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D Bruneo, A Puliafito and M ScarpaEnergy control in dependable wireless sensor networks: a modelling perspective

  

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Energy control in dependable wireless sensornetworks: a modelling perspectiveD Bruneo*, A Puliafito, and M Scarpa

Dipartimento di Matematica, Universita di Messina, Messina, Italy

The manuscript was received on 15 June 2010 and was accepted after revision for publication on 11 November 2010 .

DOI: 10.1177/1748006X10397845

Abstract: Wireless sensor networks (WSN) are composed of a large number of tiny sensornodes randomly distributed over a geographical region. In order to reduce power consump-tion, battery-operated sensors undergo cycles of sleeping–active periods that reduce their abil-ity to send/receive data. Starting from the Markov reward model theory, this paper presents adependability model to analyse the reliability of a sensor node. Also, a new dependabilityparameter is introduced, referred to as producibility, which is able to capture the capability ofa sensor to accomplish its mission. Two different model solution techniques are proposed,one based on the evaluation of the accumulated reward distribution and the other based onan equivalent model based on non-Markovian stochastic Petri nets. The obtained results areused to investigate the dependability of a whole WSN taking into account the presence ofredundant nodes. Topological aspects are taken into account, providing a quantitative com-parison among three typical network topologies: star, tree, and mesh. Numerical results areprovided in order to highlight the advantages of the proposed technique and to demonstratethe equivalence of the proposed approaches.

Keywords: wireless sensor networks, reliability, producibility, energy consumption, network

topology, Markov reward models, non-Markovian stochastic Petri nets

1 INTRODUCTION

Wireless sensor networks (WSN) are networks com-

posed of tiny sensors equipped with radio interfaces

and distributed over a geographical region. The task

of each sensor is to perform measurements and to

send data to a node collector (usually referred to as

sink node). Different network topologies can be

exploited and consequently different data routing

strategies have to be implemented (either single- or

multi-hop). The WSN application fields are numerous

and different ranging from disaster recovery to agri-

culture monitoring. Interesting applications have also

been found in industrial scenarios where a hostile

environment can preclude the human intervention

or the deployment of networking infrastructures.

In recent years, research on WSN was mainly

focused on networking aspects [1, 2] as well as on

data management [3]. However, particular applica-

tions (such as industrial applications) also have

strict dependability requirements [4, 5]. In fact,

cheap sensors do not guarantee their functioning

over time and are normally equipped with low-

voltage batteries that limit their lifetime. For this

reason, several countermeasures have to be taken in

order to increase the lifetime of the whole network

and to improve system reliability.

In order to reduce energy consumption, nodes can

be turned off in sleep mode, by deactivating the radio

equipments when no data have to be transmitted [6].

However, usually data are requested by the sink in

an on-demand fashion and then sensors in sleep

mode cannot satisfy such requests. Another compli-

cation arises if a sleeping node is involved in a multi-

hop communication, thus producing the effect of a

broken link. If on the one hand a possible solution is

to design routing protocols that self-adapt to the

*Corresponding author: Dipartimento di Matematica,

Universita di Messina, Messina, Italy.

email: dbruneo@unime.it

424

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

environmental changes [7], on the other hand, an

efficient strategy is to design the WSN with a redun-

dancy level (that is, redundant sensor nodes) in order

to reduce the effects due to the sleeping nodes [8].

The aim of this work is to analyse dependability

parameters of a WSN where single nodes undergo

cycles of active–sleep periods and taking into account

the presence of redundant nodes. A Markov reward

model is presented which is able to capture the battery

depletion process of a sensor node and to derive

important dependability parameters. In particular, a

new dependability parameter, referred to as produci-

bility, is introduced in order to capture the capability

of a sensor to accomplish its mission. Starting from

this model, an equivalent method, based on non-

Markovian stochastic Petri nets (NMSPNs), is pre-

sented in order to relax some of the assumptions

related to the analytical solution technique. After mod-

elling the sensor node behaviour, a technique to derive

dependability parameters of a whole WSN is pre-

sented. In particular, dependability models referring to

different network topologies (star, tree, and mesh) are

developed in order to analyse the influence of the net-

work topology on the reliability and producibility of

the WSN. Thanks to the proposed methodology,

important metrics can be exploited, such as the

expected lifetime of a WSN and the average produci-

bility during the functioning time. Such metrics can

help designers during the setting phase of the network

in order to estimate the required number of redundant

nodes. Finally, the equivalence between the Markov

reward model and the NMSPN model is verified, com-

paring the results obtained in a simple scenario.

