Modeling and Improving CSMA Unfairness inMobile Ad Hoc Networks

Skander Banaouas Paul Muhlethaler

INRIA Rocquencourt INRIA Rocquencourt

Le Chesnay FRANCE Le Chesnay FRANCE

Iskander.Banaouas@inria.fr Paul.Muhlethaler@inria.fr

Abstract—In this paper we study and model the unfairness inCarrier Sense Multiple Access (CSMA) Mobile Ad hoc Networks.We use a Markov model to represent the functioning of ournetwork. The Markov states correspond to the set of nodes whichtransmit simultaneously. This model allows the steady state tobe simply computed when the network is overloaded. When theCSMA back-off is small compared with the duration of a packet,we show that the most probable states are those containing themaximum number of stations, which corresponds to a maximumnumber of simultaneously transmitting nodes. In other words,the nodes in the maximum sets of the graph inferred by theCSMA rule are favored.

We study this model with several examples of ad hoc networks.We compare the predictions of this model with simulation resultsand observe that the matching is good. We show how this modelcan be extended to situations where there are hidden nodes. Wealso show that our model can be used to adjust the back-offwindows of the network nodes in order to enhance fairness.

I. INTRODUCTION

The area of Mobile Ad hoc NETworks (MANETs) is a very

challenging research field. The need for connectivity in these

networks has given rise to a large number of studies, particu-

larly regarding routing. Many proposals have been made, e.g.

[1], [2], [3], [4] and a few of them are now standards [5], [6].

The problem of the access schemes in these networks is also

extremely challenging. Many protocols have been proposed for

TDMA-based (Time Division Multiple Access-based) access

schemes in ad hoc networks e.g. NAMA [7], USAP [8], [9]

and ASAP [10]. These protocols share the idea of reserving

time-slots in the 2-hop neighborhood. This reservation rule is

used to avoid collision with neighbor nodes or with hidden

nodes.

Despite these studies for TDMA-based access which usually

target military applications, main-stream access schemes really

implemented or experimented for MANETs are CSMA-based

(Carrier Sense Multiple Access-based) protocols. The reason

for this probably lies in the dominant position of the IEEE

802.11 standard whose first version was published in 1997 and

thus has existed for more than 13 years. The existence of af-

fordable IEEE 802.11 products sometimes controlled by open

source drivers has motivated the development of numerous

products and academic studies based on IEEE 802.11 chipsets.

This work was supported by the RAF (Reseau Ad hoc haute eFficacite)project funded by the DGE (Delegation Generale des Entreprises)

However, these techniques are known to suffer from seri-

ous drawbacks such as: unpredictability, unfairness, hidden-

collisions, exposed nodes, etc. But it has also been shown

in [11] that Quality-of-Service (QoS) constraints can be suc-

cessfully handled with CSMA-based approaches if additional

mechanisms such as admission control, reservation and prior-

itization schemes are added to the network.

In this paper we propose an analytical model to predict the

unfairness of CSMA-based protocols used in MANETs. We

show how the model can be extended to the case of MANETs

where there are hidden collisions. We also prove that the

unfairness of CSMA-based protocols can be mitigated using

different values for the back-off windows.

This paper is organized as follows. Section II presents

related works. Section III describes the physical and the

network model used in this paper. Section IV describes the

analytical model which allows the throughput of the network

nodes to be computed. In section V we propose a simple

example of the analytical model for a given ad hoc network.

We compare the network nodes’ throughput computed by the

analytical model with that obtained by simulations. We also

analyze the network nodes’ throughput when there are hidden

collisions. In section VI we study how changing the sizes

of the collision windows can help to improve the fairness of

CSMA-based protocols.

II. RELATED WORKS

Many studies such as [12], [13], [14], [15] deal with the

unfairness of CSMA-based access protocols in ad hoc net-

works. Most of them use either simulations or real experiments

to quantify the unfairness of CSMA-based access protocols.

[14] especially studies the unfairness of CSMA-based access

protocols when TCP is used to maintain connections. [15]

distinguishes between short and long term unfairness.

In contrast to these studies, [16] proposes an analytical

model to capture the performance of the IEEE 802.11 access

protocol when the network is in overload. However this

analysis does not take into account the network topology; there

is no spatial reuse in this model. Simultaneous and successful

transmission can not occur, it is as if the model were for a

wired network. Nevertheless simulations show the accuracy

of the model to capture the effect of the binary exponential

back-off.

