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IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 13, NO. 2, APRIL 2005 343

Performance Analysis of Exponential Backoff

Byung-Jae Kwak, Nah-Oak Song, Member, IEEE, and Leonard E. Miller, Senior Member, IEEE

Abstract—New analytical results are given for the performance

of the exponential backoff (EB) algorithm. Most available studies

on EB focus on the stability of the algorithm and little attention has

been paid to the performance analysis of EB. In this paper, we ana-

lyze EB and obtain saturation throughput and medium access delay

of a packet for a given number of nodes . The analysis considers

the general case of EB with backoff factor ; binary exponential

backoff (BEB) algorithm is the special case with

=2

. We also

derive the analytical performance of EB with maximum retry limit

(EB- ), a practical version of EB. The accuracy of the anal-

ysis is checked against simulation results.

Index Terms—Backoff algorithm, BEB, exponential backoff,

medium access delay, performance analysis, throughput.

I. INTRODUCTION

RANDOM access schemes for packet networks featuring

distributed control require algorithms and protocols for re-

solving packet collisions that occur as the uncoordinated termi-

nals contend for the channel. A widely used collision resolu-

tion protocol is the binary exponential backoff (BEB), forms of

which are included in Ethernet [1] and Wireless LAN [2] stan-

dards. In this paper, we analyze the exponential backoff (EB)

with backoff factor , where BEB is a special case of EB

with . The analysis uses a model that closely resembles

practical network transmission schemes and therefore is useful

for system planning and analysis. We also derive the analytical

performance of EB with maximum retry limit (EB- ), a

practical version of EB.

A. Previous Work on Exponential Backoff Algorithm

Many papers study exponential backoff algorithms including

BEB, in terms of their effect on network performance as the

offered load increases. Most of the works available, which are

summarized below, focus on the stability of EB, and little work

has been conducted on its performance analysis. Furthermore,

these studies have produced contradictory results on the stability

of EB instead of a consensus: some prove instability, others

show stability under certain conditions. The mixed results are

due to the differences in the analytical models, where simpli-

ﬁed and/or modiﬁed models of the backoff algorithm are used,

and in the deﬁnitions of stability used in the analysis.

Simpliﬁed and/or modiﬁed models of the backoff algorithm

are often used to make analysis more tractable, but can lead to

Manuscript received September 26, 2002; revised November 16, 2003; ap-

proved by IEEE/ACM TRANSACTIONS ON NETWORKING Editor Z. Haas.

B.-J. Kwak and N.-O. Song are with the Electronics and Telecommunica-

tions Research Institute, Daejeon 305-350, Korea (e-mail: bjkwak@etri.re.kr;

nsong@etri.re.kr).

L. E. Miller is with the National Institute of Standards and Technology,

Gaithersburg, MD 20899-8920 USA (e-mail: lmiller@antd.nist.gov).

Digital Object Identiﬁer 10.1109/TNET.2005.845533

very different analytical results. For example, Aldous [3] proved

that BEB is unstable for an inﬁnite-node model (a simpliﬁed

model) for any nonzero arrival rate, while Goodman et al. [4]

showed using a modiﬁed ﬁnite-node model that BEB is stable

for sufﬁciently small arrival rates. Also, while modiﬁcation of

the model can make the analysis much simpler, the analytical

result may have limited relevance because it cannot be guaran-

teed that the modiﬁed model exhibits the same behavior as the

actual algorithm.

The various deﬁnitions of stability used in the studies of

backoff algorithms can be classiﬁed into two groups. One group

of studies uses a deﬁnition based on throughput and the other

uses delay to deﬁne stability. Under the throughput deﬁnition,

the algorithm is stable if the throughput does not collapse as

the offered load goes to inﬁnity [3] or is an increasing function

of the offered load [5]. Under the delay deﬁnition, the protocol

is stable if the waiting time is bounded. Systems that are stable

under the delay deﬁnition can be characterized by a bounded

backlog of packets in the queue, or the recurrent property of

Markov chains [6].

Most of the analytical and simulation studies on EB treat the

backoff algorithm in the context of a speciﬁc network medium

access control (MAC) protocol such as Ethernet [7]–[11] or

WLAN [12]. The characteristics of the speciﬁc protocol seem

to have as much effect on the network performance results as

the intrinsic behavior of EB. Thus, the results depend heavily

on which MAC protocol is used in the study and it is not pos-

sible to understand the behavior of EB from the results. Some

of the analytical works that focus on EB itself are summarized

as follows:

Kelly and MacPhee [13] prove that “for a general acknowl-

edgment based random access scheme there exists a critical

value , with the property that the number of packets

successfully transmitted is ﬁnite with probability 0 or 1 ac-

cording as or ,” where is the arrival rate of

the system. It is also shown that for any scheme with

slower than exponential backoff, and for BEB. They

use an inﬁnite-node model with Poisson arrivals, assuming that

no node ever has more than one packet arrive at it. This result

proves that BEB is unstable for , but leaves open the

stability for .

In [3], Aldous shows that, with inﬁnite-node and Poisson ar-

rival assumptions, BEB is unstable in the sense that con-

verges to zero as goes to inﬁnity for any nonzero arrival rate,

where is the number of the successful transmissions made

during the time . This result tries to solve the open problem

left in [13], but the model Aldous uses is slightly different from

Kelly and MacPhee’s model.

