Accepted Manuscript

Energy Balanced Position-based Routing for Lifetime Maximization ofWireless Sensor Networks

Vipin Kumar , Sushil Kumar

PII: S1570-8705(16)30198-6DOI: 10.1016/j.adhoc.2016.08.006Reference: ADHOC 1434

To appear in: Ad Hoc Networks

Received date: 29 January 2016Revised date: 11 June 2016Accepted date: 20 August 2016

Please cite this article as: Vipin Kumar , Sushil Kumar , Energy Balanced Position-based Rout-ing for Lifetime Maximization of Wireless Sensor Networks, Ad Hoc Networks (2016), doi:10.1016/j.adhoc.2016.08.006

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Energy Balanced Position-based Routing for Lifetime

Maximization of Wireless Sensor Networks

Vipin Kumar, vipinjnu2010@gmail.com, Sushil Kumar, skdohare@yahoo.com

School of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi, India

Abstract

Maximizing the network lifetime is the fundamental design issue in wireless sensor networks

(WSNs). The existing routing algorithms DIR, MFR, GEDIR, and Dijkstra’s algorithms select the

same set of sensors for a source-destination pair for packet transmission that results in an early dead

situation of the sensors, leading to the partition in the networks. In this paper, we present a new

position based routing algorithm to fairly use the energy of the sensors to maximize the network

lifetime. The forwarding search space (FSS) is introduced to control unnecessary transmissions. A

next forwarder selection function is designed based on the residual energy, node degree, distance,

and angle. Each time, this function selects different set of sensors for packets transmission, which is

the essence of fairly balance the energy consumption among the sensors. Further, the mathematical

expression for connectivity probability, expected one hop distance, expected distance between source

and destination node, expected hop count, and expected energy consumption are derived. The

simulation results are given to validate the analytical results. The experimental results show the

proposed algorithm outperforms the state of the art routing algorithms in terms of network lifetime.

Keywords

Wireless sensor networks; Position-based routing; Lifetime maximization; Residual energy;

Load balancing.

1. Introduction

Wireless sensor networks are formed by a group of large number of sensors deployed over a

geographical region without using any infrastructure [1, 2]. The purpose of installing the sensors in a

particular area is to sense the various kinds of phenomena and forward the sensed information to the

sink [3]. These sensors are battery dependent and have limited transmission range. The dissemination

of the information is done in a multi-hop fashion where at the same time a sensor produces the data

packets and also acts as a relay node to forward the information sensed by neighbor sensors [4].

Because of short transmission range a route is formed through various hops, and information is

forwarded towards the destination [1]. WSN can be used in many applications such as monitoring or

tracking the enemies, battlefield surveillance, nuclear, biological and chemical attack detection [1].

Sensors can be installed in cities to monitor and control the concentration of dangerous gasses, and

fire in the forest. Because of limited battery power, storage capacity, communication ability, and

computing ability, the design of routing algorithm is an important issue in order to maximize the

network lifetime [5]. In shortest path forwarding schemes, for a source-destination pair, packets are

forwarded through shortest path. This leads to early energy depletion of the sensors along this path

[6]. This quick energy depletion of some sensors in the networks creates partitions. Hence, lifetime

of the network is closely related to the energy consumed by the individual sensor. Therefore, it is

important to design a routing algorithm where each sensor should be used efficiently to improve the

network lifetime.

One important factor that affects the network lifetime is an approach used to find path between two

sensors. Another factor that affects the network lifetime is energy consumption balancing among the

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sensors. If more packets are transmitted in route selection process, more energy will be consumed.

The problem can be described in other words as to balance the load evenly in the networks and

minimizing the transmissions.

In this paper, we present a new position-based routing algorithm to address the problem of lifetime

maximization of WSN. The routing algorithm balances the energy consumption of individual sensor

by selecting the different sets of sensors for packets transmission in the network. The concept of

forwarding search space is introduced to reduce the unnecessary transmissions. By prohibiting the

unsuitable neighbor sensors from participating in the path finding process, unnecessary transmissions

are reduced, and network lifetime is maximized. The proposed routing algorithm is compared with

DIR, MFR, GEDIR, Dijkstra’s algorithms, and energy-balanced routing method based on forward-

aware factor (FAF-EBRM).

1.1. Our contributions

In large-scale WSNs, the problem of energy balance is relevant in order to maximize the network

lifetime. It is assumed that the network lifetime corresponds to the number of packets transmitted for

a source-destination pair before the first node dies. To ensure the energy balancing, we focus our

study on fairly balancing the traffic load as equally as possible among the sensors. It is assumed that

the location of the destination is known, then on this basis a forward search space is designed

accordingly. Rather involving all the neighbors of a sensor, only sensors lie in FSS participate in

routing process. This controls the overhead occurs due to unnecessary transmissions. Based on the

residual energy, node degree, distance, and angle, a next forwarder selection function is also

designed. Moreover, mathematical expression for connectivity probability, expected one hop

distance, expected distance between source and destination node, expected hop count, and expected

energy consumption are derived. We compare the proposed algorithm with other routing algorithms,

namely, DIR, MFR, GEDIR, and Dijkstra’s algorithms. The main contribution of our paper is to

efficiently balance the load inside the network so that significant energy saving can be achieved.

1.2.Organization

The rest of the paper is organized as follows. In Section 2, related work is summarized. The

proposed energy balanced position-based routing algorithm is discussed in Section 3. In Section 4,

analytical framework, experiments and discussions are described. Finally, Section5 concludes the

work done and future scope to extend the current work.

