Computer Networks 56 (2012) 521–532

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Computer Networks

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Evaluating wireless sensor node longevity through Markovian techniques

Dario Bruneo a, Salvatore Distefano b,⇑, Francesco Longo a, Antonio Puliafito a, Marco Scarpa a

a Dipartimento di Matematica, Università di Messina, Contrada di Dio, S. Agata, 98166 Messina, Italyb Dipartimento di Elettronica e Informazione, Politecnico di Milano, Via Ponzio 34/5, 20133 Milano, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 10 October 2010Received in revised form 20 July 2011Accepted 10 October 2011Available online 20 October 2011

Keywords:ReliabilityEnergy consumptionBattery discharge processMarkov reward modelsContinuous phase type distributionsKronecker algebra

1389-1286/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.comnet.2011.10.003

⇑ Corresponding author. Tel.: +39 090 3977335; fE-mail addresses: dbruneo@unime.it (D. Brun

polimi.it (S. Distefano), flongo@unime.it (F. Longo)(A. Puliafito), mscarpa@unime.it (M. Scarpa).

Wireless sensor networks are constituted of a large number of tiny sensor nodes randomlydistributed over a geographical region. In order to reduce power consumption, nodesundergo active-sleep periods that, on the other hand, limit their ability to send/receive data.The aim of this paper is to analyze the longevity of a battery-powered sensor node. A batterydischarge model able to capture both linear and non linear discharge processes is presented.Then, two different models are proposed to investigate the longevity, in terms of reliability, ofsensor nodes with active-sleep cycles. The first model, well known in the literature, is basedon the Markov reward theory and on the evaluation of the accumulated reward distribution.The second model, based on continuous phase type distributions and Kronecker algebra, rep-resents the main contribution of the present work, since it allows to relax some assumptionsof the Markov reward model, thus increasing its applicability to more concrete use cases. Inthe final part of the paper, the results obtained by applying the two techniques to a case studyare compared in order to validate and highlight the benefits of our approach and demonstratethe utility of the proposed model in a quite complex and real scenario.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Wireless sensor networks (WSN) are networks com-posed of tiny sensors equipped with radio interfaces anddistributed over a geographical region. The task of eachsensor is to perform measurements and to send data to anode collector or sink. The WSN application areas arenumerous and different, ranging from disaster recoveryto field monitoring. Interesting applications have also beenfound in industrial scenarios where hostile environmentscan preclude the human intervention or the deploymentof networking infrastructures.

In the last years, research on WSN has mainly focusedon networking aspects [1,2] as well as on data manage-ment [3]. However, many specific applications introducestrict dependability requirements [4,5] that are primarily

. All rights reserved.

ax: +39 090 393502.eo), distefano@elet.

, apuliafito@unime.it

related to data security and reliability [6] and to the lon-gevity of the single node and the WSN as a whole [7–11].In fact, in this latter case, cheap sensors do not guaranteetheir functioning over the time and they are normallyequipped with low voltage batteries that limit their life-time or longevity. From such a perspective, a WSN is apower constrained system, since nodes run on limitedpower batteries.

In order to reduce and optimize the energy consump-tion, a common practice is to switch WSN nodes into alower powered state, usually identified as sleep mode, bydeactivating the radio equipment when no data have tobe transmitted [7]. Since quite often the radio is the high-est power consumer subsystem in a WSN node, this allowsto save battery and therefore to improve the node lifetime.In this way, by associating the battery discharge to theWSN node aging process as in [7–9], the node longevitycan be expressed, in terms of reliability, as a function ofthe battery state of charge.

Several authors deal with WSN and power managementtopics, at different layers and achieving different goals. In

522 D. Bruneo et al. / Computer Networks 56 (2012) 521–532

the specific context of reliability, an interesting investiga-tion is performed in [9], where a simulation techniquefor evaluating blind flooding over WSN is proposed. Theauthors base the evaluation on a realistic and accurate bat-tery model, also considering capacity and recovery effects aswell as switching energy, instead of starting from the lineardischarge model usually assumed in literature.

The choice of the battery model is of primary impor-tance in evaluating the WSN longevity as also highlightedin [10]. Mathematical models [12] apply empirical equa-tions to address the battery charge/discharge behavior;electrochemical models [13] provide a more detailed rep-resentation of the process but, on the other hand, they usu-ally require greater efforts in their evaluation; electricalmodels [14] are instead mainly used for circuit analysis,simulation and optimization.

Mathematical battery models are usually used in thenode/WSN longevity and power management modelingand evaluation. For example, in [11] a new sensor nodedeployment scheme is proposed to increase the sensor net-work longevity assuming a non-linear battery dischargemodel. More recently, Somov et al. [10] propose a flexibleand extensible simulation framework to estimate powerconsumption of sensor network applications for arbitraryhardware platforms in which energy and timing parame-ters are partially obtained through direct measurements.

The problem of WSN longevity and power managementhas also been investigated through the use of analyticalmodels. For example, in [8] Markov models are used in or-der to evaluate WSN reliability. In such case the context isslightly different: WSN composed of several nodes are con-sidered and the final model reflects such choice, represent-ing the WSN as a whole and introducing some higher levelapproximations.