Goodman et al. prove in [4] that BEB is stable if the ar-

rival rate of the system is sufﬁciently small in the sense that

1063-6692/$20.00 © 2005 IEEE

344 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 13, NO. 2, APRIL 2005

the backlog of packets awaiting transmission remains bounded

in time. More speciﬁcally, they show that BEB is stable if the

arrival rate is smaller than , where for

some constant and is the number of nodes. They assume

that each of the ﬁnite number of nodes has a queue of inﬁnite

capacity.

In [14], Al-Ammal et al. give a greater upper bound of the

arrival rate than that given in [4] for the stability of BEB under

the delay deﬁnition of stability. By using the same analytical

model as in [4], they show that there is positive constant such

that, as long as is sufﬁciently large and the system arrival rate

is smaller than then the system is stable for the -user

system. The upper bound in [14] is further improved in [15],

where it is proved that BEB is stable for arrival rate smaller than

, where . The main point of their work is that

BEB is stable for an arrival rate that is the inverse of a sublinear

polynomial in .

Finally, in [6], Håstad et al. show, using the same analytical

model as in [4], that BEB is unstable whenever for

and , or when and

is sufﬁciently large under the delay deﬁnition of stability, where

is the system arrival rate and is the arrival rate at node .

In summary, these representative analyses indicate that BEB

is unstable for an inﬁnite-node model, and for a ﬁnite-node

model it is stable if the system arrival rate is small enough but

unstable if the arrival rate is too large. We note that they all as-

sume slotted transmissions. While these analytical results are

well established, because they are contradictory and do not rep-

resent the actual system, there remains doubt about the stability

of BEB so that this question continues to be an open problem.

As noted in [6] and [16], the inﬁnite-node model used in [13]

and [3] is a mathematical abstraction with limited practical ap-

plication.

And except for [13], all of the studies cited actually analyze

a modiﬁed BEB with -persistent backoff protocol, where the

probability is exponentially decreased on collision instead of

increasing the contention window. In BEB, after consecutive

packet transmission failures (collisions) a node selects a single

random slot from the next slots (contention window) with

equal probability for the next transmission. On the other hand,

in the modiﬁed BEB, after collisions, the probability of packet

transmission in each slot is until the transmission occurs,

which can happen after slots. Clearly, it is easier to analyze

the -persistent like modiﬁed BEB because of its memoryless

nature, but it is not guaranteed that it has the same characteristics

as BEB.

B. Approach of This Paper

As mentioned earlier, most available studies on EB focus on

the stability of the algorithm and little attention has been paid

to the performance analysis of EB. However, the performance

of EB—the characteristic behavior of the throughput and delay

with respect to the network load—is of more importance in prac-

tical situations. In this paper, we analyze the performance of EB

in steady state in terms of network load.

Network performance measures are usually given as a func-

tion of the offered load, which is the actual trafﬁc demand pre-

sented to the network. The number of nodes that are contending

for the access of the medium can be used as an offered load;

this concept is what underlies BEB [7], which indirectly esti-

mates the number of nodes contending by counting consecu-

tive collisions. Another commonly used offered load is the total

packet arrival rate of the system relative to the channel capacity.

Since the purpose of BEB is to alleviate the effects of contention

among the nodes and to adapt the system to the number of nodes,

the number of nodes contending for medium access is a more

appropriate deﬁnition of offered load for analyzing BEB, and is

used in this paper. For the same reason, the performance of EB

should be evaluated based on its effect on the measures of net-

work efﬁciency, such as throughput.

In this paper, we assume a ﬁxed number of nodes in sat-

uration conditions. Here saturation condition means that each

node always has a packet to transmit. Thus, in our analysis,

represents the offered load of the network. We also assume an

implicit ACK and no errors on the channel. The saturation con-

dition assumption is also made by Bianchi in [12] and by Wu et

al. in [17], where they used 2-D Markov chain models to analyze

the throughput of the distributed coordination function (DCF)

mode of the IEEE 802.11 wireless LAN standard. In our anal-

ysis, a 1-D Markov chain model is used instead of a 2-D Markov

chain model, which enables us to analyze medium access delay

in addition to saturation throughput.

Under this assumption, we analyze network throughput and

medium access delay in steady state for a slotted system with ex-

ponential backoff and compare the analytical results with sim-

ulation. The analysis considers the general case of EB with

backoff factor ; BEB is the special case with . Note that

all the previously cited works do not evaluate delay, even though

some of the papers deﬁne stability in terms of delay.

We consider two different versions of EB in our analysis. The

ﬁrst of the two is the original EB which is simply referred to

as “EB”in this paper. In the analysis, we derive the analytical

performance measures such as the throughput and the medium

access delay of the network in term of the network load. From

the analytical results, we also show the asymptotic behavior of

EB as the number of nodes goes to inﬁnity.

The second version considered is EB with maximum retry

limit , where a packet is dropped after transmission re-

tries. Variants of this truncated version of (binary) exponential

backoff have been speciﬁed as part of the MAC protocol in sev-

eral network standards, including the MAC layer of Ethernet

local area networks (LANs) (IEEE 802.3) and the wireless LAN

standard IEEE 802.11 [2], [18]. These truncated versions are re-

ferred to as EB in many papers [16]. We call it “EB- ”in this

paper.