2. Related work

In [7, 8], various routing protocols have been discussed. In position-based routing, each node knows

its position through the use of GPS or some other types of positioning services. A location service

provides information about the position of the nodes. In position-based routing, a node takes

forwarding decision based on the position of itself, one-hop neighbors, and the destination. The

sensor node obtains the position of its neighbors through one-hop broadcasting. In paper [9, 10],

various greedy routing techniques have been discussed. Greedy routing approaches are based on the

progress, distance, and the direction. Most forward within radius (MFR) is the first progress based

routing introduced by Takagi and Kleinrock [11], in which the node with maximum forward progress

is selected as the next forwarder. Hou and Li [12] proposed nearest with forward progress (NFP),

where, the next forwarder with forward progress is selected that is nearest to the sender. Compass

routing (DIR) is the direction based routing defined by Kranakis et al. [13 14]. In DIR, the neighbor

node that is the closest to direction of the destination is selected as next forwarder. Geographical

distance routing (GEDIR) is the variant of distance-based routing, proposed by Stojmenoic and Lin

[15]. In paper [16], Dijkstra’s algorithm has been proposed to obtain the shortest path between two

nodes.

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Location-aided routing (LAR) [7] protocol uses the location information of the nodes. LAR is

categorized into two types of algorithms, namely LAR1 and LAR2. LAR1 uses two geographical

regions to forward the path finding packets. These two regions are ExpectedZone and RequestZone

[7]. In LAR1 algorithm, the information about RequestZone is explicitly specified in the

RouteRequest packet. In LAR2 algorithm [7], the source includes distance between itself and the

destination in RouteRequest packet. This RouteRequest packet also contains the (X, Y) coordinates

of the destination node. In paper [17], a Virtual backbone scheduling (VBS), based on sleep-

scheduling technique is proposed. In VBS, to maximize the network lifetime, multiple overlapped

backbones are designed and used alternatively. Backbone sensors are responsible to forward the

traffic while other sensors turnoff their radios to conserve the energy.

In paper [18], to prolong the network lifetime, based on distance and density distribution, a load-

balanced clustering algorithm is proposed. The density and residual energy are used to form clusters.

In paper [19], to evaluate a set of link-disjoint paths, a path vacant ratio is proposed. Also, to adjust

the load, a load balancing algorithm is proposed. The packets are divided into multiple segments

using a threshold sharing algorithm and then these segments are transmitted to the destination using

multipath.

In the paper [20], FAF-EBRM is proposed which balances the energy consumption to prolong the

lifetime of WSN. The selection of the next-hop node is based on link weight and forward energy

density. To maximize the lifetime of the WSN, using ant colony optimization (ACO), an optimal-

distance-based routing scheme is proposed in the paper [21]. A General Self-Organized Tree-Based

Energy-Balance routing protocol (GSTEB) is proposed in the paper [22]. It maximize the network

lifetime through balancing the energy. It builds a routing tree where in each round; BS allocates a root

node and broadcasts this selection to all nodes.

In the paper [23], we have developed position-based beaconless routing (PBR) algorithm that

eliminates unnecessary transmissions in forwarding the data to next forwarder sensor. The process of

beaconing used to gather the information about the position of neighboring sensors consumes extra

energy due to periodic packet transmissions. PBR uses distance and angle based neighbor selection at

each forwarding step. The end-to-end delay and energy consumption along the path is computed. In

addition, the impact of packet transmissions in average energy depletion, residual energy of the

network and changes in network topology is discussed. This work was focused on conservation of

energy by reducing the unnecessary transmissions. In our new work, we focus our study on fairly

balancing the traffic load as equally as possible among the sensors to enhance the lifetime of overall

network. The balancing of traffic load is ensured through multipath routing. Rather than using the

same path, different paths are used in data packet routing.

In DIR, MFR, and GEDIR routing techniques, all the neighbors of the sender node are involved in

routing decision. This leads to unnecessary overhead, and extra energy consumption due to

participation of the nodes lying in backward direction. This extra energy consumption leads to shorten

the network lifetime. In our work, a different approach is proposed and a forward search space in

forward direction is introduced where a significant number of sensors participate in routing decision.

The drawback of state of the art algorithms is that for a source destination pair, the generated traffic is

always forwarded through the same path. This result in quick energy depletion of the sensors

contained therein. The proposed algorithm gives the solution to this problem by sending the generated

traffic through multiple paths, instead of using the same path.

3. Energy Balanced Position-Based Routing Algorithm

In this section, we present the design of energy balanced position-based routing algorithm. First the

network and energy models used in this work are presented and then forwarding search space, next

hop selection function, and packet forwarding are explained.

In this paper, we mainly focus on the lifetime maximization and energy conservation by distributing

the load evenly in the network. So, in the proposed work, the number of hops and the end-to-end

delay along the computed path may be high. This may be the overhead of the proposed work.

Because instead of using the shortest path, each time it uses multipath for the packet transmissions.

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The energy consumption is controlled by using a forward search space (FSS) and duty cycle

discussed in next sections.

3.1 Network model and Energy model

A WSN consists of sensors, uniformly distributed in a pre-defined region. It is assumed that no

two nodes have the same location, and the maximum transmission range of each node is . Each

node knows its own location and the location of the sink node. A WSN can be represented as an

undirected graph ( ) where ( ) and each represents a sensor and is

the set of links between any two neighbors. The coordinates of and are denoted as ( ) and

( ) respectively, and distance between and is calculated

as, √( ) ( )

. For any , there is an undirected edge, iff .

