Performance analysis of exponential backoff

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 analyze EB and obtain saturation throughput and medium access delay of a packet for a given number of nodes N. The analysis considers the general case of EB with backoff factor r; binary exponential backoff (BEB) algorithm is the special case with r=2. We also derive the analytical performance of EB with maximum retry limit M (EB-M), a practical version of EB. The accuracy of the analysis is checked against simulation results.

Full text

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).