Even though the analyses of EB and EB- share many

common procedures, we deliberately separate the analyses to

emphasize the differences in characteristics of the two versions

of EB. The analysis of EB has been a theoretically very inter-

esting problem as demonstrated by the numerous papers cited

in Section I-A. Our interest in the characteristics of EB-

stems from the practical aspect of it as it is used in real network

protocols successfully deployed and being used. By separate

treatment, features unique to EB- can be more clearly exam-

ined. Care is taken to minimize unnecessary repetition of the

analytical steps.

KWAK et al.: PERFORMANCE ANALYSIS OF EXPONENTIAL BACKOFF 345

This paper is organized as follows. In Section II, we analyze

EB to obtain the analytical performance measures. The analysis

is carried out in several steps, which consist of modeling of EB,

analysis of saturation throughput and medium access delay, and

analysis of asymptotic behavior. In Section III, EB with max-

imum retry limit is analyzed. Section IV presents simulation re-

sults with discussion of the applicability of the analysis model.

Finally, Section V is the conclusion of the paper.

II. ANALYSIS OF EB

In our analysis, the time is divided into time slots of equal

length, and all packets are assumed to be of the same duration,

equal to the slot time. Furthermore, all nodes are assumed to be

synchronized so that every transmission starts at the beginning

of a slot and ends before the next slot. At its ﬁrst transmission, a

packet is transmitted after waiting the number of slots randomly

selected from , where is an integer rep-

resenting the minimum contention window size. Every time a

node’s packet is involved in a collision, the contention window

size for that node is multiplied by the backoff factor and an

integer random number is generated within the contention

window for the next transmission attempt, where on a packet’s

th retransmission

(1)

where and are the integer and fractional parts of

deﬁned by and , respectively.

represents the ﬁrst transmission attempt. For integer ,

this operation is equivalent to randomly selecting a number from

with equal probability . With

, this procedure is called binary exponential backoff.

A. Analytical Model of EB

The characteristic behavior of a backoff algorithm is critical

when the channel is heavily loaded, and in fact, the very idea of

EB is to cope with the heavily loaded channel condition. Thus,

we analyze EB under saturation conditions, which represent the

largest possible load offered by the given number of nodes, a

reasonable assumption for investigating EB.

We model the operation of EB at an individual node using

the state diagram in Fig. 1(a), in which each node is in one of an

inﬁnite number of backoff states in steady state and denotes

the probability that a transmission experiences a collision. In

backoff state , , the contention window size for

a node is , where is the minimum contention

window size. As the diagram in Fig. 1(a) indicates, after a suc-

cessful transmission, which occurs with probability , from

any other state a node enters backoff state with contention

window size . While in backoff state , after an unsuc-

cessful transmission, a node enters backoff state with

probability .

Fig. 1. State transition diagram of a node in steady state. (a) EB. (b) EB with

maximum retry limit

M

.

Denote as the th state that a node enters in steady state.

Then, is a Markov chain with the transition probabilities

given as follows:

Let be a probability deﬁned as

(2)

then is the relative frequency that a node will enter state

in steady state. Since , from Fig. 1(a), can be

obtained as follows:

(3)

B. EB Throughput

The main performance measure in evaluating a network is its

throughput. We analyze the saturation throughput in steady state

by calculating the probability that there is a successful transmis-

sion in a time slot.

The probability given in (3) is the relative frequency that a

node enters state . However, the average time a node stays in a

state is different for each state and is a function of the contention

window size of the state. As illustrated in Fig. 2, if a node enters

state , an integer random variable of distribution given by

(1) is generated, and after waiting for time slots, the node

will (re-)transmit the packet, after which the success or failure

of the transmission will determine the next state of the node.

Note that the node will stay in state for time slots until

the node moves to the next state. On average, a node will stay

in state for

(4)

346 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 13, NO. 2, APRIL 2005

Fig. 2. Procedure taken by a node in state

i

.

Z

means time delay by one

time slot, and

t

- - is a decrement operation on the backoff timer.

time slots. Let be the probability that a node is in state at a

given time; then speciﬁes the distribution of nodes over the

states. Since is proportional to , it is given by

(5)

where the summation

(6)

does not exist if . In fact, is a necessary condi-

tion for the system to reach steady state. Note that is given as a

function of and . Later, we show that the value of is de-

termined when the value of and the number of nodes are

given. As the number of nodes grows, there are more trans-

mission attempts and, as a result, the probability of collision

will also increase. Consequently, as shown in Section II-D,

approaches 1 as the number of nodes goes to inﬁnity, and for

large

(7)

where it is assumed that and .

Equation (7) implies that, if the channel is crowded ,

the nodes are relatively evenly distributed over the states and the

number of nodes in state is similar to the number of nodes in

state in steady state. This is because a node has

times higher chance of entering state than entering state ,

but a node will stay times longer in state than it does in

state on average.

Deﬁne , , as the conditional

probability that a node’s backoff counter will have value

given that the node is in state . Since ,

it follows that

(8)

where is the probability that the node is in state and the

backoff timer has value . Since the backoff counter is decreased

by one with probability one at the elapse of each slot time, from

the pmf (probability mass function) of given in (1), there

exist and such that

(9)

(10)

where

(11)

or equivalently

(12)

By substituting (9), (10), and (12) into (8), we obtain

(13)

where and (4) are used. If is an integer, the

equations in (13) are reduced to

From (13), (9) can be written as

(14)

When ,wehave

(15)

where is the probability that a node is in state and the

backoff timer is expired, that is, a node will transmit a packet in

state .