978-1-4244-8435-5/10/$26.00 ©2010 IEEE

Other papers try to investigate the throughput of CSMA

protocols. In [17] the authors study how the network through-

put can be optimized by a proper tuning of the carrier sense

threshold. The network is supposed to be in overload thus all

nodes have pending packets. The interference is assumed to

result only from the nearest neighboor. The intensity of the

point process built with the simultaneous transmissions in the

network is assumed to be a Matern point process. The model

presented in [17] show that a suitable tuning of the carrier

sense threshold allows the network throughput to be very

significantly improved. This result is confirmed by simulation.

In [18] a stochastic geometry model for the performance

analysis and the planning of dense IEEE 802.11 networks is

presented. This model, also based on Matern point processes,

takes the effect of interferences and that of CSMA into account

within this dense network context. The long term throughput

obtained by end-users when the access point density increases

is studied. The model derived in [18] can be used for the

planning of managed networks and for the economic modeling

of unplanned networks.

A previous study [19] proposed a Markov analysis of

CSMA networks in the same spirit as the model presented

in this paper. However our model is simpler and gives closed

formulas for the nodes’ throughput at highload. Our model is

also extended for hidden nodes.

The next section provides the general assumptions that are

used in this article, especially for the simulation part of this

paper.

III. PHYSICAL MODEL AND TRAFFIC MODEL

A. Physical model

We use the simple power law decay to evaluate the trans-

mission power received at distance d from the transmitter. The

power at 1 meter is denoted by P0, the decay rate of the

transmission power is β. Thus the received power at distance

d is P :

P =P0

dβ. (1)

We denote by Ccs the carrier sense threshold, which governs

the carrier sensing. If the power received on the channel by

a given node is greater than this threshold, the channel is

assumed be busy and this node refrains from transmitting. If

the power received on the channel by a given node is smaller

than this threshold, the node is allowed to transmit.

Moreover a packet is assumed to be correctly received if

the corresponding value of the SINR (Signal over Interference

plus Noise Ratio) is greater than the capture threshold that we

denote by T .

B. Traffic model and fairness

We suppose an exponentially distributed traffic at each node

i of rate λi. We also assume that the duration of a transmission

is exponentially distributed of rate μi. The access scheme is

of the IEEE 802.11 type.

It is difficult to define a fairness objective in multiple ad

hoc networks. Let us assume that we have a set of connexions

between pairs of nodes in the network; these pairs of nodes

may be more that one hop away. A fairness objective should

that the throughput is equally shared between the defined set

of connexions. The study of this fairness goal is complex

because of the relayed traffic, the precise topology and the

routes to convey the connexions must be taken into account.

In this paper we use a simpler fairness objective which is that

the throughput of each node should be equally shared. We

assume that each node transmits packets evenly to each of its

neighbors.

IV. MARKOV MODEL AND EVALUATION OF THE

UNFAIRNESS

A. Markov model and computation of the nodes’ throughput

To simplify the model we assume that when a node itransmits to a node j, there is no other possible transmission

within a given radius of node i. We consider the set of nodes

N = {1, 2, 3, .., n− 1, n} and the graph of links {i, j} when

node i and j can not transmit simultaneously. We consider the

set of nodes that are transmitting at the same time t. These

sets are the Markov states s of our model. For a given state

s, we note T (s) the set of nodes that can transmit (according

to the CSMA rule) while all the nodes in s are transmitting.

Thus for any node l in T (s), we note by s ∪ {l} the Markov

state where all the nodes in s plus l are transmitting. For any

node m in s, we denote by s \ {m} the Markov state where

node m is removed from s. In a given state s a node i in T (s)may be added with probability αi and a node may be removed

with probability μi.

For αi we will use an approximation. For small values of

λi we have αi � λi, the access rate is governed by the arrival

process of packets in the node. Then when the sum of the loads

within carrier range of node i (including node i itself) exceeds

the wireless channel capacity there is a queue of packets in

node i. The transition rate for admitting a new node in state

s is then governed by the IEEE 802.11 back-off strategy. We

approximate the transmission rate by αi � δi with 1δi

being

equal to CW2 where CW is the size of the collision window

of the IEEE 802.11 access protocol1.

