Optimizing Cached Route Time-To-Live in Mobile Ad-hoc Networks

Quang-My Tran Vietnam Posts and Telecommunications Group

Ho Chi Minh City, Vietnam mytq@vnpt.vn

Arek Dadej University of South Australia

Mawson Lakes, SA 5095, Australia Arek.Dadej@unisa.edu.au

AbstractAs reactive routing protocols take advantage of using recently active routes in the route cache, determining appropriate cached route Time-To-Live plays an important role in reducing the routing overhead by avoiding misrouting overhead caused by route errors. In this paper, an analytical model is presented for quantifying the bandwidth necessary to facilitate routing. Then, a numerical model for determining the optimal cached route Time-To-Live is derived to minimize the reactive routing overhead depending on network mobility and data traffic load. We show that cached route Time-To-Live should be set far shorter than that commonly seen in or recommended for simulations to save bandwidth. This paper provides a valuable insight into the fundamental limits of routing performance in MANETs in response to mobility.

KeywordsReactive routing protocol; Routing overhead; Misrouting overhead; Mobile Ad-hoc Network.

I. INTRODUCTION In a MANET, any transmission of control or data packets

from any node will consume not only the bandwidth of that node but also other nodes bandwidth because of the multi-hop nature. The higher the level of mobility, the more bandwidth is consumed for control packets and dropped data packets. However, little is known about how to minimize the routing overhead in response to mobility and data traffic load.

In reactive routing, a newly discovered route should be cached for later use when the same route is needed to reduce the control overhead and to lower routing latency. However, as nodes in a MANET are free to move independently, it is necessary to remove stale routes timely to avoid route errors which may cause extra routing delay and misrouting overhead. Therefore, to minimize routing overhead and routing delay, the cached route expiry time should be as close as possible its real lifetime depending on the network mobility.

Several works have evaluated the routing overhead for some different classes of routing protocols by simulation or theoretical analysis. In this paper, we focus on the theoretical analysis of the routing overhead for reactive routing protocols with a flat structure. Paper [1] provides a lower bound on the routing overhead by omitting the overhead component incurred due to link failures and the analysis is conducted for the DSR protocol without the route cache option. In [2], authors propose an analytical model for analyzing the routing overhead for the reactive protocol model without the route cache option. Paper [3] presents the average number of control packets incurred in a reactive routing protocol as a function of the network size. In [4] and [5], authors analyze the overhead

of reactive routing protocols without route cache option in networks with stationary nodes but unreliable. [6] estimates the number of control packets for two reactive routing protocols AODV and DSR with the cache route option. However, the routing overheads analyzed in the abovementioned works consider only the number of control packets and omit a significant amount of misrouting overhead incurred by lost data packets due to route invalidity. In terms of optimizing cached route Time-To-Live, authors in [7] propose numerical methods to determine the optimal Time-To-Live of a newly cached route to minimize routing delay.

The contribution of this paper is twofold. First, by considering the misrouting overhead in the analysis, the resulting model captures completely and thoroughly the amount of communication bandwidth required for performing reactive routing in MANETs in response to mobility. Second, by considering a cache route option in the analysis, the minimum cost of routing can be achieved depending on the mobility and traffic load in the network.

The rest of the paper is organized as follows. Section II describes notation, mobility and a reactive routing protocol model used in this paper. Section III analyses the routing overhead incurred by reactive routing. Section IV proposes a method to determine the optimal cached route Time-To-Live to minimize routing overhead. Section V presents numerical results of the study. A conclusion is drawn in Section VI.

II. PRELIMINARIES

A. Assumptions Let us consider a MANET with N nodes.

Identical node property: All nodes in the network are assumed equal in terms of the bandwidth demand and the need to communicate with one another. Packet generation rate is thus assumed the same for all nodes in the network.

Uniform distribution of destinations: Assume that the distribution of destinations for any node is uniform over the whole network. Thus, the probability that a node i communicates with any node j of other N 1 nodes is the same as p = 1/(N 1).

Shortest route principle: Each message is delivered through the shortest route available. This assumption together with the assumption that the network is dense enough, we can approximate the route length as L = d/R, as used in [5], [8]. Here d is the distance between two nodes, and R is the transmission range.

