Analysis of Lifetime of Large Wireless Sensor Networks Based on Multiple Battery Levels

doi:10.4236/ijcns.2008.12018

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I. J. Communications, Network and System Sciences, 2008, 2, 105-206

Published Online May 2008 in SciRes (http://www.SRPublishing.org/journal/ijcns/).

Analysis of Lifetime of Large Wireless Sensor Networks

Based on Multiple Battery Levels

Ruihua ZHANG1, Zhiping JIA1, Dongfeng YUAN2

1 School of Computer Science and Technology, Shandong University, Jinan 250061, P.R.China

2 School of Information Science and Engineering, Shandong University, Jinan 250100, P.R.China

E-mail: 1ruihua_zhang@sdu.edu.cn

Abstract

Due to the limited transmission range, data sensed by each sensor has to be forwarded in a multi-hop fashion

before being delivered to the sink. The sensors closer to the sink have to forward comparatively more

messages than sensors at the periphery of the network，and will deplete their batteries earlier. Besides the

loss of the sensing capabilities of the nodes close to the sink, a more serious consequence of the death of the

first tier of sensor nodes is the loss of connectivity between the nodes at the periphery of the network and the

sink; it makes the wireless networks expire. To alleviate this undesired effect and maximize the useful

lifetime of the network, we investigate the energy consumption of different tiers and the effect of multiple

battery levels, and demonstrate an attractively simple scheme to redistribute the total energy budget in

multiple battery levels by data traffic load. We show by theoretical analysis, as well as simulation, that this

substantially improves the network lifetime.

Keywords: Wireless Sensor Networks, Energy Efficient, Network Lifetime, Battery Level

1. Introduction

Sensor network applications have recently become of

significant interest due to cheap single-chip transceivers

and micro-controllers. Because sensor nodes are battery-

powered, and their operational lifetime should be

maximized, one of the most important design criteria for

this type of network is energy efficiency.

Confirming the importance of the problem, many

aspects of the problem have been extensively studied [1–

4]. Medium access control (MAC) layer techniques [2]

[3] aim to conserve battery energy by turning the

receiver off whenever it is not needed. It is clear that the

energy problem cannot be completely solved at any one

single layer [4].

The motivation for our work stems from the

observation that in a sensor network, the sensor nodes

closer to the sink have to relay more packets than the

ones at the periphery of the network. We assume that

this increase in workload results in an increase in energy

consumption, the nodes close to the sink will die first,

leading to a premature loss of connectivity in the sensor

network. To alleviate this undesirable effect, we study

the energy consumption of different tiers, and

demonstrate a scheme to redistribute the total energy

budget for the sensor network, the lifetime of the

network can be significantly improved over the case

where all sensors have a uniform lifetime. The optimal

solution is formulated theoretically and validated via

simulations.

About this problem, in [5–9], non-uniform node

deployment is exploited as an alternative manner to get

over the effect of non-uniform energy depletion. The

basic concept is that different node densities are

assigned to different sub-regions trying to balance the

communication load of each sub-region. Some works

specifically consider the scenario for concentric sub-

regions (rings); the node density of a ring is determined

based on the number of hops to the sink, which is

roughly approximated to be the same for all locations in

a ring in [5–7]. In [8], the imbalanced energy utilization

of a WSN was analyzed based on the hop counts, and

the authors accordingly proposed a non-uniform node

distribution depending on the hop-count to improve the

long-term connectivity. In [9], the energy-hole problem

was discussed in a more general form for a rectangular

sensing area with multiple data sinks at different

ANALYSIS OF LIFETIME OF LARGE WIRELESS SENSOR NETWORKS 137

BASED ON MULTIPLE BATTERY LEVELS

locations. Another approach, heterogeneous deployment,

overcomes the communication load unbalance problem

by using two types of sensors, low energy sensors and

high energy sensors, to construct a hierarchical network

structure [10]. However, this scheme raises the

deployment and implementation complexity and cannot

be easily applied.

The remainder of the paper is organized as follows.

