on the lifetime of large scale sensor networks

On the lifetime of large scale sensor networks

Qi Xue*, Aura Ganz

Department of Electrical and Computer Engineering, University of Massachusetts at Amherst, Amherst, MA 01003, USA

Available online 16 February 2005

Abstract

Network lifetime is an important metric for battery operated sensor networks. In this paper, we study the lifetime of a large scale sensor

network with n randomly deployed sensors communicating with a base station (BS), where each sensor node has the same probability to

sense and report its data to the BS. We show how the lifetime of such kind of sensor networks is influenced by transmission schemes, network

density and transceiver parameters with different constraints on network mobility, position awareness and maximum transmission range. Our

results offer insight into the network deployment and protocol design that optimize the network lifetime.

q 2005 Elsevier B.V. All rights reserved.

Keywords: Sensor networks; Lifetime

1. Introduction

The sensor networks [1] that we discuss in this paper

consist of a large number of small battery powered devices

with wireless connectivity. The typical functionality of such

sensor networks involves sampling of environment

information, such as temperature or magnetic field, and

reporting the readings to data collectors or base stations

(BS), where the data will be made available to the end-users.

Such networks have a wide range of potential applications,

from military surveillance to habitat monitoring.

After the initial deployment, sensor networks are left

unattended for a long period of time. Consequently, one of

the most essential requirements for sensor networks is to

maximize their post-deployment active lifetime. In the

literature, there has been extensive work on improving the

lifetime of sensor networks. Some authors make the sensor

node itself as energy efficient as possible [3–5]. The others

propose energy-efficient collaborative schemes between

nodes for sensing and data delivery, including in-network

processing or data aggregation [6–9], routing [10–12] and

medium access [13,14] protocols.

0140-3664/$ - see front matter q 2005 Elsevier B.V. All rights reserved.

doi:10.1016/j.comcom.2004.12.033

* Corresponding author. Tel.: C1 413 545 4847; fax: C1 413 545 1993.

E-mail addresses: [email protected] (Q. Xue), [email protected].

edu (A. Ganz).

In this paper, we focus on a general large scale sensor

network with n nodes, randomly distributed in a disk area,

communicating with a BS located at the center of the disk.

We show how the network lifetime is influenced by the

network density, transceiver parameters and transmission

schemes with different constraints on network mobility,

position awareness, and maximum transmission range. We

define the network lifetime as the cumulative active time of

the network until the first loss of sensing or networking

coverage in the target field. It is a well known fact that the

energy consumed by communication is typically several

orders of magnitude higher than the energy consumed by

computation [15]. Therefore, in this paper, the lifetime of

the sensor networks is derived as a function of the

communication energy consumption only. Unlike other

work on the lifetime of sensor networks [16–18], our key

objective is to explore the fundamental limits of network

lifetime that these schemes strive to improve. In finding the

network lifetime, we expose its dependence on network

density, path loss factor and radio energy parameters. This

allows us to see what factors dominate the lifetime and

consequently where engineering effort should be invested.

These results on achievable network lifetime allow us to

calibrate the performance of collaborative schemes and

protocols for sensor networks.

The rest of the paper is organized as follows. To facilitate

our analysis, a general network model is introduced in

Section 2. The network lifetime for sensor networks using

uniform forwarding schemes is investigated in Section 3.

Computer Communications 29 (2006) 502–510

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Q. Xue, A. Ganz / Computer Communications 29 (2006) 502–510 503

In Section 4, we present a hybrid transmission model to

balance the energy consumption in stationary networks and

prolong network lifetime. Finally, Section 5 concludes the

paper.

2. A general network model

We consider a random sensor network where n sensor

nodes are uniformly and independently distributed in a disk

with radius of R meters on the plane. Every node has a fixed

unit battery life of 1. A base station (BS), that collects the

data from the sensors, is located at the center of the disk. We

study the lifetime of such a sensor network where each

sensor node has the same probability to report its own

readings to the BS. Due to the symmetry property of the disk

area, the results we obtain can be applied to more specific

sensing areas. Note that the energy consumption of the BS is

not included in our study.

