Randomized Energy Aware Routing Algorithms

in Mobile Ad Hoc Networks

Israat Tanzeena Haque

Computer Sci.and Soft.Eng.

Concordia University

Montr ´eal,Qu´ebec,Canada

it

haque@cse.concordia.ca

Chadi Assi

Concordia Institute for

Information Systems Eng.

Montr ´eal,Qu´ebec,Canada

assi@ciise.concordia.ca

J.William Atwood

Computer Sci.and Soft.Eng.

Concordia University

Montr ´eal,Qu´ebec,Canada

bill@cse.concordia.ca

ABSTRACT

We consider the problem of energy aware localized routing

in ad hoc networks.In localized routing algorithms,each

node forwards a message based on the position information

about itself,its neighbors and the destination.The objec-

tive of energy aware routing algorithms is to minimize the

total power for routing a message from source to destina-

tion or to maximize the total number of routing tasks that

a node can perform before its battery power depletes.In

this paper we extend our previous work on randomized lo-

calized routing algorithms that achieve high packet delivery

rates and show that they have good overall power consump-

tion.We present two diﬀerent variants of energy aware ran-

domized routing,namely “greedy” and “compass”,and we

study their performance using diﬀerent cost metrics (e.g.,

forwarding power,remaining node energy,or a combination

of both).We study their performance experimentally on

diﬀerent topologies and compare it with other existing algo-

rithms.Our simulation results show that energy aware ran-

domized algorithms achieve superior packet delivery rates

and moderate energy consumption.

Categories and Subject Descriptors

C.2.1 [Computer-Communication Networks]:Network

Architecture and Design—Wireless communication;C.2.2

[Computer-Communication Networks]:Network Pro-

tocols—Routing protocols

General Terms

Performance,algorithms

Keywords

wireless networks,mobile ad hoc and sensor networks,rout-

ing,position based routing,energy aware routing

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MSWiM’05,October 10–13,2005,Montreal,Quebec,Canada.

Copyright 2005 ACM1-59593-188-0/05/0010...$5.00.

1.INTRODUCTION

We consider the problemof energy aware routing,in which

a message has to be sent eﬃciently from a source node to a

destination node,in a sensor or an ad hoc network.Amobile

ad hoc network (MANET) is a collection of autonomous mo-

bile devices that can communicate with each other without

having any ﬁxed infrastructure.Each node in the network

can have an omni-directional antenna and communicate us-

ing wireless broadcasts with all nodes within its transmis-

sion range.A multi-hop routing protocol is needed to enable

communication between nodes that are not in transmission

range of each other.However,the absence of infrastruc-

ture in MANETs,the dynamic topology of these networks,

the autonomous heterogeneous nodes,and the resource con-

straints (battery power,bandwidth,computational power,

etc.) all contribute to make the problemof routing a tremen-

dous challenge.The constrained resources,especially the

battery power,make the routing problem a very challeng-

ing issue for MANETs.Moreover,mobile devices in an ad

hoc network need to forward packets for other nodes;this

extra activity consumes a signiﬁcant amount of the energy

of mobile devices.Therefore,it is critical to design eﬃcient

routing algorithms with the objectives of (1) minimizing the

overall energy usage in routing packets and (2) maximizing

the packet delivery rate.Applications of minimum energy

networks include soldiers deployed on a hostile terrain and

multisensor networks,where sensors communicate with each

other without having any central control or base station.

Energy is consumed at two levels during routing,namely

communication energy and the energy dissipated at the nodes.

The communication energy or the energy needed per routing

task can be optimized if nodes can adjust their transmission

power to eﬃciently select the next hop along the route.This

is equivalent to hop count if the transmission power is kept

constant [13].Routing algorithms that solely focus on the

communication energy are not eventually a good choice for

network lifetime.Some of the nodes (hot spots) in this ap-

proach will be chosen very often and this will ultimately

drain out the battery power of these nodes quickly,there-

fore partitioning the network abruptly.Routing algorithms

that minimize the energy required per routing task are called

power/energy aware algorithms [13];on the other hand,cost

aware routing algorithms ensure optimal use of node’s bat-

tery power and hence prolong network’s life time [13].

In this paper we focus on the energy aware localized rout-

ing algorithms,where each node forwards the routing pack-

ets based only on the position or geographic coordinates of

71

itself,its neighboring nodes,and the destination.We model

a MANET by a unit disk graph,where two nodes are con-

nected if and only if their Euclidean distance is at most

the transmission range [1].We classify the position based

algorithms as deterministic and randomized.In the ﬁrst cat-

egory,the current node holding the packet selects the next

node deterministically out of its neighbors,whereas in the

second case the selection is random.

Extensive work has been done on energy eﬃcient routing,

however in this paper we will mainly focus on position based

routing.We will start ﬁrst by presenting the performance

analysis of some of the existing energy aware and non aware

position based routing algorithms.Then,we will present our

new randomized algorithms that control the communication

power with high packet delivery rates.Essentially,to decide

on the next node to which the packet should be forwarded,

our algorithms pick one neighbor of the current node from

above the line passing through the current node and the

destination,and another neighbor below this line.Then the

next node is chosen randomly from these two neighbors ac-

cording to some probability distribution.The exact choice

of the neighbors and the probability distribution determines

the algorithm.

