sq,K=∞A diminishes

as the aggregate input rate λ increases, and finallyshrinks to a single point

qK=∞m =

(1+(1+1/τF )W0

(− 1

e(1+1/τF )

))/

(1−W

2n(1+1/τF )

·W0

(− 1

e(1+1/τF )

)ln(−(1+1/τF )W0

(− 1

e(1+1/τF )

)))(35)

with which the maximum stable throughput of

λmax Sq,K=∞A

=λmax=−W0

(− 1

e(1+1/τF )

)τF /τT−(1−τF /τT )W0

(− 1

e(1+1/τF )

)(36)

can be achieved. Note that qK=∞m in (35) should not

exceed 1, which requires that

W ≤ 2n

ln(−(1 + 1/τF )W0

(− 1

e(1+1/τF )

)) . (37)

Otherwise, the maximum stable throughput λmax Sq,K=∞A

is lower than λmax.Recall that it is shown in Section 2.2.1 that the desired

stable point pL exists if and only if the aggregate inputrate λ is no larger than λmax. Here we can concludefrom (22) and (36) that λmax is the maximum throughputthat can be achieved by an IEEE 802.11 DCF network

irrespective of which stable point it operates at. It is inde-pendent of the backoff parameters and solely determinedby the holding times in successful transmission andcollision states, τT and τF . The maximum throughputsof IEEE 802.11 DCF networks with the basic accessmechanism (i.e., τBasic

T =180 time slots and τBasicF =175

time slots) and the RTS/CTS mechanism (i.e., τRTST =192

time slots and τRTSF =9 time slots) can be obtained as

λBasicmax =0.9 and λRTS

max =0.97, respectively.For any aggregate input rate λ ≤ λmax Sq,K=∞

A, a

stable throughput λout = λ can be achieved at pK=∞A

if the backoff factor q is selected from the stable regionSq,K=∞A . With q /∈ Sq,K=∞

A , the network throughput fallsbelow the aggregate input rate λ and is determined bythe aggregate service rate, which can be obtained bycombining (29-30) as:

λK=∞s =

τT

(τT−τF )+(1+τF )·W0

(2n(1−q)

Wq ·exp(

2nWq

))−

2nτF (1−q)Wq

4n2(1−q)W 2q2 −

2n(1−q)Wq ·W0

(2n(1−q)

Wq ·exp(

2nWq

)).

(38)In the following subsections, we will demonstrate theabove results in the cases of the basic access mode andthe RTS/CTS mode, respectively.

3.1.1 Basic Access MechanismFig. 5a presents the stable region Sq,K=∞

A of a 50-node IEEE 802.11 DCF network with the basic accessmechanism. It can be observed from Fig. 5a that thestable region Sq,K=∞

A is substantially enlarged as theinitial backoff window size W increases from 8 to 128.Intuitively, with a larger W , HOL packets should havehigher chances to succeed as their requests are spreadin a more even way. A closer look at (29) and Fig. 3bfurther indicates that the probability of success is indeedimproved with an increase of the initial backoff windowsize W .

Nevertheless, an excessively large W may impair thethroughput performance. With W=8, 32 and 128, the sta-ble region Sq,K=∞

A rapidly shrinks as the aggregate inputrate λ increases, and finally becomes a single point qK=∞

m

when λ reaches the maximum λmax Sq,K=∞A

=λBasicmax =0.9.

With W=1024, in contrast, the maximum stable through-put λmax Sq,K=∞

Ais slightly lower than λBasic

max . In fact, ac-cording to (37), λmax Sq,K=∞

Astarts declining from λBasic

max

if the initial backoff window size W exceeds 971.

3.1.2 RTS/CTS MechanismThe stable region Sq,K=∞

A with the RTS/CTS mechanismis also illustrated in Fig. 5a. Compared to the basic accessmode, the stable region is now greatly enlarged andbecomes insensitive to the aggregate input rate λ and theinitial backoff window size W . Moreover, the maximumstable throughput λmax Sq,K=∞

Ais boosted to λRTS

max =0.97,which can be approached with a wide range of backofffactor q even with a small W . Both facts are attributedto the decrease of the holding time in collision states.

8

0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

=0.9

=0.97

maxˆBasic

maxˆRTS

RTS/CTS

Basic

q

,q K

AS !

0.1

W=8

W=32

W=1024

W=128

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

=0.9

Basic

RTS/CTS

q

=0.97

max

ˆBasic

max

ˆRTS

q

AS

K=4

K=8

K=

K=16

(b)

Fig. 5. Stable region of backoff factor q in IEEE 802.11 DCF networks. n=50. (a) Stable region with various values ofinitial backoff window size W . K=∞. (b) Stable region with various values of cutoff phase K. W=32.