To facilitate our analysis, we introduce a hexagon

tessellation for the disk plane in Section 2.1. Using this

tessellation, in Section 2.2, we define two transmission

ranges used by the sensors under different transmission

schemes. Section 2.3 defines the energy consumption model

used in the paper. A list of the symbols used in this paper is

provided in Appendix.

2.1. Space tessellation

Starting from the center of the disk, where the base

station (BS) is located, we use a hexagon tessellation to

cover the surface of the disk as shown in Fig. 1. We call each

hexagon a cell and denote the cell containing the BS as the

center cell. The lateral length for each cell is l.

Cells next to each other are adjacent cells. We define the

set of cells that are adjacent to the center cell as the center

Annulus A1 and use Ai (iO1) to represent the set of cells

Fig. 1. Hexagon tessellation of the disk.

surrounding AiK1 from the opposite side of the BS. We

observe that Annuluses hold the following properties:

(a)
The number of cells in Annulus Ai is 6i.
(b)
The total number of Annuluses, q, needed to fully cover
the disk, must satisfy the following equations:

ð3q C1Þl=2 Z R; q is odd

lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3q C1Þ2 C3

p=2 Z R; q is even

(

For q large, we have:

qlx2

3R (1)

2.2. Transmission ranges

The communication between a sensor and the BS can be

established through either multiple small hops forwarded by

intermediary sensors or a one-hop direct connection. For

each sensor, we define two transmission ranges: a uniform

location independent range for all sensors using multi-hop

forwarding schemes; and a location-dependent direct

transmission range. Note that besides the combination of

uniform hop-by-hop forwarding and direct transmission,

sensors can also forward with varying size of hops along its

way to the BS. However, such approaches are normally

unpractical due to its resulting complex network topology

and difficulties in route maintenance.

2.2.1. Uniform forwarding range

For sensor network using a multi-hop forwarding

scheme, a uniform forwarding range is used by all the

nodes. As shown in Fig. 2, to guarantee that any two nodes

in adjacent cells can always reach each other, we choose the

forwarding range to be:

rf Zffiffiffiffiffi13

pl (2)

Lemma 1. Given that ð3p=4Þl2Z ð100 log n=nÞpR2, there is

a sequence d(n)/0 such that Prob (Each cell contains at

least one node)R1Kd(n).

Proof. As shown in [2]. ,

Note that as shown in [21], for n nodes independent

uniformly distributed in a disk of area pR2, a range

rf ZRffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðlog nCgnÞ=n

pleads to probability of connectedness

converging to 1 as n/N if and only if gn/CN. Clearly,

l

rf

Fig. 2. Range of multi-hop forwarding.

Q. Xue, A. Ganz / Computer Communications 29 (2006) 502–510504

the requirement of Lemma 1 is more stringent than that of

network connectivity only. According to Lemma 1, the

minimum cell size must satisfy:

lmin Z

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi400 log n

3n

rR (3)

2.2.2. Direct transmission range

In addition to forwarding packets via multiple intermedi-

ate hops, each node can also directly transmit to the BS. Due

to the symmetry property of the disk, every node in Annulus

Ak uses the same direct transmission range rk given by:

rk Z lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ðk C1Þ2 C1

p(4)

We observe that the direct transmission range for each

node depends on the distance between the node and the BS.

Such relative location information can be found in the

network layer at each sensor in terms of hop count k to the BS

under uniform forwarding scheme or be inferred at the

application layer by relative location information available in

many sensor network applications, such as tracking and

surveillance. Clearly, as shown in Fig. 3, r1Zrf.

2.3. Energy consumption model

Due to the fact that the energy consumption required by

sensing and computation is several orders of magnitude

lower than the communication power consumption [15], we

will only consider the communication energy consumption.