When the network is stable, the steady state probability

satisfies the following equation :

P (s)(∑

i∈s

μi +∑

i∈T (s)

αi

)=

∑i∈T (s)

μiP (s ∪ {i}) +∑i∈s

αiP (s \ {i}). (2)

This equation of a Markov reversible chain has the following

solution:

μiP (s \ {i}) = αiP (s). (3)

1This approximation is obtained by assuming that in IEEE 802.11 the actualtransmission occurs in the first collision window and by writting that the twomeans are equal

Thus we obtain :

P (s) =(∏

i∈s

ui

)P (φ). (4)

where ui = αi/μi and φ denotes the state where no node is

transmitting. Let us denote by S the set of all the possible

states where at least one node is transmitting. For normalisa-

tion reason we obtain:

P (φ) =(1 +

∑s∈S

∏i∈s

ui

)−1(5)

Let us denote by R(l) the set of states where node lis transmitting. In a first step we assume that there are no

hidden collisions. Thus when a transmission is permitted by

the CSMA protocol, the transmission is a success unless

transmissions begin at the same time. We do not consider here

the effect of such an event. The throughput thl of node l is

thus:

thl =(1 +

∑s∈S

∏i∈s

ui

)−1.

∑s∈R(l)

(∏i∈s

ui

). (6)

This formula allows the links throughput to be computed and

thus the unfairness of the network to be measured.

B. Throughput at low load

We have seen in the previous subsection that at low load

we have αi � λi. Thus the formula above for thl gives

thl = αl.

Between the low load conditions and the overload condition

it is difficult to quantify the transition rate when a state gains

a new transmitting node. We have tried a few approximations

to quantify this transmission rate when the load is between

low load and overload and then to solve the equations of the

steady state. But the results were not good, mainly because

the approximations were too imprecise. For a given ad hoc

network, it is possible to evaluate the input load from which

the total load of nodes within carrier sense reach exceeds

the channel throughput. We know that when this occurs the

queues start to increase in the network nodes. Thus this load

is the starting load where a few nodes see their throughput

continuing to increase and other nodes see their throughput

decreasing. The characterisation of the throughput of the

network in overload is given below.

C. Asymptotic values of the throughput in overload

We assume that the network is in overload. All the nodes

have pending packets in their queues. The transition to a state

with a transmitting node is governed by the backoff strategy of

the CSMA protocol. We also assume that the backoff strategy

is the same for all the network nodes and that the mean back

off is small with respect to the mean packet length. We also

assume that the packet length is Poisson and has the same

mean length for all the nodes. Although the model is rigorous

only if the packet and the back-off length are exponentially

distributed, we also use the same model with a packet length

of constant size and with a back off time randomly chosen in a

collision window. In these conditions and considering that the

backoff time is small compared with the transmission time,

we have:

∀i ui ≡ C � 1 (7)

and the dominant factor in the formula giving the throughput

of a node is given by the states with the maximum number

of transmitting nodes. Let us call MS the set of states with

the maximum number of transmitting nodes, Card(MS) the

number of states in MS. For a given node l we also denote

Cardl(MS) the number of states in MS where node l is

transmitting. With these notations we obtain :

thl � Cardl(MS)Card(MS)

. (8)

This formula allows the throughput of a node to be com-

puted simply. We notice that only nodes which are transmitting

in the state with the maximum number of transmitting nodes

have a non zero throughput. When one node appears more

often than another in the state with the maximum number of

transmitting nodes, this node has a larger throughput.

Contrary to what we might think, the throughput of a node

does not depend on the number of its neighbors. However

another intuitive ideas is that the throughput of a node depends

on its location and that nodes on the border of the network

exhibit higher throughput. Although we will not explicitly

prove (or disprove) this latter idea here, we will see in the

examples given below that nodes on the border of the network

do indeed reach a higher throughput than the nodes more in

the center of the nework.

D. Extensions of the model

D.1 Throughput with hidden nodes

In the above model, we have assumed that when the carrier

sense allows a transmission this transmission is bound to be

successful. This is not always the case. For instance, with

hidden collisions, a transmission may be possible with respect

to the carrier sense rule while in fact provoking a collision.

Thus equation 6 which gives the throughput may be corrected

by adding a probability of success which corresponds to the

probability of capture. Thus we have

thl =(1 +

∑s∈S

∏i∈s

ui

)−1 ∑s∈R(l)

Psucc(l, s)(∏

i∈s

ui

). (9)

where Psucc(l, s) is the probability that the transmission of

node l is captured by its destination node say node j.