3.2 Stable Region of Backoff Factor q with CutoffPhase K<∞

With a finite cutoff phase K < ∞, the undesired stablepoint pA is an implicit function of the backoff factor q.Numerical results of the stable region Sq

A can be obtainedby combining (28) and (33), and are presented in Fig.5b. It can be seen from Fig. 5b that the maximum stablethroughput λmax Sq,K=∞

A=λmax can be always achieved

with K ≥ 4. Moreover, a large cutoff phase K leads to animproved stable region in both the basic access and theRTS/CTS modes. Intuitively, HOL packets would backoff to deeper phases if they experience collisions, and thetransmission probability quickly decreases as the phasenumber grows, until it reaches the maximum K. A largecutoff phase K implies that nodes have more room toreduce their transmission probabilities, and hence thenetwork is better capable of absorbing the mountingcontention and remaining stable as the aggregate inputrate increases.

3.3 Simulation Results and Discussions

So far we have shown that when the network operatesat the undesired stable point pA, a stable throughputλout = λ can be achieved if the backoff factor q is selectedfrom the corresponding stable region Sq

A. Otherwise,the throughput λout falls below λ and the networkbecomes unstable. The stable region Sq,K=∞

A and thenetwork throughput with q /∈ Sq,K=∞

A have been givenin (34) and (38), respectively, and are verified by thesimulation results presented in Fig. 6a. A closer lookat Fig. 6a also confirms that the stable region can besignificantly improved by properly enlarging the initialbackoff window size W if the basic access mechanism

is adopted. With RTS/CTS, however, the stable region isnot sensitive to the value of W .

With a finite cutoff phase K<∞, the numerical resultsof the stable region Sq

A under various values of cutoffphase K have been presented in Fig. 5b and are verifiedby the simulation results presented in Fig. 6b. It can beclearly observed from Fig. 6b that with the basic accessmechanism, the stable region is remarkably enlargedeven with a slight increment of K when the cutoff phaseK is small. The improvement nevertheless becomesmarginal as K increases. The stable region with K=16approaches that with K=∞. If the RTS/CTS mechanismis adopted, the cutoff phase K can be further reduced.

Figs. 6a and 6b corroborate that the stable region ofbackoff factor q can be always improved by increasingthe initial backoff window size W or the cutoff phaseK. A proper selection of W and K is especially crucialfor networks in the basic access mode. In contrast, withthe RTS/CTS mechanism, a stable throughput can beachieved within a wide range of backoff factor q evenfor small W and K, indicating that IEEE 802.11 DCFnetworks in the RTS/CTS mode are much more robustcompared to those in the basic access mode.

4 DELAY ANALYSIS

In this section, we will characterize the first and secondmoments of access delay of HOL packets and explorehow the moments of access delay vary with systemparameters at the desired and undesired stable points.

Let Yi denote the holding time of a HOL packetin State Ri, and Di denote the time spent from thebeginning of State Ri until the service completion, i =

9

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)

)

+ 2

K−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) . (42)

ˆ out

Analysis

Simulation

Simulation

Simulation

Analysis

Analysis

W=8

W=32

W=128

RTS/CTS

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Basic

q

(a)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

RTS/CTS

Basic

ˆ out

q

K=4

Simulation

K=8

K=16

K=

Analysis

Simulation

Analysis

Simulation

Analysis

Simulation

Analysis

(b)

Fig. 6. Network throughput λout versus backoff factor q in IEEE 802.11 DCF networks. λ=0.8 with the basic accessmechanism and λ=0.9 with the RTS/CTS mechanism. n=50. (a) Network throughput with various values of initialbackoff window size W . K=∞. (b) Network throughput with various values of cutoff phase K. W=32.

0, . . . ,K. According to Fig. 1, we have

Di =

{Yi + τT with probability pYi + τF +Di+1 with probability 1− p, (39)

i=0, . . . ,K−1, and

DK =

{YK + τT with probability pYK + τF +DK with probability 1− p, (40)

where τT and τF are holding times in State T and StatesFi, i = 0, . . . ,K, respectively.

Note that D0 is the service time of HOL packets(also the access delay). Let GD0(z) denote its probabilitygenerating function. It can be obtained that

G′D0

(1) = τT+1−p

pτF+

K−1∑i=0

(1−p)iG′Yi(1)+

(1−p)K

pG′

YK(1),

(41)and G′′

D0(1) is shown at the top of the page, where G′

Yi(1)

and G′′Yi(1) are given by

G′Yi(1) =

1

2α

(Wq−i + 1

)(43)

and

G′′Yi(1) =

1

3α2W 2q−2i +

1− α

α2Wq−i +

2− 3α

3α2, (44)

respectively, i=0, ...,K. Appendix C presents the detailedderivation of (41-44).