First we present the network model and the energy

consumption model in Section 2. Section 3 provides a

mathematical formulation that attempts to estimate the

average lifetime of a sensor network and analyzes the

design problems. We provide the simulation results in

section 4. We conclude the paper with a summary and

discussion of future work in Section 5.

2. Energy Consumption Model

Energy consumption models of the radio illustrated by

Figure 1. In [11] a model for radio energy consumption

is given for energy per bit single hop (eb) as:

btxrx

eee=+ (1)

tx tate

eede

=+ (1-1)

eta

Transmitter

Electronics

Amplifier

eteeta

Receiver

Electronics

K bitsK bits

erx

Figure 1. Radio energy consumption model

where etx and erx are the transmitter and receiver energy

consumptions per bit, respectively, ete is the energy per

bit needed by the transmitter electronics, eta is the energy

needed to successfully transmit one bit over one meter, d

is the distance from transmitter to receiver and

is a

constant which depends on the attenuation the signal

will suffer in that environment.

Consider a circular sensing field with the radius RL

meters, a total of N sensor nodes are uniformly

distributed within the field. The sink node is in the

center of the field. The transmission radius R of the

sensor nodes assumed to be fixed. We divide the sensing

field into T tiers, T=ceiling (RL /R). Figure 2 shows the

“doughnut-like” distribution of nodes [9]. In such a

circular field, consider the set of nodes close to the sink

at the center that can communicate directly with it. We

refer to these one-hop neighbors of the sink as the first

tier of nodes. Similarly, the two-hop neighbors of the

sink will be the second tier of nodes, etc. It is clear that

the first tier nodes relay the largest amount of data

pockets and will die first. If the spatial distribution of

nodes is close to uniform, then the traffic load is equally

distributed spatially. Each first tier node will relay

roughly the same amount of traffic, and all first tier

nodes will die at times very close to each other, after the

network is first put into operation. Once all of the first

tier nodes are dead, no other node will be able to send

data to the sink, and the lifetime of the network will be

over.

Figure 2. Sensing field

We assume that sensed data is collected in a periodic

manner; this period (interval P seconds) consists of the

sensing of the data and the transmission of one packet

containing the data sensed to the sink. We assume that

each sensor has a constant amount of raw data pocket (bs

bits) to sense and send in interval P.

3. Analysis and Numerical Results

3.1. Estimation of Network Lifetime

We define tier i as the set of the nodes that can reach the

sink in i hops (1iT

≤

≤). We consider the energy budget

E as the main cost function, Ei is the sum of the energy

available at the nodes of tier i, so

∑

(2)

Ni is the number of nodes at tier i. With the

assumption of a uniform density:

((21)) /

NiNT=− (3)

Fi is the total number of data packets, they are relayed

to the sink by all nodes at tier i; each data packet is

generated by one sensor node at tiers i+1…T.

22 2

(( ))/

FNNNT iT

=− =−

∑ (4)

138 R.H. ZHANG ET AL.

In tier i, nodes relay all data pockets generated by

nodes at tiers ()

ijT<≤ , and nodes at i tier generate and

send all data pockets , let Erelay and Egen denote their

energy consumed, respectively; es is the energy spent

sensing per bit; bs is data pocket size. So we get the

relation with energy consumption and time t.

irelay gen

EE=+

(5-1)

(/ )

relayisb

tpFbe= (5-2)

(/ )()

eni sstx

tpNbe e=+

(5-3)

Thus

(()()(21))

sbs tx

tNb eTieei

ETp

−++−

= (5)

Li is the lifetime of the nodes at tier i, it can be

determined by considering the energy consumption.

Using (5), we can get:

(()(2 1)())

sb stx

EPT

bTieie e

=−+−+

(6)

In accordance with all the above considerations, we

define the lifetime of the network as the minimum of the

lifetimes of its tiers:

1...

min i

= (7)

In this case, each node will start with the same energy,

namely E/N; then, the energy at each tier is

(2 1)

NE iE

−

== (8)

Hence, (6) becomes

()

bsstx

LTi

beNb ee

=−++

−

(9)

Since the term (T2-i2)/(2i-1) is monotonously

decreasing with i for1iT≤≤ , Li the lifetime of tier i is

monotonously increasing with i. In other words, the

lifetime of the network L is equal to the lifetime of the

first tier:

(1)()

bsstx

TbeNbee

== −+ +

(10)

It is obvious that when the lifetime of the first tier has

expired, the whole network has expired; the nodes in

other tiers have still remaining energy. Using (8) and (9),

the residual energy of other tiers Erest:

()(21)()

[(2 1)]

(1)

bstx

resi

bstx

Tiei ee

TTeee

−+−+

=−− −++

(11)

The above equations can also help us determine a

reasonable energy allocation for all nodes. One possible

criterion is to let the all tiers have the same-targeted

lifetime. Thus by balancing energy allocation, by using

(2) and (6), we can maximize the network lifetime for a

given fixed amount of energy E.

12 T

=⋅⋅⋅= (12)

So we get the relation:

()(21)()

(1)

bstx

Tiei ee

EE Teee

−+−+

=−++

(13)

Using (2) and (12), we get the relation:

132 2

(( 1))

(43)/ 6()

bstx

TeeeE

ETTTe Tee

−++

=−−+ +

(14)

If we allocate the energies at different tiers according

as (13) and (14), we can maximize the network lifetime;

all tiers’ energies will be depleted at the same time.

3.2. Design Problems

From the above, how to allocate the energies in different

tiers is practical importance to the network designers. A

reasonable alternative problem specification would

provide, instead of the total budget E, the uniform

battery level bu for each node. In that case, the lifetime

of the network would be independent of the total number

of nodes N. The only topology relevant quantity would

be the number of tiers T. It will be useful in nodes

energy allocation.

According to (10), it is obvious that when the lifetime

of the first tier and, therefore, the whole network has

expired, the nodes in other tiers have still not expended

their battery energy. The energy consumption for nodes

at different tiers can be easily obtained from a

consideration of (10) and (11). The hypothetical lifetime

of tier i is Li; but, it is actually expending energy for only

the lifetime L of the network. After that, communication

stops, and there is no more expenditure of energy. Thus,

the ratio of battery energy i

bactually consumed by a

node in tier i to the uniform battery level bu that it started

with is:

()

L21

L(1)

bstx

eee

CbTeee

−++

−

=== −++

(15)

We call the ratios Ci the ideal allocation ratios,

because ideally each node would be allocated only the

amount of battery it would actually consume during the

lifetime. The farther out a node, the lower the fraction Ci

of its battery that it has consumed. The unconsumed

energy is wastage of the total energy budget. The broad

design goal is to redistribute this energy budget non-

uniformly so as to increase the lifetime of the whole

ANALYSIS OF LIFETIME OF LARGE WIRELESS SENSOR NETWORKS 139

BASED ON MULTIPLE BATTERY LEVELS

network.

3.2.1. Problem 1

In many practice applications, a designer would have

several known battery capacities to choose from e.g.,

AAA, AA, C and D. Then, the following problem can be

formulated.

Given k available battery levels b1>b2>…>bk>0,

assign the battery level for each tier of nodes in a sensor

network, such that the total battery budget E is

minimized subject to maximizing the lifetime L of the

network.

Two alternative flavors of the above problem can be

articulated: (a) all nodes in any given tier are constrained

to be assigned the same battery level, and (b) different

nodes of the same tier can be assigned different battery

levels. Thus, we are required to obtain one unique

battery level bi for tier i in Problem A; every node in this

tier should be assigned this battery level. For problem B,

we can provide more than one battery level for each tier

accompanied by the proportions of the total number of

nodes in that tier that should be assigned each battery

level.

Below, we address problem 1A first.

The maximum network lifetime is obtained when all

nodes have the maximum battery level b1. Thus, the first

tier of nodes should have batteries of capacity b1, and the

maximum lifetime of the network will be

()

Lb= (16)

where, by Li(b) we denote the lifetime of tier i (given by

(6)) if each node in tier i initially has a battery capacity b.

The same maximum lifetime as in (16) may be

obtained with a lower budget by assigning lower battery

levels to nodes in the higher tiers (in tiers i with 1<i<T);

however, if the next smallest capacity size is too small,

then the lifetime of the network would be reduced

because nodes in higher tiers will deplete their batteries

before the nodes in the first tier.