To evaluate the energy consumption along the path, a simple model is used [24]. Free space model

channel model ( power loss) and multipath channel model ( power loss) can be used depending

on the distance between the transmitter and receiver. Free space (fs) model is used if the distance is

less than a threshold value , otherwise the multipath channel (mp) model is used. The

transmission energy ( ) consumed by delivering -bits data from node to is given by

( ) ( ) ( )

{

(1)

and the receiving energy to receive the data is given by

( ) (2)

is the electronics energy spent by transmitter electronics and it depends on various factors such as

modulation, digital encoding, and filtering. Amplifier energy, or depends on the

acceptable bit-error rate and the distance to the receiver. Therefore, total energy required to

transmit a packet from a source to the destination, if path length is , can be defined as

∑ {( ) }

, where, is the transmitting amplifier, and is the propagation path

loss exponent [25].The first order radio model is used to evaluate the energy consumption.

Definition 1. For any sensor node , its neighbors are defined as follows

( ) { }

Definition 2. For any sensor node , the neighbor nodes lying in forward direction is defined

as ( ) { ( ) } and the neighbor nodes lying in backward direction is

defined as ( ) { ( ) }

Definition 3. For any sensor node , the nodes lying in forwarding search space (FSS) is defined as

follows

( ) { }

Definition 4. Let a sensor node has initial energy , and in each round i.e., transmission and

reception process, energy is dissipated. After the one round, the residual energy of a node can be

defined as

.

Definition 5. At any instant, if the residual energy of any sensor is less than , is termed as dead

node.

( ) {

}

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Definition 6. In this paper, lifetime of the network is defined as the number of packets transmitted

from a source to the destination before any sensor node died.

3.2 Forwarding search space

In this section, to control the transmission overhead, a forwarding search space for the sender is

defined. Let the distance between the sender and the destination node is and transmission range or

radius is . The angle ∠DSB is β and ∠ASB is (cf. Fig. 1). Then, by using the value of R and d,

value of can be calculated as

(3)

Fig. 1 shows the FSS as the segment SAB of the circular region of the sender S. A circular region is

drawn with transmission range R by making the center point S. Now two tangents DA and DB are

drawn from the destination D. Now two perpendicular radius lines AS and BS are drawn through the

point of contacts. The circle segment SAB is called as FSS.

S

D

R

d

θ

A

B

β Forwarding search

space

Fig. 1.Forwarding search space formation

3.3 Next forwarder selection (NFS) function and Packet forwarding

Let the sender sensor has neighbors where, is the number of neighbors lies in the

FSS. In proposed routing scheme, next hop is selected based on the four parameters namely, , ,

, and where, is the node degree, is the distance of the sensor node from destination,

is the angle between the lines and , and is the residual energy of the node . To become

the next forwarder , all neighbors of sender do not require participating in the next forwarder

selection process. Only the nodes lying in the FSS participate in the selection process. Each node

uses the indicated four parameters to evaluate the value of function defined as

( )

( )

( ) ( ) (4)

Where , , and are the adjustable weights of the parameters , and respectively.

These weights can be adjusted as per the requirement of the applications. Hence, the selection of

next forwarder may be turned with the different values of , , and . In the scenario when the

network is sparse or sensors are mobile, a higher value of is preferred so that the node with more

number of link could have high probability to become as next forwarder. A high value of

increases the chance of selecting a node with the highest residual energy. To balance the energy

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distribution among the sensors, only the residual energy of the sensors may be taken into account, so

that each time the node with high residual energy could get chance to become the next forwarder.

Start

Sender has Data to send

Is destination node in

transmission range of

current sender ?

Send Data directly to

destination

Find FSS

Select next forwarder in FSS

No

Yes

Stop

Transmit Data packet

S

D

A

B

Fig. 2. Flow Chart. Fig. 3. Packet forwarding from source to destination.

Fig. 2 shows the flow chart illustrating the procedure of forwarding of data packets. Fig. 3 shows the

hop-to-hop forwarding of the data packets from the source S to the destination D. When the source

node has data to send, it finds its FSS. Each neighbor node lying in FSS calculates the value of .

Sender node selects a neighbor node having the maximum value of as next forwarder and then

sends the data packets. Each forwarder repeats the same process until the data packet reach at the

destination. If the destination node is in the direct transmission range of the sender, it sends data

directly to the destination without finding the FSS.

The complexity of the algorithm can be calculated in terms of number of communication required in

transmission of data packet from source to destination node. For a sender node, let is the number

of nodes lying in its FSS, and is the number of communication required to find the next forwarder

node. The complexity ( ) can be expressed as the number of communication required to find and

transmit the data packet to the next forwarder node multiply by the expected hop count ( ),

( ) ( ) ( ).

In the proposed scheme, duty cycle is used which is most effective energy-conserving mechanism to

put the sensor nodes in the sleep mode (low-power) until it required for communication again. Duty

Cycle is defined as the fraction of time sensors nodes is in active mode during lifetime of network.

Here, on-demand active/sleep mechanism [26] is used for energy conservation that reduces the

energy consumption in listening/monitoring state as well as limit the time for active state in data

transfer phase. Wakeup radio channel for wakeup the nodes and data radio channel for data transfer

are used. It is assumed that at any instant of time there is only one sender node that has data to send

to a destination node. In proposed scheme, all the sensor nodes do not require to be in active mode.