Assuming a fixed unit packet size, we use the following

models to calculate the energy consumption for

communication:

†
Energy consumption for one packet transmission:
etx(r)ZaCbra.

†
Energy consumption for one packet reception: erxZc.
where a, b and c are constants determined by the sensor

node’s transceiver, aR2 is the path loss factor and r is the

transmission range defined in Section 2.2. It has been shown

that, due to the distance-independent static energy overhead

of a and c, direct transmission is more energy-efficient than

multi-hop transmission for short distances [20]. According

to [19], sample values for both a and c are 50 nJ/bit. Due to

the fact that sensors considered in this paper are small

BS

Cell inAnnulus k

kr

...

Fig. 3. Range of direct transmission.

devices scattered on the ground, we set the height of

transmission and receiving antenna above ground to 0.15 m,

instead of 1.5 m in [19], and have bZ13 pJ/bit/m4 for aZ4.

Given a fixed battery life for each sensor, the lifetime of a

sensor network is determined by the sensor nodes with the

maximum energy consumption per node, emax. Based on the

above network model, we calculate emax in the network for

delivering one packet from each sensor node to the BS, and

find the network lifetime as: LZ1/emax.

Using the definitions presented in this section, in

Sections 3 and 4 we compute the network lifetime for

uniform and hybrid forwarding schemes, respectively.

3. Network lifetime for uniform forwarding schemes

In Section 3.1, we give a general routing model for the

uniform forwarding schemes. Optimal cell sizes for

maximum lifetimes are given in Section 3.2. The asymptotic

network lifetimes are given in Section 3.3.

3.1. Routing model

Consider a general cluster-based model [12] for multi-

hop forwarding in the sensor network. In each cell, a node is

randomly chosen as the cluster head. All the cluster heads

will form a communication backbone for the entire sensor

network. Only the cluster head in each cell will receive and

forward a packet from the outside Annuluses. Other nodes

in the cell can sleep most of the time and wake up only to

sense the environment or transmit their own packets. To

balance the power consumption within each cell, the cluster

head can be re-elected periodically. A detailed study on the

network lifetime increase based on an idealized cell-based

energy conservation technique is provided in [16]. Please

note that in our calculation we ignore the energy consump-

tion of overhearing because of the cluster-based approach.

For any sensor node Xi located in a cell of Annulus Ai, we

connect Xi to the BS with a straight line, intersecting iK1

Annuluses in the disk. As shown in Fig. 4, the route for

packets between the BS and Xi is chosen in a sequence of

hops from one Annulus to another to approximate the

straight-line. In each of these intersecting Annuluses, we

choose one cell which intersects with the line as the relaying

cell for Xi. If the line intersects two cells in a same Annulus,

either cell can be chosen at random. Therefore, packets from

Xi will be forwarded from one cell to another towards the BS

in exactly i hops.

3.2. Optimal cell sizes

Since each cell has an area of 3ffiffiffi3

p=2

� �l2, according to the

law of large numbers, the number of nodes in Ai can be

Fig. 4. Route for packet forwarding.


given as:

ni Z 6i3ffiffiffi3

p

2nl2=ðpR2Þ Z 9

ffiffiffi3

pinl2=ðpR2Þ

with probability approaching 1 as n/N. Due to the need

for traffic forwarding towards the BS, sensors that reside

closer to the BS have much higher energy consumption than

those far away from the BS. As a result, the energy

consumption in a stationary network is unbalanced and

depends on the relative location of each sensor to the BS.

Since nodes in Ak need to relay packets generated from

Annuluses outside of Ak and transmit its own packets, the

total energy consumption of nodes in Ak can be given as:

Ek Z etxðrfÞXq

iZk

9ffiffiffi3

pinl2 Cerx

Xq

iZkC1

9ffiffiffi3

pinl2

" #=ðpR2Þ

Due to the symmetry property of the disk, we assume that

the energy consumption per node within each Annulus is

balanced. The average energy consumption per node in Ak,

ek, can be given by:

ek ZEk

nk

Z etxðrfÞq Ck

2ðq Kk C1ÞCerx

q Ck C1

2ðq KkÞ

�.k

(5)

Clearly, ek is a monotonously decreasing function of k.