The aim of this work is to analyze the longevity of a sin-gle WSN node undergoing cycles of active-sleep periods.Both linear and non linear battery models are taken intoconsideration, thus proposing and comparing two differentanalytical techniques. The former is based on a Markov re-ward model able to capture the linear battery depletionprocess of a WSN node and to derive the reliability param-eters under specific assumptions. The latter is based oncontinuous phase type distributions (CPHs) and Kroneckeralgebra and is able to relax some of the assumptions re-lated to the Markov reward solution technique. In particu-lar, non linear battery discharge model and differentactive-sleep modes sojourn time distributions can be takeninto account. Moreover, some numerical problems thatcould affect the implementation of the first techniquecan be overcome by our approach.

From an operational/practical viewpoint, the proposedtechnique can be exploited to perform parametric evalua-tions on the WSN also taking into account the applicationrequirements, as for example: finding out the appropriateduty cycle to be used for achieving the expected lifetimerequired by the application, deriving the expected node’slifetime, depending on its assigned duty cycle, etc.

The paper is organized as follows: Section 2 introducesand discusses the battery discharge phenomena, also pro-posing a possible model and its validation. Section 3 de-scribes WSN nodes characterizing the problem under

evaluation. In Section 4 a Markov reward model of aWSN node is described, while in Section 5 a techniquebased on CPHs and Kronecker algebra is detailed. Fromthe evaluation of such models specific results are obtained,compared and discussed in Section 6, while Section 7 con-cludes the paper with final remarks and future work.

2. Battery discharge model

As stated above, WSNs can be considered as power-con-strained systems in which the power management prob-lem has to be adequately addressed. It is thereforerequired to use adequate techniques and models in orderto satisfy and optimize the power management, also takinginto account the application requirements and specifica-tions. With regards to batteries, that usually provide theWSN power supply system, it is necessary to define ade-quate models to describe their discharge process.

The battery lifetime or time-to-failure is the time to dis-charge. Once the battery is exhausted the powered system,in the specific the WSN node, shuts down; therefore, max-imizing the battery lifetime is an important goal to beaddressed.

Starting from the existing literature [9,15], one of themost adopted analytical models to represent the dischargeprocess of a battery subjected to a constant load is basedon the Peukert’s law [16] that expresses the capacity of abattery in terms of the rate at which it is discharged: ifthe rate increases, the available charge capacity decreases.Generally such an effect can be quantified by the formula:

T ¼ H � CH � I

� �g

; ð1Þ

that analytically formalizes the relationships among thetime of discharge T, expressed in hours, the rated capacityof the battery C at the specified hour rating H (hours), ex-pressed in Ampere � hours and the (constant) discharge cur-rent I, expressed in Ampere. In (1) g is the Peukert’sexponent or constant, ranging in the interval [1,2], depend-ing on the chemical process characterizing the battery. ThePeukert constant increases with age for any of the batterytypes, but generally ranges from [1.05,1.15] for lead–acidbatteries, [1.1,1.25] for gel, [1.2,1.6] for flooded batteries,[1.06,1.13] for lithium ion batteries while [1.2,1.4] charac-terizes alkaline batteries. A value of g equal to 1 identifiesthe ideal case, in which the discharge process is linear, i.e.,the battery capacity C linearly decreases with the current I.In such case the actual capacity would be independent ofthe current.

The Peukert’s law expresses the battery lifetime givenan initial capacity. In the proposed modeling technique,we need to know the trend of the battery discharge processwith respect to the time. To this end, by inverting Eq. (1)we can obtain the relationship for the capacity C asfollows:

C ¼ I � H � TH

� �ð1=gÞ: ð2Þ

Then, considering C as a function of the time variable t, wecan express its evolution in time as c(t) from Eq. (2):

D. Bruneo et al. / Computer Networks 56 (2012) 521–532 523

cðtÞ ¼ c0 � I � H � tH

� �ð1=gÞ; ð3Þ

where c0 is the initial capacity of the battery.By Eq. (3) we can argue that, when a constant load is ap-

plied, a time-varying discharge rate commonly character-izes the batteries discharge process. A typical trend of thedischarge process in time c(t) is shown in Fig. 1.

Since such trend is not linear, it is necessary to recur toan adequate model to represent it. Let us identify the bat-tery useful charge range as [c0,cmin], that we split into ncontiguous intervals [ci,ci+1] of equal size c0�cmin

n , withi = 0, . . . ,n � 1. In this way, we discretize the battery capac-ity into n + 1 charge levels with generic value ci (i = 0, . . . ,n),where cn = cmin. Since c(t) is usually strictly decreasing, thetime instants ti such that c(ti) = ci, with i = 0, . . . ,n, can beunivocally identified. Let si = ti+1 � ti, with i = 0, . . . ,n � 1,

Fig. 1. Discharge curve of a battery subject to a constant load.