When a sensor node has data packets, it changes its state listening mode to active mode (i.e., turn

“on” its data radio) and sends the beacon message to its neighbor sensors lying in FSS on

wakeup channel. This beacon message contains the sender’s and destination’s position information.

After receiving the message, neighboring nodes calculate the value of function given in e.q. 4

and sent back the message containing the value of , to the sender. Now, sender node selects a

next forwarder and sends a notification message to . Now node changes its state to active

mode and receives the data packet from source on data channel and simultaneously sends the beacon

message to its neighbors lying in FSS on wakeup channel. This process saves the energy and reduces

the wakeup latency as well. The remaining sensor nodes except which have not been received any

notification message or data packets from sender automatically go to sleep state after timeout of

listening period and remain into sleep state until next cycle time . This current forwarder node

repeats the same process and forwards the data packet to next node. As the data packet is being

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forwarded fromone sensor node to another, the neighbor sensors lying in backward direction of a

sender S go to sleep mode. In this way, energy is conserved, first using FSS and second using duty

cycle. Fig. 4 depicts the transition states of a sensor node.

Listening/

Monitoring state

Sleep State

Active State

Node has

data to send

Dat

a tra

nsfe

r is

com

plet

ed

After tim

eout of

listening mode

Wakeup after expire

cycle time T

c

Fig.4 State transition diagram

V1 V2 V3

Fig.5 Source node V1 sends data to node V2 through node V3

V1 sends beacon

to its Neighbors

V2 sends beacon to

its neighbors

V3 sends beacon to

its neighbors

V1 sends data packet to

V2

V2 sends data packet to

V3

T1 T2 T3

V1 notifies V2

V2 acknowledge V1

V2 notifies V1

V3 acknowledge V2

T4 T5

Wakeup Channel

Data Channel

Fig.6 Pipelined wakeup procedure

In Fig. 5, node V1 wants to transfer the data to node V3 via node V2. At time T1, node V1

broadcasts the beacon in wakeup channel to its neighbors lying in FSS. After receiving the

message from its neighbor nodes, node V1 selects the node V2 as . At time T2, node V1 notifies

node V2 that next data packet is designated to him. Now node V2 sends back acknowledgement

to node V1 and turn on its data channel to receive data from node V1. At time T3 node V2

starts receiving data on data channel as well as it sends beacon message on wakeup channel to select

next node among the nodes lying in its FSS. These both task are done concurrently which

minimize the energy as well as wakeup latency which results into prolonging the network lifetime.

Let active time for a sensor node is , then it can be expressed as

(5)

Where, is the time required to acknowledge the sender that it is ready to receive data,

is the time required to send/receive the data, and is the time required for broadcasting beacon

message. As receiving of data and beacon message broadcasting are done concurrently the time

taken for broadcasting the beacon message can be ignored, now e.q.5 can be written as

(6)

Whereas, listening time for node that only receives beacon message and reply with need to active

for very less time which is maximum to active time .

(7)

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Where, is time required to reply for , and is time after which a node goes to sleep

state. Duty cycle of a node is defined as ratio of total duty time of a node with total time

(cycle time), where, is the total time required to transfer the data from source node to

destination node.

Duty cycle = ⁄ (8)

Therefore, asynchronous mode of on-demand based sleep/wakeup protocol saves the energy as it

minimizes the wakeup latency and it also provides higher duty cycle.

4. Analytical framework, experiments and discussions

In this section, various mathematical models and analysis are presented. The connectivity

probability, number of transmissions, and expected energy consumption under varying parameters

are presented. Further, extensive experiments are conducted to evaluate the performance of the

proposed algorithm using MATLAB platform. In order to compare the performance, various routing

algorithms such as DIR, MFR, GEDIR , Dijkstra's algorithm , and FAF-EBRP are also simulated.

The parameters used in the simulation are listed in Table 1.

Table 1. Parameter Settings

Parameter Value

500-700

50-200 nodes

80-100

512 bit

50nJ/bit

100pJ/bit/

2

Let is the area, is the node density and is the total number of nodes in a circular region. The

transmission area of a sender node with transmission radius is and . By using

the area and the angle shown in fig. 1, area of FSS can be calculated as

( ⁄ ) ⁄⁄ (9)

It is assumed that the sensor nodes are distributed using Poisson distribution [27] with random

variable that represents the number of nodes in FSS. The probability of nodes present in FSS is

given by

( ) ( )

( ( ⁄ ) ⁄ ) ( ( ⁄ ) ⁄ )

(10)

( ) ( )

( ( ⁄ ) ⁄ ) (11)

The probability of at least one node in FSS can be defined as

( ) ( )

( ( ⁄ ) ⁄ ) (12)

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Figure 4 and 5 show the probability of k nodes present in FSS for different values of N and R,

respectively. It can be seen that as the network size and transmission range increase, the probability

of occurrence of number of node in FSS increases. This is because when network size increases

consequently, node density increases which result in more number of nodes in FSS. When

transmission range R increases, area of FSS also increases because the area depends on the value of

R, consequently, more number of nodes lies in the FSS. In Fig. 7, for example, N=100, the

probability of 5 nodes in FSS is high, as the network size increases to 150, the probability of number

of nodes lie in FSS increases to 7. Similarly, from the Fig.8 , it can be noticed that when R=80 m, the

maximum number of nodes in FSS is 3, as the transmission range increases to 100m, the probability

of 5 number of nodes is high.