The nodes in the center Annulus A1 are the most heavily

loaded by forwarding almost all the packets in the network

to the BS, and we have:

emax Z e1 Z bffiffiffiffiffi13

pl

�a qðq C1Þ

2C ða CcÞ

qðq C1Þ

2Kc

(6)

Comparing eqZetx(rf) with (6), we observe that the

energy consumption in a stationary network is extremely

unbalanced. The nodes close to the BS die out much faster

than those far from the BS, leaving most of the nodes in the

sensor network disconnected with plenty of battery power

left.

Given (1), taking differentiation on both sides of (6) with

regard to q and letting demax/dqZ0, the optimal cell size

that maximizes stationary network lifetime is given by:

lstatic x

1ffiffiffiffiffi13

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ða CcÞ

bða K2Þ

a

r; aO2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4Rða CcÞ39b

3

q; a Z 2

8><>: (7)

Assuming the ideal case where the energy consumption

of each node is perfectly balanced such that every node runs

out of battery at the same time, we have:

emax ZXq

kZ1

Ek=n

Z etxðrfÞXq

kZ1

9ffiffiffi3

pl2k2 Cerx

Xq

kZ1

9ffiffiffi3

pl2kðkK1Þ

" #�ðpR2Þ

(8)

In the ideal case, the network lifetime is maximized when

the total network energy consumptionPq

kZ1 Ek is mini-

mized. Taking differentiation on both sides of (8) with

regard to q and letting demax/dqZ0, we have the optimal

value of l that minimizes the total network energy

consumption:

loptx1ffiffiffiffiffi13

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaCc

b aK1ð Þ

a

r(9)

Note that the result in (9) is congruent with the result in

[18]. However, due to the unbalanced energy consumption

caused by asymmetric traffic forwarding towards the BS, the

optimal cell size for the total network energy consumption

does not necessarily lead to the maximum network lifetime

in stationary networks.

According to Eqs. (6) and (8), Fig. 5 depicts emax versus

the cell size l. In agreement with Eq. (6), for the stationary

case, emax demonstrates a ‘V’ shape in Fig. 5. For l!lstatic,

ðaCcÞðqðqC1Þ=2ÞKc dominates in Eq. (6) and emax is a

decreasing function of l. For lOlstatic, bffiffiffiffiffi13

pl

� �aðqðqC1Þ=2Þ

dominates in Eq. (6) and emax is an increasing function of l.

Similar behavior can be observed for the optimal case

described in Eq. (8). We observe that emax in the optimal

case is much smaller than that of the stationary network. As

the cell size increases towards 100 m the difference between

two cases decreases. For cell sizes large enough such that q

becomes 1, the network becomes single hop and emax for

both cases will become the same.

Given the radio energy parameters in Section 2.3, we have

lstatic x2:6 m and lopt x2 m for aZ4, regardless of n and R.

However, a dense sensor network with one sensor per square

meter in an area of RZ100 m necessitates lmin x13:8 m by

Eq. (3). According to Lemma 1, all possible cell sizes must

Fig. 5. Maximum energy consumption per node emax versus cell size (aZ4

and RZ100 m).


satisfy lRlmin. As a result, the maximum achievable network

lifetime depends on the relationship of lopt and lstatic to lmin,

which is determined by network population n and target area

range R. Given a fix disk size, RZ100 m, Fig. 6 shows the

relationship between the size of the network n and the

maximum achievable lifetimes. For both stationary and

optimal cases, the maximum network lifetime has two phases:

an increasing phase and a saturation phase. When the network

is not very dense and lminOlstaticOlopt, the lifetime is

maximized by using the minimum cell size lmin. In such

case, the maximum lifetime increases with the sensor network

population size n as shown on the left hand side of Fig. 6. Only

for high density networks where lstaticRlmin and loptRlmin, the

maximum achievable network lifetime saturates by using

fixed cell sizes of lstatic and lopt, respectively.