In our simulation,we consider both the uniform and clus-

ter distributions of nodes in a given area.Our simulation

results show that on both data distributions our algorithms

have a substantially higher delivery rate than the determin-

istic algorithms in unit disk graphs.The rest of the paper is

organized as follows.The next section reviews some of the

enery aware routing related to our work.Section 3 gives a

brief description of the exsisting position based routing al-

gorithms.Section 4 describes the system model and other

relevant preliminaries.Randomized energy aware routing

algorithms are presented in Section 5.The simulation en-

vironment is given in Section 6.In Section 7 we present

the empirical results of our simulations and provide an in-

terpretation of the behavior of the algorithms.We conclude

with a discussion of the results and future directions of this

research in Section 8.

2.OVERVIEWOF ENERGY AWARE

ROUTINGALGORITHMS

In [12],the authors propose several power aware routing

metrics to increase the lifetime of the nodes and the net-

work.Conventional routing protocols in ad hoc networks

use delay or hop count to calculate the path to the destina-

tion.This approach might accelerate the battery drainage of

some speciﬁc nodes,which forward packets for many source-

destination pairs.The eﬀect would be early node failure and

network partition.Following a longer path of a set of nodes

with plenty of energy would be a better choice [12].The

energy-aware metrics proposed by Singh et al.are as fol-

lows.

Minimum energy consumed per packet:This metric is

used to minimize the total communication energy of a packet

regardless of the available energy at the nodes.Assume a

packet j traverses a set of nodes n

1

,n

2

,........,n

k

,where n

1

is

the source and n

k

is the destination.Let P(n

i

,n

i+1

) be the

power needed to forward j from the node n

i

to n

i+1

.The

total energy consumed by packet j to reach the destination

is then the sum of the energy over the entire path.The

optimization of this metric,which is called power metric in

[13],is then subject to

min

∀j

e

j

,e

j

=

k−1

i=1

P(n

i

,n

i+1

) (1)

Minimum cost per packet:This metric tries to prolong

the lifetime of the nodes and networks through the careful

selection of next route node with plenty of energy.Let f

i

(x

i

)

be a function that denotes the cost or weight of node i,

where x

i

represents the total energy that node i already

expended.Hence,the total cost c

j

of sending a packet j

from the node n

1

to n

k

is sum of the cost of the entire

route.The optimization of this metric is then subject to,

min

∀j

c

j

,c

j

=

k−1

i=1

f

i

(x

i

) (2)

Another way of using the cost metric is normalizing f

i

to

reﬂect remaining battery lifetime of a node.Then,f

i

(x

i

)

can be modiﬁed as f

i

(z

i

) = 1/1 − g(z

i

),where z

i

is the

measured voltage and g(z

i

) is the normalized (between 0

and 1) remaining lifetime of node i.Singh et al.consider a

static random graph,where two nodes are connected with a

ﬁxed probability p,and use nonlocalized Dijkstra’s shortest

path algorithm to evaluate the performance of their pro-

posed metrics compared to the hop count metric.

In [13],the authors proposed localized power,cost,and

power*cost algorithms based on the observation that if ad-

ditional intermediate nodes are placed at desired positions

between the source and destination that are at distance d

apart,then transmission power may be linearly dependent

on d,rather than d

α

(α ≥ 2).The authors ﬁrst deﬁned two

optimal algorithms in terms of communication energy and

battery lifetime.The SP −power algorithm that minimizes

the total communication energy of a packet is obtained by

applying Dijkstra’s shortest path algorithm with the weight

of the edge as ad

α

+ C,where a,α,and C are constants

that depend on the radio models.On the other hand,the

SP −cost algorithm,which maximizes the node’s lifetime,

can be obtained by applying Dijkstra’s shortest path algo-

rithm with an edge-weight of f(n

i

) = 1/g

i

.Stojmenovic et

al.use these optimal energy aware algorithms to compare

the performance of their proposed algorithms.

Let c,N

i

,and d be the current node holding the packet,

set of neighbors of c,and destination of the packet,respec-

tively.The distance between c and one of its neighbors is x

and the remaining distance to the destination is d−x,where

d is the total distance between c and d.The localized loop

free power and cost aware algorithms are next deﬁned:In

the power algorithm,c selects the next neighbor such that

min

∀j∈N

i

e

j

,e

j

= E

c→x

i

+E

x

i

→d

(3)

Ec→x

i

= ax

α

+C (4)

E

x

i

→d

= C(d−x)[

a(α−1)

C

]

1/α

+C(d−x)[

a(α−1)

C

]

(1−α)/α

(5)

Here,E

c→x

i

and E

x

i

→d

are the energy needed for direct

communication.This process continues untill the packets

reach the destination or the next node is one from which c

received the packet,in which case algorithm fails to deliver

the packet.In the cost algorithm,c selects the next neighbor

72

x

i

such that

min

∀j∈N

i

c

j

,c

j

= f(x

i

) +t(d −x)/r,(6)

Here,r is the transmission radius of the nodes and t is a

network dependent constant.For example,t = f(x

i

) means

the rest of the nodes have the same cost of the node x

i

,

which is not realistic.Also,the other values of t,(f

(x

i

) and

1/g

(x

i

)),where f

(x

i

) and 1/g

(x

i

) represent the average

cost and remaining battery power of x

i

and its neighbors,

respectively,and do not consider actual cost of the nodes.In

our algorithms (presented in Section 4),nodes periodically

broadcast their remaining battery power to their neighbors

to make routing decision based on accurate energy level of

nodes.