Fig. 7. Probability of number of nodes Fig. 8 Probability of number of nodes

in FSS with different network size. in FSS with different transmission range

In Figure 6 and 7, the probability of at least one node in FSS for a certain sensor density and

distance, respectively is depicted. From Fig.9, it can be observed that the probability increases as the

sensor density and transmission range increase. Similarly, when distance and the transmission range

increase, the probability of at least one node in FSS also increases as shown in Fig.10. For a

transmission range over 80m and sensor density of 0.0007 sensors/ m2, the probability is close to 1.

But as the transmission range increases to 100m, for sensor density of 0.0004 sensors/ m2, the

probability is close to 1. The reason is that as the transmission range increases, the area of FSS

increases. Consequently, more number of nodes will lie in the FSS. Similarly, when transmission

range and distance d increase, the area of FSS also increases, which results in more number of

sensors in FSS.

Fig.9. Probability of at least one node Fig.10. Probability of at least one

with different node density and node with varying transmission

transmission range. range and distance.

5 10 15 200

0.05

0.1

0.15

0.2

Number of nodes (k)

Pro

babi

lity

of k

nod

es in

FS

S

N=100

N=150

N=200

5 10 150

0.05

0.1

0.15

0.2

0.25

Number of nodes (k)

Pro

babi

lity

of k

nod

es in

FS

S

R= 80m

R= 90m

R=100m

2 4 6 8

x 10-4

0.8

0.85

0.9

0.95

1

Sensor density ()

Pro

babili

ty o

f atleast

one n

ode in F

SS

R= 80 m

R= 90 m

R=100 m

200 250 300 350 4000.94

0.95

0.96

0.97

0.98

0.99

1

Distance (d)

Pro

babi

lity

of a

tleas

t on

e no

de in

FS

S

R= 80 m

R= 90 m

R=100 m

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4.1 Expected Hop Count ( )

To compute the expected hop count analytically, first the expected one hop distance ( ) between

source S and the next forwarder neighbor is calculated. Also, the expression for expected distance

( ) between S and D is derived. it is assumed that there are neighbors of

distributed over the FSS. The distances from to the neighbor nodes are , where . Let

be potential forwarder sensor situated at maximum distance from that is used to forward the

packets. To calculae the expected value ( ) of first, cumulative density function (CDF) ( ) and

probability density function (PDF) ( ) are derived.

( ) ∏ (

)

The PDF of can be expressed as

( )

( )

( )

(

)( )

By definition, the expected value of is

( ) ∫ ( )

∫

( )

( ) (13)

Here, it is assumed that the sensing region is square in shape with side length of . If S and D are

located at corners of the diagonal, can be the maximum distance between S and D. Let ( )be

the expected value of distance between S and D. The CDF ( ) and PDF ( ) of can be

expressed as

and

, respectively.

By definition, the expected value of is

( ) ∫ ( )

[( ) ] (14)

For simplicity, it is assumed that . Now, eq. 14 can be written as

( )

[( ) ] (15)

Therefore, expected hop count ( ) can be calculated as

( ) ( )

( )

( ) ( )

(16)

4.2 Connectivity probability ( )

Let ( ) be the connectivity probability of a route from source S to destination D. For any two

sensor nodes and a link will exist if the distance between them lies within the transmission

range. In other words, at least one sensor node exists in FSS, the connected path from to can be

obtained. Therefore, the connectivity probability of the route from S to D can be defined as

( ) ( ( ⁄ ) ⁄ )

( ) ∏ ( ) ( )

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∏ ( ( ( ⁄ ) ⁄ )) ( )

( ( ( ⁄ ) ⁄ )) ( )

(17)

Fig. 11 presents the connectivity probability with different parameters transmission range and sensor

density. It can be observed that as the sensor density and transmission range increase, the

connectivity probability increases. In Fig.12, when distance and network size increase the

connectivity probability increases.

Fig. 11. Connectivity probability with different Fig. 12 connectivity probability with different

sensor density and transmission range. network size and distance.

From Fig. 11, for R=80 m and sensor density of 0.0007 sensors/ m2, the connectivity probability is 1

and for R=100 m and sensor density of 0.0005 sensors/ m2, the connectivity probability is 1. From

Fig.12, it can be noted that when N=150 and d=400 m the connectivity probability is close to 1

while, for N=200 and d=250 m, connectivity probability is 1. It can be concluded that, to deliver the

data packet successfully to the destination, there should be a connected path from S to D. Therefore,

the sensor density, and the transmission range can be adjusted as per the requirements.

4.3 Impact of network size on the expected distance, and expected hop count

To see the variation in the expected one hop distance, and hop counts against network size, the

resulting function for ( ) and ( )are presented in Fig.13 and Fig. 14, respectively. It can be seen

that as the values of both N and R increase, expected one hop distance ( )increases. This is because

number of neighbor nodes in FSS will increase.

2 4 6 8

x 10-4

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Sensor density()

Conn

ectiv

ity p

roba

bilit

y Pr

c(l)

R= 80 m

R= 90 m

R= 100 m

200 250 300 350 4000.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Distance ( d )

Conn

ectiv

ity p

roba

bilit

y Pr

c(l)

N=100

N=150

N=200

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Fig. 13. Expected distance with different Fig. 14 Hop counts with different

network size and transmission range. network size and side length.