Fig. 2. CTMC representing the batte

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35

c(t),

c- (t)

Time (h)

c-(t) with n=25c-(t) with n=100

c-(t) with n=1000c(t)

(a)Fig. 3. Battery discharge function (a) and corresponding total de

be the duration of the ith time interval, in which the chargeassumes values ranging into [ci,ci+1]. By discretizing the va-lue assumed by the charge in [ti, ti+1] with ci, we can repre-sent the discharge phenomenon through the n + 1-statecontinuous time Markov chain (CTMC) shown in Fig. 2and defined by the stochastic process

B ¼ fBðtÞ; t P 0g; ð4Þ

where the state bi encodes the ith charge interval. Since"t 2 [ti, ti+1]) c(t) 2 [ci,ci+1], si can be considered as the so-journ time into the state bi and, as a consequence, the tran-sition rate between states bi and bi+1 has to be set to ki ¼ 1

si.

The introduced model is a CTMC with an absorbingstate whose absorption time is a random variable that rep-resents the time-to-discharge of the battery. As a conse-quence, the distribution of the time to absorption of theCTMC represents the distribution of the time-to-discharge.The CTMC thus obtained describes a single dischargebehavior corresponding to a constant load; if differentloads have to be considered, as a number of CTMCs equalto the number of different loads have to be specifiedaccordingly. Anyway, all the CTMCs modeling the batterydischarge in the different load conditions are characterizedby the same number of states if the discretization levels arethe same; considering the same battery this is not a restric-tive assumption, since the initial charge level c0 and itsminimum cmin are fixed for a given battery. The sojourntime into each state of the chain of Fig. 2 has been consid-ered as exponentially distributed because our aim is tocharacterize the function describing the discharge processof the sensor node battery through a CPH distribution.Such a characterization allows us to approximate a non-Markovian stochastic process with an expanded CTMC thatcaptures the discharge process of the sensor node.

ry discharge process of Fig. 1.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

F(t),

F- (t)

Time (h)

F- (t) with n=25F- (t) with n=100

F- (t) with n=1000F(t)

(b)pletion time distribution (b) varying the number of states.

Fig. 4. CTMC representing the active-sleep cycles of a WSN node.

524 D. Bruneo et al. / Computer Networks 56 (2012) 521–532

2.1. Validating the battery discharge model

In order to validate the CPH battery discharge modelproposed we solved the CTMC depicted in Fig. 2 and com-pared the results against the battery discharge function c(t)obtained by the Peukert’s law expressed in Eq. (3). Moreformally, we obtained an estimation �cðtÞ of the c(t) func-tion as the expected value of the stochastic process B ofEq. (4) at time t:

�cðtÞ ¼Xn

i¼0

ci � PrfBðtÞ ¼ big: ð5Þ

We also evaluated an estimation of the total batterydepletion distribution FðtÞ as the probability the CTMC isin the absorbing state, i.e., the probability for the batteryto be depleted at time t. Such function is compared withthe F(t) that is defined as:

FðtÞ ¼1 if t 6 T;

0 if t > T;

�ð6Þ

where T is the battery lifetime expressed in Eq. (1).Fig. 3 shows the obtained results. We referred to an

alkaline battery (AA/R6) with Peukert constant g = 1.3, ini-tial capacity c0 = 2500 mAh, constant load I = 100 mA and,consequently, H is 25 h. As expected, by increasing thenumber n of states of the CTMC, we obtained a betterapproximation. In particular, it can be observed that witha 1000 states-CTMC we obtain a very good approximationof the battery discharge process.

3. Longevity of WSN nodes with active-sleep cycles

In typical WSN applications, wireless sensor nodes arerandomly scattered in large geographical regions and it isnot always possible to perform node maintenance afterthe WSN deployment by intervening on the single node.For this reason, sensor nodes have to adapt their behaviorto the environmental changes. In particular, each node hasto reduce its power consumption in order to increase theWSN lifetime. Sensor nodes are usually powered by lowvoltage batteries and they are composed of a processingunit, a wireless communication unit, and a sensing unit.Wireless communications represent the most expensivetask a node has to accomplish [7], but even if no data areactually transmitted, the energy consumed by the wirelesscommunication unit is significant. Then, in order to reduceenergy consumption, a good strategy is to deactivate theradio equipments when no data have to be transmitted.According to this strategy, nodes periodically go throughtwo different functioning states: active and sleep. Whenthe node is in the active state, it is able to send/receive datawhile, when sleeping, it is only able to perform off-linetasks such as sensing and data processing.

In order to analyze the battery discharge process of anode with active-sleep cycles, it is possible to associateto each operating mode the corresponding battery dis-charge function c(t). Being the discharge process strictly re-lated to the battery type, functions c(t) associated todifferent operating modes have the same value of g andH of Eq. (3) and only differ in the discharge current I.

Assuming that the only failure node condition is due tobattery depletion, we can study the node longevity byassociating the node reliability with the battery charge.Let us assume that, when the battery charge goes belowa certain threshold cmin, the node cannot fulfill its tasksand it can be considered failed. Then, it is possible to definethe reliability of a sensor node at time t, R(t) as:

Definition 1. The reliability of a sensor node at time t, R(t),is the probability that its battery is not depleted in the timeinterval [0, t].

RðtÞ ¼ PrfcðtÞP cming: ð7Þ

In the following, starting from the battery dischargemodel presented in Section 2 and knowing the energy con-sumption parameters associated to the two operatingmodes and the period of the active-sleep cycles, we showhow the reliability function R(t), and then the node longev-ity, can be evaluated.