The paper is organized as follows. In section 2 a

model to evaluate dependability parameters of a

sensor node is provided. In section 3 an analytical

technique to solve the model is described and an

equivalent model based on the non-Markovian sto-

chastic Petri nets is presented. In section 4 the

dependability analysis is extended to the whole net-

work and topological aspects are discussed in sec-

tion 5. A numerical example is shown in section 6,

while section 7 concludes the paper with a discus-

sion about possible future works.

2 DEPENDABILITY MODEL OF A SENSOR NODE

The system to be analysed is a sensor node that

periodically goes through two different function-

ing states: active and sleep. During the active period

the node is able to send/receive data, while during the

sleep mode the radio interface is turned off. Sensors

in sleep mode can continue to collect sensed data but

they are not able to communicate with the sink.

It is possible to model the node activity with the

continuous time Markov chain (CTMC) depicted in

Fig. 1 and defined by the stochastic process

X = fX(t), t ˜ 0g (1)

where states 0 and 1 are associated to the active and

sleep conditions, respectively, and l and m represent

the active–sleep and the sleep–active transition

rates.

During its functioning, the node consumes

energy and then its battery charge c decreases.

When the battery charge goes under a certain

threshold cmin the node cannot fulfil its tasks and

then it can be considered to have failed. Assuming

that the only node failure condition is due to the

battery depletion, the node reliability can be defined

in terms of its battery charge.

Definition 1 The reliability of a sensor node at

time t, Rnode(t), is the probability that its battery is

not depleted in the time interval ½0, t�.That is

Rnode(t) = Prfc(t) ˜ cming (2)

where c(t) is the battery charge level of the node at

time t.

To derive the expression of Rnode(t) from the anal-

ysis of the CTMC of Fig. 1, the Markov reward

model theory is returned to [9]. Then, indicating

with S = f0, 1g the state space associated to the

CTMC of a sensor node, a reward function r : S! <can be defined, where for each state i 2 S, r(i) = ri

represents the reward obtained per unit time by the

process X in that state.

The reward can be associated with the battery

charge consumption by defining r as

ri =qa if i = 0qs if i = 1

�(3)

where qa and qs represent the charge absorbed by

the sensor per unit time in the active and sleep

states, respectively.

Fig. 1 CTMC representing the active–sleep cycles of asensor node

Energy control in dependable wireless sensor networks 425

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

If the charge absorbed by the node is not constant

over time [10], it is possible to consider reward func-

tions r(i, t) that are also time dependent [9].

Let Z(t) = rX(t) be the system reward rate (that is, the

node charge absorption rate) at time t. It is then possi-

ble to define the accumulated reward Y (t) in the inter-

val ½0, t) as

Y (t) =

ðt

0

Z(t)dt (4)

Y (t) can be interpreted as the charge consumed

by the node from its activation until time t.

Indicating with C the initial battery charge of a sen-

sor node, the battery charge level c(t) at time t can

be expressed as

c(t) = C � Y (t) (5)

An interesting measure is the distribution of the

accumulated reward that can be expressed as

F(t, y) = PrfY (t) < yg (6)

In fact, from equations (2), (5), and (6) the expres-

sion of Rnode(t) can be derived as

Rnode(t) = Prfc(t) ˜ cming= PrfC � Y (t) ˜ cming= PrfY (t) < C � cming= F(t, C � cmin) (7)

Once the reliability function of a node has been

derived, let us investigate the influence of the

active–sleep cycles on the fulfillment of the node

mission. A new dependability measure is then intro-

duced, referred to as producibility.