.

When N=100 and R=100 m, the expected one hop distance is 71.96 m, as the value of N increases to

100, the expected one hop distance becomes 83.69 m. As shown in Fig. 14, an increase in the

network size decreases the hop counts because of increasing the one hop expected distance (c.f., eq.

16). Also, when L increases the value of ( ) increases, this is because of increasing the ( ). For

a given L=500m, and N=50, the value of expected hop count is 4.91, while, for the same value of L,

and N=200, the expected hop count decreases to 3.99.

4.4 Expected energy consumption ( ) in data packet delivery

In this section, expected energy consumption in the delivery of data packet along the route from S to

D is derived. In transmitting and reception of data packet of size of bits through a link of length

between two nodes and , the energy consumption is defined as

( ) (18)

Now the PDF of can be calculated as

( ) ( )

( ) (19)

where, and

∫ ( )

[(

) (

)]

(

) (

)

By putting the value of , e.q.19 can be written as follows

( )

(

) (

) ( )

Now, the expected value ( ) of can be calculated as

( ) ∫ ( ) ( )

50 100 150 20050

55

60

65

70

75

80

85

90

95

Network size (N)

Exp

ecte

d di

stan

ce E

(q)

R= 80 m

R= 90 m

R= 100 m

50 100 150 2003

4

5

6

7

8

9

Network size (N)

Expecte

d h

op c

ount

E(C

h)

L= 500 m

L= 600 m

L= 700 m

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∫( )

(

) (

)

(

) (

)[

]

(

) (

)[(

) (

)](20)

For simplicity, it is assumed that , because it is very small value. Now, e.q. 20 can be

represented as

( )

(

) (

)[(

) (

)]

In the delivery of bits data through a routing path from to , expected energy consumption can

be expressed as

( ) ( ) ( )

(

) (

)[(

) (

)] ( )(21)

4.5 Impact of sensor density and sensing region size on the expected energy consumption

To see the variation in the expected energy consumption with respect to sensor density and sensing

region size, the analytical results obtained from e.q.21, are presented in Fig. 15 and Fig. 16. From the

figures, it can be observed that as the sensor density increases, expected energy consumption

decreases.

Fig. 15.Expected energy consumption Fig. 16. Expected energy consumption

with different sensor density and with different network size and

sensing region size sensing region size.

This is because, as the value of ρ increases, the probability of more number of sensors in FSS

increases. Consequently, a more suitable next hop situated at the optimal position from the sender,

can be selected. Also, if the value of ρ increases, the expected hop counts decreases. On the other

hand, when the sensing region size L increases, the expected value of energy consumption increases,

because for a fixed network size, the expected hop counts increases. Consequently, more energy is

required in transmission because the energy consumption depends on the path length between the

sender and destination.

1 2 3 4

x 10-4

1

1.5

2

2.5

3

3.5

Sensor density ()

Exp

ecte

d en

ergy

con

sum

ptio

n(m

J)

L=500m

L=600m

L=700m

500 550 600 650 7001

1.5

2

2.5

3

3.5

Square region size (L)

Exp

ecte

d en

ergy

con

sum

ptio

n(m

J)

N= 50

N=100

N=150

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4.6 Comparison with the DIR, MFR, GEDIR and Dijkstra's routing algorithms

This section presents the routes searched by the proposed algorithm and other greedy algorithms.

The routing paths computed by the proposed algorithm with different value of adjusting

parameters are presented in Table 2, and Table 3. To compute the routes, initially, in a

square region of area 500m×500m, a random network topology (c.f. Fig.17) is generated, and the

coordinates of each sensor node are assigned. Then, links are established between sensor nodes. If

the distance between two nodes is less than or equal to the radio transmission range, then a link is

created.

Fig.17. Network topology of 100 nodes

Table 2, various routes are listed, computed by the proposed algorithm. Here, the source node, and

the destination nodes are taken 25 and 41, respectively. From Table 2, it is observed that when the

network size is 100, and the initial energy of each sensor is 10 units, the computed routes are

different for packet transmissions. Here, 20 packets are transmitted from source node number 25 to

destination node number 41. For load balancing in the networks, a higher value of is preferred.

Each time next forwarder is selected, the residual energy has preference over other parameters

i.e., , , and , so that the load can be distributed to a maximum number of nodes.Consequently,

for a source-destination pair, the same path will not be selected for the transmission of several

packets. Therefore, energy load is distributed among the sensors that result in lifetime maximization

of the network. The same behavior of the proposed algorithm can be seen when the network size is

100 and . For these parameter values, the various routes found by the

proposed algorithm are listed in Table 3. So, the values of the indicated adjusting parameters can be

tuned as per the requirement of applications. Table 4 presents the path computed by the DIR, MFR,

GEDIR and Dijkstra's algorithms. It can be noticed that in packet transmission between a source-

destination pair, greedy algorithms always select the same route. Consequently, the energy of the

sensor nodes belonging to the routes depletes early. This results in more number of dead nodes in the

networks. Therefore, the network becomes disjoint which results in shortening the networks lifetime.

From the above study, it can be concluded that the proposed algorithm provides better results in

terms of load balancing and network lifetime.

