ANALYTICAL STUDY OF BROADCASTING IN 802.11 AD HOC

NETWORKS

Andrey Lyakhov and Vladimir Vishnevsky

Institute for Information Transmission Problems, RAS

B. Karetny 19, Moscow, 127994, Russia

Pavel Poupyrev

ATR Adaptive Communications Research Laboratories

2-2-2 Hikaridai, Keihanna Science City, Kyoto, Japan

Keywords:

Broadcasting, IEEE 802.11, Markov models, performance study.

Abstract:

IEEE 802.11 technology becomes attractive with the implementation of various broadcast applications. For

these applications, one of the main performance indices is the mean notiﬁcation time namely the interval be-

tween consecutive successful receipts of the same source’s packets. In this paper, we develop an analytical

method to study the ad hoc 802.11 network performance with broadcasting stations. The method based on a

Markov model allows the estimation of the mean notiﬁcation time and the optimization of the packet genera-

tion time. The developed method has been validated by simulations and has demonstrated a high accuracy of

notiﬁcation time estimation as well as the its efﬁciency in broadcasting optimization.

1 INTRODUCTION

IEEE 802.11 (IEEE 802.11, 1999) is one of the most

popular technologies for wireless ad hoc and mobile

networking. The fundamental access mechanism in

the IEEE 802.11 protocol is the Distributed Coordi-

nation Function (DCF), which implements the Car-

rier Sense Multiple Access with Collision Avoidance

(CSMA/CA) method.

In the majority of previous related papers, the

802.11 DCF performance was studied with unicast

transmissions: see (Bianchi, 2000; Cali et al., 2000;

Vishnevsky et al., 2002), for example. However, the

802.11 standard speciﬁes both unicast transmissions,

and multicast or broadcast ones. Broadcasting in ad

hoc 802.11 networks is of our main interest in the pa-

per.

As an example of such broadcast applications, we

can consider the Whizbe project (Poupyrev et al.,

2002) developed in ATR, Japan. Whizbe is an appli-

cation for discovering resources such as users, prod-

uct information discounts and etc. in ad hoc networks

using broadcasting. The main goal of the application

is to provide mutual presence awareness for acquain-

tances that are in close physical proximity. Whereas

an online friend list application tells if a friend or ac-

quaintance is available for online chat, Whizbe lets

the user know if a friend or acquaintance is in physi-

cal proximity to facilitate a physical meeting or other

interaction. A number of PDAs supporting such sort

of message broadcasts and located within the same

line-of-sight area can be large and reach hundreds and

thousands (for example, in a stadium).

Another type of application, where 802.11 ad hoc

broadcasting can be successfully used, is wireless

sensor networks with distributed control. For exam-

ple,

• Wireless home temperature sensors, each of them

transmitting information to receivers of a climatic

system, ﬁre ﬁghting service and the home owner

work station (Cheng et al., 2002);

• Gas escape sensors in gas pipelines, where mea-

sured data are to be delivered to repair service con-

trollers. To improve reliability, several data re-

ceivers are exploited in these systems, what makes

broadcasting attractive.

In such broadcast systems, the key point is to opti-

mize the mean message generation time, that is, the

time interval separating the instances when a data

source puts its data packets into the broadcast device’s

queue. As the optimization criterion, we can con-

sider the mean notiﬁcation time namely the interval

between consecutive successful receipts of the same

source’s packets. Obviously, the 802.11 DCF pecu-

liarities described below will affect the performance

index.

In the 802.11 DCF, no ARQ mechanisms are used

Lyakhov A., Vishnevsky V. and Poupyrev P. (2005).

ANALYTICAL STUDY OF BROADCASTING IN 802.11 AD HOC NETWORKS.

In Proceedings of the Second International Conference on e-Business and Telecommunication Networks, pages 86-91

DOI: 10.5220/0001410000860091

 SciTePress

with packet broadcasting. A device starts transmitting

a DATA frame containing a broadcast packet with the

following conditions:

1) The channel was free for DIFS (Distributed In-

terFrame Space) since the latest transmission in the

network;

2) The device’s backoff time has been expired;

3) There are packets to be transmitted in the de-

vice’s queue.