Finally,the authors combine power and cost metrics in

additive and multiplicative forms to obtain a single metric

that considers both the communication energy and the bat-

tery power of the nodes.When multiplication of two metrics

is used,the energy needed to forward a message from c to

next node x is f(x)E

c→x

.The additive form of the met-

rics is αE

c→x

+βf(x),where α and β may be ﬁxed by the

source node s as f

(s) and average of E

s→N

i

,respectively.

The corresponding optimal energy consumption algorithms

are called SP −power +cost and SP −power ∗cost.The lo-

calized version of these algorithms select the next node that

not only minimizes the communication energy but also se-

lects the next node with plenty of remaining battery power.

On the other hand,[3] proposed a distributed non-localized

shortest cost routing algorithm for sensor networks,where

the cost takes into account both the communication energy

consumption and the residual energy level at nodes.Their

objective is to maximize the network life by increasing the

number of routing tasks,before the ﬁrst node is dead (does

not have enough energy to forward message).Their link cost

can be formulated as follows.

cost

ij

= (e

t

ij

)

x

1

E

−x

2

r

E

x

3

i

+(e

r

ij

)

x

1

E

−x

2

r

E

x

3

i

,(7)

Where E

i

and E

r

are initial and remaining energy at nodes,

respectively.x

1

,x

2

,and x

3

are some positive weighting

factors.For example,if x

1

= x

2

= x

3

= 0,then the resulting

path is the minimum hop path,however,if x

2

= x

3

= 0,

then the path is the minimum energy consumption path.

In [15] Xue et al.proposed a location aided or position

based energy aware routing algorithm,called LAPAR,under

node mobility and unidirectional links.In particular,each

node constructs relay regions based on the position of the

neighbors.The next node is then chosen such that its relay

region covers the destination.In case of multiple neighbor-

ing nodes that covers the destination,the greedy approach is

applied.This algorithm is loop free,but might fail if there

is no neighbor whose relay region covers the destination.In

this case LAPAR is combined with the perimeter routing.

A planar subgraph of the original topology is required in

perimeter routing,and the packet traverses those faces of

the subgraph that are intersected by the source and desti-

nation nodes.

Kuruvila et al.in [10] proposed another set of power

and cost aware routing algorithms that guarantee progress

In power progress algorithm the current node forwards the

packet to one of its neighboring nodes that is closer to the

destination than itself,such that it minimizes (d

α

cx

+C)/d

t

−

d

r

,where d

cx

,d

t

,and d

r

are the distances between the cur-

rent and neighboring node,current and destination node,

and neighboring node and destination,respectively.Simi-

larly,in projection power progress,the next node is the one

that minimizes (d

α

cx

+C)/cd.cx,where cd.cx is the dot prod-

uct of two vectors.

3.POSITION BASED ROUTING

ALGORITHMS

In position based routing algorithms,each node makes

a decision about which neighbor to forward the message

to based solely on the location of itself,its neighboring

nodes,and the destination.Although routing-table-based

solutions merely keep the best neighbor information on a

route toward the destination,the communication overhead

for maintenance of routing tables due to node mobility and

topology changes is quadratic in network size [6].On the

other hand,position-based algorithms do not require route

establishment and maintenance,hence these algorithms may

eﬃciently utilize the scarce memory resources and the rel-

atively low computational power available to the wireless

nodes.More importantly,given the numerous changes in

topology expected in ad-hoc networks,no reconﬁguration of

the routing tables is needed and therefore we expect a signiﬁ-

cant reduction in the route maintenance overhead.Position-

based routing algorithms are classiﬁed in diﬀerent ways in

the literature,and we will describe some of those related to

our work.

In [6],the progress of a node x,with a given transmitting

node c and the ﬁnal destination node d,is deﬁned as the

projection of |cx| on the

cd line.A neighbor of c is in the

forward direction if the progress is positive;otherwise,it is

said to be in the backward direction.

The Most Forward within Radius (MFR) [14] routing al-

gorithm maximizes the progress toward the destination by

forwarding the packets to one of the neighboring nodes,

whose projection onto the line between the current node

and the destination is closest to the destination.However,

it also consumes maximum power to cover maximum dis-

tance,which in turn increases collisions with other nodes.

Hence,instead of forwarding packets to the farthest neigh-

boring node,the Nearest with Forward Progress (NFP) [8]

scheme sends the packet to the node closest to the sender.

Recently in [13],Stojmenovic et al.have introduced another

new routing method called Nearest Closer (NC),which is a

variation of NFP method.In this method,c selects one of

its closest neighbors that are closer to d than c.The trans-

mission can thus be accomplished with minimum power;

hence the interference with the other nodes is minimized,

while the probability of a successful transmission is maxi-

mized.In Greedy routing [5,6],a node forwards a packet

to the neighbor that is closest to the destination.Compass

or directional routing [9] moves the packet to a neighboring

node such that the angle formed between the current node,

the next node,and the destination is minimized.All of the

above mentioned algorithms (except NC) choose the next

node from among all the neighbors of the current node c,

and fail to deliver the packet if the chosen next node is the

one from which c receives the packet.Whereas,in NC,c

considers only the neighbors closer to the destination than

itself,and drops the packet if no such neighbor is available.