If the hidden collisions for node l may only occur due to

the transmission of nodes in s, a simple analysis determines

Psucc(l, s). Depending on the destination node j, the trans-

mission may or may not be captured and thus Psucc(l, s) can

be easily computed.

In contrast, if for some node l a hidden collision may result

from other transmissions than those in s then the computation

of the throughput is more complex and we have to assume

that the network is in the states s ∈ MS 2. For a given node

l we must consider all the simultaneous transmissions at the

beginning of node l’s transmission and the possible changes in

transmitting nodes when a transmission in s\{l} stops. Taking

into account all the possible cases and determining if node l’stranmission is still captured allows Psucc(l, s) to be computed.

An example of such a computation is given in section V-C.

D.2 Link throughputs

In some networks, it may be useful to consider transmission

between two given nodes rather than only considering the

transmission of a given source node without specifying the

destination. To model the effect of the CSMA, this leads to

considering a graph which has links as vertices. In this graph

two vertices (thus links of the ad hoc network) are linked by

an edge when the two links of the ad hoc network can not

transmit simultaneously because of the carrier sense effect.

Using this graph, we can use the same methodology as that

used in the previous sub-section. We have to find independent

sets of links, i.e. pairs of source-destination nodes which can

transmit simultaneously according to the CSMA rule. The

equations remain the same with the same product form. Instead

of computing the throughput of nodes, the model naturally

involves computing the throughput of a link.

V. USING THIS MODEL WITH GIVEN AD HOC NETWORKS

A. Presentation of the model in a simple example

Below we give an example of the utilization of this model

on the topology presented in Figure 1.

1 2 3 4

5

6(0,0) (200,0) (400,0) (600,0)

(773,100)

(773,-100)

Fig. 1. Network topology 1

The graph of the nodes within carrier sense range is given

in Figure 3. The values of the physical parameters are given

in Figure 2. These values are used in the analytical model and

in the simulations for the network topology 1.

P0 β Ccs T1.0 4.0 7.7× 10−12 2.0

Fig. 2. Physical parameters used for the network topology 1

In this network and with the physical parameters used, the

carrier sense only allows node 1 and node 5 or node 1 and

node 6 to transmit simultaneously. Any possible simultaneous

transmissions are forbidden according to the CSMA rule.

Moreover with the capture threshold T set to 2 there are no

hidden collisions.

2This is true in overload as seen in section IV-C

Fig. 3. Graph of the nodes within carrier sense range for network topology1

All the possible states are presented in Figure 4 with the

transitions between states. In this example we assume that

μi ≡ 1. We use equations 5 and 6.

0.0.0.0.0.0

1.0.0.0.0.0

0.1.0.0.0.0

0.0.1.0.0.0

0.0.0.1.0.0

0.0.0.0.1.0

0.0.0.0.0.1

1.0.0.0.1.0

1.0.0.0.0.1

μ1

α1

μ2

α2

μ3

α3

μ4 α

4μ5

α5

μ6

α6

μ1

μ1

α1

α1

μ1

μ1

α1

α1

α1

α1

μ1

α1

Fig. 4. The state transmission diagram of the Markov chain of thetransmissions for network topology 1

We have :

P (φ) =(1+α1 +α2 +α3 +α4 +α5 +α6 +α1α5 +α1α6

)−1.

For the throughput of the 6 nodes, we obtain:

thr1 =α1 + α1α5 + α1α6(

1 + α1 + α2 + α3 + α4 + α5 + α6 + α1α5 + α1α6

) .

thr5 =α1α5(

1 + α1 + α2 + α3 + α4 + α5 + α6 + α1α5 + α1α6

) .

thr6 =α1α6(

1 + α1 + α2 + α3 + α4 + α5 + α6 + α1α5 + α1α6

) .

We also have ∀i ∈ {2, 3, 4}

thri =αi(

1 + α1 + α2 + α3 + α4 + α5 + α6 + α1α5 + α1α6

) .

We find the following values :

thr1 = 0.96, thr5 = thr6 = 0.48

and

i = 2, 3, 4 thri = 0.008.

The asymptotic values at very high load are the following:

thrasyp1 = 1.0, thrasyp

5 = thrasyp6 = 0.5

and

i = 2, 3, 4 thrasypi = 0.