Accordingly, the mean access delay E[D0] (in the unitof time slots) can be obtained as

E[D0] = G′D0

(1) = τT +1−p

pτF +

1

α·

(1

2p+

W

2

(1

1−1−pq

+

(1

p− 1

1− 1−pq

)·(1−p

q

)K))

. (45)

The second moment of access delay E[D20] is given by

E[D20] = G′′

D0(1) +G′

D0(1), (46)

which can be obtained by substituting (41-44) into (46).Equations (45-46) indicate that the delay performance

crucially depends on p, the probability of successfultransmission of HOL packets given that the channel isidle. Section 2 has revealed that an IEEE 802.11 DCFnetwork operates at the desired stable point p = pL if itis unsaturated, and shifts to the undesired stable pointp = pA if it becomes saturated. In the following sections,we will analyze the delay performance at the bi-stablepoints pL and pA, respectively.

10

4.1 Access Delay at the Desired Stable Point pL(20) has shown that the desired stable point pL is de-termined by the aggregate input rate λ and the holdingtimes in successful transmission and collision states, τTand τF . It is further illustrated in Fig. 3a that pL is closeto 1 within a wide range of aggregate input rate λ inboth the basic access and the RTS/CTS modes. WithpL ≈ 1, the first and second moments of access delaycan be obtained as

E[D0,p=pL ] ≈ τT +1 +W

2, (47)

and

E[D20,p=pL

] ≈ τ2T + (1 +W )τT +1 + 3W + 2W 2

6, (48)

respectively, according to (45-46). It is clear from (47-48) that both E[D0,p=pL ] and E[D2

0,p=pL] increase with

the holding time in the successful transmission stateτT and the initial backoff window size W . They areinsensitive to the backoff factor q, the cutoff phase K,and the holding time in collision states τF because theHOL packets hardly encounter any collisions if pL isclose to one. They are also invariant to the number ofnodes n as pL is independent of n.

The analysis is verified by simulation results presentedin Fig. 7. It can be seen from Fig. 7 that better delay per-formance is achieved in the basic access mode, becausethe holding time in the successful transmission state islarger in the RTS/CTS mode due to the overhead ofRTS/CTS frames.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910

2

103

104

105

106

W=1024

q

W=128

W=16

Basic

Analysis

(n=10, K=1)Simulation

Simulation

Simulation

(n=50, K=1)

(n=50, K= )

RTS/CTS

Analysis

Simulation

Simulation

Simulation

(n=10, K=1)

(n=50, K=1)

(n=50, K= )

0,[ ]Lp p

E D W=1024

W=128

W=16

2

0,[ ]Lp p

E D

Fig. 7. Mean access delay E[D0,p=pL ] and second mo-ment of access delay E[D2

0,p=pL] versus backoff factor q

in unsaturated IEEE 802.11 DCF networks. λ = 0.1.

4.2 Access Delay at the Undesired Stable Point pAThe undesired stable point pA is a function of 1) the cut-off phase K, 2) the backoff factor q, 3) the initial backoff

window size W and 4) the number of nodes n. The firstand second moments of access delay at the undesiredstable point pA can be obtained by substituting (28) into(45-46). Specifically, with an infinite cutoff phase K=∞,the mean access delay at pA can be written as

E[DK=∞0,p=pA

] = τT +1− pApA

τF +

(1

2pA+

Wq

2(pA + q − 1)

)·(1 + (1− pA)τF +

2n(pA + q − 1)

Wq(τT − τF )

), (49)

which is minimized at

minq

E[DK=∞0,p=pA

]=n

τT−

1+1

W0

(− 1

e(1+1/τF )

) τF

.

(50)

The minimum mean access delay is achieved when thebackoff factor is set to be q=qK=∞

m according to (35-36).Fig. 8a illustrates how the mean access delay per-

formance varies with the backoff parameters when thenetwork is saturated at the undesired stable point pK=∞

A .As we can see from Fig. 8a, with W=16 and n=50,E[DK=∞

0,p=pA] sharply increases as the backoff factor q

departs from qK=∞m , indicating that q should be carefully

selected to optimize the delay performance. As n/Wdeclines, nevertheless, pK=∞

A becomes less sensitive tothe backoff factor q, and hence only a slight incrementof E[DK=∞

0,p=pA] can be observed as q increases if W=128 or

n=10. The minimum mean access delay linearly growswith the number of nodes n. Note that pA is a monotonicincreasing function of cutoff phase K according to (28),implying that the mean access delay with a finite Kshould be higher than E[DK=∞

0,p=pA]. As we can see from

Fig. 8b, the mean access delay declines as the cutoffphase K increases, and quickly converges to E[DK=∞

0,p=pA].

The second moment of access delay E[D20,p=pA

] is alsoclosely determined by the above backoff parameters. Incontrast to the mean access delay, however, Fig. 8b showsthat with W=16, E[D2

0,p=pA] rapidly grows as the cutoff

phase K increases. A closer look at (41-42) suggests thata dominating component of the second moment is givenby

W 2

3α2

(K−1∑i=0

(1− pAq2

)i

+1

pA·(1− pAq2

)K), (51)

which sharply increases with the cutoff phase K, andeventually becomes infinite as K→∞, if (1−pA)/q

2 > 1.A large second moment indicates that the access delayperformance drastically varies from node to node. In thatcase, some node may capture the channel and producea continuous stream of packets, while others have towait for a long time. The short-time unfairness thereforearises. This irregular behavior has long been observedand referred to as the “capture phenomenon” [44] [45].