Definition 2 The producibility x(t) of a system at

time t is the capability of the system to accomplish

its mission at time t.

During its functioning time, if a node is sleeping,

it is not able to send/receive data and then it cannot

communicate with the sink. Then, from the sink

point-of-view, sleeping nodes are not accomplishing

their mission.

Definition 3 The producibility of a sensor node at

time t, xnode(t), is the probability that the node is

able to communicate with the sink at time t.

Obviously if a node is not functioning (i.e. its bat-

tery is depleted) its producibility is zero, then

xnode(t) can be expressed as

xnode(t) =Anode(t) if c(t) ˜ cmin

0 if c(t) \ cmin

�(8)

where Anode(t) is the probability that the CTMC of

Fig. 1 is in the state 0 at time t. Such quantity corre-

sponds to the node availability during the function-

ing time, and its expression can be derived by

solving the Chapman–Kolmogorov equations associ-

ated with the CTMC, thus obtaining

Anode(t) = PrfX(t) = 0g=m

l + m+

l

l + me�(l + m)t

(9)

Finally, to obtain xnode(t) the total probability theo-

rem can be applied

xnode(t) = Prfxnode(t)jc(t) ˜ cming � Prfc(t) ˜ cming+ Prfxnode(t)jc(t) \ cming � Prfc(t) \ cming= Anode(t) � Rnode(t)

(10)

3 MODEL EVALUATION

To perform a quantitative analysis of Rnode(t) and

xnode(t), it is necessary to evaluate the distribution

of the accumulated reward F(t, y). Such distribution

can be computed starting from the analysis of the

CTMC described in Fig. 1.

A method for evaluating F(t, y) during a finite

mission time and assuming the reward rates to

be time independent is provided in reference [11]. If

A = ½aij� is the infinitesimal generator of the process

X , it is possible to indicate with P = ½pij�= A=g + I the

stochastic matrix of the associated Poisson process,

where g ˜ maxf aij

�� ��g and I is the identity matrix.

According to reference [11], F(t, y) can be then

expressed as

F(t, y) =X‘

k = 0

½a(k)0 u(y � r0t) +

Xk

h = 1

X1

w = 0

k

h� 1

� �

3b(k)0 (w, h)

y � rwt

t

� �k�h + 1

u(y � rwt)� (gt)ke�gt

k!

(11)

where

u(x) =0 if x \ 01 if x ˜ 0

�(12)

and coefficients a(k)i and b(k)

i (w, h) are recursively

defined by

a(k)i =

0 if k = 0P1j = 0rj = ri

pija(k�1)j if k . 0

8<: (13)

426 D Bruneo, A Puliafito, and M Scarpa

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

b(k)i (w, h) =

�P1

j = 0rj = rw

pija(k�1)j

rw�riif w 6¼ i, h = k

b(k)i

(w, h + 1)

rw�ri�P1j = 0

pijb(k�1)j

(w, h)

rw�riif w 6¼ i, h \ k

�P1

w = 0w 6¼i

b(k)i (w, 1) if w = i, h = 1

P1j = 0

pijb(k�1)j (i, h� 1) if w = i, h . 1

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

(14)

Such explicit expression of F(t, y) is derived using

the randomization technique (see reference [11]

and references therein). To solve equation (11)

numerically, the infinite sum has to be truncated to

a value k� that depends on the required error toler-

ance. An estimation of the error truncation is given

by

e(k�) < 1� e�gtXk�k = 0

(gt)k

k!(15)

3.1 Equivalent model

Another interesting solution able to overcome some

numerical issues related to the evaluation of equa-

tion (11) and also to release the assumption of time-

independent rewards, is to use an equivalent model

of a sensor node based on NMSPNs. NMSPNs

are an extension of generalized stochastic Petri

nets where the associated stochastic process is not

a Markov chain [12]. More formally, an NMSPN is

a tuple

MNMSPN = (P, T , I ,O,H,M0,Y,A)

where P is the set of places; T is the set of transi-

tions; I ,O,H : T ! Bag(P) are the set of input, out-

put, and inhibition functions, respectively;M0 is the

initial marking of the net, that is, the initial distribu-

tion of tokens among the places in P; Y : T ! F( � )is a function that assigns a random variable charac-