Let be the probability that a given node will transmit in

an arbitrary time slot. Then, since , , are the

probabilities of mutually exclusive events,

(16)

From (15), , . Thus,

(17)

Note that is a function of and , but also related to

through the value of as will be shown later. As we shall see

in the following discussion, since goes to as goes to

inﬁnity, converges to zero as the number of nodes goes to

inﬁnity.

KWAK et al.: PERFORMANCE ANALYSIS OF EXPONENTIAL BACKOFF 347

Fig. 3. Plots of

p

as a function of

p

; dashed lines:

p

in (17), solid lines:

p

in (19), dotted lines:

p

in (42) with

M

=6

.

r

=2

.

As noted in [12], the numerical value of is also constrained

by the fact that can be expressed in terms of , that is

(18)

Solving (18) for ,wehave

(19)

Equation (19) is a continuous monotonically increasing function

of in the range [0,1], and at and at

. Equation (17) is also a continuous function of in the

range . Furthermore, since

(20)

(17) is a monotonically decreasing function in the interval with

at and at . Therefore, the

curves in (17) and (19) have a unique intersection, which can

be easily calculated numerically. Since (17) and (19) are two

constraining equations for as a function of , the unique

intersection of these two equations gives us the values of and

for given and . Note that is always less than ,

which is the requirement for the existence of the summation in

(6). Fig. 3 shows plots of as a function of given in (17)

(dashed lines) for and various values of , and in (19)

(solid lines) for various values of . The probability of collision

and the probability of transmission , obtained numerically

from (17) and (19) by calculating the intersection, are plotted in

Fig. 4(a). The plot shows and converging to

and zero, respectively, as the number of nodes increases. The

symbols drawn along the curves are simulation results obtained

for , , and , respectively. Fig. 4(a)

shows that the analytical and simulation results agree extremely

well. (More discussion on the simulation is given in Section IV.)

Fig. 4. Plots of the probability of collision

p

, and the probability of

transmission

p

.

r

=2

. (a) EB. (b) EB with maximum retry limit

(

M

=6)

.

Since the channel is busy for a given time slot when there is at

least one transmitting node in the time slot, the probability that

the channel will be busy in a time slot is

(21)

where is the probability that a time slot is idle. On the

other hand, a successful transmission occurs when there is only

one transmitting node. Thus, the probability that there will be a

successful transmission in a time slot is deﬁned as

(22)

where is the number of ways of choosing one out of

nodes. Note that a collision occurs if there are multiple nodes

transmitting in the same time slot. Thus, the probability that a

collision will occur in a time slot is given by

.

If we normalize the slot time as the unit time, in any given unit

time duration, the average number of frames that are success-

fully transmitted is . Thus, if we ignore the packet over-

head which may consist of MAC/PHY header and explicit ACK,

the normalized throughput is simply . In the notation of

[3], . Fig. 5(a) shows plots of ,

the normalized throughput, for various values of . Note that

348 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 13, NO. 2, APRIL 2005

Fig. 5. The probability of successful transmission in a slot (normalized

saturation throughput).

r

=2

. (a) EB. (b) EB with maximum retry limit

(

M

=6)

.

converges to a nonzero constant ( to be precise

when ), which does not depend on , as the number of

nodes increases. Even when there are many nodes contending

for the medium access, EB manages to control the transmis-

sion attempts in a slot to guarantee sustained probability of suc-

cessful transmission. In fact, it is shown in Section II-D that the

average number of nodes that transmit in a slot converges to

a constant less than 1 as the number of nodes goes to inﬁnity.

Note that, for a small number of nodes and large , in-

creases as the number of nodes increases. This behavior occurs

because of the large number of idle time slots on the channel,

and increasing the number of nodes increases the efﬁciency of

the channel usage leading to a higher .

C. EB Expected Medium Access Delay

Delay is another key element in evaluating the performance

of a network. We deﬁne the medium access delay as the time

from the moment a packet is ready to be transmitted to the mo-

ment the packet starts its successful transmission. The medium

access delay is obtained by analyzing the expected total number

of backoff time slots.

deﬁned in (3) gives information regarding the behavior of

anode. But the state transition information of a packet trans-

mitted by a node is necessary to analyze the backoff proﬁle of

the packet. Let be the probability that a packet enters state ,

, in steady state. Then because every packet

starts at state , and since a packet enters

state when it experiences a collision in state . Thus, by

mathematical induction, is given by

(23)

Note that is numerically identical to after normalization,

that is .

Deﬁne as the probability that a packet will be successfully

transmitted on exactly the th retransmission, then

(24)

where is the probability that a packet will be successfully

transmitted without retransmission. Let be a random vari-

able representing the number of retransmissions until success;

then is the probability mass function of , and the average

number of retransmissions per packet is given by

(25)

On average, it requires

(26)

transmissions per packet for successful transmission. If a packet

is retransmitted times, then the packet will be delayed by

time slots, where is an integer

random variable deﬁned in (1). Thus, the expected number of

time slots of backoff per packet is given by

(27)

where is an ensemble average over the random vari-

able . The derivation of (27) can be found in Appendix II.