This paper has been focused on the reliability distribu-tion of a single sensor node. Once such distribution is ob-tained it is possible to extend the investigation to theWSN as a whole. Two possible cases can be identified. Ifthe failure events (in this case the battery discharge) arestatistically independent the reliability of the whole WSNcan be obtained using combinatorial techniques such asfault trees. We focused on this aspect on [17,18]. If the fail-ure events are statistically dependent the reliability ofWSN can be studied using dynamic reliability tools suchDRBD [19] or modeling techniques able to capture theinteractions among sensors, such as Markovian Agents[20]. We plan to investigate such a second aspect in futureworks.

4. Markov reward model

Let us assume that the WSN node sojourn times in theactive and sleep states are exponentially distributed. Undersuch assumption, we can model the node activity with theCTMC depicted in Fig. 4, defined by the stochastic process

X ¼ fXðtÞ; t P 0g; ð8Þ

where states Sa and Ss are associated to the active and sleepconditions, respectively, and k and l represent the active-sleep and the sleep-active transition rates.

To derive the expression of R(t) from the analysis of theCTMC of Fig. 4 we can recur to the Markov reward theory[21]. Let us associate to states Sa and Ss indices i = 0 andi = 1, respectively, and let us indicate with S = {0,1} thestate space associated to the CTMC of the WSN node. Then,we can define a reward function r : S! R in which, for each

D. Bruneo et al. / Computer Networks 56 (2012) 521–532 525

state i 2 S,r(i) = ri represents the reward obtained per unittime spent by process X in that state.

We can associate the reward to the battery consump-tion by defining r as:

ri ¼qa if i ¼ 0;qs if i ¼ 1;

�ð9Þ

where qa and qs represent the charge absorbed by the sen-sor per unit time in the active and sleep states, respec-tively. If the charge absorbed by the node is not constantover the time [22], it is possible to consider reward func-tions r(i,t) that are also time-dependent on time [21].

Let Z(t) = rX(t) be the system reward rate (i.e., the nodecharge absorption rate) at time t. We can define the accu-mulated reward Y(t) in the interval [0, t) as:

YðtÞ ¼Z t

0ZðsÞds: ð10Þ

The value of Y(t) can be interpreted as the charge con-sumed by the node from the activation instant till time t.By and indicating with c0 the initial battery charge of theWSN node, from (10) we can express the battery charge le-vel c(t) at time t as:

cðtÞ ¼ c0 � YðtÞ: ð11Þ

An interesting measure to compute is the distribution ofthe accumulated reward that can be expressed as:

Fðt; yÞ ¼ PrfYðtÞ 6 yg: ð12Þ

In fact, from (7), (11), and (12) we can derive the expres-sion of R(t) as:

RðtÞ ¼ PrfcðtÞP cming ¼ Prfc0 � YðtÞP cming¼ PrfYðtÞ 6 c0 � cming ¼ Fðt; c0 � cminÞ: ð13Þ

Then, to perform a quantitative analysis of R(t), we need toevaluate the distribution of the accumulated reward F(t,y).Such distribution can be computed starting from the anal-ysis of the CTMC described in Fig. 4.

4.1. Closed-form solution

A method for evaluating F(t,y) during a finite missiontime and assuming the reward rates to be time indepen-dent is provided in [23]. If A = [aij] is the infinitesimal gen-erator of the process X, it is possible to indicate withP = [pij] = A/c + I the stochastic matrix of the associatedPoisson process, where c P max{j aijj} and I is the identitymatrix. Then, F(t,y) can be then expressed as:

Fðt; yÞ ¼X1k¼0

aðkÞ0 uðy� r0tÞ þXk

h¼1

X1

w¼0

k

h� 1

� �"

�bðkÞ0 ðw;hÞy� rwt

t

� �k�hþ1

uðy� rwtÞ#� ðctÞke�ct

k!;

ð14Þ

where

uðxÞ ¼0 if x < 0;1 if x P 0

�ð15Þ

and coefficients aðkÞi and bðkÞi ðw;hÞ are recursively definedby:

aðkÞi ¼

0 if k ¼ 0;P1j¼0

rj¼ri

pijaðk�1Þj if k > 0;

8>><>>: ð16Þ

bðkÞi ðw; hÞ ¼

�P1j¼0

rj¼rw

pijaðk�1Þj

rw�riif w – i; h ¼ k;

bðkÞiðw;hþ1Þ

rw�ri�P1j¼0

pijbðk�1Þj

ðw;hÞrw�ri

if w – i; h < k;

�P1

w¼0w–i

bðkÞi ðw;1Þ if w ¼ i; h ¼ 1;

P1j¼0

pijbðk�1Þj ði;h� 1Þ if w ¼ i; h > 1

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:ð17Þ

To numerically solve Eq. (14), the infinite sum has to betruncated to a value k⁄ that depends on the required errortolerance. An estimation of the error truncation is given by:

eðk�Þ 6 1� e�ctXk�k¼0

ðctÞk

k!: ð18Þ

5. The CPH distributions and Kronecker algebra model

Eq. (14) cannot be used to compute the WSN node reli-ability in case of time dependent reward rates. However,even in the simple case of constant reward rates, the algo-rithm that can be used to evaluate Eq. (14) and to computethe recursive coefficients aðkÞi and bðkÞi ðw;hÞ could be subjectto numerical problems (overflow, underflow, stiffness).Such issues could prevent Eq. (14) to be used for the com-putation of the node reliability R(t) for large values of t.