terized by a particular cdf F( � ) to each timed transi-

tion; A : T ! fprd, prsg is a function that assigns a

preemption memory policy to each transition. An

age memory variable is associated to each timed

transition to keep track of the time during which the

transition has been enabled. The age memory vari-

able models the clock associated with the transition,

and its value is increased while the corresponding

transition is enabled. The way the age memory is

related to the past history determines the different

preemptive memory policies. According to the

preemptive repeat different (prd) memory policy,

each time a transition fires, the age variables of the

disabled transitions are set to zero. According to the

preemptive resume (prs) policy, if a transition is dis-

abled, the value of the age variable, accumulated up

to that point, is maintained.

From a graphical point of view, transitions with a

non-exponentially distributed random delay are

drawn as filled rectangles. Extensive work has been

carried out to define techniques for the analytical

solution of such models [13].

The sensor node equivalent model is depicted in

Fig. 2.

Points P0 and P1 model the active and the sleep

states, respectively. Exponentially distributed transi-

tions T0 and T1 model the active–sleep cycle and

their rates are equal to l and m, respectively.

Transition T2 is a generally distributed transition

with a prs memory policy that, maintaining the

value of its age variable, and then keeping trace of

the time spent by the token at P0, is able to capture

the reward process behaviour. In fact, being qa � qs

the battery charge consumed in the sleep state can

be neglected, assuming qs = 0. Finally, the absorbing

place P2 models the energy failure condition.

The firing time distribution of transition T2 repre-

sents the battery charge absorption model [10].

For example, if a constant charge absorption with

rate qa is considered, transition T2 has to be a

deterministic transition with firing time equal to

(C � cmin)=qa.

With reference to the model of Fig. 2, Rnode(t) and

xnode(t) can be evaluated as follows

Rnode(t) = Prf#P2(t) = 0g (16)

xnode(t) = Prf#P0(t) = 1g (17)

Fig. 2 The non-Markovian stochastic Petri net equiva-lent model of a sensor node

Energy control in dependable wireless sensor networks 427

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

where #Place name(t) indicates the number of

tokens in a given place at time t.

4 WIRELESS SENSOR NETWORK WITH

REDUNDANT NODES

Starting from the above results, in this section the

dependability analysis is extended to a whole WSN.

A network composed of n sensor nodes and one

sink is assumed. A scenario is considered in which

sensors send measured data (temperature, luminos-

ity, etc.) to the sink in order to extract information

related to a given location. From the sink point of

view, sensors are assumed to be equivalent.

Moreover, the network can contain a redundant

number of sensors. In fact, to extract the required

information properly, the sink has to receive data

from at least k sensors, with k < n.

Having in mind a WSN respecting such assump-

tions, the following dependability parameters can

be defined.

Definition 4 The reliability Rwsn(t, k, n) of a WSN

at time t is the probability that at least k of the n

nodes are still functioning at time t.

Definition 5 The producibility xwsn(t, k, n) of a

WSN at time t is the probability that at least k of the

n nodes are able to send data to the sink at time t.

Starting from the above indexes it is possible

derive important parameters that are strictly related

to the design of a WSN. From Rwsn(t, k, n), it is possi-

ble to estimate the expected WSN lifetime (T k, n) as

the mean time to failure (MTTF) of the network

T k, n = MTTF =

ð‘

0

Rwsn(t, k, n)dt (18)

With respect to the WSN producibility, it can be

noticed that being the system subjected to the bat-

tery depleted condition, the producibility at the

steady state is null, that is

limt!‘

xwsn(t, k, n) = 0 (19)

However, if the observation time is restricted to the

network functioning period, the time averaged pro-

ducibility of the network in the interval ½0, T � can be

estimated as

xk, n(T ) =1

T�ðT

0

xwsn(t, k, n)dt (20)

Such value, computed over significant time intervals

(for example T = MTTF or T = 23MTTF), can be

used to estimate the number of redundant nodes

required to obtain the desired value of producibility.