Table 2. Experimental results of computed routing paths when N=100,

Packets Routes Hop

count

0 100 200 300 400 5000

50

100

150

200

250

300

350

400

450

500

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

2627

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

4748

49

50

51

5253

54

55

56

57

58

59

6061

6263

64

65

66

67

68

6970

71

72 73

74

75

7677

78

79

80

81

82

83

84

85 86

87

88

89

90

9192

9394

95

96

97

9899

100

Source

Sink

X-axis

Y-a

xis

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1 25 97 78 83 80 79 76 65 41 9

2 25 87 69 70 77 60 47 32 9 41 10

3 25 50 95 72 67 36 74 96 71 6 12 9 41 13

4 25 50 68 57 61 3 56 12 9 41 10

5 25 87 69 48 35 28 90 23 41 9

6 25 50 51 49 33 98 64 66 52 71 6 12 9 41 14

7 25 97 78 7 70 47 14 41 8

8 25 50 44 19 10 77 43 90 23 41 10

9 25 87 69 16 19 48 60 14 65 41 10

10 25 97 57 16 19 10 80 56 6 12 9 41 12

11 25 87 69 44 16 19 7 61 3 47 41 11

12 25 50 39 29 22 75 46 54 11 66 52 71 6 12 9 41 16

13 25 50 30 2 88 67 4 38 96 52 71 6 12 9 41 15

14 25 87 69 44 16 19 35 28 40 43 90 23 41 13

15 25 97 78 7 83 76 32 65 41 9

16 25 50 39 72 16 36 4 46 54 64 66 52 71 6 12 9 41 17

17 25 87 69 44 48 70 76 41 8

18 25 50 30 68 44 19 36 33 38 11 66 52 71 6 12 9 41 17

19 25 50 51 39 49 22 75 46 74 79 32 41 12

20 25 97 57 44 16 60 14 41 8

Table 3. Experimental results of computed routing paths when N=100,

Packets Routes Hop count

1 25 97 78 83 76 41 6

2 25 97 78 83 76 41 6 3 25 87 69 70 76 41 6

4 25 97 78 83 76 41 6

5 25 87 69 70 76 41 6 6 25 87 69 70 76 41 6

7 25 97 78 83 76 41 6

8 25 50 44 48 60 32 41 7

9 25 87 69 70 76 41 6

10 25 87 97 78 83 76 41 7

11 25 50 44 48 77 14 41 7 12 25 87 69 70 76 41 6

13 25 87 97 78 83 76 41 7

14 25 50 44 48 60 32 41 7 15 25 87 69 70 76 41 6

16 25 87 97 78 77 14 41 7

17 25 50 44 10 80 47 41 7 18 25 87 69 35 77 14 41 7

19 25 87 69 78 35 77 14 41 8

20 25 50 44 48 60 32 41 7

Table 4. Comparison of routing paths obtained by routing algorithms when network size N=100.

Algorithm Routes (For packets 1 to 20) Hop count

DIR 25 97 78 83 76 41 6 MFR 25 97 78 60 14 41 6

GEDIR 25 97 78 60 32 41 6

Dijkstra's 25 97 69 57 10 61 60 47 32 41 10

4.7 The Lifetime and number of dead nodes comparison with the DIR, GEDIR, and Dijkstra's

algorithms

In this section, experiments are performed with different source and destination pairs and the average

value is taken to compare the performance of the proposed algorithm with that of DIR, GEDIR and

Dijkstra’s algorithm. In Fig. 18, the variation in the lifetime of the networks with thedifferent initial

energy of the sensor nodes is presented. From the Fig.18, it can be seen that as the initial energy of

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the sensor nodes increase, for all algorithms, the lifetime of the network increases. It is also noticed

that in our case, the lifetime of the network increases as compared to FAF-EBRM, DIR, GEDIR and

Dijkstra’s algorithm. For example when initial energy is 20 units, the lifetime of the network for the

proposed algorithm, FAF-EBRM, DIR and Dijkstra’s algorithm are about 50, 46, 40 and 30

respectively. The reason is that, each time a packet is sent, DIR, GEDIR, and Dijkstra’s algorithms

always select the same route for each packet transmission that results in an early dead situation of the

nodes. But in our case, proposed algorithm selects a different route to transmit the packets. Thus, the

load is distributed among the maximum number of sensor nodes in the network. Consequently,

network lifetimeis maximized.

Fig.18.Lifetime comparison with Fig. 19. Comparison of dead nodes

varying initial energy of sensors. with varying number of packet sent.

Fig.19 shows the number of dead nodes varying with the number of packets sent. From Fig.19, it is

seen that as the number of packets increases, the number of dead nodes increases because more

energy is dissipated when more number of packets are transmitted. Further, it is also noticed that in

our case, the number of dead nodes is less as compared with that of FAF-EBRM , DIR, GEDIR, and

Dijkstra’s algorithms. This is because, in our case, the different route is selected for different packets

transmission. Thus, more number of sensor nodes participates in packets transmissions.

Consequently, the number of dead nodes is minimized.

4.8 Expected energy consumption and hop count comparisons of simulation and analytical

results

In this section, the simulation result of the proposed algorithm is validated by comparing with

analytical results. A simulation scenario comprises a square region with side length of L=500 m and

network size that varies from 50 to 200 nodes are considered. Fig. 20(a) and Fig.21(a) present the

analytical and simulation results of expected energy consumption, and expected hop count obtained

from e.q.21, and e.q. 16, respectively. First, for a network size, a random source-destination pair is

selected and energy consumption for transmission of 20 packets from source to destination are

calculated. Then, the expected value of energy consumption is computed by taking the average of the

energy consumption for different source-destination pair by generating 100 random topologies. This

process is repeated for varying network size. The comparison of simulation results with that of

analytical results shows the accuracy of the proposed analysis.