Since it takes on average time slots for a packet until suc-

cessful transmission, is the medium access delay in time

slots. Fig. 6(a) illustrates the expected medium access delay in

time slots for various values of obtained by analysis as well

as by simulation. It shows that the medium access delay in-

creases almost linearly with the number of nodes . It is shown

in Section II-D that the medium access delay approaches a linear

function of as the number of nodes goes to inﬁnity.

D. Asymptotic Behavior of EB

Now we investigate the asymptotic behavior of the analysis

model for EB observed when the number of nodes goes to in-

ﬁnity. As shown in Fig. 4(a), converges to zero as the number

of nodes increases, due to the increased contention window sizes

which causes a smaller probability of transmission in a given

time slot. The following theorem describes the asymptotic be-

havior of .

KWAK et al.: PERFORMANCE ANALYSIS OF EXPONENTIAL BACKOFF 349

Fig. 6. Medium access delay in the number of time slots.

r

=2

. (a) EB. (b) EB

with maximum retry limit

(

M

=6)

.

Theorem 1: Deﬁne as the expected number

of nodes that will transmit in an arbitrary time slot. Then,

converges to the nonzero value as the number of

nodes goes to inﬁnity.

Proof: We ﬁrst show that converges to , and con-

verges to zero as goes to inﬁnity. From (17) and (19), we have

(28)

Taking an inﬁnite limit of , and noting that

we have

which implies . Thus,

(29)

Now, rewrite (17) to yield as a function of as follows:

(30)

By equating (18) and (30), we have

(31)

Since , by multiplying both sides by and

taking the limit as ,wehave

(32)

By taking the natural logarithm of both sides, (32) can be written

as

which concludes the proof.

This theorem tells us two very important facts. First, con-

verges to a ﬁnite positive constant. In fact, with , con-

verges to . Thus, no matter how many nodes the network

contains, it can be expected that, on average, less than one node

will try to transmit in any time slot, which in turn guarantees

nonzero throughput of the network regardless of the number of

the nodes in the network as shown in the following corollary.

Secondly, is not a function of . Thus, the min-

imum contention window size does not affect the asymptotic

behavior of the network. These facts, however, do not hold for

EB with maximum retry limit.

Fig. 7(a) shows the plots of versus along with simula-

tion results. With a larger minimum contention window size ,

the expected number of transmitting nodes in a slot is smaller

because of the longer average backoff by each node. But as the

number of nodes increases, all curves converge to the asymptote

, which is shown with a thin line in Fig. 7(a).

Corollary 2: The probability that channel is busy, and

the probability that there will be a successful transmission

in a time slot converge as the number of nodes goes to inﬁnity

as follows:

(33)

(34)

The proof of Corollary 2 is straightforward from Theorem 1.

As noted in Section II-B, represents the normalized

throughput. Thus, (34) in Corollary 2 shows that the normal-

ized throughput of EB converges to a nonzero constant as the

load of the network goes to inﬁnity. The asymptote in (34) is

drawn in Fig. 5(a) with thin solid line. Note that even with a

large number of nodes, the channel is idle about 50% of the time

(for ), which guarantees sustained nonzero probability of

350 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 13, NO. 2, APRIL 2005

Fig. 7. The expected number of nodes

Np

that will transmit in an arbitrary

time slot.

r

=2

. (a) EB. (b) EB with maximum retry limit

(

M

=6)

.

successful transmission. This is due to the backoff mechanism

controlling transmission attempts by the nodes. Consequently,

as the number of nodes increases, the medium access delay

also increases; each node has to wait longer to have its turn.

In Section II-C, , the expected medium access delay was

derived. To see the asymptotic behavior of , note that (27)

can be written as

(35)

From (18) and (22), (35) can be written as

which is a variation of Little’s result. Since,

and , approaches an

asymptote, that is a linear function of ,as goes to inﬁnity

(36)

Note that (36) is not a function of . The thin solid line in

Fig. 6(a) shows a plot of the asymptote (36).

As shown in (34), the throughput in the limit (the asymptotic

throughput) is a function of the backoff factor . The optimum

backoff factor that maximizes the asymptotic throughput,

and the asymptotic throughput when are as follows:

III. ANALYSIS OF EB WITH MAXIMUM RETRY LIMIT

EB works well as far as the throughput of the network is con-

cerned. Assume, however, that a node is trying to send a packet

to another node and the destination node is not reachable. Then

the source node will continue to retransmit the packet, and other

packets in the queue destined to other reachable nodes will not

be able to get through even when the channel is idle. Further-

more, without a transmission retry limit, the packet transmission

delay can grow beyond reasonable range. To avoid this kind of

problem, practical network protocols use truncated BEB, where

the contention window is given by

for

for

when , and

when , where is the maximum retry limit and

determines maximum contention window size. We analyze the

latter case where , which we call EB with maximum

retry limit (EB- ) in this paper. The case of can be

readily analyzed using the technique in this section.

A. Analytical Model of EB With Maximum Retry Limit

Fig. 1(b) illustrates the state transition diagram of EB- .As

shown in the ﬁgure, there is a limit, , to the maximum retries

per packet and after reaching state , a node goes to state 0

with a new packet even if the transmission was unsuccessful.