2015 IEEE 29th International Conference on Advanced Information Networking and Applications

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B. Notation and Definition Node i is said to select a node j to communicate with if

they are the source and the destination, respectively, of a packet transmission.

Definition: The total routing overhead is defined as the total amount of bandwidth consumed by control packets incurred in route discovery and route maintenance processes and by data packets lost due to misrouting.

Besides the bandwidth consumed by control packets, routing overhead in reactive routing includes the bandwidth consumed by the data packets that have been sent through the network by the sources but could not reach the destinations because of route errors.

Notation that will be used in this paper is summarized in Table I and Table II. C. Mobility Metric

The mobility of nodes in a MANET whose area is larger

(from down state to up state or the opposite). The state of any link at time t in the network can be represented by a random variable m(t) = {0,1}, where 0 represents the link down (d(t) > R), and 1 the link up (d(t) R) states.

The probability of link change event (ProbLC) within a time interval [t, t+), q(t), is defined as follows:

RtdtmtmRtdtmtm

tq )(}1)(|0)(Pr{)(}0)(|1)(Pr{)(

TABLE I. TABLE OF NOMENCLATURES I

Network area

Packet generation rate per node (s-1) u Link break rate (s-1)

Time-to-Live of a cached route opt Optimal Time-to-Live of a cached route in terms

of routing overhead n Average number of neighbours, n = R21

perPacketRI

Total bandwidth required for routing per node per packet generated

IR Total bandwidth required for routing per node per second

L Average route length (in hops) between two communicating nodes

N Number of nodes in the network NRREQ Number of RREQ transmissions for a single route

discovery NRREP Number of nodes generating RREP packets for a

single route discovery R Radio transmission range of a node SD Data packet size (bits) SRREQ RREQ packet size (bits) SRREP RREP packet size (bits) SRERR RERR packet size (bits)

TABLE II. TABLE OF NOMENCLATURES II

p Probability that node A selects node B to communicate with, p = 1/(N 1)

pab Probability that node A selects node B to

communicate with or selects other node to communicate with and the communicating route is going via node B.

pa Probability that node A (or B) selects node B (or

A) to communicate with or selects a node other than node B (or A) to communicate with and the communicating route is going via node B (or A).

pb

Probability that node A and node B are two intermediate nodes on the same communicating route between two other nodes

pB*(t)

Probability that node B floods the network to search for a route within a time interval [t, t+).

pc

Probability that node A selects a node other than node B to communicate with and the communicating route is not going via node B.

pd(t)

Probability that node B floods the network within a time interval [t, t+) to search a route to a node other than node A and the communicating route is not going via node A.

p1(t)

Probability that node A knows a route to node B within a time interval [t, t+)

p0(t)

Probability that node A does not know a route to node B within a time interval [t, t+)

p11(t)

Probability that the cached route to node B is valid within a time interval [t, t+)

p10(t)

Probability that the cached route to node B is invalid within a time interval [t, t+)

q(t) Probability of link change within a time interval [t, t+)

According to [9], [10] and [11] the route duration can be approximated with an exponential distribution with mean r = 1/r, i.e.

tretp 1)(10 Generally, the exponential distribution of route duration

does not mean that the link duration can also be approximated with exponential distribution. However, for simplicity (to make the analysis possible), assume that the link duration can be approximated with an exponential distribution with mean u = 1/u, i.e

tuetq 1)(

The exponential distribution of link duration has also been proposed and verified by experimental data in [12] and [13] under the Random Walk Mobility Model and semi-Markov smooth mobility model, respectively. More specifically, the authors in [13] found that the link duration can be effectively approximated by an exponential distribution with parameter

eRV / , where V is the average speed and Re is the effective transmission range of a mobile node. The exponential

194

distribution has also been used for link duration in a number of research works, such as [7] and [14].

The ProbLC has been proposed in [15] as a generalized mobility metric for MANETs as it thoroughly represents network topology changes independent of any mobility model and the network size. In this paper, the ProbLC and u will be used interchangeably to represent mobility in MANETs. D. Reactive Routing Protocol Model

The reactive routing protocol modeled here is based on the concept of source routing. It is composed of two major mechanisms, i.e. Route Discovery and Route Maintenance.