B. Comparisons between the model and simulation results

Our aim is now to compare the results obtained using the

model with simulation results. We built our own simulator to

efficiently implement the IEEE 802.11 medium access scheme.

The inter-arrival of the traffic is exponentially distributed

and we increase the input load simultaneously in all the net-

work nodes. Each node randomly selects one of its neighbors

as the destination node for its pending packets.

The result of this simulation is shown in Figure 5. We can

observe that there is a good matching between the model at

overload and the simulation results. In the Figure we have not

presented the value of the throughput foreseen by the model

for node 2 (i.e 0.008) . Node 2, which is in the carrier sense

of all the other network nodes, should in theory start to starve

for an input load of 16%. We observe this phenomenon for a

slightly larger input load of around 23%.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Nor

mal

ized

thro

ughp

ut

λ

Normalized throughput, β=4

n1n2n5n6

n1 - modn5 n6 - mod

Fig. 5. Throughput of nodes 1, 2, 5, 6 and for ad hoc network topology 1,no hidden collisions

The results of the model may appear rather intuitive for

network topology 1. Only node 1 and node 5 or node 1 and

node 6 can transmit simultaneously. These are the only two

states with the maximum number of transmitting nodes. Thus

applying the formula given in section IV-B gives the result we

obtain.

2

3

45 6

1

7

(0,0)

(0,200)

(0,400)

(-173,-100) (173,100)(-173,-100)(-346,-200) ( 346,200)

Fig. 6. Ad hoc network topology 2

P0 β Ccs T1 4.0 3.5× 10−11 6

Fig. 7. Physical parameters used for network topology 2

2 3

4

5

6

1

7

Fig. 8. Graph of the nodes within carrier sense range for network topology2

The results are less intuitive with ad hoc network 2

whose topology is shown in Figure 6. We use the physi-

cal parameters given in Figure 7 for the analytical model

and for the simulations. With these parameters and these

figures the graph of nodes within carrier sense range is

given in Figure 8. In this network, up to three simultane-

ous transmissions can occur. There are four possible states

{1, 4, 7}, {1, 5, 7}, {1, 5, 6}, {2, 5, 7}. There are also nine

states with two simultaneous transmissions (left to the reader)

and, of course, seven states with only one transmission.

We find the following values using the value of the IEEE

802.11 collision window adopted in our simulation :

thr1 = thr5 = thr7 = 0.736

thr2 = thr4 = thr6 = 0.248, thr0 = 0

The asymptotic values are the following :

thrasyp1 = thrasyp

5 = thrasyp7 = 0.75,

thrasyp2 = thrasyp

4 = thrasyp6 = 0.25, thrasyp

0 = 0.

We note that these values are very close to the exact values.

In Figure 9 we present the nodes’ throughput for ad hoc

network 2. We use the same back-off window of 32 slots.

We observe that in overload the simulation results match the

predictions of the analytical model very well.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Nor

mal

ized

thro

ughp

ut

λ

Normalized throughput, β=4

n 1n 2n 3

n 1 - modn 2 - mod

Fig. 9. Throughput of stations 1,2,3 for ad hoc network 2 , no hiddencollisions

C. Comparisons between the model and simulation resultswith hidden nodes

In this section we study the performance of CSMA with

hidden nodes and we compare the results of the prediction of

the model extended for hidden collisions presented in IV-D.

C.1 Study of ad hoc topology 3

1 (0 , 0 )

2 (170 , 0 )

3 (400 , 0 )

4(400 , 50 )

5(600 , 100 )

6 (600 , 0 )

7 (600 , 50 )

8 (570 , 100 )

Fig. 10. Ad hoc network topology 3.

P0 β Ccs T1.0 4.0 2.5× 10−11 15.0

Fig. 11. Physical parameters used for network topology 3

The topology of ad hoc network 3 is presented in Figure 10.

There are eight nodes in this network whose topology is close

to the topology of ad hoc network 1. The physical parameters

are given in Figure 11. With these figures node 1 and node

6 can transmit simultaneously as can node 1 and node 7 and

also node 1 and node 8. Thus {1, 6}, {1, 7}, {1, 8} are the

three maximum independent sets. In this configuration the

asymptotic value of the throughput when there are no hidden

collisions is thus 1 for node 1 and 1/3 for nodes 6, 7 and 8. We

assume that node 6, 7 and 8 transmit with an even probability

to nodes 3, 4 and 5 as node 1 only transmits to node 2. Yet

without hidden collisions but with a random back-off of 31

slots, the throughput is 0.956 for node 1 and 0.318 for nodes

6, 7 and 8.