Moreover, the described technique cannot be usedwhen the relationship between the charge of the batteryand the reliability of the node is more complex than thatof Eq. (7): for example if we assume that there is a non-nullprobability for the node to fail, even if the battery charge isgreater than cmin. Finally, the stochastic process X could bemore complex than a simple CTMC, being the sojourntimes of the node in active and sleep states non-exponen-tially distributed. As an example, the sojourn times coulddepend on the load of the node in terms of number of pack-ets it is processing at a specific time instant.

For such reasons we propose an alternative analyticaltechnique based on CPH distributions and Kronecker alge-bra. It can be used for evaluating the WSN node reliabilityrelaxing some of the assumptions specified in Section 4.

The state space model depicted in Fig. 5 represents thebehavior of a sensor node also considering its batterydepletion. As in Fig. 4, states Sa and Ss are related to the ac-tive and the sleep modes of the WSN node under exam,respectively, while the absorbing state Sf represents thenode failure due to the battery depletion. Events ea and es

are related to the transitions between active and sleep

Fig. 5. State space model representing the sensor node lifecycle.

526 D. Bruneo et al. / Computer Networks 56 (2012) 521–532

modes, while event ef represents the failure of the WSNnode.

Let us assume that the cumulative distribution func-tions (CDFs) Fa(t) and Fs(t) are associated to events ea andes, while two different time-to-failure distributions Fa

f ðtÞand Fs

f ðtÞ are associated to event ef in states Sa and Ss,respectively. Fa

f ðtÞ and Fsf ðtÞ are the node lifetime distribu-

tions in the two functioning states in isolation, as specifiedin Section 3. They are obtained from the battery dischargefunctions of states Sa and Ss as detailed in Section 2, so bothof them are represented by n + 1 states CTMCs.

Since the WSN node cyclically changes its functioningstate triggered by events ea and es, the distribution associ-ated to event ef changes from Fa

f ðtÞ to Fsf ðtÞ and vice versa.

Moreover, at the generic change point �t the battery chargehas to be preserved. For example, the changing from Fa

f ðtÞto Fs

f ðtÞ at �t means that the battery discharge process ofthe node continues according to Fs

f ðtÞ, thus preserving thebattery charge level reached so far, i.e., Fa

f ð�tÞ. In terms ofthe battery model introduced in Section 2, the batterycharge level is preserved by the CTMC state reached whenFa

f ðtÞ is disabled at �t, since each state corresponds to a spe-cific charge level.

Based on these considerations, in the following wedescribe a technique based on CPH distributions andKronecker algebra for evaluating such model. In thisway, the unreliability and the sojourn time distributionof the WSN node in the two functioning modes arerepresented by CPHs, moving the problem towards thesolution of an expanded CTMC. Kronecker algebra istherefore used in order to deal with the well-known statespace explosion problem, since this allows the infinitesi-mal generator matrix of the expanded CTMC is nevergenerated nor stored as a whole, but it can be algorithmi-cally evaluated on-the-fly in a block-wise fashion duringthe model solution. In this way, only few informationabout the stochastic process is permanently stored, withconsequent memory saving.

5.1. Representing events through CPHs

The use of phase type (PH) distributions dates back tothe pioneering work of Erlang on congestion in telephonesystems at the beginning of the last century [24] and waspopularized in 1981 by Neuts [25] who formally defined

a PH distribution as the distribution of the time untilabsorption in a finite state Markov chain with a singleabsorbing state. In particular, the class of CPH distributionsis defined over CTMCs.

More specifically, let us consider a CTMC v with ‘ tran-sient states and a single absorbing state (labeled ‘ + 1)whose infinitesimal generator matrix bG 2 R‘þ1 � R‘þ1 is inthe form:

where G 2 R‘ � R‘ describes the transient behavior of theCTMC and U 2 R‘ is a vector grouping the transition ratesto state ‘ + 1. Moreover, let us suppose that the chain isstarted with an initial probability vector p(0) = [a,a‘+1]such that a‘þ1 ¼ 1�

P‘i¼1ai. We say that a random variable

T is distributed according to the CPH distribution with rep-resentation (a,G) and order ‘if its CDF FT(t) is the probabil-ity to reach the absorbing state of v; FT(t) can be expressedas:

FTðtÞ ¼ 1� a � eGt � 1; t P 0: ð19Þ

The theory of PH distributions found several applica-tions in reliability (see Neuts and Meier [26], Pérez-Ocónet al. [27–29] and references therein) and, in general, inthe analysis of stochastic models where non-exponentiallydistributed events are considered. Other interesting andpowerful techniques exist for the analysis of such class ofnon-Markovian models such as supplementary variables,semi-Markov processes, and Markov regenerative models[30], but they usually present severe restrictions on thestructure of the manageable systems. On the contrary,the use of PH distributions for the representation of non-exponentially distributed events (state space expansion ap-proach [25]) allows to manage a more general class ofmodels.