1) Route Discovery When a source node in the ad-hoc network attempts to

send a packet to a destination but it does not already have a route to that destination in its route cache, it initiates a route discovery process by broadcasting a Route Request (RREQ) packet. This RREQ packet contains the source node address, the destination node address, a unique sequence number, and an empty route record. Each intermediate node, upon receiving a RREQ for the first time, will check in its own route cache if there is an available route to the destination. If it has no route to the destination, the intermediate node will add its own address to the route record and rebroadcast the RREQ. If it has a route to the destination in its route cache, the intermediate node will append the cached route to the route record and initiate a Route Reply (RREP) packet back to the source node. The RREP contains the complete route record from the source to the destination. The intermediate node ignores the RREQ if it has already seen the packet by examining the sequence number and if its address is present in the route record of the packet. If the node receiving the RREQ is the destination node, it will copy the route record from the RREQ and send a RREP back to the source.

2) Route Maintenance Due to the node movement, the routes discovered may no

longer be valid over time. The route maintenance is accomplished by sending Route Error (RERR) packets. When a link is found broken, a RERR packet is sent back from the node that detects the link failure back to the source node. Each node, upon receiving the RERR packet, removes from its cache all routes that contain the broken link.

The HELLO packets used to ensure the local connectivity are not considered in our reactive routing protocol model. Thus, once a route enters the cache, the failure of the route can only be detected when it is actually used to transmit a packet and the reception by the next hop is not confirmed (acknowledged).

3) Route Cache Strategy Route caches used in reactive routing protocols are to

reduce control overhead and to lower routing latency by avoiding unnecessary requests for a route that was recently discovered or used. The route cache for the purpose of our analysis is modelled to contain only one route for each destination. The main reason for this choice is simplicity. Moreover, since our goal is to evaluate the routing overhead, a single route cache is more suitable than a multiple route cache.

Node A will cache a route to other node B if a) node A (or B) has communicated with node B (or A) or communicates with other node via node B (or A); b) node A and node B have

been two intermediate nodes on the same communicating route between two other nodes; c) node B has flooded the network to find a route to a node other than node A and the communicating route is not going via node A.

III. OVERHEAD ANALYSIS Suppose node A has a packet to send to node B. Node A

will look in its cache if there is an available route to node B. There are two possible cases:

1. There is no available route to node B in the cache. The routing overhead will be the one required for route discovery by flooding.

2. There is a cached route (unexpired) to node B in the cache. There are two possible cases:

If the cached route is valid, no extra routing overhead is incurred.

If the cached route is invalid, then the overhead required for routing is the sum of the misrouting overhead caused by route invalidity and the one for discovering a new route to B by flooding.

The above description can be illustrated in Fig. 1. The total bandwidth

perPacketRI required for routing per node

per packet generation can be estimated as

RIFL

FLRIFLR

ItptpItptptp

IItptpItpIperPacket

)()()()()( ))(()()(1011010

1010

(1)

Let be the packet generation rate per node; the total bandwidth IR required for routing per second per node is then estimated as

RIFLRR ItptpItptptpII perPacket )()()()()( 1011010

(2)

We are now going to determine the expressions for p1(t), p0(t), p10(t), IRI, and IFL. A. Preliminary Quantities of Interest

According to [16], the average distance d between two nodes in a square area having sides of a is equal to 0.5214a. The average route length between two nodes can be estimated as follows

Fig. 1. Illustration of how routing overhead is incurred by reactive routing

p11(t) p10(t)

p0(t) p1(t)

Node A wants to send a packet to node B

A has a route to B in its cache

A has no route to B in its cache

Invalid cached route

Flooding

Flooding

Valid cached route

195

RaRdL /5214.0/

(3) Lemma 1: Probability that node A selects node B to

communicate with or selects a node other than node B to communicate with and the communicating route is going via node B is given by

Lppab (4) Proof: A route of length L is going through (L 1)

intermediate nodes. The probability that node A selects node B to communicate with is p. The probability that node A selects a node other than node B to communicate with is (N 2)p. The probability that the communicating route is going via node B is equal to (L 1)/(N 2). Therefore, the probability that node A selects node B to communicate with or selects a node other than node B to communicate with and the communicating route is going via node B is given by

LpNLpNppab

21)2(

From (7), it follows that Corollary 1: The probability that node A (or B) selects

node B (or A) to communicate with or selects a node other than node B (or A) to communicate with and the communicating route is going via node B (or A) is given by