Let us consider the example given in Figure 1 to illustrate

the successful operation of each of the above mentioned al-

gorithms,where the source and the destination are s and

73

d1,respectively.In this example Greedy and MFR choose

b,NFP and NC select a,whereas Compass routing picks c

as the next node.

d2

s

d1

a

b

c

e

f

h

g

i

Figure 1:A sample network topology to illustrate

the operation of the algorithms.

In most cases,MFR and Greedy require the same number

of hops to reach the destination.However,Compass routing

needs a few extra hops compared with the Greedy routing,

while the delivery rate is similar.All these methods have

high delivery rates for dense graphs,but low delivery rate

for sparse graphs.However,the performance of MFR and

Greedy routing come close to matching the path length given

by the shortest path algorithm in case of successful delivery

[6].Hence,we might expect similar energy consumption for

MFR,Greedy,and Compass algorithms.NC has fewer el-

igible next nodes compared to the other algorithms,which

in turn reduces its delivery rate.However,when this algo-

rithm succeeds,the probability of achieving the best power

dilation is high.The energy consumption of NFP may have

the power dilation in between NC and the other algorithms.

Now we will present another example given in Figure 1,

where the above mentioned algorithms fail to deliver the

packet.In this example,the destination is d2.The next

node is b in MFR,Greedy,and Compass algorithms.Then,

b selects f which forwards the packet to the next node g.

At this point,g forwards back the packet to f since its the

only closest neighbor to the destination.According to the

deﬁnition,all these algorithms will drop the packet at g.In

the Greedy algorithm,if a node does not have any neighbors

closer to the destination than itself,the packet gets stuck at

the local maximum that reduces the delivery rates.NFP

and NC also face a similar problem,when the packet arrives

at f both the algorithms will drop the packet,where f does

not have any neighbors closer to the destination than itself.

All of these algorithms are routing loop free except Compass

algorithm.

4.SYSTEM MODEL AND PRELIMINAR-

IES

In our MANET model,which is adapted from [1],a set

of mobile hosts are spread out in an environment that is

modeled by the Euclidean plane:each mobile host with x-

and y-coordinates is represented by the point (x,y).All

distances are Euclidean distances in the plane.We use the

following hypotheses and notation:

1.Any mobile host knows the coordinates (x,y) of its

position.

2.The transmission range of each mobile host is r,that

is,two hosts can directly communicate with each other

if their distance is at most r.

3.Each mobile host has an omni-directional antenna,which

covers a circular area of radius r.

4.Communication links are bidirectional,that is,if a

mobile host u is able to receive signals from a mobile

host v,then v is also able to receive signals from u.

Based on the above hypotheses,we can represent a MANET

as a geometric undirected graph G = (S,E),where vertices

represent mobile hosts and edges represent a link through

which a pair of mobile hosts can communicate directly.The

set of vertices S is thus a set of points in the Euclidean plane.

Let d(u,v) be the distance between the points u and v in the

plane.The set of edges E {{u,v}:u,v ∈ S,d(u,v) ≤ r},

that is,E contains all the pairs of mobile hosts at a distance

of at most r [1].The resulting graph UDG(S) is called a unit

disk graph.For node u,we denote the set of its neighbors

by N(u).

Given a unit disk graph UDG(S) corresponding to a set

of points S,and a pair (s,d) where s,d ∈ S,the problem of

energy eﬃcient position-based routing is to construct a path

in UDG(S) from s to d,where in each step,the decision of

which node to go to next is based only on the coordinates of

the current node c,N(c),and d.At the same time,the en-

ergy consumption both at the nodes and for communication

must be minimized to maximize the network lifetime.Here,

s is termed the source and d the destination.Frequently,we

will also refer to the line

cd passing through c and d.

An algorithm is deterministic if,when at c,the next node

is chosen deterministically from N(c),and is randomized if

the next step taken by a packet is chosen randomly from

N(c).

The routing algorithm may or may not succeed in ﬁnd-

ing a path from s to d.We use the following notion of

a graph defeating an algorithm from [2].A deterministic

algorithm is defeated by a graph G = (S,E) if there is a

pair (s,d) ∈ S such that a packet using the algorithm never

reaches the destination d when beginning at the source s [2].

A randomized algorithm is defeated by a graph G = (S,E)

if there is a source/destination pair (s,d) ∈ S such that a

packet using the algorithm and originating at source s has

probability 0 of reaching destination d in any ﬁnite number

of steps.

We are interested in the following performance measures

for routing algorithms:the delivery rate,that is,the per-

centage of times that the algorithm succeeds and the power

dilation,the average ratio of the total communication power

consumption by the algorithm to the energy consumption of

the shortest path in the graph.

5.RANDOMIZED ENERGY AWARE

ROUTINGALGORITHMS

The deterministic algorithms that follow a path,constructed

based on a speciﬁc heuristic,might face the local maxi-

mum or routing loop,or lack of eligible neighbors during

the packet forwarding process.Therefore,this may reduce

their packet delivery rate,which is the primary objective of

a routing algorithm.The deterministic face and GFG algo-

rithms successfully overcome the above mentioned problems

74

and always guarantee packet delivery at the price of follow-

ing a long path.