3.3. Asymptotic network lifetime

Unfortunately, according to Eq. (3), nearly a million

nodes are required to have lminZlstatic for RZ100 m.

Fig. 6. Maximum lifetime versus number of sensors n (aZ4 and

RZ100 m).

Therefore, most sensor networks will operate in their

increasing phase in Fig. 6, where the network is not dense

enough to make lstatic or lopt feasible. In such case, given a

fixed disk size R, the lifetime is maximized by using lmin. To

understand the relationship between the network lifetime

and the sensor population size n during the increasing phase,

in this section we focus on the asymptotic behavior of the

network lifetime as a function of the population size, n.

Lemma 2. For a stationary sensor network that uses

uniform forwarding schemes, the maximum lifetime can be

given as Q½ðn=log nÞðaK2Þ=2� for n/N, where aR2.

Proof. Given lminOlstatic, bffiffiffiffiffi13

pl

� �aðqðqC1Þ=2Þ dominates

in Eq. (6), we have:

emax Z e1 Z bffiffiffiffiffi13

pl

�a

Ca Cc � qðq C1Þ

2Kc Z Q½laK2�

Given Eq. (3), the lifetime L can be given as:

L Z1

emax

Z Qn

log n

� �aK2=2 �(10)

,

Assuming the ideal case where the energy consumption

of each node is perfectly balanced in the network, we find

the asymptotic network lifetime as follows.

Lemma 3. In sensor networks that use uniform forwarding

schemes, the maximum lifetime is upper bounded by O½ðn=

log nÞaK1=2� for n/N, where aR2.

Proof. Given lminOlopt and Eq. (8), we have

emax R3ffiffiffi3

p

2pR2l2½etxðrfÞCerx�Q½q3� Z Q½laK1�

where equality holds only when the energy consumption per

node is perfectly balanced, i.e. all nodes run out of battery at

the same time. Thus:

L Z1

emax

Z On

log n

� �aK1=2 �(11)

,

Note that the lifetime upper bound in Lemma 3 is only

achievable when the traffic load and the resulting energy

consumption per sensor are perfectly balanced. Under

uniform forwarding schemes, this can be achieved when

the sensor nodes or the BS move randomly within the disk

during the network lifetime such that the energy consump-

tion per sensor is perfectly balanced over time. Unfortu-

nately, in most deployments the sensors are not capable of

moving. Therefore, to improve the network lifetime, it may

be beneficial for the BS to move randomly in the target area.

Compared to Lemma 2, we observe that network mobility

significantly improves the lifetime of sensor networks by a

factor of Qffiffiffiffiffiffiffiffiffiffiffiffiffiffin=log n

p� �.


4. Network lifetimes for hybrid transmission schemes

Given the optimal lifetime upper bound in Lemma 3, we

are interested to know how close we can approach to this

bound in practical sensor networks where the sensors are

stationary. Since the total energy consumed at a sensor

node is determined by energy consumption per packet

transmission and the amount of traffic transmitted. To

prolong the system lifetime without assuming network

mobility, we should even out the energy consumptions

between different Annuluses by balancing their traffic loads.

To achieve this, we consider hybrid transmission schemes

where both location-dependent direct transmission and

uniform multi-hop forwarding can be used by the sensor

nodes. Such location information is usually available at a

sensor node at both network layer and application layer.

Note that the direct transmission range only introduces

an alternative route from each node directly to the BS,

requiring no additional route maintenance. As a result, the

existence of direct transmissions in hybrid transmission

schemes will not influence the performance of the

underlying routing protocols for multi-hop forwarding.

4.1. Protocol model

As shown in Fig. 7, the hybrid transmission model can be

summarized as follows:

†
Nodes in Annulus Ak, where qRkOm, use the uniform
forwarding scheme.