In particular, when a packet arrives a device ei-

ther starts the packet transmission immediately, if

the channel is free and the device is in the idle

state at the arrival time, or switches to the backoff

state otherwise. We call these immediate transmis-

sions the asynchronous ones to distinguish them from

other (synchronous) transmissions carried out after

the backoff. After passing to the backoff state, the

backoff counter is reset to the initial value b, which

is called the backoff time, measured in units of back-

off slots of duration σ, and chosen uniformly from a

set (0, . . . , W − 1). W is called the contention win-

dow and it does not depend on the number of retries

(contrary of unicast transmissions), because broadcast

transmissions are not acknowledged and hence there

are no retries at the MAC layer.

Backoff intervals are reckoned only as long as the

channel is free: the backoff counter is decreased by

one only if the channel was free for the entire previous

slot. Counting the backoff slots stops when the chan-

nel becomes busy, and backoff time counters of all

stations decrement only when the channel is sensed

idle for DIFS. When the backoff counter attains its

zero value, the station starts transmission. A collision

happens when backoff counters of two or more sta-

tions are zeroed simultaneously. All packets of sta-

tions involved in the collision are lost.

A device that has completed a packet transmission

returns to the backoff state. If the device’s queue ap-

pears to be empty at the end of the backoff time, the

device becomes idle.

Further, in Sections 2 and 3 we develop an analyt-

ical model, which considers all signiﬁcant DCF fea-

tures and provides the mean notiﬁcation time estima-

tion. In Section 4, we give some numerical research

results obtained by both our analytical method and

simulations, which allows us to validate the devel-

oped method. The obtained results are summarized

in Section 5.

2 ANALYTICAL MODEL

Let us consider an ad hoc 802.11 network of N sta-

tistically homogeneous stations generating broadcast

packets of the same ﬁxed length with an identical rate

λ ≪ (Nσ)

−1

. Generation intervals are assumed to be

distributed exponentially. Every station’s queue can

contain no more than B packets. The channel is ideal,

that is, there are no noise-induced distortions. Signal

propagation time is assumed to be negligible.

The analytical model to be developed is intended

for the determination of the optimal generation rate

opt

, when the mean notiﬁcation time T

not

is mini-

mal.

As in (Bianchi, 2000) and (Vishnevsky et al.,

2002), let us subdivide the time of the network op-

eration into non-uniform virtual slots such that:

• Every station with non-empty queue changes its

backoff counter at the start of a virtual slot and

begins a synchronous transmission if the counter

value becomes zero.

• If a transmission happens within a given virtual

slot, the slot is ended by the DIFS closing the trans-

mission.

• If nobody transmits during σ since the current slot

beginning, this slot duration is equal to σ.

Such a virtual slot is either (a) an “empty” slot

σ in which no station transmits, or (b) a “synchro-

nous” slot when one or more stations transmit syn-

chronously or (c) an “asynchronous” slot when a sta-

tion transmits asynchronously. Note that an asynchro-

nous transmission can be considered as always suc-

cessful since, ﬁrstly, it can be performed only in a slot

when no other stations transmit synchronously (oth-

erwise it will sense the channel busy at the slot begin-

ning and defer from its transmission), and secondly,

we can neglect the probability that two or more pack-

ets arrive to queues for σ because λ ≪ (N σ)

−1

Following (Bianchi, 2000), we describe a station

state by the couple (i, k), where i = 0, 1 is an indi-

cator of non-empty queue and k = 0, . .., W -1 is a

backoff counter value, and a station behavior by the

Markov chain shown in Fig.1. In the chain, all transi-

tions happen at virtual slot borders.

Figure 1: Markov chain.

State (0,0) corresponds to the idle state of the con-

sidered station. In states (0, 1)...(0,W-1), the station

is in the backoff state, but its queue is empty. At last,

(1, 0),... , (1,W -1) are the states when the station’s

queue is not empty.

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Let us introduce the following notation:

P {i

, k

} is the probability of one-step tran-

sition from (i

, k

) to (i

, k

is the probability that a queue becomes empty

after completing its synchronous transmission.

is the probability that at least one packet arrives

in the considered station’s queue for a virtual slot un-

der the condition that the queue was empty at the slot

beginning. Obviously, the probability consists of two

items: P

= P

+ P

, where P

and P

are the

arrival probabilities for non-empty and empty slots,

respectively, under the same condition as above.

is the probability that at least one packet arrives

in the considered station’s queue for a packet trans-

mission time, including the closing DIFS.

To ﬁnd stationary probabilities α (i, k) of the states

in Fig. 1, let us deﬁne non-null transition probabili-

ties:

P {1, k|1, k + 1} = 1, k ∈ [0, W − 2] : backoff

counter decrement, the queue is not empty.