Another simple and eﬃcient way to avoid these problems

might be using randomization when choosing a neighbor.A

position-based routing algorithm is randomized if the next

node is chosen randomly out of the neighbors of the cur-

rent node.In [2],Bose and Morin proposed a randomized

algorithm called Random Compass in the context of trian-

gulations.In Random Compass the next node is chosen

uniformly at random from the two nodes that satisfy the di-

rectional heuristic going in the clockwise and the anticlock-

wise directions.The algorithm has a higher delivery rate

than the deterministic algorithms at the price of a longer

path length.In [4],the authors proposed a new set of ran-

domized algorithms,called AB algorithms,to increase the

packet delivery rate with the control over the path length.

In AB algorithms,the current node selects two neighboring

nodes (candidate nodes) from above and below the

cd line

based on either the greedy or compass heuristic.The next

node x is then chosen from these two nodes based on some

probability distribution.For example,in uniform distribu-

tion,the next node is chosen uniformly at randomout of the

candidate nodes.Let n

a

and n

b

be the candidate nodes from

above and below the

cd line,and dis

n

a

d

and dis

n

b

d

be their

distances to the destination d,respectively.Furthermore,

let θ

n

a

=

n

a

cd and θ

n

b

=

n

b

cd be the angles formed by

n

a

and n

b

with c and d,respectively.In case of distance

based biasing,the candidate nodes n

a

and n

b

have weights

of dis

n

b

d

/(dis

n

a

d

+dis

n

b

d

) and dis

n

a

d

/(dis

n

a

d

+dis

n

b

d

),re-

spectively.The next node is chosen such that the probability

to pick the candidate node closest to the destination is high.

However,in angle based biasing,the weights of n

a

and n

b

are θn

b

/(θn

a

+θn

b

) and θn

a

/(θn

a

+θn

b

),respectively.The

next node is the one that minimizes the angle to the direc-

tion of the destination with high probability.The simulation

results in [4] shows that AB algorithms not only have high

packet delivery rates but also have good stretch factor.

However,the above mentioned randomized algorithms do

not consider the energy constraints while routing packets,

and as a result might not be directly applicable in an en-

ergy constrained environment.In this paper we extend the

above schemes and propose new energy aware randomized

algorithms that enable higher packet delivery rates and eﬃ-

cient utilization of energy in the network.Below,we present

variant of our algorithms:

Let n

a

and n

b

be the neighbor of c from above and below

the

cd line,respectively.Furthermore,P

ca

= dis

2

cn

a

+C and

P

cb

= dis

2

cn

b

+C are the power needed to forward one bit

information from c to n

a

and n

b

,respectively.Here,dis

cn

a

and dis

cn

b

are the Euclidean distance between c and n

a

and

n

b

.Finally,the cost at n

a

and n

b

are Cost

n

a

= 1/g(n

a

) and

Cost

n

b

= 1/g(n

b

),respectively.Our randomized algorithms

can then be deﬁned as follows:

1.PowerGreedy:Let n

a

be the neighbor of c from

above the

cd line such that P

ca

∗ dis

n

a

d

is minimized

among such neighbors.Similarly,let n

b

be the neigh-

bor of c from below the

cd line such that P

cb

∗dis

n

b

d

is

minimized among such neighbors.The next node x is

chosen fromn

a

and n

b

with probability dis

n

b

d

/(dis

n

a

d

+

dis

n

b

d

) and dis

n

a

d

/(dis

n

a

d

+dis

n

b

d

),respectively.

2.CostGreedy:Let n

a

be the neighbor of c from above

the

cd line such that Cost

a

∗dis

n

a

d

is minimized among

such neighbors.Similarly,let n

b

be the neighbor of c

from below the

cd line such that Cost

b

∗dis

n

b

d

is min-

imized among such neighbors.The next node x is cho-

sen from n

a

and n

b

with probability dis

n

b

d

/(dis

n

a

d

+

dis

n

b

d

) and dis

n

a

d

/(dis

n

a

d

+dis

n

b

d

),respectively.

3.Power*CostGreedy:Let n

a

be the neighbor of c

from above the

cd line such that (P

ca

∗Cost

a

)dis

nad

is

minimized among such neighbors.Similarly,let n

b

be

the neighbor of c frombelow the

cd line such that (P

cb

∗

Cost

b

)dis

n

b

d

is minimized among such neighbors.The

next node x is chosen from n

a

and n

b

with probability

dis

n

b

d

/(dis

n

a

d

+dis

n

b

d

) and dis

n

a

d

/(dis

n

a

d

+dis

n

b

d

),

respectively.

4.PowerCompass:Let n

a

be the neighbor of c from

above the

cd line such that P

ca

∗ θ

n

a

is minimized

among such neighbors.Similarly,let n

b

be the neigh-

bor of c from below the

cd line such that P

cb

∗ θ

n

b

is

minimized among such neighbors.The next node x is

chosen fromn

a

and n

b

with probability θ

n

b

/(θ

n

a

+θ

n

b

)

and θ

n

a

/(θ

n

a

+θ

n

b

),respectively.

5.CostCompass:Let n

a

be the neighbor of c from

above the

cd line such that Cost

a

∗ θ

n

a

is minimized

among such neighbors.Similarly,let n

b

be the neigh-

bor of c from below the

cd line such that Cost

b

∗θ

n

b

is

minimized among such neighbors.The next node x is

chosen fromn

a

and n

b

with probability θ

n

b

/(θ

n

a

+θ

n

b

)

and θn

a

/(θn

a

+θn

b

),respectively.