Given b1, the ideal level of battery that each node of

tier i should have is obviously Cib1, where Ci is the ideal

allocation ratios given by (15), so that tier i will have

exactly the same lifetime as tier 1. In short, for each tier i,

in order to maximize the lifetime of the network and

then conditionally minimize the total battery budget, we

need to make sure that we assign the battery size bj to

tier i such that bj+1<Cib1<bj. If the ideal level is less than

the minimum provided battery level bk, then that

minimum level must be used in that tier instead; in this

case some battery will indeed remain unconsumed at the

end of the lifetime.

Now, if we consider problem 1B, we have the added

flexibility of mixing the provided battery levels. First,

we pick the highest battery level b1 for tier 1 as before:

this maximizes the network lifetime. Now we predict the

ideal battery levels for each tier i as before using (15);

but now, instead of picking the next highest available

level from the available ones, we aim at attaining this

ideal level exactly as the effective battery level by

mixing the two provided battery levels in the appropriate

ratio. In case the desired battery level is exactly equal to

one of the provided battery levels, no mixing is needed.

How to mix the same tier nodes with the different

battery levels? Usually, we would provide two battery

levels to a tier. If we pick battery level b1 for tier 1, the

ideal level of battery that each node of tier i should have

been Cib1. We pick two battery levels bi and bi-1 for tier i

from available battery levels such that b

i<Cib1<bi-1. If

we specified proportions f1 and f2 for the two levels with

f1+f2=1, then the effective battery level of tier i would be

given by bif1+bi-1f2, such that,

1121ii i

bfbfCb

−

= (17)

So we get the relation:

bCb

fbb

−

=−

(18)

Cb b

fbb

−

(19)

3.2.2. Problem 2

In the above problem, minimizing the battery budget is a

secondary goal of the optimal design. It is also possible

that the battery budget may be strictly a constraint for

the design. If the total cost of the batteries used for a

particular network is upper bounded by a total battery

budget E and the battery capacities are fixed and given,

we can formulate the following problem:

Given k available battery levels b1>b2>…>bk>0, and

a total energy budget E, assign the battery level for each

tier of nodes in a sensor network, such that the total

lifetime L of the network is maximized.

As with Problem 1, we can conceive of two alternate

problems, Problems 2A and 2B, in which the nodes in

any tier are constrained to have the same battery level, or

mixing is allowed, respectively.

We consider Problem 2A first.

As the solution to Problem 1A, we assign the battery

levels at different tiers with the maximum lifetime

unconstrained by total battery budget. Assume that Li(bj)

is pre-computed lifetime for each tier i and each battery

level bj. Initialize “current network lifetime” Lc to L1(b1).

Whenever the total batteryij

iNb

∑

becomes less than or

equal to the battery budget E, the algorithm is terminated.

If the current total battery exceeds E, repeatedly

perform the following. Find the tier i such that the

difference Lc–Li(

+1) is minimized. The tier iwill be

assigned the next lower battery level from the one it is

currently assigned, that will result in the minimum

reduction of the lifetime. Note that this minimum

140 R.H. ZHANG ET AL.

difference may well be zero at some iteration. Increment

the current level for tier i by 1 and the update the

network current lifetime to the new lifetime of tier i.

Recalculate the total battery used. When the total battery

first falls below the budget E, the algorithm will stop

with an optimal solution.

Problem 2B turns out to be a modification of Problem

1B, where the total battery budget E is now a hard

constraint. Accordingly, we take the following approach

to solve it: first solve Problem 1B on the same

parameters, ignoring the battery budget. If the total

battery budget Et of this solution does not exceed E, then

this is the desired solution. If instead Et>E, then obtain a

new set of effective battery levels for each tier by scaling

the battery levels by a factor of E/Et: these are the new

desired battery levels. A special case arises when some

of these new desired battery levels are less than the

minimum provided battery level bk. In this case the

battery levels for the inner tiers have to be reduced to

allow the outer tiers to have battery level b

k, further

reducing the lifetime.