P {1, k|1, 0} =

1−

, k ∈ [0, W − 1] : syn-

chronous transmission of the station, after which

its queue remains to be non-empty. Here,

−λDIFS

is the probability that a queue becomes

empty after completing its synchronous transmission

and keeps to be empty after DIFS closing the trans-

mission.

P {0, k|1, 0} =

, k ∈ [0, W − 1] : synchro-

nous transmission of the station, after which its queue

becomes empty.

P {1, k|0, k + 1} = P

, k ∈ [0, W − 2] : back-

off counter decrement, and a packet arrives in the sta-

tion’s empty queue.

P {0, k|0, k + 1} = 1 − P

, k ∈ [0, W − 2] :

backoff counter decrement, the queue remains to be

empty.

P {1, k|0, 0} =

, k ∈ [0, W − 1] :

the station switches from the idle state to the backoff

state. The switch takes place, if either a packet arrives

when the channel was busy, or one more packet ar-

rives during the station’s asynchronous transmission.

P {0, k|0, 0} =

(1−P

)

, k ∈ [1, W − 1] : the

transition corresponds to the station’s asynchronous

transmission, after which the station’s queue is empty

and the backoff counter b = k > 0.

P {0, 0|0, 0} = 1 − P

(1−P

)

: there are no

packet arrivals in the queue, or the station’s asynchro-

nous transmission takes place, after which the sta-

tion’s queue is empty and the backoff counter b = 0.

Now we can determine stationary probabilities

α(i, k). Using global balance equations and the nor-

malizing condition

(i,k)

α(i, k) = 1, we obtain:

α (0, 0) =



1 − P

W + 1

× (1)





W P

1−(1−P

)

− P

(1 − P

)











−1

α(1, 0) =

W P

1−(1−P

)

− P

(1 − P

)

· α(0, 0),

(2)

α(0, j) = P

1 − (1 − P

)

W −j

1 − (1 − P

)

· α(0, 0), (3)

j = 1, . . . , W − 1,

α (1, j) =

W − j

· α (0, 0) + α (1, 0)]−α (0, j) ,

(4)

j = 1, . . . , W − 1,

Let τ be the probability of the station’s synchro-

nous transmission for a virtual slot. Similarly to

(Bianchi, 2000; Cali et al., 2000), we assume that

the probability depends neither on the previous his-

tory, nor on the behavior of other stations. Obviously,

τ = α(1, 0). Moreover, the probability of synchro-

nous transmission failure due to a collision is P

1 − (1 − τ )

N−1

, and the probability of the station’s

asynchronous transmission for a virtual slot is τ

α(0, 0)P

, where P

= (1−τ )

N−1

[1−exp{−λσ}]

since we have neglected the case, when two or more

packets arrive for σ.

Let us ﬁnd durations of non-empty slots:

• “Synchronous” slot: t

= t

+DIFS, where t

is a

DATA frame transmission time;

• “Asynchronous” slot: the mean duration is t

+ t

+DIFS.

The probability P

that at least one packet arrives

in the station’s queue for t

is P

= 1 − exp(−λt

To ﬁnd the probability P

, we need to know prob-

abilities of all possible kinds of slots under the con-

dition that the considered station does not transmit

in the slot. Under the condition, the “empty” slot

probability is Q

= (1 − τ − τ

)

N−1

, the “syn-

chronous” slot probability is Q

= 1 − (1 − τ)

N−1

and the “asynchronous” slot probability is Q

1 − Q

− Q

. So we have

= (Q

+ Q

) P

To complete the model deﬁnition, it remains to ﬁnd

probability P

that a queue becomes empty after ﬁn-

ishing the current packet service. A packet service

starts at the instance of either the packet arrival in an

empty queue or completing the transmission of the

previous packet in the queue. If a packet arrives in

an empty queue and is transmitted asynchronously,

ANALYTICAL STYDY OF BROADCASTING IN 802.11 AD HOC NETWORKS

we deal with the asynchronous service case, when a

packet is served for the ﬁxed time t

. With synchro-

nous service, the service time is random.

Let us describe a device’s queue change process by

the auxiliary model shown in Fig. 2. With the model,

we assume that a packet arriving at a device not en-

gaged in servicing other packets is serviced asynchro-

nously (and hence successfully) with probability p

Otherwise, the packet is put into a queue of size B.