6.Power*CostCompass:Let n

a

be the neighbor of c

from above the

cd line such that (P

ca

∗ Cost

a

)θ

n

a

is

minimized among such neighbors.Similarly,let n

b

be

the neighbor of c frombelow the

cd line such that (P

cb

∗

Cost

b

)θ

n

b

is minimized among such neighbors.The

next node x is chosen from n

a

and n

b

with probability

θ

n

b

/(θ

n

a

+θ

n

b

) and θ

n

a

/(θ

n

a

+θ

n

b

),respectively.

This process continues until the packet reaches the desti-

nation or traverses a number of hops equal to three fourths

of the total number of nodes in the network.In the later

case,we drop the packet where randomized algorithms fail.

Also,in cost aware routing,current node c needs to know

the exact remaining power available at its neighbors.Hence,

in our work,nodes periodically broadcast the battery power

information to their neighbors.It is obvious that the smaller

the broadcast period,the more accurate the battery power

information.However,if the period is too small,it will in-

crease the communication overhead.On the other hand,a

larger period might lead the nodes to use inaccurate battery

power level.Hence,we choose the broadcast period in be-

tween two extremes.When a node forwards 3 consecutive

routing packets,it broadcast its remaining battery lifetime

to the neighbors.

Our algorithms ﬁrst select two candidate nodes according

to the deﬁnition to control both the energy requirement and

distance or direction (ensure progress) to the destination.

The diﬀerence between our algorithms and the AB algo-

rithms is that in AB algorithms the communication and/or

battery power is not taken into account to select the candi-

date neighbors.In its new version we expect a performance

that is similar to ABalgorithms in terms of the packet deliv-

ery rates;however,since we control the energy consumption

both at communication and node levels,we expect a better

75

overall energy consumption when routing packets between

source and destination.

Let us again consider the second example in Figure 1,

where the deterministic algorithms fail to deliver the packet.

The PowerGreedy will follow c −b −f,whereas PowerCom-

pass will follow a − b − f with higher probability.At this

point the algorithms may pick g and forward the packet to

it.However,at some point h will be chosen as the next node,

and the packet may reach the destination through the node

i.Hence,it is clear that our randomized algorithms still

oﬀer higher packet delivery rates.In addition,we consider

energy consumption of routing packets and nodes to pick the

next node,this in turn ensures moderate power dilation.

An immediate application of this class of algorithms is as

follows:Mobile nodes may not always be distributed uni-

formly.For example,two sessions of a conference on wire-

less communications are going on in two diﬀerent rooms of a

building.People from these two rooms might need to com-

municate with each other through some of the intermediate

mobile devices placed in between them.The resulting topol-

ogy forms two distinct clusters of mobile devices.In this

topology,only a few mobile devices might be placed between

the clusters,we call this sparse part of the topology as a hole.

Another such topology may appear in sensor networks.In

sensor networks there are thousands of tiny sensors with

constrained resources that are placed in an inaccessible en-

vironment.These sensors monitor the target area and send

back the gathered information to one or more sinks (devices

with more energy and computational power),which are ac-

cessed by the end users.Although the sinks are not usually

energy constrained,direct communication between sinks and

sensors might dissipate their energy abruptly.In [13] the au-

thors mentioned that multihop routing might achieve better

performance than direct communication to save the energy

of sensors.However,in multihop routing,nodes close to the

sink become hot spots and may lose their energy quickly be-

cause they are continuously forwarding messages for other

sensors.This will create a hole around the sink.Hence,

energy aware randomized algorithms are well suited for this

particular situation.

6.SIMULATION ENVIRONMENT

In the simulation experiments,a set S of n points (where

n ∈ {75,100,125,150}) is randomly generated on a square

of 100m by 100m.For the transmission range of nodes,we

use 15m (experiments showed that with lower transmission

radii,the graph was too often disconnected,and with higher

transmission radii,the generated graphs were so dense that

the delivery rate of all algorithms approached 100%).Each

node has initial energy level between 3M and 4M,where

M = 10

6

,which is assigned randomly.After generating

a fully connected UDG(S),a set of 100 source-destination

pairs is randomly chosen.A ﬁxed size data packet of length

16 bytes is used in addition to a 6 byte control packet that

contains the IDand current battery level of nodes.This con-

trol packet is periodically broadcast by a node to its neigh-

bors to advertise the current energy levels (nodes broadcast

their energy level after forwarding every three consecutive

routing packets).All the routing algorithms are then applied

in turn on the chosen source-destination pairs.Clearly,an

algorithm succeeds if a path to the destination is discovered.

The deterministic algorithms are deemed to fail if they enter

a loop,while the randomized algorithms are considered to

fail when the number of hops in the path computed so far

exceeds three fourths of the number of nodes in the graph.

To compute the average packet delivery rate,this process

is repeated with 100 random graphs and the percentage of

successful deliveries determined.Additionally,the average

power dilation is computed.

There are diﬀerent radio models for energy aware routing.