5 TOPOLOGICAL ASPECTS

The evaluation of the above-described indexes

strictly depends on the network topology. With

respect to the application field, the WSN topology

can be set during the network design and installa-

tion. In some particular applications, the topology is

randomly obtained after the node deployment (for

example, by launching sensor nodes from an air-

plane). However, also in this case, the topology can

be retrieved after the WSN deployment by keeping

information from the routing layer. In this section,

some network examples are presented which exploit

three typical WSN topologies: star, tree, and mesh.

5.1 Star topology

Let us consider a WSN composed of n sensor nodes

connected to one sink through a star topology (see

Fig. 3). Each node has a direct connection to the

sink and the node–sink communication does not

involve other nodes. If reliable communication

channels are considered, the reliability and produci-

bility of a WSN with a star topology are related only

to the number of functioning nodes notwithstand-

ing their position.

Then, assuming node failures are statistically

independent, the Rwsn(t, k, n) and xwsn(t, k, n) can be

derived as functions of Rnode(t) and xnode(t) in terms

of a k-out-of-n configuration

Rwsn(t, k, n) =Xn

i = k

ni

� �Ri

node(t)(1� Rnode(t))(n�i) (21)

xwsn(t, k, n) =Xn

i = k

ni

� �xi

node(t)(1� xnode(t))(n�i)(22)

Fig. 3 A WSN with 8 sensor nodes and 1 sink con-nected through a star topology

428 D Bruneo, A Puliafito, and M Scarpa

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

5.2 Tree topology

The main drawback of the star topology is the lim-

itation on the network size. In fact, in order to have

direct communication links between any node and

the sink, the maximum node–sink distance has to

be compared with the radio signal power. A tech-

nique to reduce such limitation is to realize a tree

topology where each node still maintain a single

route toward the sink but it can use other nodes to

send data in a multi-hop fashion. An example of a

network with a tree topology is depicted in Fig. 4.

In such topology, the failure of a node has a

strong impact on the reliability and producibility of

the WSN, mainly if the node is involved in a multi-

hop path toward the sink. Assuming node failures

are statistically independent, the dependability

analysis of a WSN with tree topology can be con-

ducted using analytical techniques, such as fault

trees. The fault tree associated to the WSN depicted

in Fig. 4 is shown in Fig. 5, where leafs correspond

to node failure events and considering a number of

required nodes (k) equal to 4.

Fixing the number k of required nodes, the fault

tree can be automatically obtained through an anal-

ysis of the sample space by repeating the following

steps for each of the 2n network configurations

obtained by considering each node as functioning

or failed.

1. Construct the graph corresponding to the net-

work configuration by removing the failed nodes.

2. Visit the graph starting from the sink node and

count the number of reachable nodes.

3. If the number of reachable nodes is lower

than k, consider the configuration as a failure

condition.

At the end of this procedure the fault tree is con-

structed by removing all the redundant failure con-

ditions, for example if the failure of node 3

corresponds to a failure condition all the other fail-

ure conditions containing node 3 (for example

(2, 3)) are removed from the fault tree.

From the analysis of the fault tree of Fig. 5, the

expression of Rwsn(t, k, n) can be derived for such

specific network configuration. The analysis of the

fault tree requires attention due to the fact that

there are repeated events on different gates; for such

reason the classical relationships to the gates cannot

be used alone and the technique called factoring

has to be applied [9]. The derived expression is

Fwsn(t, 4, 8)

= 1� Rwsn(t, 4, 8)

= R2(t) R4(t)� R3(t)R4(t) + F1(t)F5(t)R3(t)R4(t)½+ F4(t)(F5(t) + R5(t)� R3(t)R5(t)

+ F1(t)F6(t)F8(t)R3(t)R5(t))�+ F2(t) (R4(t)(F5(t)½+ R5(t)� R3(t)R5(t) + F6(t)F8(t)R3(t)R5(t))

+ F4(t)(F5(t) + R5(t)(F6(t) + R6(t)� R3(t)R6(t)