When node 6 transmits simultaneously with node 1, there

is a hidden collision when node 6 transmits to nodes 4 and

5. When node 6 transmits to node 3 the packet is captured

even during node 1’s transmission. In the simulations we

have assumed that node 6 transmits with equal probability

to nodes 3, 4 and 5. Thus in the formula of IV-D we shall use

Psucc(6, {1, 6}) = 13 . When 7 transmits the packet is captured

by node 4 and 5 but not by node 3. Since the simulation

assumes that 7 transmits evenly to nodes 3, 4 and 5 we shall

use Psucc(7, {1, 7}) = 23 in the model. When 8 transmits the

packet is captured by nodes 3, 4 and 5. Thus we shall use

Psucc(8, {1, 8}) = 1 in the model.

The results of simulations are compared with the results of

the analytical model at overload in Figure 12. We observe that

the matching between the two approaches is rather good. In

order not to waste bandwidth with hidden collision there is a

single transmission attempt in the simulation. The maximum

throughput of node 1 is not shown in Figure 12 to improve the

readability of the Figure. The throughput of node 1 is found

to be 0.93 by simulation whereas the model predicts 0.956.

The prediction here is also good.The discrepency between the simulation and the model

probably results from the collisions of transmissions starting

nearly at the same time. These collisions are ignored in

the analytical model whereas the simualtions take them into

account. With 8 nodes and a back-off window of 32 slots the

probability of a collision due to two transmissions starting

nearly at the same time (thus not within carrier sense reach)

is not very small.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Nor

mal

ized

thro

ughp

ut

λ

Normalized throughput, β=4

n6n7n8

n6 modeln7 modeln8 model

Fig. 12. Ad hoc network topology 3.

C.2 Study of the ad hoc topology 2The analysis of hidden collisions is simple in ad hoc

network topology 3. The hidden collisions may only result

from node 1. In ad hoc network topology 2 if the capture

threshold is greater than 8 then node 1 is in collision when it

sends a packet to node 2 as soon as node 4 and 6 transmit.

Thus node 1 transmits successfully to node 2 only when the

network is in the state {1, 5, 7} however this condition is

not sufficient. Since the transmissions are asynchronous it is

necessary that the states after state {1, 5, 7} do not include

node 4 or 6 and the formula given in section IV-D can used

if we compute Psucc(1, {1, 5, 7}). We have to assume that

the back-off window and the packet duration are such that

ui � 1. The network is in the four more probable states

{1, 4, 7}, {1, 5, 7}, {1, 5, 6}, {2, 5, 7}. Node 1 successfully

transmits to node 2 if and only if

• the network is in state {1, 5, 7} when node 1 starts

transmitting; this occurs with probability 14

• when node 5 stops transmitting the next transmission

which can be either from node 4 or 5 should be from

node 5; this occurs with probability 12

• when node 7 stops transmitting the next transmission

which can either come from node 6 or 7 should be from

node 7; this occurs with probability 12 .

Thus

Psucc(1, {1, 5, 7}) =12

12

=14

and

thr1 =14Psucc(1, {1, 5, 7}) = 0.0625.

For node 2 the reasoning is simpler. Node 2 successfully

transmits if and only if the network is in state {2, 5, 7} when

node 2 starts transmitting and node 2 transmits to node 1. If

node 2 transmits to node 3 the transmission always results in

a collision. Thus Psucc(2, {2, 5, 7}) = 12 and thr2 = 1

412 =

0.0125. The term 14 comes from the fact that the network must

be in state {2, 5, 7} and we have four states with the maximum

number of transmitting nodes.

In Figure 13 we present the nodes’ throughput for ad

hoc network 2. We still use the same back-off window of

32 slots but the capture threshold is now 10. With this

capture threshold, there are hidden collisions. For instance,

node 1 transmitting to node 2 enters in collision with node

4 transmitting to node 3 but in the same conditions node 4transmits successfully to node 5.

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Nor

mal

ized

thro

ughp

ut

λ

Normalized throughput, β=4

n1n2n3

n1 modeln2 modeln3 model

Fig. 13. Throughput of nodes 1, 2, 3 for ad hoc network 2 and hiddencollisions

VI. MITIGATING THE UNFAIRNESS

In this section we study how the unfairness can be mitigated

using different values of the back-off windows for the different

nodes.