Denote as the th state that a node enters. Then, is

a Markov chain with the transition probabilities given as

follows:

Let be a probability deﬁned as

(37)

KWAK et al.: PERFORMANCE ANALYSIS OF EXPONENTIAL BACKOFF 351

then is the relative frequency that a node will enter state in

the steady state. Since , from Fig. 1(b), can be

obtained as follows:

(38)

B. Throughput for EB With Maximum Retry Limit

For EB- , the expected time that a node will stay in state

is the same as that of EB shown in (4). Thus, the probability that

an arbitrary node is in state is

(39)

Note that is not required for the existence of the sum-

mation in (39). In fact, has a value greater than when the

number of nodes is large enough.

For large , (39) can be approximated as

(40)

where it is assumed that and

. Since is possible, (40) implies that there can

be more nodes in state than in state , while it was always

for EB in Section II.

Substituting and (39) into (14),1we have

(41)

Since (note that the summation is from 0

through ), we obtain

(42)

The dotted lines in Fig. 3 are the plots of (42) for various values

of . Note that when is small, the curves for (42) and (17)

(dashed lines; as a function of for EB) follow almost the

same trajectories. This is because most of the packets are suc-

cessfully transmitted within a small number of retries and thus

affected little by the maximum retry limit. As gets larger,

however, packets experience more collisions and the character-

istics of the system are determined by the maximum retry limit,

and the curves for (42) and (17) follow different trajectories.

Furthermore, while (17) is deﬁned only for , (42)

is deﬁned for . In EB, the mechanism of the backoff

algorithm and the unbounded contention window size make

upper bounded by .2On the other hand, in EB- , in (42)

is always greater than zero due to the upper bounded contention

window size, and thus as goes to inﬁnity, goes to inﬁnity

1Equation (14) is valid for both EB and EB-

M

.

2If we assume

rp

1

, then (42) converges to zero as

M

goes to inﬁnity.

Since there are only ﬁnite number of nodes in the network, this means there is

no transmission in the network. Thus the conditional probability of collision

p

converges to zero as

M

goes to inﬁnity, which contradicts with the assumption.

As a result, (17) for EB is deﬁned only for

0

p <

1

=r

.

and converges to unity. Note that it is easily shown that (42)

converges to (17) as goes to inﬁnity when .

in (42) is also a continuous monotonically decreasing func-

tion of in the range of [0,1] as shown in Appendix I. Thus

(19) and (42) have a unique intersection, and the intersection

determines the probability of collision , and the probability of

transmission for given values of and . Fig. 4(b) shows

the probability of collision and the probability of transmis-

sion , obtained numerically from (42) and (19) by calculating

the intersection, along with the simulation results represented

by circles and bullets. Unlike for EB, converges to 1 as the

number of nodes increases. Furthermore, does not converge

to zero. This is due to the maximum retry limit, which limits the

backoff.

For EB- , and (the normalized throughput) are

the same as for EB as deﬁned in (21) and (22), respectively.

However, since for EB- is different from that of EB, the be-

havior of and show big differences. See Fig. 5(b) for

plots of for various values of . The most noticeable dif-

ference from EB is that diminishes to zero as the number

of nodes goes to inﬁnity as shown in Section III-D. This is be-

cause of the excessive collisions caused by the limited backoff.

Also note that when the number of nodes is small, for

EB- is slightly larger than the value for EB.

C. Expected Medium Access Delay for EB With Maximum

Retry Limit

Let be the probability that a packet enters state ,

, in steady state. Then, as in the case of EB, ,

.Deﬁne as the probability that a node will

successfully transmit on th retransmission, and let be

the probability that a packet will be dropped after maximum

allowed transmission retries, then ,

, , and ,

which results in

(43)

(44)

where is the probability that a packet will be successfully

transmitted without retransmission. Since a packet is either

successfully transmitted within tries or dropped,

.

Let be a random variable representing the number of re-

transmissions per packet, then the average number of retrans-

missions per packet for the successfully transmitted packets is

given by

(45)

Thus, it requires

(46)

transmissions per packet on average for successful transmission.

Note that (46) converges to (26) as goes to inﬁnity. At ﬁrst

352 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 13, NO. 2, APRIL 2005

glance, in (46) seems to have a smaller value than in (26).

However, since for EB- is greater than for EB, (46) has

a larger value under the same conditions.

For EB- , as the number of nodes increases, approaches

unity and most packets will be dropped after transmission

retries, and only a few packets will be successfully transmitted

within transmission tries. Since (46) represents the av-

erage number of required transmission attempts of the success-

fully transmitted packets, the value is bounded by .In

fact, it can be shown that

(47)

The expected medium access delay for EB- is obtained

similarly to EB as

(48)

where is the expected time delay per packet in slots. Note

that in (48) and in (45) are deﬁned only for the packets

transmitted successfully. But, in (42) is relevant to all packets

including the dropped ones. So, a relationship similar to (27)

does not exist for EB- . Equations (47) and (48) are derived in

Appendix II.

As discussed above, for EB- the number of retries is

bounded by , and thus the medium access delay also has

an upper bound. This property is well illustrated in Fig. 6(b),

where the expected medium access delay is plotted for various

values of along with the corresponding asymptotes, which

are calculated in Section III-D. However, to guarantee the upper

bound of the medium access delay, EB- drops packets that

fail retries, and eventually the probability of dropping a

packet will converge to unity as goes to inﬁnity.