+ F7(t)F8(t)R3(t)R6(t)))� (23)

where Fi(t) and Ri(t) = 1� Fi(t) represent the failure

distribution and the reliability function of node i,Fig. 4 A WSN with eight sensor nodes and one sink

connected through a tree topology

AND AND AND AND AND AND AND

3 2 5 2 4 6 2 4 7 8 1 5 1 4 6 8 4 5 2 6 8

OR

Fig. 5 Fault tree corresponding to the WSN of Fig. 4 when k = 4

Energy control in dependable wireless sensor networks 429

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

respectively. Assuming nodes have the same relia-

bility properties (that is, Fi(t) = 1� Rnode(t), i = 1,

. . . , 8), equation (23) can be written as

Rwsn(t, 4, 8) = 1 + R4node(t)3 2Rnode(t)� 3½ �

3 3 + Rnode(t)(Rnode(t)� 3)½ � (24)

Similarly, the expression of xwsn(t, k, n) can be

derived using the same fault tree and substituting

Rnode(t) with xnode(t), thus obtaining

xwsn(t, 4, 8) = 1 + x4node(t) � 2xnode(t)� 3½ �

� 3 + xnode(t)(xnode(t)� 3)½ � (25)

5.3 Mesh topology

Using a tree topology, when a node fails all the chil-

dren of the failed node are not able to communicate

with the sink any longer. In order to increase the relia-

bility of the network, a strategy is to create alternative

paths by introducing redundant links on the network,

thus obtaining a mesh topology. Figure 6 shows an

example of a network with a mesh topology.

Using redundant links, nodes can find alternative

routes when a failure event occurs. To this end, the

routing protocol has to be characterized by an auto-

nomic behaviour.

Using the same methodology adopted for the tree

topology, it is possible to build the fault tree of the

mesh network and derive its reliability and

Rwsn(t, k, n) and producibility xwsn(t, k, n). The fault

tree is depicted in Fig. 7; since the derived expres-

sions for Rwsn(t, k, n) and xwsn(t, k, n) are cumber-

some, they are not written down.

6 A NUMERICAL EXAMPLE

To illustrate the advantages of the proposed tech-

nique, some numerical results obtained solving the

sensor node Markov reward model presented in sec-

tion 2 and the equivalent model of Fig. 2 are pre-

sented. The latter has been analytically solved using

the WebSPN tool [14]. Due to some numerical

issues encountered in the implementation of the

Markov reward model technique and specifically of

4

Fig. 6 A WSN with eight sensor nodes and one sinkconnected through a mesh topology

AND

AND AND AND AND AND AND AND AND AND

AND AND AND AND AND AND AND AND

2

4 5 7 3 5 6 7 3 4 7 3 4 6 8 2 5 6 7 2 4 7 8 2 4 6 8 2 4 6 7 2 4 5

3 1 4 6 8 1 5 6 1 3 6 8 1 3 5 7 8 1 3 4 1 2 6 8 1 2 5 7 8 1 4 5

OR

Fig. 7 Fault tree corresponding to the WSN of Fig. 6 when k = 4

430 D Bruneo, A Puliafito, and M Scarpa

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

equation (11), it is not possible to evaluate through

such technique the WSN node reliability for values

of t greater than 20 s. Therefore a WSN node charac-

terized by values of qa and C � Cmin corresponding

to a linear battery discharge of 20 s is considered.

The WSN node lifecycle is regulated by the CTMC

depicted in Fig. 1 with l = 1:0 s21 and m = 1:0 s21.

Figure 8 reports the comparison between the WSN

node reliability function Rnode(t) computed with the

Markov reward model and with the NMSPN tech-

nique. It can be observed that the two curves are

totally overlapped and little differences are present

only in the two knees of the curves, thus demon-

strating the effectiveness of the NMSPN technique.

In order to make an exhaustive study of a real sce-

nario, the following presents the results obtained

solving the NMSPN model using typical parameters

representing the battery depletion process of a sen-

sor node. In particular, the sensor nodes under exam

will be characterized by a constant charge absorp-

tion and they will have the following parameters:

C � cmin = 1800 mAh, qa = 150 mA.