After standing in the queue, it is serviced synchro-

nously for a random time distributed exponentially

with the average value T

Synchronous

transmission

Asynchronous

transmission

Queue

Figure 2: Queue change process.

Under these assumptions, we can use a birth-and-

death process to describe how the length i of the syn-

chronously serviced packets queue changes. Since the

birth rate is λ

= (1 − p

) λ with i = 0 and λ with

0 < i < B, while the the death rate is 1/T

, we ob-

tain the following formula for the steady-state proba-

bilities:

= π

i−1

, i = 1, . . . , B. (5)

Obviously, π

is the probability of packet rejection

because of buffer overﬂow. Therefore, the probability

that a queue becomes empty after completing its syn-

chronous transmission is equal to P

= π

/(1 − π

Moreover, since

1 + (1 − p

)

i=1

(λT

)

, (6)

we obtain

i=1

(λT

)

i−1

1 − λT

1 − (λT

)

. (7)

In the next section, we will estimate the probability

and the average time T

of synchronous service,

as well as the main performance index T

not

3 ESTIMATION OF

PERFORMANCE INDICES

Let us subdivide all packets, which arrive at some sta-

tion A during all possible virtual slots identiﬁed by

couple (i, k) and are serviced synchronously, into the

following 4 categories:

1. Packets arriving for slots (1, k);

2. Packets arriving for slots (0, k > 0);

3. Packets arriving for slot (0,0), when a device dif-

ferent from A transmits;

4. Packets arriving during an asynchronous transmis-

sion of device A.

For each of the categories, we will count the aver-

age number n

of these packets arriving for a virtual

slot and the average number n

of these packets ar-

riving for a virtual slot in empty queue. It is easy to

show that

∗

+ DIFS)

i=1



− n



i=1

, (8)

= τ

i=1

, (9)

where

∗

W − 1

V S

+ t

and t

V S

= Q

σ +Q

is the average dura-

tion of a virtual slot, when the given device A does not

transmit. Moreover, T

is the average service time for

category i packets arriving in empty queue.

To estimate T

and p

, using (8) and (9), it re-

mains to ﬁnd n

, n

and T

for each of introduced

categories. Let us do it.

Category 1. In the case, packets can arrive in

empty queue of the considered station A only for the

DIFS closing the station’s transmission and only, if

the transmitted packet was the last in the queue. So

we have



1 − e

−λ·DIFS



· α(1, 0),

= λt

V S

W −1

k=1

α(1, k) + λt

· α(1, 0),

and

= T

∗

DIFS

Category 2. Since only the ﬁrst of packets arriving

in the queue for each of the slots can ﬁnd the queue

empty, we have

= Q

∗

W −1

k=1

α(0, k),

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= λt

V S

W −1

k=1

α(0, k),

and

= t

V S

∗

W −1

k=1



k −



α(0, k),

where

∗

= Q



1 − e

−λσ



+ Q



1 − e

−λt





1 − e

−λt



Category 3. For this case, we have





1 − e

−λt



+ Q



1 − e

−λt



α(0, 0),

= λ (Q

+ Q

) α (0, 0) ,

= T

∗

+ Q

2 (1 − Q

)

Category 4. As for Category 2, in this case we

should distinguish the ﬁrst of these arriving packets

and others. So

= P



1 − e

−λt



α(0, 0) = P

= τ

λt

, T

= T

∗

Finally, taking into account of asynchronous trans-

missions, we obtain the mean notiﬁcation time:

not

λ [π

+ (1 − π

) (1 − P

) (1 − π

)]

(10)

We adopt the following iterative procedure to cal-

culate this performance index.

Step 0. Deﬁne an initial value for P

Step 1. Deﬁne initial values for transmission prob-

abilities τ and τ

Step 2. Calculate, ﬁrstly, probabilities P

and P

and then the steady-state probabilities α(i, k). Us-

ing α(0, 0) and α(1, 0), calculate modiﬁed values for

transmission probabilities. If differences of initial and

modiﬁed values are greater than a predeﬁned limit ∆

repeat this Step, using new initial values for τ and τ

– the half-sums of their old initial values and the mod-

iﬁed values.

Step 3. Calculate, ﬁrstly, T

and p

, using (8) and

(9), secondly, the steady-state probabilities π

and

then the modiﬁed value of probability P

. If the dif-

ference of initial and modiﬁed values of P

is greater

than a predeﬁned limit ∆

, return to Step 1.