222 pLLppa (5) Lemma 2: Probability that node A and node B are two

intermediate nodes on the same communicating route between two other nodes is given by

pLLpb )2)(1( (6) Proof: Without loss of generality, suppose node C

communicates with any other node D. Except node A and node B, there are possible N 3 nodes can be a destination (node D) of node C. Thus, the probability that node C communicates with any node D is (N 3)p. The probability that node A is an intermediate node on the C-to-D route is equal to (L 1)/(N 2). The probability that node B is another intermediate node on the C-to-D route is equal to (L 2)/(N 3). There are N 2 nodes which can be a source node like node C. Therefore, the probability that node A and node B are two intermediate nodes on the same communicating route between two other nodes is determined as

pLLNL

NLpNNpb )2)(1(3

221)3)(2(

Lemma 3: Probability that node B floods the network to search for a route is given by

)()()( 1010 tptptppB (7) Proof: Probability that node B floods the network to search

for a route is equal to the sum of the probability that node B does not know a route and the probability that the route being used is found invalid. From Fig. 1, we have (7).

Lemma 4: Probability that node B communicates with a node other than node A and the communicating route is not going via node A is given by

pLNpc )1( (8)

Proof: The probability that node B selects a node other than node A to communicate with is (N 2)p. The probability that the communicating route is not going via node B is equal to 1 (L 1)/(N 2). A product of these two probabilities yields what needs to be proved, i.e.

pLNNLpNpc )1(2

11)2(

Corollary 2: Probability that node B floods the network to search a route to a node other than node A and the communicating route is not going via node A is given by

pLNtptptppd )1()()()( 1010 (9) Proof: The probability that node B floods the network to

search for a route to a node other than node A and the communicating route is not going via node A is the product of (7) and (8), i.e.

pLNtptptpppp cBd )1()()()( 1010

B. p1(t) and p0(t) Theorem 1: Let be the packet generation rate and be

the cached route TTL. The probability that node A has a cached route to node B can be computed as

1 ,)(1min)( 111

c

cbaptpppp

tp (10)

Proof: The probability that node A will have obtained information about a route to B for caching is given by

dba ppptp )(1 (11)

where is the packet generation rate, is the cached route TTL, and is the number of packets generated within the duration of cached route TTL.

Substituting dp (9) into (11) yields cba ptptptppptp )]()()([)( 10101

! cba ptptptppptp )]()()(1[)( 10111 ! ccba ptptpppptp )()()( 1111

c

cbaptpppp

tp )(1)( 111

Many routes may be found or available to the destination B and one cached route may be refreshed more than one time within the duration of cached route TTL. However, node A can cache only one route for the destination B at a time. Therefore, in order to avoid overlapping routes the probability that node A has a cached route to node B should be computed as given in (10).

From (10), it follows that

1 ,)(1min1)( 110

c

cbaptpppp

tp

(12)

196

C. p11(t) and p10(t) Corollary 3: The probabilities that a route of length L stays

up or goes down within a time interval [t, t+) are given, respectively, by

Ltqtp )(1)(11 (13)

Ltqtp )(11)(10 (14) Proof: Here we assume the probability of link change

within a time interval [t, t+) is the same for all links and is denoted by q(t). The probability that a link remains in up state within a time interval [t, t+) is thus equal to 1 q(t). Therefore, for a route of length L, p11(t) and p10(t) are as given in (13) and (14). D. Misrouting Overhead due to Route Invalidity IRI

Theorem 2: Let SD and SRERR be the sizes of a data packet and a RERR packet, respectively. The average misrouting overhead IRI due to route invalidity is given by

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Proof: A link is found broken only when it is actually used for sending a data message, but the sender fails to receive a confirmation from the next hop. The node detecting a link failure will then send a RERR message to the source node. Therefore, the overhead required for detecting route invalidity depends on which link of the route is the first broken link.