First of all, the node reliability and producibility

functions have been derived analysing the effects

due to the duration of the active–sleep period by fix-

ing l equal to 1.0 s21 and by varying m in the range

½0:1, 1:0� s21. Figure 9 shows the node reliability

function Rnode(t) and the node producibility function

xnode(t), obtained using different values of m. From

the analysis of the plotted curves it can be argued

that both the node reliability and producibility are

strongly affected by the active–sleep period. In par-

ticular, it is possible to observe that, as expected,

decreasing the value of m (i.e. increasing the sojourn

time in the sleep mode) the node reliability increases

while the node producibility decreases. Then, a

trade-off between two such parameters can be found

with respect to the design specifications.

6.1 Evaluating the effects of redundant nodes

Once the node dependability properties have been

analysed, let us investigate the dependability of a

whole WSN with redundant nodes. At the start of

the analysis a WSN with a star topology is used. The

effects of node redundancy have been tested setting

the minimum number of nodes required (k) to 10

and increasing the overall number of sensor nodes

(n) from 10 (no redundancy) to 100. A value of l

and m equal to 1.0 s21 has been chosen to character-

ize the active–sleep activity.

Figure 10 shows the reliability function Rwsn when

the WSN has no redundant nodes (n = 10). Such

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Nod

e R

elia

bilit

y

time (h)

Markov reward approachNMSPN equivalent model

Fig. 8 Node reliability function Rnode(t) computedwith the Markov reward approach and theequivalent NMSPN model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120 140

Nod

e P

rodu

cibi

lity

time (h)

µ =1s-1

µ =0.5s-1

µ =0.2s-1

µ =0.1s-1

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200

Nod

e R

elia

bilit

y

time (h)

µ =1s-1

µ =0.5s-1

µ =0.2s-1

µ =0.1s-1

(a) (b)

Fig. 9 (a) Node reliability function Rnode(t) and (b) node producibility function xnode(t), withvarying m

Energy control in dependable wireless sensor networks 431

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

function is compared with the reliability function of

a single node. As expected, the reliability of the

whole WSN is lower than the node reliability.

However, the two corresponding MTTFs are compa-

rable: 23:88 h for the WSN and 24:01 h for the node.

Such a point can be explained by considering the

deterministic behaviour of the battery discharge

process that produces a quasi-concurrent failure

event of all the nodes composing the WSN.

Moreover, increasing the redundancy level of the

network (see Fig. 11(a)) it can be observed that the

effects on the network reliability rapidly become less

evident for a value of n greater than 20, thus demon-

strating that the network lifetime is mainly related

to the period of the active–sleep cycle and not to the

number of redundant nodes.

With respect to the network producibility,

Fig. 11(b) shows the influence of the number of

redundant nodes on the xwsn trend. As expected, the

value of xwsn becomes 0 when the network reaches

its lifetime. However, restricting the observation

interval to the functioning time, it is noted that

when the network does not have redundant nodes

(curve n = 10) the network producibility is almost 0

while it reaches a value of about 0.99 when n = 50,

thus demonstrating the importance of the redun-

dancy levels on the fulfillment of the required levels

of producibility.

6.2 Evaluating the effects of network topology

The focus is now on evaluating how the network

topology influences the WSN reliability and produ-

cibility functions defined in section 4. The three

WSN topologies shown in Figs 3, 4, and 6 are taken

into consideration: star, tree, and mesh topology,

respectively. The number of redundant nodes is set

to 4 (that is, n = 8 and k = 4) and the value of l and m

is set equal to 1:0 s21. Figure 12 shows the values

obtained by plotting the Rwsn(t, 4, 8) and xwsn(t, 4, 8)

functions derived in section 5 for the three topolo-

gies under examination.

From the analysis of Fig. 12(a) it is possible to

evaluate the variations on the WSN reliability when

different topologies are adopted. The star topology

is the topology that guarantees the higher reliability

and it is also possible to highlight the differences

between the tree and mesh topologies. However, for

the specific WSN under exam the differences in

terms of expected lifetime are negligible, as can be

noticed in Table 1 where the values of Tk, n are pre-

sented for the three topologies.