To prevent the capture phenomenon, backoff param-eters should be carefully selected to ensure that (1 −pA)/q

2 < 1. According to (28) and Fig. 3b, (1 − pA)/q2

11

E[D2,K=∞0,p=pA

]=W 2

α2(1−1−pA

q2

) ·1

3+

1−pA

q

2(1−1−pA

q

)+ W

α(1− 1−pA

q

) ·(τT+1−pApA

τF+

1−pA

q

1−1−pA

q

(τF+

1

2α

)−1

2

+1+pA2αpA

)+

1

pA·(2(1−pA)τF

(τT+

1−pApA

τF

)+1

α

(τT+

2(1−pA)

pAτF−

1

2

)+

1

α2

(1

2pA+1

6

))+τ2T+

1−pApA

τ2F . (53)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

1.5

2

2.5

3x10

4

W=16

Analysis

Simulation

W=128

Analysis

Simulation

0,

[]

A

K

pp

ED

!

q

K

mq

!

n=10

n=50

(a)

1 2 3 4 5 6 7 8 9103

104

105

106

107

108

109

1010

1011

W=1024

0,[ ]

Ap pE D

2

0,[ ]

Ap pE D

W=16

W=1024

W=16

Simulation

Analysis

Simulation

q=0.4 q=0.5 q=0.6

Analysis

Simulation

Analysis

K

(b)

Fig. 8. Access delay of saturated IEEE 802.11 DCF networks with the basic access mechanism. (a) Mean accessdelay E[DK=∞

0,p=pA] versus backoff factor q. K=∞. (b) Mean access delay E[D0,p=pA

] and second moment of accessdelay E[D2

0,p=pA] versus cutoff phase K. n=50.

can be effectively diminished by choosing a small ratioof the number of nodes n and the initial backoff windowsize W . Specifically, with K=∞, the second momentE[D2,K=∞

0,p=pA] is finite if and only if

W >2n

−(1 + q) ln(1− q2). (52)

As we can see from Fig. 8b, by increasing the initial back-off window size W from 16 to 1024, the second momentof access delay is drastically reduced, and converges toE[D2,K=∞

0,p=pA] (which is shown on the top of the page), as

the cutoff phase K goes to infinity.We can conclude from Figs. 7-8 that the delay perfor-

mances at the bi-stable points are drastically differentfrom each other. When the network operates at thedesired stable point pL, both the first and the secondmoments of access delay are insensitive to the numberof nodes n, the cutoff phase K and the backoff factor q,and increase with the initial backoff window size W . Ifthe network is saturated at the undesired stable point pA,on the other hand, the delay performance significantlydeteriorates, and becomes closely dependent on backoffparameters. With a small initial backoff window size W ,for instance, the second moment of access delay maygrow unboundedly as the cutoff phase K increases, and

the capture phenomenon occurs eventually. A large ini-tial backoff window size W is therefore much desirableto improve the delay performance at the undesired stablepoint pA, which is in sharp contrast to that at the desiredstable point pL.

5 PERFORMANCE ANALYSIS OF BINARY EX-PONENTIAL BACKOFF

Note that in current IEEE 802.11 DCF networks, BinaryExponential Backoff (BEB) is widely adopted where thebackoff factor q is fixed to be 1/2. In this section, we willapply the analysis presented in Sections 3-4 and demon-strate how to optimize the performance of IEEE 802.11DCF networks with BEB by properly choosing systemparameters. We only focus on the undesired stable pointpA, because the throughput and delay performance isinsensitive to the backoff factor q when the networkoperates at the desired stable point pL.

5.1 Saturation ThroughputSection 3 demonstrates that for any aggregate input rateλ ≤ λmax, a stable throughput λout = λ can be achievedat the undesired stable point pA if the backoff factor q isproperly selected. With BEB where q is fixed to be 1/2,

12

the initial backoff window size W should be carefullytuned to achieve a stable throughput.

Specifically, define SW,q=1/2A as the stable region of the

initial backoff window size W with BEB, within which astable throughput λout = λ is achieved at the undesiredstable point. Similar to (33), it can be written as

SW,q=1/2A = {W |pS ≤ p

q=1/2A ≤ pL}. (54)

With an infinite cutoff phase K=∞, SW,q=1/2,K=∞A can

be obtained as

SW,q=1/2,K=∞A =

[4npS − 2n

−pS ln pS,4npL − 2n

−pL ln pL

](55)

by combining (54) and (29). SW,q=1/2,K=∞A decreases as

the aggregate input rate λ increases, and eventuallyshrinks to a single point

W q=1/2,K=∞m =2n

(1+2

(1+

1

τF

)W0

(− 1

e(1+1/τF )