Step 4. Calculate the mean notiﬁcation time by

(10).

We do not prove exactly the convergence of this it-

erative technique due to its complexity. In practice,

numerous examples of adopting the technique with

various values of network and trafﬁc parameters have

shown that this technique provides very fast conver-

gence to the solution and high speed of T

not

calcula-

tion.

4 NUMERICAL RESULTS

To validate our model, we have compared its results

with that obtained by GPSS (General Purpose Sim-

ulation System) simulations (Schriber, 1974; GPSS

World, 1998). The object of our numerical investiga-

tions was an ad hoc 802.11b (with Short Preamble)

network (IEEE 802.11b, 1999) of N statistically ho-

mogeneous stations generating broadcast packets of

the same length L. The every station’s buffer size B

was chosen equal to 100 packets. In our simulation

model, we took into account of all real features of the

802.11 MAC protocol and did not adopt the assump-

tions used in analytical modelling and described in

Section 2. In each run (it took about an hour on the

average) of the simulation model, we observed values

of the measured performance indices and stopped the

simulation when their ﬂuctuations became quite small

(within 0.5%).

In Fig. 3 we show how the mean notiﬁcation time

depends on the average generation time T

g en

−1

with the packet length L = 1000 bytes that

corresponds to the DATA transmission time t

850 µs. The dependency has been obtained analyti-

cally (the “Math” curve) and validated by simulation

(the “GPSS” circles). In particular, the comparison

has shown a high accuracy of the analytical model:

the errors never exceed 5% with the mean notiﬁcation

time estimation.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Math

y=x

GPSS

Figure 3: Mean notiﬁcation time T

not

(s) versus the average

generation time T

gen

(s) with L = 1000 bytes and N = 50:

Math – analytical method; GPSS – simulation.

The curve T

not

g en

) in Fig. 3 has a clear mini-

mum at T

g en

= T

opt

g en

≈ 0.48 s. When T

g en

exceeds

opt

g en

, most of packets are transmitted asynchronously

(and hence, successfully) and the mean notiﬁcation

time is approximately equal to T

g en

. However, in this

case there are wide gaps between packet arrivals, and

the network capacity is used ineffectively. With high

ANALYTICAL STYDY OF BROADCASTING IN 802.11 AD HOC NETWORKS

generation rates (T

g en

< T

opt

g en

), most of packets are

transmitted synchronously, what causes frequent col-

lisions and hence T

not

growth. Moreover, there is a

threshold (≈ 0.015 s in Fig. 3) of T

g en

, below which

the network becomes saturated and buffers are almost

always overﬂowed. In this saturation area, T

not

does

not change.

Fig. 4 shows dependencies T

not

g en

) for various

packet lengths. It proves that the optimal generation

rate depends essentially on the packet length (as well

as on the number N of stations – these results are not

shown). Concerning the form of this dependency, it

appears that the optimal average generation time de-

pends on both the packet length and the number N of

stations almost linearly.

0 0.1 0.2 0.3 0.4

0.1

0.2

0.3

0.4

0.5

L=500 byte

L=1000 byte

L=2000 byte

y=x

Figure 4: Mean notiﬁcation time T

not

(s) versus the average

generation time T

gen

(s) with N = 50 and various packet

lengths L: (a) L = 500 bytes; (b) L = 1000 bytes; (c)

L = 2000 bytes.

5 CONCLUSION

In this paper, we have developed a novel analytical

method to study the performance of an ad hoc 802.11

network of stations generating broadcast packets. The

method based on Markov models allows estimating

the main performance index of such a network: the

mean notiﬁcation time namely is the interval between

consecutive successful receipts of the same source’s

packets. Using the method, we succeeded in optimiz-

ing the packet generation time to minimize the mean

notiﬁcation time.

According to numerical results obtained by both

the developed method and simulations, our method

is quite exact: the errors never exceed 5% with the

notiﬁcation time estimation. This method provides a

high speed of calculating the values of performance

indices, which has allowed us to perform the exhaus-

tive search of optimal packet generation time.

As a future research activity in this direction, we

propose extensions of this method to take into account

the cases when (a) the wireless channel is not ideal;

and (b) broadcasting stations can be hidden from each

other.

ACKNOWLEDGMENT

This work was partially supported by Russian Science

Support Foundation.

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