In considering these problems, any such algorithm

would be executed not by the nodes themselves but

offline during network design. In all realistic cases of

deployment, most nodes will have positions in the

appropriate annular regions, but some randomness will

be introduced; lifetime will be then somewhat reduced

from that achieved in the ideal case, but the ideal

lifetime is a good indicator of actual lifetime in such

cases.

4. Simulation Results

To validate the results presented in the previous section,

we decided to simulate the wireless sensor network in

several scenarios. For simplicity reasons, we will assume

that perfect scheduling is achieved at the MAC layer and

routing layer. Any two nodes at a distance less than the

transmission radius can communicate with no errors; any

nodes at a distance larger than the transmission radius

cannot communicate. We considered a network of N =

500 nodes, which corresponds to a five-hop route for the

peripheral nodes (T=5), we assume that the total energy

of the network is bounded by E=40000 joules in each

case. Each node generates one data packet every minute;

the size of pocket is 1024 bits.

Table 1. Simulation parameters

Parameter Value

Transmitter circuitry, ete 2.34 µЈ/bit

Receiver circuitry, erx 2.34 µЈ/bit

Transmit one bit over one meter, eta 7.8 nЈ/bit/m2

Sensing energy per bit, es 1.75µЈ/bit

Bits sense per sensor, bs 1024bits

Table1 shows the values of the parameters in this

sample circuit and the propagation environment [12].

4.1.

Base Case

Figure 3. The energy consumption at defferent tiers

Figure 4. The residual energies at different tiers

Figure 5. The allocating energies ratio at different tiers

with maximizing the network lifetime

Using (5), we get Figure 3. It depicts the energy

consumption at different tiers, because the first tiers

relay the most data pockets, and consume the maximum

energies. The energy consumption of the second tier, the

third tier … the fifth tier is ordinal decrease. The energy

ANALYSIS OF LIFETIME OF LARGE WIRELESS SENSOR NETWORKS 141

BASED ON MULTIPLE BATTERY LEVELS

consumption of the fifth tier is the minimum; nodes at

the fifth tier don’t relay other data pockets. If each node

will start with the same energy (namely E/N), nodes at

the first tier depleted their energies full out. Though

nodes at other tiers have the residual energies, but their

data pockets can’t be relay to the sink, the network

becomes disconnected; the lifetime of the network is the

end. Using (11), we get Figure 4. The residual energies

at different tiers aren’t the same. The residual energies at

fifth tier are the most; the residual energies at second tier

are the least. For maximizing the network lifetime for a

given fixed amount of energy E, we will balance energy

allocation at different tiers. Using (13), we get Figure 5.

It depicts the allocating energies ratio at different tiers. If

we allocate the energies at different tiers according as

the radio with Figure 5, we can maximize the network

lifetime; all tiers’ energies will be depleted at the same

time. In the same parameters case, the network lifetime

is 226.8427 seconds and 1494.8 seconds by the two

energy allocation schemes, respectively. The lifetime of

the network can be significantly improved.

4.2. The Effect of Multiple Battery Levels

On the assumption that we can provide 5 battery levels,

they are 5178mAh, 2000 mAh, 1000 mAh, 500 mAh,

250 mAh, respectively.

For the base case, each node will be assigned with the

same battery level 5178 mAh, the whole networks

energy budget is 7767J.

For the problem 1A and maximizing network lifetime,

the first tier of nodes should have batteries of capacity

5178mAh. According to the ideal allocation ratios Ci,

the second tier, the third tier, the fourth tier, the fifth tier

of nodes should have batteries of capacity 1656 mAh,

868.7 mAh, 472.1 mAh, 205.7 mAh, respectively, but as

available battery levels limit, they have batteries of

capacity 2000 mAh, 1000 mAh, 500 mAh, 250 mAh in

application, respectively. Temporality, the whole

networks energy budget is 1315.7J.