If the first (i 1) links are up, but the ith link is down: 1

10 )](1)[(])1([)( iRERRDRI tqtqSiiSItp where i L. The average misrouting overhead due to route invalidity is thus given by

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L

i

iRERRDRI tqSiiStqItp

1

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After some manipulations, it yields (15). E. Lengths of RREQ, RREP, and RERR messages

Claim 1: Assume the size of a RREQ message is dynamic. The average length of a RREQ message can be estimated as

NLSRREQ log]2/)3[( (bits) (16) Proof: Originally, a RREQ message from the source node

contains two addresses (one for the source and one for the destination). The number of addresses in a RREQ message increases by one after each transmission by an intermediate

neighbor to the destination contains (L+1) addresses for a route of length L. The average number of addresses in a RREQ message is thus equal to [(L+1)+2]/2. Every address has a length of logN (bits), therefore, the average length of a RREQ message is as given in (16).

Claim 2: The average length of a RREP message is given by

SRREP = (L + 1)logN (bits) (17) Proof: A RREP message generated by the node that has a

cached route to the destination follows the shortest route of length L to reach the source. A RREP message contains the route record consisting of all on-are (L+1) nodes on the route of length L, the length of a RREP message is thus given as in (17).

Claim 3: The length of a RERR message is given by SRERR = H(q(t)) + logN (bits) (18)

Proof: A RERR message contains the amount of information needed to notify the source about the broken link. It should contain at least the information about the reporting node and the information describing the change in the state of the broken link. The amount of information required for identifying a node among N nodes is equal to logN. The amount of information required for describing the state of a link at a next time step is H(q(t)). Summing these two values yields (18). F. Route Discovery Overhead by Flooding IFL

1) Overhead incurred by RREQ messages When a source node does not have a route to a destination

or when it receives a RERR message notifying about the invalidity of the current route, it will flood the network with a RREQ message. The RREQ message is then retransmitted by intermediate nodes which do not have a route to the destination.

Lemma 5: The number of RREQ transmissions in a single route discovery can be estimated as

)()2()(1 2/100 tpnNtnpN LRREQ (19) Proof: !

retransmitting the RREQ message is equal to np0(t). For a connected network, any node is connected with at least one other node. Therefore, the probability that a node, which is i-hop away from the source, retransmits the RREQ message is at least )(0 tpi , where i " # . Except the neighbors, other (N n 2) nodes, on average, are (L+)-hop away from the source. The number of RREQ transmissions in a single route discovery can be estimated as

)()2()(1 2/100 tpnNtnpN LRREQ Theorem 3: The bandwidth consumed by RREQ messages

in a single route discovery is given by

NLtpnNtnpI LRREQ log23)()2()(1 2/100

(20)

Proof: The result is the product of (16) in Claim 1 and (19) in Lemma 5.

2) Overhead incurred by RREP messages Upon receiving a RREQ message, a node will generate a

RREP message if it knows a route to the destination. The destination is always generates a RREP message if it receives a RREQ message.

Lemma 6: The number of RREP transmissions in a single route discovery can be estimated as

197

)()2)(()()2/1()( 10121

01 tLpnNtptpLtnpNLL

RREP

(21) Proof: The number of nodes generating RREP messages

for a single route discovery is equal to the number of nodes that receive a RREQ message and have a route to the destination. The number of the a RREP message is equal to np1(t). For a connected network, any node is connected with at least one other node. Therefore, the probability that a node, which is i-hop away from the source, generates a RREP message is at least )()( 110 tptpi , where i "#$%neighbors, other nodes, on average, are (L+)-hop away from the source. The probability that the destination generates a RREP message is at least )(10 tpL . The number of nodes generating RREP messages in a single route discovery can thus be estimated as

)()2)(()()( 1012/101 tpnNtptptnpN LLnodesRREP (22) The RREP message generated from the destination

requires L transmissions, while the RREP messages generated !hbors require L+ transmissions, on average, to reach the source. This statement and (22) leads to (21).

Theorem 4: The bandwidth consumed by RREP messages in a single route discovery is given by

NLtLp

nNtptpLtnpIL

LRREP

log)1()( )2)(()()2/1()(

10

12/1

01

(23)

Proof: The result is the product of (17) in Claim 2 and (21) in Lemma 6.

From (20) and (23), it follows that Corollary 4: The information required for flooding can be

estimated as

RREPLL

RREQL

FL

StLpnNtptpL

tnpStpnNtnpI

)()2)(()()2/1( )()()2()(1

101

2/10

12/1

00

(24)

Substituting (10), (12), (14), (15), and (24) into (1) and (2), we can obtain the average routing overhead I per node per packet generation and the average routing overhead IR per node per second. G. Special Cases

1) Routing protocols without the route cache option In this case, the TTL of cached routes is set to 0, = 0.

