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All content in this area was uploaded by Khaled Elleithy

Content may be subject to copyright.

Abstract-The capacity of mobile ad hoc network (MANET) is

typically determined by the size of network, routing protocol,

mobility and the interactions that occur between the nodes.

Moreover, these critical parameters cause the loss rate that has

severed impact on the performance of the MANET. This situation

even becomes worst when these critical parameters are chosen

inappropriately. This paper presents an analytical model that

incorporates most of the critical parameters that can influence the

capacity of MANET. Based on the analytical model, an efficient 3-

phase algorithm is designed to optimize the performance of

MANET in terms of an increased capacity and the reduced

transmission delay. The proposed 3-pahse algorithm considers

both delay-tolerant and delay-sensitive network traffics. In

addition, the 3-phase algorithm can be used to approximate both

the best and worst case capacities of MANET with the relaying

and non-relaying nodes.

I. INTRODUCTION

Much research has already been done for improving the

performance of MANET [1, 2, 5]. The capacity of a fixed

wireless network decreases as the number of nodes increases

when all the nodes share a common wireless channel [1]. This

scheme was shown to increase the capacity of the MANET in

such a way that it remains constant as the total number of

nodes increases in the system. More recently, [3] has proposed

a two phase packet relaying technique. The main intention of

this technique was to reduce the overall packet transmission

delay. However, the delay experienced by packets under this

strategy was shown to be large and it can be even infinite for a

fixed number of nodes in the system, which has prompted

more recent work presenting analysis of capacity and delay

tradeoffs. However, the proposed 3-phase technique is highly

efficient in such a way that it shows that the capacity of

MANET can be increased at the expense of comparatively

small increased in the transmission delay.

According to the analysis of Gupta and Kumar [1], the

capacity region of network is defined as n(n-1) where n

represents the number of nodes. Two types of capacities are

typically measured: transport and throughput. The transport

capacity is determined by multiplying bits and the distance per

second where as the throughput capacity is expressed simply

by bits per second. In a MANET, the transport capacity is

approximated as O(√n) bit-meter per second whereas the

throughput capacity of each node is O(1/√n) bit per second.

However, this analysis does not include the mobility of nodes.

The proposed analytical model and 3-phase algorithm not only

accounts the transport and throughput capacities of MANET

but also considers the node mobility at one specific point. Our

numerical and simulation analysis demonstrates that the

capacity of MANET can be improved significantly if the

critical parameters are set intelligently.

II. PROPOSED ANALYTICAL MODEL AND THE 3-PHASE

ALGORITHM

Our proposed analytical model is based on the method

proposed by Gupta and Kumar [1]. They considered a static

model where all nodes are fixed and relaying is an allowable

property of MANET. The node positions Xi are independent

and identically-distributed in the open disk of a unit area. The

destinations are chosen independently and the destination for

each source node is chosen with respect to the node closest to a

randomly chosen point on the disk. For such a static model, the

upper and lower bounds on the asymptotically feasible

throughput reaches to infinity for each pair of source and

destination (S-D) [1]. However, the capacity of mobile nodes

without relaying can also be computed [5]. This requires

considering a model that consists of n nodes in an open disk of

unit area where the radius can be approximated as 1/√π.

Although, [3, 4] proved that the capacity of MANET is

constant, but they did not determine the particular capacity at

one certain point of time. In other words, the proposed

algorithm considers the transmission time which involves one

particular pair of S-D. However, we use these 2 basic models

to develop the proposed algorithm that considers both the delay

time and the broadcasting between the nodes of MANET.

A. Proposed 3-Phase Algorithm

We use special scheduling policy [3], called as “π,” for our

3- phase transmission algorithm. This scheduling policy (π)

selects S-D pairs randomly in each time slot t. Our

transmission algorism is divided into three phases. In the first

phase, the transmission occurs only between the sender that

has packets for transmission and the relay nodes as shown in

Fig. 1. The relay nodes may have some part of packets which

must be sent to the destination or between sender and

The Worst and Best Case Capacity Analysis of

Mobile Ad Hoc Networks (MANET) Using a 3-

Phase Algorithm

Syed S. Rizvi1, Aasia Riasat2, Mustafa A. Khan3, and Khaled Elleithy4

Computer Science and Engineering Department, University of Bridgeport1, 3, 4, Bridgeport CT, 06604

Department of Computer Science, Institute of Business Management2, Karachi, Pakistan