The state space expansion approach consists of the rep-resentation of a non-Markovian stochastic process bymean of a Markov chain defined over an augmented statespace [31]. Each state in the non-Markovian process is ex-panded into a set of states within the Markov chain, calledmacro-state, with the purpose to capture the evolution ofthe non-exponentially distributed events within it. Inparticular, when CPH distributions are used, the non-Markovian process is represented via an expanded CTMC.

By applying the state space expansion approach to theWSN node under evaluation, we have to representFaðtÞ; FsðtÞ; Fa

f ðtÞ, and Fsf ðtÞ through CPH distributions. In

such model, special care has to be devoted to the represen-tation of the node unreliability in the two operating modes,since the charge level has to be preserved at change points.

5.2. From CPHs to Kronecker algebra

Let us consider a discrete-state discrete-event non-Markovian model and let S be the system state space ande the set of system events. Following the state spaceexpansion approach, the events are represented by CPHdistributions, and the expanded CTMC thus resulting is

D. Bruneo et al. / Computer Networks 56 (2012) 521–532 527

composed of kSkmacro-states, characterized by a kSk � kSkblock infinitesimal generator matrix Q in which:

� the generic diagonal block Qii (1 < i < kSk) is a squarematrix that describes the evolution of the CTMC insidethe macro-state related to state i, and it depends onthe possible events occurring in that state;� the generic off-diagonal block Qij (1 < i < kSk,1 < j < kSk)

describes the transition from the macro-state related tostate i to that related to state j, and it depends on theevent occurred in state i and the possible events ableto still occur in state j.

The main drawback of the state space expansion ap-proach is the explosion of the state space. This limits theapplicability of the approach thus reducing its benefitsand potentialities. A possible solution to overcome theproblem is to recur to Kronecker algebra [25,32,28], a wellknown and effective technique able to provide a compactrepresentation of CTMCs. Kronecker algebra is based ontwo main operators [33]: Kronecker product (�) and Kro-necker sum (�). Given two rectangular matrices A and Bof dimensions m1 �m2 and n1 � n2, respectively, their Kro-necker product A � B is a matrix of dimensionsm1n1 �m2n2. More specifically, it is a m1 �m2 block matrixin which the generic block i,j has dimension n1 � n2 and isgiven by ai,j � B. On the other hand, if A and B are squarematrices of dimensions m �m and n � n, respectively,their Kronecker sum A � B is the matrix of dimensionsmn �mn written as A � B = A � In + Im � B where In andIm are the identity matrices of order n and m, respectively.

By exploiting Kronecker algebra, matrix Q does notneed to be generated and stored as a whole but can besymbolically represented through Kronecker expressionsand algorithmically evaluated on-the-fly when needed[34,35]. As detailed below, the matrix Q blocks have thefollowing form:

Q ii ¼ �1<e<kek

Q e; ð20Þ

Q ij ¼ �1<e<kek

Q e: ð21Þ

Diagonal blocks are computed as Kronecker sums (off-diagonal blocks are computed as Kronecker products) ofa series of matrices Qe each of which is associated to oneof the system events and depends on the behavior of suchevent in the involved states. In this context, operators �and � assume particular meanings. Let us consider twoCTMCs v1 and v2 with infinitesimal generator matricesQ1 and Q2. The matrix resulting by the Kronecker sum ofthese matrices (Q1 � Q2) is usually interpreted as the infin-itesimal generator matrix of the CTMC that models theconcurrent evolution of v1 and v2. It is important to re-mark that the resulting CTMC is composed of a set of statesthat derives from the combination of all the states of thetwo initial CTMCs.

On the other hand, let us consider a CTMC v1 with a sin-gle absorbing state and let U be the column vector contain-ing the transition rates to such state. Moreover, let usconsider a CTMC v2 and let P be a matrix that containsthe probabilities to enter into the v2 states. The matrix ob-

tained by the Kronecker product of U and P(U � P) isusually interpreted as the matrix containing the transitionrates to the CTMC resulting from the combination of theabsorbing state of v1 with all the states of v2. For acomplete treatment of all the possible forms of matricesQe, in the case of discrete-time models, see [35]. Here, weapply such technique to the evaluation of a WSN nodereliability in order to derive the corresponding Qe

matrices.In order to represent the WSN node with active-sleep

cycles, let us consider the three states model of Fig. 5. Interms of Kronecker algebra terms, the expanded CTMC, ob-tained by introducing the sojourn time distributions instates Sa and Ss and the battery discharge models, is repre-sented by the infinitesimal generator matrix Q, that is the3 � 3 block matrix:

where states Sa, Ss and Sf are identified as 0, 1 and 2,respectively.

In the following, (aa,Ga) and (as,Gs) are the CPH repre-sentations of the node sojourn time CDFs Fa(t) and Fs(t) in

the two functioning states, and aaf ;G

af

� �and as

f ;Gsf

� �are

the CPH representations of the node time-to-failure CDFsFa

f ðtÞ and Fsf ðtÞ.