0

0.2

0.4

0.6

0.8

1

23.6 23.8 24 24.2 24.4

Rel

iabi

lity

time (h)

Rnode(t)

RWSN(t)

Fig. 10 A comparison between the node reliabilityfunction Rnode(t) and the WSN reliability func-tion Rwsn(t, 10, 10) of a WSN with a star topol-ogy and without redundant nodes

Fig. 11 (a) WSN reliability function Rwsn(t, 10, n) and (b) WSN producibility function xwsn(t, 10, n),with respect to the overall number of nodes

432 D Bruneo, A Puliafito, and M Scarpa

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

A stronger impact due to the network topology can

be observed with respect to the WSN producibility. As

depicted in Fig. 12(b), switching from the star to the

tree topology a reduction on the WSN producibility of

about 66 per cent is obtained, while comparing the

tree and the mesh topologies it is possible to observe

an increase on the WSN producibility of about

95 per cent when the latter topology is adopted.

It can be concluded that during the WSN design,

the network topology plays a crucial role in the

expected WSN producibility.

6.3 WSN design

Finally, it is demonstrated how the proposed meth-

odology can be applied in a typical WSN design.

Again a WSN with a star topology is considered. In

Fig. 13, the value of the expected WSN lifetime

(T k, n) and of the time averaged WSN producibility

computed over an interval time equals to the MTTF

(xk, n(MTTF)) are plotted under different value of n

and m, fixing the value of l to 1.0 s21. From the anal-

ysis of Fig. 13(a), once a value for the expected life-

time is chosen, the corresponding value of m that

determines the active–sleep behaviour can be

Fig. 12 WSN reliability function Rwsn(t, 4, 8) (a) and WSN producibility function xwsn(t, 4, 8) (b)corresponding to the network topologies depicted in Figs 3, 4, and 6

Table 1 Expected WSN lifetime T4, 8 corresponding to

the network topologies depicted in Figs 3, 4,

and 6

Star topology Tree topology Mesh topology

24.03 h 23.97 h 24.00 h

Fig. 13 (a) Expected WSN lifetime T k, n versus m without network redundancy and (b) time aver-aged WSN producibility up to MTTF xk, n(MTTF) versus n with m = 0:3 s21

Energy control in dependable wireless sensor networks 433

Proc. IMechE Vol. 225 Part O: J. Risk and Reliability

found. For example, considering a scenario where

nodes are powered by solar energy and the goal is to

have an expected lifetime able to guarantee the

functioning during the night and the cloudy days

(for example, two days), it is found that the required

value of m is 0.3 s21. Starting from the obtained

value of m (with respect to the application require-

ments) the acceptable value of producibility can be

fixed (for example, 0.8). Then, analysing the data

plotted in Fig. 13(b), showing the value of

xk, n(MTTF) when m = 0:3 s21, the corresponding

required level of redundancy can be found. In the

examined case, a value of n greater than 50 is

required to properly set up the WSN.

7 CONCLUSIONS AND FUTURE WORK

This paper has studied dependability parameters, in

terms of reliability and producibility (a new defined

parameter able to capture the capability of a system

to accomplish its mission) of wireless sensor nodes

characterized by active–sleep cycles. A Markov

reward model and its equivalent non-Markovian

Stochastic Petri net model have been presented to

perform a quantitative analysis of the sensor

dependability. The study was also extended to a

WSN composed of several sensor with a star topol-

ogy and with the presence of redundant nodes.

Other topologies (for example, tree and mesh) have

been studied providing the corresponding depend-

ability models and carrying out a quantitative com-

parison. Important parameters such as the expected

network lifetime and the average producibility have

been derived with respect to the number of redun-

dant nodes and to the active–sleep period duration.

Future works will include the investigation of the

influence of unreliable wireless links on the pre-

sented dependability parameters and the adoption

of non-linear battery discharge processes.

� Authors 2011

FUNDING

This work was supported by MIUR (PRIN 2007)[grant number 2007J4SKYP].

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