D. Asymptotic Behavior of EB With Maximum Retry Limit

From (42), we can see that is always greater than zero for

all . As a result, instead of converging to a constant,

goes to inﬁnity as goes to inﬁnity. Since converges to

unity as goes to inﬁnity [see (18)], from (42),

(49)

and thus approaches the following linear function of as

goes to inﬁnity

(50)

In Fig. 7(b), the asymptotes are drawn with thin lines in the same

dash patterns as the corresponding plots of .

The divergence of that we have noted implies that there

will be inﬁnitely many nodes transmitting in a slot as goes to

inﬁnity. In fact, since and

for , and converge to unity and zero,

respectively, as goes to inﬁnity. That is, because of collisions,

the channel is always busy and a successful transmission will be

achieved with probability zero.

Finally, the delay asymptotes in Fig. 6(b) can be obtained by

taking the limit of (48) as follows:

IV. SIMULATION

In Figs. 4–7, simulation results, which are represented by

symbols, are added to the curves of analytical results. The sim-

ulator is written in the C++ programming language, and simula-

tion results were obtained by running 5 000 000 time slots after

1 000 000 time slots of warming up. The backoff factor

was used for the simulations included in the paper.

In the simulation of EB, minimum contention window sizes

16, 32, and 64 were used. Note that, in the IEEE 802.11

speciﬁcation, 16, 32, and 64 are used as the minimum con-

tention window sizes for frequency hopping spread spectrum

(FHSS), direct sequence spread spectrum (DSSS), and infrared

(IR) physical layers, respectively. The simulations were run for

. The simulation results in Figs. 4(a)–(7a)

agree with those obtained from our analysis. However, when

the steady state assumption does not hold true, the analysis and

simulation produce different results. See Section IV-A for the

discussion of the applicability of the analysis model.

In the simulation of EB- , was used as in the IEEE

802.11 speciﬁcation. Additional cases with a larger number of

nodes ( 70, 100, 150, 200) were considered for a better

comparison of the simulation results with the asymptotes. Fur-

thermore, minimum contention window sizes 4 and 8

were also considered in addition to 16, 32, and 64, to dis-

play that a capture effect does not occur even for small in the

case of EB- (see Section IV-A). Consequently, the simulation

results for EB- give much better match with the analytical re-

sults for a wide range of and , as shown in Fig. 4(b)–(7b).

A. Applicability of the Analysis Model

Our analysis model is based on the assumption that the

system is in steady state. Even well-designed analysis models

cease to represent the real system correctly under inordinate

operating conditions. Our simulation study shows that our

analysis model represents the exponential backoff algorithm

accurately over a wide range of operating conditions. However,

when is too small, a capture effect was observed. A capture

effect makes only a few nodes consume the whole transmission

channel, and results in a higher throughput by making most of

the nodes starve (no transmission for a prolonged time period).

When , an absolute capture occurs, that is, a single node

captures the channel with probability one, which is observed

KWAK et al.: PERFORMANCE ANALYSIS OF EXPONENTIAL BACKOFF 353

Fig. 8.

N

versus

W

, where

N

is the number of nodes for which the

simulation results begin to depart from the analytical results.

both in EB and EB- . In EB, when is greater than one

but too small, the nodes randomly take turns to capture the

channel temporarily. As a result, the network cannot reach

steady state and our analysis model does not apply. Also for a

large number of nodes , a system may not reach steady state,

depending on the size of the minimum contention window size

. For given , when the number of nodes is too large, a

signiﬁcant number of the nodes stay in the states of extremely

large contention window sizes, where the exact distribution of

nodes over the states is given by deﬁned in (5) for EB, and

in (39) for EB- . While the nodes in the states of extremely

large contention window sizes backoff for prolonged periods,

other nodes may have many chances of transmission, causing

the network to adapt locally in time excluding the nodes

with prolonged backoff. Eventually, the nodes with prolonged

backoff will transmit packets and these transmissions will

disturb the network, preventing it from reaching a steady state.

An investigation of the transmission history of an individual

node shows that the nodes experience temporary starvation and

capture throughout the simulation when there are discrepancies

between the analysis and the simulation. In the case of EB- ,

the maximum contention window size is limited by , and

thus this problem is less likely to occur, and the steady state

assumption holds true for much wider operating conditions. In

our simulation, is used and no noticeable discrepancy

between the analysis and simulation was observed. For larger

, however, it is expected that EB- will also exhibit the

problem described above.

Let be the number of nodes for which the simulation re-

sults begin to depart from the analytical results. In the case of

EB, simulation results with show larger compared

to the analytical results for the same given . Fig. 8 shows a

plot of with respect to for , where for various

are manually obtained from extensive simulation results. As

shown in the ﬁgure, increases with , and appears to con-

verge to an asymptote , where is a constant.

Finding the exact remains an open problem, but the simula-

tion study shows that a larger value of produces a smaller value

of .

V. C ONCLUSION

The contribution of this paper is that we provide a new and

efﬁcient analytical means to evaluate the performance of a net-

work with an exponential backoff algorithm. Using the proposed

analytical model, we analyze the performance of EB and EB-

to obtain the saturation throughput and the medium access delay.

The asymptotic behaviors of EB and EB- are also shown.

To validate the analytical results, the simulation results are pro-

vided. The results indicate that EB- provides the nodes fairer

service even for very small values of . EB- also bounds

the medium access delay. These beneﬁts are accomplished by

limiting the number of transmission tries for a packet and thus

giving a chance of transmission to the next packet waiting. But

the maximum retry limit also causes the throughput to diminish

as increases.