))/

(−(1

+1

τF

)W0

(− 1

e(1+1/τF )

)ln

(−(1+

1

τF

)W0

(− 1

e(1+1/τF )

))),

(56)

with which the maximum stable throughput ofλmax S

W,q=1/2,K=∞A

= λmax can be achieved. It can beeasily shown from (56) that the optimal initial backoffwindow sizes in the basic access mode (i.e., τBasic

F =175time slots) and the RTS/CTS mode (i.e., τRTS

F =9 timeslots) are given by

Wq=1/2,K=∞m,Basic ≈ 17.3n, (57)

andW

q=1/2,K=∞m,RTS ≈ 2.66n, (58)

respectively.With W /∈ S

W,q=1/2A , the network throughput falls

below the aggregate input rate λ and is determined bythe aggregate service rate, which can be obtained bycombining (28) and (30). With an infinite cutoff phaseK=∞, it can be explicitly written as

λq=1/2,K=∞s =

τT

(τT−τF )+(1+τF )·W0(2n/W exp(4n/W ))−2nτF /W8n2/W 2−2n/W ·W0(2n/W exp(4n/W ))

.

(59)Note that λ

q=1/2s is usually referred to as saturation

throughput in previous studies [2]–[13]. In particular, theaggregate input rate λ is intentionally raised until thenetwork throughput falls below λ. In that case, thenetwork becomes unstable, and the throughput startsto vary with backoff parameters including the initialbackoff window size W and the cutoff phase K.

Figs. 9a-9b present the curves of λq=1/2s versus initial

backoff window size W under various values of cutoffphase K in the basic access mode and the RTS/CTSmode, respectively. As we can see from both figures,with a small W , the throughput with K=6 is lower thanthat with K=∞. The gap, however, diminishes as W

increases, and the throughput in both cases is maximizedwhen W is set to be W

q=1/2,K=∞m , i.e., W q=1/2,K=∞

m,Basic =865

and Wq=1/2,K=∞m,RTS =133 for n=50 according to (57) and

(58), respectively. It can be clearly seen from Fig. 9a thatan appropriate value of W is of great importance tothe throughput performance of BEB if the basic accessmechanism is adopted. Considerable throughput loss isincurred if the initial backoff window size W and thecutoff phase K are both small.

Note that numerical and simulation results of the satu-ration throughput of BEB with a finite cutoff phase K<∞are also presented in [2]. The throughput considered in[2] is the fraction of time that the payload is transmitted,which is slightly different from the one defined in thispaper. Nevertheless, let ξ denote the ratio of the packetpayload size L and the holding time in the successfultransmission state τT . The effective throughput used in[2] can be then expressed as ξ · λq=1/2

s . It can be easilyshown that with the packet size L=164 time slots, ξis around 91.1% with the basic access mechanism (i.e.,τBasicT =180 time slots) and 85.4% with the RTS/CTS

mechanism (i.e., τRTST =192 time slots). The curves of

effective throughput with K=6 plotted in Figs. 9a and9b are consistent with Figs. 9 and 10 in [2], respectively.

Fig. 10 illustrates the effect of cutoff phase K on thethroughput performance of BEB. According to (59), withan initial backoff window size of W=32, the throughputλq=1/2,K=∞s in the basic access and the RTS/CTS modes

are given by 0.73 and 0.97, respectively. As we can seefrom Fig. 10, to approach the throughput with K=∞,the cutoff phase K should be large enough, i.e., K=16for basic access and K=6 for RTS/CTS. The effectivethroughput plotted in Fig. 10 is consistent with Fig. 13in [2].

In the current IEEE 802.11 FHSS standard, the initialbackoff window size W and the cutoff phase K are fixedto be 16 and 6, respectively [46]. Fig. 11 illustrates thatin the basic access mode, the throughput achieved withthe standardized parameters is significantly lower thanλmax, and the gap is further enlarged as the network sizen increases. To achieve the maximum throughput λmax,our analysis has revealed that one option is to select alarge cutoff phase K and set the backoff factor q as q =qK=∞m according to (35). If q is fixed to be 1/2, the initial

backoff window size W should be adjusted according to(56). Both cases lead to an optimized throughput, as Fig.11 shows.

In the RTS/CTS mode, the throughput performance ofBEB is substantially enhanced and becomes insensitiveto backoff parameters such as the initial backoff windowsize W and the cutoff phase K. It can be clearly seenfrom Fig. 11 that with the standard setting, λ

q=1/2s is

quite close to the maximum throughput λmax even witha large network size n. Although the throughput can stillbe improved by dynamically tuning the backoff factor qor the initial backoff window size W , the gains becomemarginal.

13

2 4 8 16 32 64 128 256 512 1024 2048

0.4

0.5

0.6

0.7

0.8

1

W

0.3

=0.9maxˆBasic

1/ 2ˆqs

!

Simulation

Analysis

K !