†
Nodes in Annulus Ak, where mRkR1, use the uniform
forwarding scheme with probability rk; and transmit

directly to the BS, using range rk, with probability 1Krk.

The existence of threshold m allows us to take into

account the maximum transmission range of a sensor, which

is determined by its physical constraints. For example,

Fig. 7. The hybrid transmission scheme.

the transmission range of Berkeley Motes [3] can only vary

from 10 to 300 ft.

In hybrid schemes, the total energy consumption of

Annulus Ak can be written as

Ek Z etxðrkÞNdk CetxðrfÞN

fk CerxNf

kC1

where Nfk represents the number of packets forwarded to

AkK1 and Ndk represents the number of packets directly

transmitted to the BS. Since nodes in Annuluses outside of

Am use only a forwarding scheme, the Annulus with the

maximum energy consumption per node must reside

between A1 and Am, inclusively. Thus, in the following

calculations we focus on the threshold region, where

mRkR1. For Annulus Ak, mRkR1, the following

equations hold:

Nfk Z ðNf

kC1 C9ffiffiffi3

pnl2k=ðpR2ÞÞrk

Ndk Z ðNf

kC1 C9ffiffiffi3

pnl2k=ðpR2ÞÞð1 KrkÞ

(

Given the initial value

Nfk Z

Xq

iZk

9ffiffiffi3

p

pR2inl2

Ndk Z 0

for qRkOm:

8><>:we have

Ndk rk ZNf

Akð1KrkÞ

Nfk Z

9ffiffiffi3

pnl2

pR2

Xm

jZk

jYj

iZk

ri

!C

ðqCmC1ÞðqKmÞ

2

YmiZk

ri

" #8>><>>:

(12)

for mRkR1. The average energy consumption per node in

Annulus Ak can be given as:

ek ZEk

nk

Z1

nk

etxðrkÞNfk

1Krk

rk

CetxðrfÞNfk CerxNf

kC1

�(13)

To find the maximum lifetime of all hybrid transmission

schemes, we need to find the optimal value of q, m and rk

such that emaxZmin1%m%qfmax1%k%m½ek�g and the network

lifetime is maximized.

4.2. Asymptotic lifetimes of hybrid transmission schemes

From Eq. (13), a lifetime curve similar to Fig. 6 can be

plotted. However, it is difficult to derive an optimal cell size

for the hybrid schemes in Eq. (13). Due to the fact that most

sensor networks are not dense enough to make such an

optimal cell size feasible, most networks will operate in

their increasing phase and their lifetimes are maximized by

using the minimum cell size lmin. In this section, we focus on

the asymptotic behavior of the network lifetime as a

function of n when the network is not dense enough to make

its optimal cell size feasible.

k C1%m

k Z m

:


Lemma 4. For the proposed hybrid transmission scheme, an

achievable network lifetime can be given as Q½ðn=

log nÞðaK2CxÞ=2� for n/N, where xZQ½ðlog n=nÞðaK1Þ=2�.

Proof. Due to the complex relationship expressed in

Eq. (13), it is hard to find the optimal expression of rk

that minimizes emax. To remove the nested coefficientQjiZk ri in Nf

k, we use a heuristic value for rk

rk Zk

k C1

� �x

(14)

where xO0 (xZ0 leads to the uniform scheme discussed

in Section 3). Note that (14) is used due to its

mathematical simplicity. It does not necessarily lead to

the optimal lifetime of all possible hybrid schemes.