For the problem 1B and two battery levels,

uniformity, the first tier of nodes should have batteries

of capacity 5178mAh. Using formula (18) and (19), the

optimum is achieved when 66% of the nodes have 2000

mAh batteries, and 34% of the nodes have batteries with

1000 mAh capacity in the second tier; 74% of the nodes

have 1000 mAh batteries, and 26% of the nodes have

batteries with 500 mAh capacity in the third tier; 89% of

the nodes have 500 mAh batteries, and 11% of the nodes

have batteries with 250 mAh capacity in the fourth tier;

All nodes have 250 mAh batteries capacity in the fifth

tier. Temporality, the whole networks energy budget is

1202.6J.

Using above battery deployment and the parameters

in table 1, we can get figure 6 and Figure 7. They show

the network lifetime of the different tiers. As can be seen,

in three deployment schemes, the first tier lifetime is the

shortest with 7.341 P intervals, so the whole networks

lifetime is 7.341 P intervals. In base case deployment

scheme, energy efficiency is 15.2%; there is much

residual energy at different four tiers when the whole

network has expired. In question 1A deployment scheme,

energy efficiency is 89.6%. But available battery levels

limit, there are some residual energies at different tiers

when the whole networks has expired. In question 1B

deployment scheme, energy efficiency is 98%. From the

curve of question 1B, we can know the lifetime of nodes

in the first, the second, the third, the forth is the same.

Namely, the four tiers energy all has expired when the

networks lifetime has expired.

Figure 6. The lifetime compare between the three schemes

There is some residual energy in fifth tier. Because

available battery levels limit, we can’t deploy the ideal

appropriate battery level. If offering more battery levels,

we can win the energy efficiency with 100%. All nodes

in the different tiers have expired at the same time.

Just from the lifetime of networks, the problem 1B

deployment scheme is ideality, but calculating the

proportion of the mixing nodes is slightly complex. In

the practice application, we can select one of the two

schemes.

Figure 7. The lifetime compare between the two schemes

142 R.H. ZHANG ET AL.

The network lifetime is further increased if more than

two battery levels are considered, as seen in Figure 8.

According to the scheme in problem 2, battery levels

were used for the simulation. The same total energy

budget 4000J was used for each simulation. Figure 8

show that the network lifetime will increase with the

increase in the number of battery levels. In the same

energy budget, the network lifetime with different

battery levels will increase 188%, 356%, 487%, 559%

relative to one with one battery level, respectively. This

indicates that most of the increase in the lifetime of the

network can be achieved with a relatively more number

of battery levels.

Figure 8. The increase in network lifetime with the increase

in the battery levels by the same energy budget at each

battery level

4.3. Dependency on Number of Nodes

According to the scheme in problem 2, battery levels

were used for the simulation. The same total energy

budget 4000J was used for each simulation. Figure 9

shows the dependency of the network lifetime on the

initial number of nodes. The density of the network was

kept constant, so the area of the network was

proportionally increased with the number of nodes.

Figure 9. Dependency of the network lifetime on the initial

number of nodes for three battery levels (constant density)

The lifetime of the network decreases with the

increase in the initial number of nodes. This is expected,

as we already know a larger number of tiers results in

more battery wastage.

4.4. Dependency on Node Density

In this section, we study the effect of increasing the

density of the network on the lifetime of the network,

while keeping the network size constant. Figure 10

shows the dependency of the network lifetime on the

node density. As can be seen in Figure 10, the node

density has no influence on the network lifetime as long

as it remains uniform. The reason is that the number of

nodes in each tier will increase in the same proportion

with the node density; hence, the number of load flows

carried by each node does not change. The lifetime of

the network remains constant even if the density of the

network doubles or triples.

Figure 10. Dependency of the network lifetime on the

network density for three battery levels (constant network

area)

5. Conclusions and Future Work

In this paper, we have addressed a problem expected to

occur in large, multi-hop wireless sensor networks. The

nodes closer to the sink will die before the nodes at the

periphery of the network. The main disadvantage of the

expiration of the nodes close to the sink is that the

network becomes disconnected while most of the nodes

still have a considerable amount of energy left. To

alleviate this undesirable effect, we proposed an energy

allocation scheme, allocating different energy at

different tiers by traffic. With this strategy, we have

shown that the lifetime of the network can be

significantly improved. Future work would explore

similar issues, but MAC protocol will be considered.

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