From (10), we have p1(t) = 0, and p0(t) = 1. Substituting these values into (2), yields

])1[()0()0( RREPRREQFLR LSSNII

(25) The routing overhead in this case is the one required for

route discovery by flooding. 2) Routing protocols with the never-expire route cache

option Never-expire route cache option means the TTL of cached

routes is set to infinity, = '. From (10), we have p1(t) = 1, and thus p0(t) = 0. Substituting these values into (2), yields

))(( )(10)( '' FLRIR IItpI (26) where

RREPRREQFL nSSI ' )(

The routing overhead in this case is the one incurred by route invalidity. H. Total Routing Overhead

As mentioned in subsection 2.3, the ProbLC at time t after the link has been connected can be approximated as an exponential distribution with mean u = 1/u, i.e.

tuetq 1)( If we assume that the distribution of destinations for any

node is uniform over the whole network, then the average route request inter-arrival interval for the same destination is given by

Ta = (N 1)/ where is the packet generation rate.

Suppose a data packet is sent from a source at time tp after the route has been cached. If the TTL of the cached route is less than T, tp is uniformly distributed on the interval [0, ], otherwise tp is uniformly distributed on the interval [0, Ta], i. e. the average value of tp, pt , is given by

TTT

ta

p 2/2/

As the routing overhead incurred at time tp after the route has been cached, the overhead required for routing per node per second is given by

})()()]()()({[ 1011010 RIppFLpppR ItptpItptptpI (27) where

pu tLLpp etqtp

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IV. CACHED ROUTE TIME-TO-LIVE OPTIMIZATION While taking advantage of using recently active routes,

there is a need to remove stale links timely to avoid route errors. To do that, every route is assigned an appropriate TTL value when it enters the cache, and it will be removed from the cache when its lifetime is over. To minimize routing overhead and routing delay, the cached route expiry time should be as close as possible its real lifetime. If the TTL value is assigned too small, routes are discarded before they really break, more costly new route searches would have to be performed. On the other hand, if the TTL value is set too large, invalid routes are likely to be used which results in extra routing overhead and routing delay for detecting the broken links.

To derive the optimal Time-To-Live opt, at which the cost of routing is minimal, we take the derivative of the overhead IR (27) with regard to , &IR'&, and opt is one of the roots of the equation &IR'& = 0. To solve the equation, we use ())%! IR() plot. The plot of IR() gives an approximate starting root, based on which the $) ! ()hod.

V. NUMERICAL RESULTS

A. The behaviour of the overhead in response to mobility Consider a network with 20 nodes moving in a square area

of side a = 1,000 m. Other parameters are set as follows: R = 250 meters, SD = 512 bytes, = 1 pkt/s, and = 300, 3, 1, 0.5, 0.001 seconds,

From [16], the average distance between the source and the destination is given by

d = 0.5214a = 0.5214)1,000 = 512.4 meters The lower bound on the average number of hops in a route

can be computed as d/R [16], i.e. L ( d/R = 512.4/250 = 2.05

It is hard to determine theoretically the exact value of the average path length. In [5] and [7], the authors simplified their analysis by estimating L to d/R, hence we will use this approximated value (L = d/R = 2.05 = 3) for our numerical analysis in this section.

The relationship between the overhead and the network mobility is shown in Fig. 2 and Fig. 3 for different TTLs. As the mobility increases, the overhead increases to the maximum value and starts decreasing to a certain level. The constant portion of the overhead curves takes place when the ProbLC is equal to one.