6K

Effective throughput

Simulation

Analysis

1/ 2ˆqs

! "#

x

1/2,

,

q K

m BasicW

!

(a)

2 4 8 16 32 64 128 256 512 1024 20480.75

0.8

0.85

0.9

0.95

1

W

maxˆRTS =0.97

K !

6K

1/ 2ˆqs !

Effective throughput

K !

6K

1/ 2ˆqs

! "#

Simulation

Analysis

Simulation

Analysis

1/2,

,

q K

m RTSW

!

x

(b)

Fig. 9. Saturation throughput λq=1/2s versus initial backoff window size W in IEEE 802.11 DCF networks with BEB

(q=1/2) under various values of cutoff phase K. n=50. (a) Basic access mechanism. (b) RTS/CTS mechanism.

0 2 4 6 8 10 12 14 160.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1

RTS/CTS

Basic

Effective throughput

1/ 2ˆqs !

1/ 2ˆqs ! "#

=0.9

max

ˆRTS =0.97

max

ˆBasic

K

Simulation

Analysis

Simulation

Analysis

Fig. 10. Saturation throughput λq=1/2s versus cutoff phase

K in IEEE 802.11 DCF networks with BEB (q=1/2). n=50and W=32.

5.2 Access DelaySection 4.2 has shown that when the network shifts to theundesired stable point pA, the access delay performancebecomes crucially dependent on backoff parameters.With BEB, the mean access delay at the undesired stablepoint with an infinite cutoff phase K=∞ can be writtenas

E[Dq=1/2,K=∞0,p=pA

] = τT+1−pApA

τF+

(1

2pA+

W

2(2pA − 1)

)·(1+(1−pA)τF+

2n(2pA − 1)

W(τT−τF )

), (60)

10 20 30 40 50 60 70 80 90 1000.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.95max

ˆRTS =0.97

=0.9maxˆBasic

Basic

RTS/CTS

1/ 2ˆq

s ! with W=16, K=6

Simulation

Analysis

n

1/ 2ˆq

s !

with

ˆKmq q

s !"

!with W=K=16

Simulation

Analysis

Simulation

Analysis

1/ 2,q K

mW W

! , K=16

Fig. 11. Saturation throughput versus number of nodes nin IEEE 802.11 DCF networks.

by substituting q=1/2 into (49). It can be easily shownthat (60) is consistent to Lemma 1 in [15]. The minimummean access delay can be further obtained as

minW

E[Dq=1/2,K=∞0,p=pA

]=n

τT−

1+1

W0

(− 1

e(1+1/τF )

) τF

,

(61)

which is achieved when the initial backoff window sizeis set to be W=W

q=1/2,K=∞m .

Fig. 12a presents the curves of mean access delayE[D

q=1/20,p=pA

] versus initial backoff window size W undervarious values of cutoff phase K. As we can see from Fig.

14

50n

10n

16 32 64 128 256 512 1024 20480

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 10

4

AnalysisSimulation

AnalysisSimulation

AnalysisSimulation

AnalysisSimulation

Basic

RTS/CTS

6K K

q=

1/2

0,

[]

Ap

pE

D

W

1/2, , 10

,

q K n

m RTSW 1/2, , 10

,

q K n

m BasicW

1/2, , 50

,

q K n

m RTSW

1/2, , 50

,

q K n

m BasicW

(a)

1 2 3 4 5 6 7 8 9108

109

1010

W=16, 512AnalysisSimulation

W=128, 1024AnalysisSimulation

AnalysisSimulation

AnalysisSimulation

Basic

RTS/CTS

K

2,

1/2

0,

[]

A

q pp

ED

16W

128W

512W

1024W

(b)

Fig. 12. Access delay of saturated IEEE 802.11 DCF networks with BEB (q=1/2). (a) Mean access delay E[Dq=1/20,p=pA

]

versus initial window size W . (b) Second moment of access delay E[D2,q=1/20,p=pA

] versus cutoff phase K. n=50.

12a, with a small W , the mean access delay with K=6 islarger than that with K=∞ due to a lower probability ofsuccess of HOL packets. The mean access delay in bothcases declines as W increases, and is optimized whenW=W

q=1/2,K=∞m . A closer look at (61) further indicates

that the minimum mean access delay in the basic accessmode (i.e., τBasic

T =180 time slots and τBasicF =175 time

slots) and the RTS/CTS mode (i.e., τRTST =192 time slots

and τRTSF =9 time slots) is approximately given by 200n

and 198n, respectively. Both of them linearly increasewith the number of nodes n.

Section 4.2 also demonstrates that the second momentof access delay at the undesired stable point may sharplygrow with the cutoff phase K if the backoff factor q or theinitial backoff window size W is not properly selected.With BEB, a huge second moment of access delay canbe observed from Fig. 12b when W is small, i.e., W=16.The second moment is significantly reduced by enlargingW to 128. Nevertheless, it still increases with the cutoffphase K and becomes infinite as K→∞.