The analysis presented in this paper is an analysis of EB and

EB- in steady state (equilibrium). A sudden change of the of-

fered load (the number of contending nodes) will cause change

of equilibrium and there will be a transition to the new equi-

librium. The dynamic behavior of the backoff algorithm when

there is a sudden change of offered load is another interesting

problem. Note that the result of the steady state analysis and the

dynamic behavior are very closely related because the equilib-

rium provides the limit that the transient response converges to.

In that regard, the steady state analysis may be considered as a

prerequisite for an analysis of the dynamic behavior.

APPENDIX I

PROOF OF CONTINUITY AND MONOTONICITY OF (42)

To show the continuity of (42) in the range [0,1], we need to

establish the continuity of at .If , the nodes

will always stay in state , and it can be shown that ,

from which we obtain

(51)

To establish continuity, we also need to show that converges

to (51) as goes to zero

(52)

(53)

(54)

Thus, is continuous in the range .

Now, we show that (42) is a monotonically decreasing func-

tion by showing that

(55)

354 IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 13, NO. 2, APRIL 2005

is a monotonically increasing function of in [0,1], which is

equivalent to showing . In order for the inequality

to hold, the following equation needs to be true:

(56)

But

(57)

Since inside the summation, (57) is always positive. This

concludes that in (42) is a continuous and monotonically de-

creasing function in the range [0,1].

APPENDIX II

DERIVATION OF EQUATIONS (27), (47), AND (48)

A. Derivation of Equation (27)

Noting that and , , are both random

variables

From (4)

(58)

where is used. Since in (24) is the probability

mass function of

(59)

Substituting (59) and [(25)] into (58),

and using (17), we have

B. Derivation of Equation (47)

Since (see Section III-D), from (46), we

have

Using L’Hospital’s rule

C. Derivation of Equation (48)

The derivation of (48) is similar to the derivation of (27) in

Appendix II-A. Since and , , are both

random variables, as in the case of (27)

(60)

From (43) and (44)

(61)

Substituting (61) and (45) into (60), we have

KWAK et al.: PERFORMANCE ANALYSIS OF EXPONENTIAL BACKOFF 355

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Byung-Jae Kwak received the B.S and M.S. degrees

in electronic engineering from Yonsei University,

Seoul, Korea, in 1989 and 1991, respectively, and the

Ph.D. degree in electrical engineering and computer

science from the University of Michigan, Ann Arbor,

in 2000.

From 1991 to 1992, he was with the Engineering

Research Institute, Yonsei University, where he was

involved in the development of sonar systems. In

2000, he joined the Telecommunication R&D Center,

Samsung Electronics, Korea, where he participated

in the standardization effort of third-generation mobile communication systems

(3GPP). From 2001 to 2004, he was with the Advanced Network Technologies

Division at the National Institute of Standards and Technology, Gaitherburg,

MD, as a visiting scholar. Since 2004, he has been with the Digital Home Re-

search Division at the Electronics and Telecommunications Research Institute,

Korea. His research interests include wireless communications, mobile ad hoc

networks, distributed MAC protocols, and adaptive signal processing.

Nah-Oak Song (M’03) received the B.S. and M.S.

degrees from Yonsei University, Seoul, Korea, in

1989 and 1993, respectively, and the Ph.D. degree

from the University of Michigan, Ann Arbor, in

1999.

From 1989 to 1991, she was with the Application

Speciﬁc Integrated Circuit Research Institute, Yonsei

University, where she was involved in a project on

echo cancellation systems. From 1999 to 2001, she

was with the Telecommunication R&D Center, Sam-

sung Electronics, Korea, and participated in the de-

velopment of CDMA 2000 system. From 2001 to 2004, she was with the Ad-

vanced Network Technologies Division at the National Institute of Standards

and Technology, Gaitherburg, MD, as a visiting scholar. Since 2004, she has

been with the Digital Home Research Division at the Electronics and Telecom-

munications Research Institute, Korea. Her main interest is in wireless networks

with emphasis on mobile ad hoc networks and wireless LAN. Her other interests

include medium access control protocol, quality of service, stochastic sched-

uling, and MPLS.

Leonard E. Miller (S’63–M’64–SM’92) received

the B.E.E. degree from Rensselaer Polytechnic

Institute, Troy, NY, in 1964, the M.S.E.E. degree

from Purdue University, West Lafayette, IN, in 1966,

and the Ph.D. degree from The Catholic University

of America, Washington, DC, in 1973.

From 1964 to 1978, he was with the Naval Sur-

face Warfare Center, Silver Spring, MD, where he

was a member of the Signal Processing Branch. From

1978 to 2000, he was with J. S. Lee Associates, Inc.,

Rockville, MD, as Vice President for Research, and

was involved initially in analyzing the survivability and performance of mili-

tary communications and electronic support systems, and later in modeling of

propagation in the mobile environment and the design and analysis of cellular

and personal communication systems. Since 2000, he has been a member of the

Wireless Communication Technologies Group, National Institute of Standards

and Technology, Gaithersburg, MD, where he is responsible for analysis and

simulation of wireless ad hoc networks, wireless standards, and public safety

wireless applications. He is a coauthor (with Dr. J. S. Lee) of CDMA Systems

Engineering Handbook (Boston, MA: Artech House, 1998).