For example,in [11],Rodoplu et al.proposed a power con-

sumption radio model based on the observation that direct

transmission is more power consuming than relaying mes-

sages through the intermediate nodes.Also,transmission

power is related to the path loss as 1/d

n

,where n is the

path loss exponent.Finally,transmit and receive circuitries

are subject to the energy consumption.Hence,the energy

required to transmit a message from node A to node B at

distance d is d

n

+c/t,where t and c are the required energy

at the transmitter and receiver,respectively.In their simu-

lation,they adopt the values of n,t,and c as 4,10

−7

mW,

and 20 mW.The power needed to transmit one bit of infor-

mation is then d

4

+2∗10

8

.In [13] Stojmevovic et al.call this

model as RM model.Another radio model is,called HCB

model in [13],proposed by Heinzelman et al.in [7].In their

model,the energy needed to transmit one bit of information

between two nodes is

amp

d

n

+ 2E

elec

,where n = 2 is the

path loss exponent.The radio dissipates at transmitter and

receiver circuitry is E

elec

= 50nJ/bit and transmitter ampli-

ﬁer is

amp

= 100pJ/bit/m

2

.We can further normalize the

energy requirement by setting E = E

elec

/

amp

.Hence,the

ﬁnal expression becomes (d

2

+1000) to transmit one bit of a

message.We use HCB model to evaluate the performance

of the algorithms.

In addition to the uniform distribution,we also consider

cluster distribution.In such distribution,60 nodes are dis-

tributed randomly on the above mentioned two dimensional

plane,and the remaining 27 nodes form three clusters A,B,

and C.They are centered at (15,15),(25,25),and (75,75)

each with 9 nodes.Uniformly distributed nodes are used

to ensure connectivity among clusters.The ﬁrst two clus-

ters are overlapped and C is disjoint.We randomly pick the

source from A and the destination from C.We mainly con-

sider 100 source-destination pairs from 100 diﬀernt topolo-

gies and compute the average packet delivery rates and power

dilation of all the algorithms.

7.DISCUSSION OF RESULTS

Detailed simulation results for all the routing algorithms

are given in Tables 1 and 2 for the case when the trans-

mission radius is 15m.In particular,we are interested in

the performance of our proposed randomized routing algo-

rithms compared to the previously published routing algo-

rithms Greedy,Compass,PowerProgress,and Projec-

tionPowerProgress.

The randomization helps us to avoid the local maximum

or routing loop and oﬀers high packet delivery rate.How-

ever,this also means that the extra paths found can be

long,and this contributes to the higher power dilations of

the randomized algorithms.Our simulation results give us

the exact expected results.It is immediately evident from

the results given in Tables 1 to 2 that all the deterministic

algorithms have the worst delivery rates but the best power

dilations.All the randomized algorithms improve on the de-

livery rates of the four deterministic algorithms.Among all

the algorithms,PowerProgress has the best power dila-

76

Algorithms

n = 75

n = 100

n = 125

n = 150

Greedy

61.17

72.33

84.52

92.44

Compass

63.08

73.24

85.81

93.97

PowerProgress

52.82

64.21

78.68

88.67

ProjecPProgress

57.98

69.79

84.12

91.53

PowerGreedy

76.59

86.06

94.82

98.81

CostGreedy

70.99

85.51

95.54

99.08

Power*CostGreedy

69.18

82.93

94.29

99.04

PowerCompass

79.19

88.46

96.59

99.13

CostCompass

78.46

88.11

96.14

99.05

Power*CostCompass

78.99

87.32

96.29

99.17

Table 1:Average packet delivery rate in terms of

percentages on UDG,for transmission radius r =

15m.

Algorithms

n = 75

n = 100

n = 125

n = 150

Greedy

1.02

1.03

1.03

1.03

Compass

1.05

1.07

1.08

1.09

PowerProgress

1.01

1.02

1.02

1.02

ProjecPProgress

1.07

1.09

1.10

1.12

PowerGreedy

2.24

2.19

1.97

1.68

CostGreedy

2.38

2.39

2.08

1.69

Power*CostGreedy

2.46

2.51

2.27

1.87

PowerCompass

1.74

1.64

1.54

1.34

CostCompass

1.75

1.66

1.55

1.37

Power*CostCompass

1.75

1.70

1.58

1.38

Table 2:Average power dilation on UDG,for trans-

mission radius r = 15m.

tion and the worst packet delivery rate.In this algorithm,

candidates nodes are closer to the destination than the cur-

rent node.Hence,if there is no such candidate node to

forward the packets,the algorithm fails,however,when it

succeeds the resultant path minimizes the total energy con-

sumption of the packet.The greedy algorithm has power

dilation close to the PowerProgress algorithm with much

better delivery rate.The other two deterministic algorithms

also have similar performance.

We can divide our randomized algorithms in two groups

based on their biasing strategy,namely greedy and com-

pass based algorithms.The compass based algorithms dom-

inate the other group both in terms of packet delivery rate

and power dilation.PowerCompass,CostCompass,and

Power*CostCompass algorithms select candidate nodes

that are close to the direction of the destination in addition

to minimizing the power and/or cost.After that the next

node is chosen with higher probability such that it again

minimizes the angle formed between that node,the cur-

rent node,and the destination.In greedy based algorithms,

PowerGreedy,CostGreedy,and Power*CostGreedy

algorithms select the candidate nodes in a slightly diﬀer-

ent way.The candidate nodes minimize both the distance

to the destination and energy requirement.The next node

is picked with higher probability such that it reduces the

distance to the destination.Our simulation results show

that the ﬁrst group has better performance in terms of both

packet delivery and power dilation compared to the other

one.Following the direction helps the packet to reach the

destination early.On the other hand,in greedy based al-

gorithms,candidate nodes are chosen such that these nodes

minimize the distance to the destination and the energy re-

quirement.However,choosing farthest neighbor of the cur-

rent node is more energy consuming than picking a node

that minimizes the angle formed between that node,the

current node,and the destination.This contributes higher

power dilation in our randomized greedy based algorithm.