{1srizvi, 3mustafak, 4elleithy}@bridgeport.edu, 2aasia.riasat@iobm.edu.pk

594978-1-4244-1777-3/08/$25.00 ©2008 IEEE

Authorized licensed use limited to: University of Bridgeport. Downloaded on February 24,2010 at 12:05:57 EST from IEEE Xplore. Restrictions apply.

destination if the nearest node of destination is sender node. In

the second phase, these nodes move around in unit circle area

with the radius of 1/√π. During the second phase, no

transmission is carried out among mobile nodes until the relay

nodes approach an appropriate destination node. Although, this

mobility of nodes may increase a delay time, this can not be

influenced to the total throughput. Lastly, the processing of

third phases begins when the location of mobile relay nodes is

near the destination node as shown in Fig. 2. It should be noted

in Fig. 2 that the first and the third phases of the proposed

algorithm are interleaved since the processing of first phase

occurs in even time slots whereas the processing of the third

phase occurs in odd time slots.

B. Implementation of the Proposed 3-Phase Algorithm

In order to implement the proposed algorithm, we use

scheduling model (π) that randomly selects one sender node

that has packets for transmission with one destination node that

exists within the unit circle area of n mobile nodes. Moreover,

we assume that there are two or more mobile nodes which are

located within the close proximity of the sender node. This

second assumption implies that the transmissions can occurs at

a distance of order of 1/√n. In first phase, sender distributes

each packet to 10 mobile nodes, so each mobile node (relay

node) has a part of the sender packets. In the second phase,

these mobile nodes move around the unit circle. If some relay

node(s) which has the part of the sender’s packet enters within

the close proximity of destination node, the transmission

occurs immediately. Finally, in the third phase, the relay

node(s) transmits the part of the sender’s packet to the

appropriate destination node. Both second and the third phases

of the proposed algorithm are repeated until the destination

receives the entire packet.

III. THE BEST AND WORST CASE CAPAC ITIES FOR MANET

In this section, we present the analysis of the best and the

worst case capacities for a MANET using the proposed 3-

phase algorithm.

A. The Worst Case Capacity

For the worst cast capacity, we assume that the transmission

occurs for the largest time. Also, we assume that Pi is the

maximum number of packets that can be transmitted from the

ith source node to a destination node. The size of a transmitted

packet between each pair of S-D can not exceed to LP. In the

given scenario, each relay node may start transmission at

different time. Based on the above assumptions, the packet

transmission in MANET can be estimated as:

1

0

2

M

Transmission

P

iP

i

SD L P L

−

=

−⎯⎯⎯⎯→ +

∑ (1)

where, M represents the number of relay nodes.

Moreover, if relay nodes transmission occurs at different

times, then Ti

L

represents the transmission time between the

ith source and the other relay nodes. In addition, the source and

the relay nodes are assumed to have the largest number of

packets for transmission. Based on the above argument, the

total transmission time can be approximated as follows:

Transmission Time

1

0

M

Tt Ti

i

L

T

−

=

=+

∑ (2)

Using (1) and (2), we can derive a closed form expression

for estimating the worst case capacity (WC capacity) such as:

1

0

2

M

capacity P Tt Ti

i

WC L L L

−

=

⎛⎞

+

⎜⎟

⎝⎠

∑

(3)

where the total delay time is set to 0

B. The Best Case Capacity

For the best case capacity, we consider the shortest

transmission time between a pair of S-D. The shortest time is

generated when the transmission starts simultaneously between

the relay nodes and the destination node. This transmission

time can be considered as a time period in which the relay node

has the largest packet size for transmission towards a

destination. This leads us to an expression for the best case

capacity: capacity

BC =PTt

L

L. Finally, we can combine the

result of worst and the best case capacities. When the

Fig. 2 Processing of the first and the third phase. The parameter Lp is set to 2,

L

Tt is the transmission-time ≤ 2 packets, M is set to 3, P0 is set to 2, P1 is set to

1, P2 is set to 2, and Tt0 , Tt1, Tt2 represents the required transmission time

Fig.1. Processing of the first phase. The biggest rectangle is unit rectangle

which is 1㎡. Small circle presents the distance at which nodes ca

n

communicate. This model has 12 mobile nodes and 3 relay nodes

595

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characteristics of property (3) are true, the total transmission

throughput reaches to

()

i

Oj between node i and j at distance

of order 1/√n. One can approximate the best and the worst case

capacities ( wb

C−) such as:

()

()

()

1

0

lim 2 ( )

ti

M

P

Tt PTt

wb ti

CLLTtLL

λ

−

−−>∞ =

⎛⎞

=+≤≤

⎜⎟

⎝⎠

∑ (4)

where )(t

λ

is the throughput

C. Reducing Delay Time

The delay time is always generated with a certain probability

with respect to a certain complexity such as O(1/n). In the

given scenario, when the transmission occurs, relay nodes

approach to appropriate destinations. The second phase of the

proposed algorithm (see Fig. 2) can be effectively used to

reduce the delay time by improving the probability (see Fig. 3).