Given (14), we have:Qj

iZk riZ ðk=jC1Þx and

ð1KrkÞ=rk Z ð1C ð1=kÞÞx K1xx=k. Applying these results

to Eq. (13), we have (15)

ek Z etxðrkÞx

kCetxðrfÞ

h ikxK1

Xm

jZk

j

ðj C1Þx

"

Cðq Cm C1Þðq KmÞ

2ðm C1Þx

�Cerx

NfkC1

nk

(15)

where

NfkC1

nk

Z

ðk C1Þx

k

Xm

jZkC1

j

ðj C1ÞxC

ðq CmÞðq Km C1Þ

2ðm C1Þx

" #;

ðq CmÞðq Km C1Þ

2k;

8>>><>>>:

Letting mZQ[qy]%q, where 1RyR0, we have:

ek Z Q½la�ðQ½kaK1x�C1ÞkxK1Q½q2Kxy�

Clearly, for different k, the optimal values of x and y are

different:

ek ZQ½laq2Kxy�ðQ½x�C1Þ; k Z Q½1�

Q½laq2Ky�ðQ½qyðaK1Þx�C1Þ; k Z Q½m�

((16)

For xZQ[q1Ka] and yZ1, we have:

min1%m%q

max1%k%m

½ek�

!Z Q½laK2Cx�

Based on Lemma 1, an achievable sensor network

lifetime L can be given as:

L Z 1= min1%m%q

max1%k%m

½ek�

!Z Q

n

log n

� �ðaK2CxÞ=2 �(17)

where xZQ½ðlog n=nÞðaK1Þ=2�. ,

Clearly, for any given mZQ[q], the required maximum

transmission range for the hybrid schemes is

rm Z lffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ðmC1Þ2C1

pZQ½1�, i.e. the required maximum

transmission range for optimal network lifetime in the

hybrid scheme remains constant as n increases. Thus, by

choosing a certain value m such that rm is within the

physically feasible range, the position-aware hybrid

schemes can prolong the network lifetime compared to the

uniform forwarding schemes. This improvement occurs due

to fact that the traffic load and associated energy consump-

tion is more balanced across multiple Annuluses.

Lemma 5. In a network where every node can transmit

directly to the BS, i.e. mZq, the lifetime of hybrid schemes

is lower bounded by U½ðn=log nÞðaK2CxÞ=2� for n/N, where

xZQ½ðlog n=nÞðaK1Þ=2�.

Proof. Consider the special case of mZq in Eq. (15), we

have:

ek Z etxðrkÞx

kCetxðrfÞ

h ikxK1

Xq

jZk

j

ðj C1Þx

" #Cerx

NfkC1

nk

Following a similar approach to Lemma 4, we have:

Q½laq�%ek %Q½laq2Kx�; k Z Q½1�

ek Z Q½laq�ðQ½qaK1x�CQ½1�Þ; k Z Q½q�

(

When xZQ[q1Ka], emax Zmax1%k%qfekg is minimized

as:

Q½laq�%emax %Q½laq2Kq1Ka

�

From Eq. (1) and Lemma 1, we have

L Z 1=emax Z On

log n

� �ðaK1Þ=2 �(18)

L Z 1=emax Z Un

log n

� �ðaK2Cx=2Þ �(19)

where xZQ½ðlog n=nÞðaK1Þ=2�. ,

Compared to Eq. (17), we observe that without the

constraint on maximum transmission range for each sensor,

the network lifetime of hybrid schemes can be further

improved due to better balancing of the traffic load and the

resulting energy consumption.

4.3. Simulation study

Note that as n/N, ðn=log nÞx=2 approaches 1. Therefore,

for n very large, the asymptotic lifetime given in Eq. (17)

approaches that in Eq. (10). To evaluate the effectiveness of

energy balancing in hybrid schemes under real network

scenarios, we compare the energy consumption ek specified

in Eqs. (5) and (13) for different network settings. For both

uniform and hybrid transmission schemes, the average node

Fig. 8. Comparison of ek between uniform and hybrid transmission schemes

(RZ100 m).


density of one sensor per square meter is used with

the minimum cell size lmin given in Eq. (3), and aZ4.

The maximum transmission range for a sensor is 100 m.