From the Fig.20(a), the mean value of energy consumption obtained through simulation and

theoretical analysis are 1.266 and 1.262, respectively. The standard deviation in the simulation and

theoretical results are 0.138 and 0.082, respectively and the mean square error (MSE) is 0.0088.

From the Fig.21(a), the mean value of hop count obtained through simulation and theoretical

analysis are 4.20 and 4.18, respectively. The standard deviation in the simulation and theoretical

5 10 15 20 25 300

20

40

60

80

100

Initial Energy (mJ)

Lifetim

e

Proposed algorithm

FAF-EBRM

DIR

Dijkstra's algorithm

GEDIR

5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

Number of packet sent

Num

ber

of

dead n

odes

Proposed algorithm

FAF-EBRM

Dijkstra's algorithm

DIR

GEDIR

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results are 0.362 and 0.283, respectively and the MSE is 0.111. So, the above study validates, and

fully justifies the proposed approach.

Fig.20. (a).Expected energy consumption comparison of simulation and analytical results obtained

with varying network size. (b) Simulation results comparison with mean value.

Fig.21. (a) Expected hop count comparison of simulation and analytical results obtained with varying

network size. (b) Simulation results comparison with mean value.

Fig. 20(b) depicts how far the data values lie from the mean. In this figure, we take the mean and

move standard deviation in either direction. This means that most of the simulation data values lie

between 1.128 and 1.404. Similarly, from the Fig. 21(b), it can be seen that most of the simulation

data values lie between 3.837 and 4.562. It can be concluded that most of the simulation data values

are close to the mean value.

4.9 Expected one hop, and source to destination distance comparison of simulation and

analytical results

In this section, the obtained simulation results of expected one hop distance, and expected distance

between source and destination node are compared with that of analytical results. Fig. 22 and Fig.

23show the analytical results obtained from e.q.13, and e.q. 15, respectively. The mean value

obtained though simulation and analytical for expected one hop distance are 83.89 and 84.94,

respectively . The standard deviations for simulation and analytical results are 5.609 and 5.415,

respectively. Similarly,the mean value obtained though simulation and analytical for expected

distance between source and distance are 402.7 and 406.7, respectively . The standard deviations for

simulation and analytical results are 32.79 and 32.15, respectively.

50 100 150 2001

1.2

1.4

1.6

1.8

2

Network size (N)

Expecte

d e

nerg

y c

onsum

ption(m

J)

Simulation results

Analytical results

50 100 150 2001

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Network Size (N)

Exp

ecte

d en

ergy

con

sum

ptio

n (m

J)

Simulation results

Mean

Mean+standard deviation

Mean - standard deviation

50 100 150 2002

3

4

5

6

7

Network size (N)

Expecte

d h

op c

ount

Simulation resuls

Analytical results

50 100 150 2003

3.5

4

4.5

5

5.5

Network Size (N)

Expecte

d h

op c

ount

Simulation results

Mean

Mean+ standard deviation

Mean - standard deviation

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Fig.22. Expected one hop distance comparison. Fig.23. Comparison of expected distance

between source and destination node.

5. Conclusion and Future work

In this paper, we introduced a position based routing algorithm to address the lifetime maximization

problem of the networks. The proposed routing algorithm ensures the balancing of energy

consumption of individual nodes throughout the network operation. It uses four parameters namely,

residual energy of the sensor, distance from destination, node degree, and angle in next forwarder

selection at each routing step. Further, a next hop selection function is designed accordingly. The

transmission overhead is reduced by introducing the concept of forwarding search space where only

a subset of sensor nodes participate in the selection process of next forwarder. The mathematical

expression for expected one hop distance, expected hop count, connectivity probability, expected

distance between source and destination node, and expected energy consumption in data packet

delivery are derived. Through simulations, it is verified that the proposed routing algorithm

significantly improves the network lifetime. Thus, the use of the proposed routing scheme is fully

justified. For future work, we plan to work on evolutionary approaches such as genetic algorithm,

and particle swarm optimization to maximize the network lifetime.

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Vipin Kumar is currently a Ph.D. research scholar at School of Computer and

Systems Sciences, Jawaharlal Nehru University, New Delhi, India. His research

interests include Wireless Sensor Networks and Mobile Ad-hoc Networks. He

received his M. Tech degree in Computer Science and Technology from School

of Computer and Systems Sciences, Jawaharlal Nehru University, New Delhi,

India in 2012, and B.Tech degree in Computer Science and Engineering from

Uttar Pradesh Technical University, India in 2010. Mr. Vipin has published

papers in International Journals andConference including Springer.

Sushil Kumar received his Ph.D., M. Tech and MCA degrees in Computer

Science from School of Computer and Systems Sciences, Jawaharlal Nehru

University, New Delhi, India in 2014, 1999 and 1997 respectively, and B. Sc.

degree in Mathematics from Kanpur University, India in 1993. He is currently

working as Assistant Professor at School of Computer and Systems Sciences,

Jawaharlal Nehru University, New Delhi, India. His research interest includes

vehicular ad hoc networks, mobile ad hoc networks and wireless sensor

networks. Dr. Kumar has published papers in International Journals and

Conferences including IEEE, Springer, Inderscience, and Hindawi Publishing

Corporation.