As can be seen from the plots, the longer the cached route TTLs , the less the routing overhead incurred at low mobility and the more the routing overhead incurred at high mobility. When the cached route TTLs is very long (300s), no overhead for route discoveries is required. At low network mobility, the routes are less likely to be broken, and thus the overhead incurred by misrouting due to route invalidity is low. If the network mobility is high, the misrouting overhead is more likely to be incurred due to route invalidity, and as a result, the total overhead is significant. On the other hand, when the cached route TTLs is very short (0.001s), the nodes initiate route discoveries for every packet generation

0 0.5 1 1.5 2 2.5 30

1000

2000

3000

4000

5000

6000

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ad

(bits/

s/nod

e)

N = 20 nodes; = 1 pkts/s; SD = 512 bytes

TTL = 300 (s)TTL = 3 (s)TTL = 1 (s)TTL = 0.5 (s)TTL = 0.001 (s)

Fig. 2. Routing overhead vs. Mobility

0 0.5 1 1.5

x 10-3

0

50

100

150

200

250

300

350

Mobility, u

Rea

ctiv

e ro

utin

g ov

erhe

ad

(bits/

s/nod

e)

N = 20 nodes; = 1 pkts/s; SD = 512 bytes

TTL = 300 (s)TTL = 3 (s)TTL = 1 (s)TTL = 0.5 (s)TTL = 0.001 (s)

Fig. 3. Routing overhead vs. Mobility for low mobility levels

independent of the network mobility, and, as can be seen from the plots, the overhead curves keep nearly constant over the network mobility.

B. Optimal cached route Time-To-Live (opt) For a given packet generation rate, the optimal TTL opt

depends on network mobility. The higher the mobility, the shorter the optimal TTL. This relationship can be seen on Fig. 4 which shows three optimal TTLs for three different mobility levels u % * ! ()method are shown in Table III.

TABLE III. RELATIONSHIP BETWEEN OPT AND

(pkts/s) u (s-1) opt (s) IR (bits/s/node) 2.0 0.01 1.4284 528.5417 2.0 0.03 0.9424 938.5112 2.0 0.05 0.7748 1,216.0

For a given network mobility, the optimal TTL opt varies depending on traffic activity. Fig. 5 plots three routing overhead curves for three different packet generation rates as

199

0 0.5 1 1.5 2 2.5500

1000

1500

2000

2500

3000

3500

4000

4500

Time-To-Live (s)

Rout

ing

over

head

(bits

/s/no

de)

N = 100 nodes; = 2 pkts/s; SD = 512 bytes

u = 0.01 s-1

u = 0.03 s-1

u = 0.05 s-1

Fig. 4. Overhead vs. Cached route TTL (for various mobility levels)

0 0.5 1 1.5 2 2.5 30

500

1000

1500

2000

2500

3000

3500

4000

4500

Time-To-Live (s)

Rou

ting

over

head

(bits

/s/no

de)

N = 100 nodes; u = 0.03 s-1; SD = 512 bytes

= 0.5 pkts/s = 1 pkt/s = 2 pkts/s

Fig. 5. Overhead vs. Cached route TTL (for various traffic loads)

the cached route TTL increases. As can be seen, the higher the packet generation rate, the shorter the cached route TTL. As the packet generation rate increases, more data packets are likely to experience misrouting. Therefore, the TTL needs to be shorter to ensure the routes being used are valid to minimize the total routing overhead. +()the optimal TTLs can be determined as shown in Table IV.

TABLE IV. RELATIONSHIP BETWEEN OPT AND U

u (s-1) (pkts/s) opt (s) IR (bits/s/node) 0.03 0.5 2.1651 463.5650 0.03 1.0 1.4424 665.6036 0.03 2.0 0.9424 938.5112

VI. CONCLUSION In this paper, we have proposed a theoretical model for

analyzing the control overhead incurred by the reactive routing algorithm in MANETs in response to mobility. We have then introduced a numerical method for determining the optimal cached route Time-To-Live to minimize the routing overhead.

The final result of this study shows the minimum communication bandwidth in bits necessary to facilitate routing for reactive approach. Due to the page limit, the simulation results are not shown in this paper.

One application of the derived model is that it allows network designers to determine the link bandwidth for each node that can support the routing overhead for different possible levels of mobility. Also, such fundamental knowledge of routing performance as a function of mobility allows predicting network performance and evaluating network resources necessary for specific requirements of given network scenarios.

The two most important practical implications of the findings of this study are 1) for most scenarios with realistic mobility levels and traffic loads in MANETs, the dominating part of overhead is the misrouting overhead, and 2) the long Time-To-Live does not save bandwidth it needs to be set far shorter than that commonly used or recommended for simulations.

Similar work for proactive routing is required to thoroughly understand the cost of routing in MANETs. Nevertheless, this work on reactive routing provides a first insight into the fundamental limits of routing performance in response to mobility.

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