Also,in these algorithms,the packets may deviate from the

direction of the destination,and this may oﬀer low packet

delivery rates.

We also examine the behavior of the variants of our algo-

rithms.In these variants,c picks the candidate nodes just

as the above mentioned algorithms,however,the next node

is chosen out of two candidate nodes that minimize power,

cost,or power*cost to reach the destination.These algo-

rithms also have delivery rates similar to our above men-

tioned algorithms.However,they have very high (almost

three times larger than the deterministic algorithms) power

dilations.This might be happening due to the fact that

their corresponding biasing does not provide any progress

(either following the direction or reducing the distance to

the destination).

The three compass based algorithms have similar perfor-

mance.This is expected because three of the algorithms use

the same biasing method to reach the destination.Hence,

they have similar delivery rates.The power dilation is also

similar with PowerCompass having slightly better results

compared to the other two algorithms.We also expect it

since PowerCompass always tries to minimize the power

requirement during routing while following the direction to

the destination.However,the CostCompass algorithmmight

follow a slightly longer path than PowerCompass,which

mainly considers remaining lifetime of a node to pick the

candidate nodes.Finally,Power*CostCompass algorithm

just combines both power and cost to pick next nodes and

has performance close to both methods.The greedy based

algorithms also show similar behavior with PowerGreedy

having the highest packet delivery rate as well as lowest

power dilation.The other two algorithms have similar per-

formance.

Algorithms

Delivery Rate

Power Dilation

Greedy

52.17

1.04

Compass

51.01

1.10

PowerProgress

37.20

1.02

ProjecPProgress

45.31

1.19

PowerGreedy

68.03

2.31

CostGreedy

66.73

2.78

Power*CostGreedy

66.74

2.86

PowerCompass

77.03

1.95

CostCompass

75.73

2.09

Power*CostCompass

74.74

2.10

Table 3:Average packet delivery rate and power

dilation on UDG of 87 nodes under cluster distribu-

tion,for transmission radius r = 15m.

The results of our simulation on cluster distribution are

given in Table 3.We investigate the performance of all

the algorithms on cluster node distribution.The random-

ize algorithms still dominate the deterministic algorithms

in terms of packet delivery rate though the performance on

power dilation is just opposite.Furthermore,compass based

algorithms again have better performance than greedy based

algorithms.

77

7.1 Effect of Node Density

As the number of nodes grows,the delivery rate of all

the algorithms increases.However,we notice the signiﬁcant

change in delivery rates in the case of deterministic algo-

rithms that performwell in dense networks.In contrary,our

randomized algorithms show the same behavior in sparse to

dense networks.This gives another explanation of the ap-

plicability of our algorithms in a sparse network with the

presence of local maximum.

In terms of power dilation,however,the results are dif-

ferent for the deterministic and randomized algorithms.For

the deterministic algorithms,the power dilations increase

very slightly as the number of nodes increases.Whereas,

our randomized algorithms show the opposite trend.The

power dilations in the UDG decrease as the number of nodes

increases.For larger values of n,the number of possible

paths available to the randomized algorithms increase,so

the power dilation can be expected to decrease.For in-

stance,in a denser graph,the algorithm can recover from a

bad path earlier,which leads to a lower power dilation for

the algorithm.The deterministic algorithms on the other

hand might need to traverse some extra length in dense net-

works,which leads to slightly worse dilations.

8.CONCLUSIONS

In this paper,we extended our previous work [4] and pro-

posed a set of new randomized energy aware routing algo-

rithms for mobile ad hoc and sensor networks.In our algo-

rithms,the current node holding the packet always forwards

the packet to the next node based on the position of itself,

its neighbors,and the destination.The compass based al-

gorithms are called PowerCompass,CostCompass,and

Power*CostCompass algorithms.In particular,to deter-

mine the next node at any point,these algorithms pick one

candidate above and one below the line between the current

node to the destination,by using the heuristic that mini-

mizes both the power and/or cost and the angle (formed

between the current node,candidate node,and the destina-

tion).The next node is then chosen with higher probability

to be closer to the direction of the destination.On the other

hand,the greedy based algorithms PowerGreedy,Cost-

Greedy,and Power*CostGreedy ﬁrst minimize the en-

ergy requirement and the distance to the destination for the

chosen candidate notes.Then,the next node is the one that

is closest to the destination,which is picked with higher

probability.

Our simulation results demonstrate that our randomized

energy aware algorithms yield a deﬁnite improvement over

all deterministic algorithms studied in terms of the delivery

rate.The best power dilations are achieved by the determin-

istic algorithms,however,by using weighted randomization

based on the angles created by the candidate neighbors and

the

cd line,we can maintain the improved delivery rates

while greatly reducing the power dilations of the random-

ized algorithms.Our algorithms retain their performance

even under cluster node distribution while deterministic al-

gorithms lose their performance.

9.ACKNOWLEDGMENTS

The anonymous referee’s comments are gratefully acknowl-

edged.This research is supported in part by the Natural

Sciences and Engineering Research Council,Canada.

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