Let us start with the description of the diagonal blocksof matrix Q. In state Sa, events es and ef are enabled andcan evolve concurrently according with their CPH distribu-tions while event ea is disabled. For such a reason, matrixQ00 is:

Q 00 ¼ Gs � Gaf � ½0; ð22Þ

where matrix [0] (i.e., a matrix with a single elementequal to 0) associated to event ea is the neutral elementfor the Kronecker sum operator. Similarly, in state Ss

events ea and ef are enabled and can evolve concurrentlyaccording to their CPH distributions while event es isdisabled, thus:

Q 11 ¼ ½0 � Gsf � Ga: ð23Þ

On the other hand, in state Sf no event is enabled, thereforeQ22 is a 1 � 1 null matrix.

With regards to off-diagonal blocks of matrix Q, let usstart with matrix Q01 associated to the transition from Sa

to Ss. Event es is enabled in Sa and causes the state transi-tion, becoming disabled in state Ss. Thus, the stochasticprocess leaves the phases of the CPH associated to eventes with the rates specified in Us. On the other hand, ea isdisabled in Sa and enabled in Ss. As a consequence, the sto-chastic process enters the phases of the CPH associated toea with the probabilities given by vector aa. Finally, ef is en-abled in both Sa and Ss but changes its distribution fromFa

f ðtÞ to Fsf ðtÞ.

Since, as stated above, the CPHs representing Faf ðtÞ and

Fsf ðtÞ have been specified by coding the battery charge lev-

els in their states, and the charge level has to be preservedchanging from Sa to Ss, the (jth) state reached by the CPH

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

R(t)

Time (sec)

CPH approach with n=25CPH approach with n=100

CPH approach with n=1000Markov reward approach

Fig. 6. Node reliability R(t) computed with the Markov reward approachand the CPH–Kronecker approach.

528 D. Bruneo et al. / Computer Networks 56 (2012) 521–532

corresponding to Faf ðtÞ has to be saved by restarting from

the same state (j) of the CPH corresponding to Fsf ðtÞ. Such

behavior can be implemented by using an n � n identitymatrix Ia?s:

Q 01 ¼ Us � Ia!s � aa: ð24Þ

Similarly Q10, representing the transition from Ss to Sa, hasthe form:

Q 10 ¼ as � Is!a � Ua: ð25Þ

The matrix representing the transition from Sa to Sf trig-gered by the firing of ef is:

Q 02 ¼ es � Uaf � ½1; ð26Þ

since es is enabled in Sa but it is disabled in Sf without firingand it is not required to keep memory of the state reached(the corresponding matrix is the column vector es whoseelements are equal to 1). Such vector represents the factthat, regardless of the probabilities to be in a stage of theCPH associated to the event es, in Sf the failure event dis-ables all the other events.

Matrix [1] (i.e., a matrix with a single element equal to1) is the neutral element for the Kronecker product opera-tor and it is associated to ea since it does not contribute tothe state transition, being disabled in both the involvedstates and having no memory in the reached state. Eventef is associated to Ua

f since its firing causes the state transi-tion. Similarly, Q12 is given by:

Q 12 ¼ ½1 � Usf � ea: ð27Þ

Finally, since Sf is an absorbing state and no event is en-abled in it, Q20 and Q21 are null matrices.

In this way, by applying the CPH–Kronecker algebratechnique, the stochastic process underlying the WSNnode lifecycle considering non-Markovian sojourn timesand time-to-failure distributions is reduced to a CTMC thatcan be easily solved by applying a classical transient solu-tion method (e.g., the uniformization method) for the com-putation of the WSN node reliability R(t).

6. Numerical results

In this section, we present some numerical results ob-tained by using the described techniques. First of all, weprovide a comparison between the Markov reward modeland the CPH–Kronecker model in order to demonstratethe equivalence of the two approaches. Then, by evaluatingthe WSN node reliability function R(t), we show how to ap-ply the CPH–Kronecker technique in order to derive impor-tant parameters that can be used during the setup phase ofa WSN.

6.1. Techniques comparison

In order to demonstrate the equivalence between themodeling techniques described in the previous sections,we need to take into consideration a scenario that can beanalyzed by mean of the Markov reward model. As wehave already discussed, only a linear battery model canbe evaluated through the accumulated reward distribution

closed-form Eq. (14), meaning that constant reward rateshave to be associated to the WSN node active and sleepmodes. Moreover, due to some numerical issues encoun-tered in the implementation of the Markov reward modeltechnique and specifically of Eq. (14), it is not possible toevaluate the WSN node reliability for large values of tand therefore it is not possible to validate the CPH–Kronecker algebra technique by the previously analyzedscenario.

We thus consider a WSN node characterized by a linearbattery discharge of 10 s in the active mode of and 20 s inthe sleep one. The WSN node lifecycle is regulated by theCTMC depicted in Fig. 4 with k = 1.0 s�1 and l = 0.2 s�1.In order to apply the CPH–Kronecker technique the WSNnode battery linear discharge process has been approxi-mated by a 1000-phase CPH.