First, we consider a disk size of RZ100 m, where all

nodes can transmit directly to the BS, i.e. mZq. For

the hybrid schemes, we use a forwarding probability of

rkZ(k/kC1)x with varying x. As shown in Fig. 8, compared

to the uniform forwarding scheme, the average energy

consumption ek is better balanced in the hybrid schemes. For

the hybrid schemes with xZ0.52, the critical energy

consumption, emaxZmax1%k%q ek, is reduced by about

33.3% from the uniform scheme. As a result, the hybrid

transmission schemes improve the network lifetime, given

as LZ1/emax, by 50%.

Next, we expand the disk size to RZ1000 m. Due to the

maximum transmission range limit of 100 m, we have

mZq/10. For the hybrid scheme, we use rkZ(k/kC1)x with

xZ0.3. In agreement with Eq. (16), we observe from Fig. 9

Fig. 9. Comparison of ek between uniform and hybrid transmission

schemes (RZ1000 m).

that the energy consumption of the hybrid scheme within

the threshold region, 4RkR1, displays a ‘V’. This is due to

our choice of rk (see Eq. (14)) which does not necessarily

lead to the optimal energy balancing within the threshold

region. However, as shown in Fig. 9, even with such an

imperfect setting, compared to the uniform schemes, the

hybrid schemes reduce emax Zmax1%k%q ek by about 24.1%.

As a result, the hybrid schemes lead to a lifetime

improvement of 31.8% even though only a small part

(last four hops to the BS) of the network uses both direct

transmission and multi-hop forwarding.

5. Conclusions

In this paper, we have investigated the lifetime of a large

scale sensor network, with n randomly distributed sensor

nodes communicating to a base station. For stationary

sensor networks using uniform forwarding schemes, as

found in most deployments, there exist optimal cell sizes

lstatic and lopt (and the corresponding transmission ranges)

that lead to maximum network lifetime and minimum total

energy consumption, respectively. Unfortunately, depend-

ing on the path loss, radio transceiver parameters and

network density, these optimal cell sizes are not always

achievable. For network not dense enough to use the optimal

cell sizes, the lifetime is maximized by using the minimum

cell size given in Eq. (3). Under such situation, we have

provided relationships between the maximum network

lifetime and population size n for both stationary and

mobile networks. The lifetime of a mobile sensor network is

longer than the lifetime of a stationary network. This is due

to the fact that network mobility can balance the sensors’

energy consumption thus prolonging the sensor network

lifetime.

To prolong the lifetime of a stationary network, we

have introduced location-aware hybrid transmission

schemes that balance the network energy consumption.

For network not dense enough to use the optimal cell size

for the hybrid scheme, the relationship between the

maximum network lifetime and population size n is

given. We have shown that in practical settings, hybrid

schemes can significantly improve the network lifetime

over uniform forwarding schemes. Since the hybrid

schemes do not require any additional route maintenance,

they are suitable for optimizing the lifetime of location-

aware sensor networks.

Acknowledgements

This project was supported in part by the following

grants: NSF-ANI-0319871, NSF-ANI-0230812, NSF-EIA-

0080119, ARO-DAAD19-03-1-0195, DARPA-F33615-02-

C-4031.


Appendix. Symbol definitions

Symbol Definition

n Number of sensors in the disk

R Radius of the disk area

a Path loss factor

l Lateral length of each cell

q Number of Annuluses needed to cover the entire disk area

lmin Minimum cell size for Lemma 1

lstatic Optimal cell size for maximum lifetime of stationary network

lopt Optimal cell size for minimum total network energy

consumption

Ak Annuluses k: set of cells with the same hop count to the BS

rf Forwarding range for all sensors

rk Direct transmission range for sensors in Ak

Nfk

Number of packets forwarded by nodes in Ak

Ndk

Number of packets directly transmitted to the BS by nodes in Ak

etx(r) Energy consumption for one packet transmission using range r

erx Energy consumption for one packet reception

Ek Total energy consumption of nodes in Ak

ek Average energy consumption per node in Ak

rk Probability of a node in Ak uses forwarding scheme

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