It is observed that the second phase of the proposed algorithm

improves this probability by asymmetrically distributing the

packets that each relay node may transmit. As the number of

relay node increases that carry the identical packets, the

probability of constructing the pairs of S-D nodes also

increases. The improvement in the probability is in the order of

O (c/n), where c > 0.

D. Numerical and Simulation Results

We assume that the location of a mobile node may change

randomly as shown in Fig. 4. In addition, each node has

enough buffer size with the maximum capacity of 80 kbps

where as the packet size is 8 bits. For the sake of simulation,

we consider a network of 100 nodes where each node may

transmit 10 packets per second. The simulation is setup in ns-2

and the results are presented in Table I. The proposed

algorithm reduced the over all delay time as shown in Table I.

IV. CONCLUSION

This paper presented an analytical model that uses special

scheduling policy for the random selection of the S-D pairs.

Based on the analytical model, we designed an efficient 3-

pahse algorithm that can be effectively used to analyze the

capacities of MANET. The proposed algorithm considers the

random selection of S-D pair which is essential in order to

produce the correct approximation of best and worst case

capacities. Our results have shown that the capacity of

MANET can be improved by using the proposed algorithm.

Also, the numerical results suggest that the transmission delay

can be reduced even in the presence of node mobility.

REFERENCES

[1] Piyush Gupta and P.Kumar, “The capacity of wireless networks,” IEEE

Transactions on Information Theory, Vol. 46, pp. 388-404, 2000.

[2] L. Jinyang, C. Blake, S. Douglas, D. Couto, I. Lee, and R. Morris,

“Capacity of Ad Hoc Wireless Networks,” In the proceedings of the 7th

ACM International Conference on Mobile Computing and Networking,

pp. 61 – 69, Rome, Italy, July 2001.

[3] M. Grossglauser and T. David, “Mobility increases the capacity of Ad-

hoc wireless networks,” IEEE/ACM Transactions on Networking,

Vol.10, no. 4, pp. 477 – 486, 2002.

[4] C. Schindelhauer, T. Lukovszki, S. Rührup, K. Volbert, “Worst case

mobility in Ad Hoc networks,“ Proceedings of the fifteenth annual ACM

symposium on Parallel algorithms and architectures, pp. 230 – 239,

2003.

[5] C.-K. Toh, “Maximum Battery Life Routing to Support Ubiquitous

Mobile Computing in Wireless Ad Hoc Networks,” IEEE

Communications Magazine, Vol. 39, no. 6, pp. 138 - 147 June 2001.

[6] A.J. Goldsmith and S.B. Wicker, "Design challenge for energy

constrained ad-hoc wireless networks," IEEE Wirel. Commun., vol.4,

pp.8–9, Aug. 2002.

Fig.3. Biggest rectangle is a unit rectangle (1㎡) and the small circle re

p

resents

the distance at which nodes can communicate. This model has 12 mobile nodes,

3 relay nodes and 3 reproduced relay nodes.

Fig.4. Location of mobile nodes in the simulation. The unit rectangle is 1㎡.

The red, blue, and green nodes represent the source, destination, and the rela

y

node, respectively. Simulation consists of 3 relay nodes and 100 regular nodes

TABLE I

THE RESULT OF RELAY MOBILE NETWORK WITHOUT REPRODUCTION

NRN Ttime (sec) Delay (sec) Throughput

1 3 0.0112 3406232 14285.70

2 4 0.0096 5056653 16666.66

3 3 0.0112 3442331 14285.70

4 3 0.012 3515001 13333.32

5 5 0.0096 4160049 16666.66

6 8 0.0096 4758960 16666.66

7 6 0.012 5544597 13333.32

8 2 0.012 1378757 13333.32

NRN = THE NUMBER OF RELAY NODE, TTIME = TRANSMISSION TIME

596

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