According to (52), the second moment of access delayof BEB, E[D

2,q=1/2,K=∞0,p=pA

], is finite if and only if the initialbackoff window size W satisfies

W >4n

3 ln 43

≈ 4.63n. (62)

As we can see from Fig. 12b, with a large W , i.e., W>232for n=50, the second moment of access delay convergesas K goes to infinity, and steadily decreases as the initialbackoff window size W increases. The second momentin the RTS/CTS mode is slightly better than that in thebasic access mode when W is small. The gap, however,diminishes for W≥512.

We can conclude from Fig. 12 that the delay perfor-

mance of BEB at the undesired stable point significantlydeteriorates if the initial backoff window size W is small.As Fig. 13 illustrates, with the current FHSS standardsetting (K=6 and W=16), the second moment of accessdelay of BEB is much higher than that with a large W ,i.e., W=1024, and the gap is further widened as the net-work size n grows. Note that a high second moment alsoindicates serious short-term unfairness among nodes.It is therefore of great importance to choose a largeinitial backoff window size when the network becomessaturated at the undesired stable point pA.

10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

7

8

9

10x 10

9

n

Basic RTS/CTS

Analysis

Simulation

Analysis

Simulation

16W

1024W

2,1/2,

6

0,

[]

A

qK

pp

ED

Fig. 13. Second moment of access delay E[D2,q=1/2,K=60,p=pA

]versus number of nodes n in saturated IEEE 802.11 DCFnetworks with BEB (q=1/2). K=6.

15

6 CONCLUSION

This paper presents the stability, throughput and delayanalysis of buffered IEEE 802.11 DCF networks. It isrevealed that an IEEE 802.11 DCF network has twosteady-state points. It operates at the desired stable pointpL if it is unsaturated, and a stable throughput can bealways achieved at pL. If it becomes saturated, it shifts tothe undesired stable point pA, and a stable throughputcan be achieved if and only if the backoff parametersare properly selected from their corresponding stableregions. Both the maximum stable throughput and thestable region of backoff factor q are derived, and shownto be crucially dependent on the holding times of HOLpackets in successful transmission and collision states,τT and τF . With a decrease of τF , the stable region isenlarged and becomes less sensitive to system parame-ters, which justifies the observation that an IEEE 802.11DCF network in the RTS/CTS mode is more robustthan that in the basic access mode. The delay analysisfurther shows that for unsaturated networks, both thefirst and second moments of access delay grow with τTand the initial backoff window size W , implying thatbetter delay performance can be achieved in the basicaccess mode with a small W . If the network is saturatedat the undesired stable point pA, on the other hand, alarge W becomes desirable in both modes to prevent thecapture phenomenon.

Our analysis provides plenty of insight for practicalnetwork design. In current IEEE 802.11 DCF networks,the backoff factor q is fixed to be 1/2, and small valuesof initial backoff window size W and cutoff phase Kare usually selected. It is shown that in the basic accessmode, there exists a huge gap between the maximumthroughput that can be achieved with the default stan-dard setting and the optimum throughput. To eliminatethe gap, a large cutoff phase K should be selected, andthe initial backoff window size W should be linearlyadjusted with the network size n according to (56).Moreover, the delay analysis shows that with the defaultstandard setting, a saturated IEEE 802.11 DCF networksuffers from serious short-term unfairness due to a highsecond moment of access delay no matter which modeis chosen. To improve the delay performance at the satu-rated point, a large initial backoff window size W shouldbe adopted, which nevertheless may cause unnecessarilylong delay if the network is unsaturated.

It should be noted that throughout the paper, we as-sume that each node is equipped with a buffer of infinitesize, and each HOL packet has no limit on the number ofretransmission attempts. In practical networks, however,packets may be dropped due to 1) a finite buffer size, and2) a limit on the number of retransmissions. Intuitively,the buffer size does not affect the contention processof HOL packets, and hence the analysis presented inthis paper remains valid for the finite buffer size case.The effect of retry limit, in contrast, could be prominentwhen the network becomes saturated. How to refine the

proposed analytical framework to include more practi-cal assumptions is an interesting issue, which deservesmuch attention in the future study.

APPENDIX ADERIVATION OF (5) AND (16)The Markov chain shown in Fig. 2 illustrates the transi-tion process of {Gi

t}, where Git denotes the state of the

backoff counter of a State-Ri HOL packet at time slot t,i = 0, . . . ,K.