A. Example with ad hoc network 1

For ad hoc network 1, we use a back-off window of 775

slots for nodes 1, 5 and 6, a back-off window of 62 slots for

nodes 2 and 4, and a back-off window of 31 slots for node 3.

The results we obtain with such back-off windows are shown

in Figures 14 and 15. Figure 14 shows the throughput for

nodes 1, 5 and 6. These three nodes capture all the bandwidth

when the back-off windows are evenly distributed for all the

network nodes. With the new lengths of the back-off windows

we obtain a throughput of 0.08 for node 1 (predicted by the

model to be 0.1 ) and of 0.05 for nodes 5 and 6

We observe that the choice of back-off windows greatly

enhances the fairness in the network. Moreover the analytical

model offers a good estimate of the nodes’ throughput in

overload.

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Nor

mal

ized

thro

ughp

ut

λ

Normalized throughput, β=4

n1n5n6

n1 modeln5 or n6 model

Fig. 14. Mitigating the unfairness in ad hoc network topology 1.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Nor

mal

ized

thro

ughp

ut

λ

Normalized throughput, β=4

n2n3n4

n2 or n4 modeln3 model

Fig. 15. Mitigating the unfairness in ad hoc network topology 1.

B. Example with ad hoc network 2

For ad hoc network 2, we use a back-off window of 310

slots for nodes 1, 5 and 7, a back-off window of 155 slots

for nodes 2, 4 and 6, and a back-off window of 31 slots for

node 3. The results we obtain with such back-off windows are

shown in Figure 16.

We observe that the choice of the back-off windows greatly

enhances the fairness in the network. Moreover the analytical

model offers a good estimate of the nodes’ throughput in

overload for nodes 1 and 2. For node 3 the estimation of the

model is average.

We can further improve the fairness of the network by using

a back-off window of 1550 slots for nodes 1, 5 and 7, a back-

off window of 155 slots for nodes 2, 4 and 6, and a back-off

window of 31 slots for node 3. In overload conditions and with

these figures, simulations provide the following results: node

1’s throughput is 0.115, node 2’s throughput is 0.152 and node

3’s throughput is 0.53. In these conditions the model foresees

0.26 for node 1’s throughput, 0.22 for node 2’s throughput and

0.23 for node 3’s throughput. The predictions are not accurate.

The difference between the simulations and the model can

probably be explained by the fact that the model assumes an

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Nor

mal

ized

thro

ughp

ut

λ

Normalized throughput, β=4

n 1n 2n 3

n 1 - modn 2 - modn 3 - mod

Fig. 16. Throughput of stations 1,2,3 for ad hoc network 2 , no hiddencollisions

exponentially distributed back-off time. This is not the case for

the IEEE 802.11 because first the initial back-off is linearly

distributed and secondly when there is a defer due to the carrier

the back-off is frozen and resumes when the channel is free

again.

Although the predictions are not always completely accu-

rate, they nonetheless provide orders of magnitude. Thus, our

analytical model can be used to foresee suitable values of the

collision windows to improve the network fairness when the

initial unfairness is overwhelming.

VII. CONCLUSION

In this paper we have presented a model to evaluate the

throughput in a CSMA ad hoc network. We have shown that

the throughput of each node can be easily evaluated at both

low load and in overload conditions. We have shown that a

few nodes capture the bandwidth while the other nodes starve.

The nodes which capture the bandwidth are actually the nodes

which in the maximum sets of the graph induced by the carrier

sense rule. We have also shown how the model can take into

account the effect of hidden collisions. We have tested this

extension of the model on two examples. The prediction of

the model in these cases is fairly good. We have studied ad

hoc networks where the sizes of the back-off windows depends

on the node. We have shown that the model can predict the

nodes’ throughput rather well. Thus it is a good tool to evaluate

the sizes of back-off windows which offer a better fairness for

nodes’ throughput.

We believe that the model presented above can be applied to

other ad hoc networks than the two simple examples given in

this paper; the key to the analysis remains the computation of

maximum sets of the graph induced by the carrier sense rule.

We think that this model can be used to foresee suitable values

of the collision windows to enhance fairness in networks where

the discrepancy between the nodes’ throughput is large.

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