1) Let Yi denote the holding time of a HOL packet inState Ri. When a HOL packet enters State Ri, it randomlyselects a number x from {0, . . . ,Wi − 1} as the initialvalue of its backoff counter. According to Fig. 2, Yi isthe sum of the sojourn time at states Bx, Bx−1, . . . , andB0, which can be written as

Yi =x∑

k=0

JBk, (63)

where JBkis the sojourn time at state Bk, which follows

a geometric distribution with parameter α as t → ∞,and x follows a uniform distribution with state space{0, . . . ,Wi−1}. The mean holding time τRi is then givenby

τRi = E[Yi] = E[x+ 1] · E[JBk], (64)

and (5) can be obtained accordingly.2) The limiting state probabilities of the Markov chain

in Fig. 2 can be obtained as

fBk=

2

1 +Wi· Wi − k

Wi, (65)

k = 0, . . . ,Wi − 1. Given that the channel is idle, atransmission request is made if the backoff counter iszero. The conditional probability of a State-Ri HOLpacket making a transmission request given that thechannel is idle, ri, is then equal to fB0 , and (16) canbe obtained according to (65).

APPENDIX BDERIVATION OF (6)The channel has three states: 1) Idle, 2) Successful Trans-mission and 3) Collision. Accordingly, the probability ofsensing the channel idle at time slot t+1, αt+1, can bewritten as

αt+1 = Pr{idle at t+1|success at t} · Pr{success at t}+Pr{idle at t+1|collision at t} · Pr{collision at t}+Pr{idle at t+1|idle at t} · Pr{idle at t}. (66)

In IEEE 802.11 DCF networks, a successful trans-mission and a collision last for τT and τF time slots,respectively. As a result, if the channel is sensed busy attime slot t, the probability of sensing the channel idle atthe next time slot t+1 is given by 1/τT if the transmissionis successful, and 1/τF if a collision occurs. We have

Pr{idle at t+ 1|success at t} =1

τT, (67)

16

andPr{idle at t+ 1|collision at t} =

1

τF. (68)

On the other hand, if the channel is sensed idle at timeslot t, the probability that it is idle at t+1 is indeedpt+1, which is the probability that all the nodes do nottransmit at t+1 given that the channel is sensed idle att. We have

Pr{idle at t+ 1|idle at t} = pt+1. (69)

The unconditional probability of sensing the channelin successful transmission at time slot t can be writtenas

Pr{success at t} =

τT∑i=1

Pr{success at t− i+ 1|idle at t− i}

· Pr{idle at t− i}. (70)

Let ωt denote the probability that a node has a requestat time slot t given that the channel is sensed idle at t−1.We have

pt = (1− ωt)n−1 for large n

≈ exp(−nωt). (71)

The probability that the channel has a successful trans-mission at time slot t − i + 1 given that the channel isidle at time slot t− i can be then written as

Pr{success at t− i+ 1|idle at t− i} = nωt−i+1 · pt−i+1

≈ −pt−i+1 ln pt−i+1, (72)

according to (71). By substituting (72) into (70), we have

Pr{success at t} = −τT∑i=1

αt−i · pt−i+1 ln pt−i+1. (73)

Finally, by combining (66-69) and (73), the dynamicequation of αt+1 can be obtained as

αt+1=− 1

τT

τT∑i=1

αt−ipt−i+1 ln pt−i+1+1

τF

(1+

τT∑i=1

αt−ipt−i+1

· ln pt−i+1 − αt

)+ αtpt+1. (74)

As t → ∞, we have

α = −αp ln p+1

τF(1 + τTαp ln p− α) + αp. (75)

(6) can be obtained by solving (75).

APPENDIX CDERIVATION OF (41-44)The probability generating functions of Di, i = 0, . . . ,K,can be obtained from (39) and (40) as{

GDi(z) = pzτTGYi(z) + (1− p)zτFGYi(z)GDi+1(z), i<KGDK

(z) = pzτTGYK(z) + (1− p)zτFGYK

(z)GDK(z),

(76)where Yi is the holding time of a HOL packet in StateRi, i=0, . . . ,K.

According to (63) in Appendix A, the probabilitygenerating function of Yi can be written as

GYi(z) =1

Wi

GJBk(z)−GJBk

(z)Wi+1

1−GJBk(z)

, (77)

whereGJBk

(z) =αz

1− (1− α)z. (78)

(41-42) and (43-44) can be then obtained from (76) and(77-78), respectively.

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Lin Dai (S’00-M’03) received the B.S. in elec-tronic engineering from Huazhong University ofScience and Technology, Wuhan, China, in 1998and the M.S. and Ph. D. degrees in electrical andelectronic engineering from Tsinghua University,Beijing, China, in 2000 and 2002, respectively.She was a postdoctoral fellow at the Hong KongUniversity of Science and Technology and Uni-versity of Delaware. Since 2007, she has beenwith City University of Hong Kong, where she isan assistant professor. Her research interests in-

clude wireless communications, information and communication theory.She received the Best Paper Award at IEEE WCNC 2007 and the IEEEMarconi Prize Paper Award in 2009.

Xinghua Sun received the B.S. degree in com-munications engineering from the Nanjing Uni-versity of Posts and Telecommunications, Nan-jing, China, in 2008, and is currently pursuingthe Ph.D. degree at City University of HongKong. His current research interests include per-formance evaluation of wireless networks anddistributed systems.