Fig. 6 reports the comparison between the WSN nodereliability function R(t) computed with the Markov rewardmodel and with the CPH–Kronecker technique varying thenumber of the CPH phases n. It can be observed that, in thecase of n = 1000, the two curves are totally overlapped andsmall differences are present only in the two knees of thecurves, thus demonstrating the effectiveness of the CPH–Kronecker technique and also validating it.

6.2. Reliability analysis

Once the reliability model has been validated, it is pos-sible to use it in the evaluation of the WSN sensor nodelongevity as specified in Section 3 and more specificallyby Eq. (7). We apply the CPH–Kronecker technique de-scribed in Section 5. The WSN node is therefore character-ized by the active and sleep states, and it periodicallyswitches between the two operating modes. The proposedtechnique is not strictly related to a particular applicationor communication protocol, and it is general enough to beadopted at large. For this reason, we do not focus on aspecific duty cycle period and the values of parameters kand l are specified in order to provide a full numericalexample. In particular, to derive the expression of R(t),we assume that the stochastic processes associated toevents ea and es of Fig. 5 are exponentially distributed with

100

120

140

160

180

200

0 2 4 6 8 10

Expe

cted

life

time

(h)

1/λ (sec)

Fig. 8. Expected lifetime varying the active-sleep transition rate k.

D. Bruneo et al. / Computer Networks 56 (2012) 521–532 529

rates k = 3600 h�1 and l = 60 h�1, meaning that the nodeswitches from active to sleep with a mean time of 1 s,while from sleep to active with a mean time of 60 s.

The battery discharge process has been characterized byusing the same battery parameters listed in Section 2.1. Todifferentiate the discharge processes in the active andsleep modes we have to evaluate the corresponding energyconsumptions in terms of the discharge current I. Startingfrom empirical data obtained by experiments on MeshNet-ics MeshBean2 boards, we observed that such WSN nodesin the active mode absorb a current Ia = 100 mA, while inthe sleep mode a current Is = 20 mA.

With regard to the CPH–Kronecker model, we can con-sider the representation of the battery discharge processshown in Fig. 2 as a specific CPH. As discussed in Section2, the two CTMCs corresponding to the discharge fromboth active and sleep modes have the same number ofstates or phases identifying and discretizing the batterycharge level reached. They differ only in the rates betweenthe phases. Thus, to the same ith phase of the two CPHs. Insuch a way the CPH–Kronecker technique described inSection 5 can be applied, keeping memory of the chargelevel reached by saving the corresponding CPH phase inthe transitions between macrostates involving active-sleepand sleep-active mode switching.

The results obtained by evaluating the reliability func-tion R(t) through a 1000 phases CPH model are shown inFig. 7, in the specific the non linear discharge curve. Theresulting function has been compared against the one ob-tained by considering a linear discharge, using the samebattery parameters of the non linear discharge case andapplying the ideal Peukert constant g = 1. This latter func-tion corresponds to the linear discharge curve plotted inFig. 7. The two curves show a similar trend is not really astep-shaped function, due to the stochastic nature of themodel evaluated, corresponding to a Markov modulatedactive-sleep cycles process.

Comparing the curves, we can argue that the two mod-els are really different. In numerical terms we can computethe mean time to failure (MTTF) that can be considered asan estimation of the expected lifetime of a sensor node.We obtain that the expected lifetime of the linear case is117.291 h, while in the non linear case it is 181.435 h. This

0

0.2

0.4

0.6

0.8

1

0 50 100 150 200 250 300

R(t)

Time (h)

non linear dischargelinear discharge

Fig. 7. Node reliability R(t).

allows us to appreciate the differences of a linear dischargemodel approximation with respect to the more realisticPeukert model. Even though the two trends are similar,we can certainly state that in such specific case the linearapproximation is not the right one. This is an importantgoal achieved by our technique that encourages its use asa more accurate alternative to the evaluation of the accu-mulated reward distribution.

In order to highlight how the knowledge of the reliabil-ity function R(t) can help a designer during the settingphase of a WSN, we evaluate the trend of the expected life-time by varying the active-sleep transition rate k in therange [360,9000] h�1, using the non linear Peukert dis-charge model.

The results thus obtained, shown in Fig. 8 considering 1/k 2 [0.4,10] s = [1/9000,1/360] h, demonstrate that 1/k (themean time of the active-sleep CDF) has a significant impacton the WSN node power consumption. Moreover, a sub-linear trend can be observed, meaning that increasing 1/khas not a proportional effect on the power consumption,and therefore has to be carefully evaluated when specify-ing a WSN power management strategy.

7. Conclusions and future work

In this paper, we have investigated the longevity of abattery-powered WSN node characterized by active-sleepcycles. In particular, the reliability function and the ex-pected lifetime have been taken into consideration as eval-uation parameters. First of all an analytical model for thebattery discharge process has been specified. Such modelis able to describe both linear and non-linear dischargeprocesses and has been validated using the well knownPeukert’s law. Then, a Markov reward model has been de-scribed for the analytical evaluation of a WSN node reli-ability in the case of linear discharge. Finally, as the maincontribution of the present work, a new technique basedon the use of CPH distributions and Kronecker algebrahas been introduced in order to relax some of the assump-tions on which the Markov reward model relies. Inparticular, such technique allows to characterize a sensornode through non-linear battery discharge processes and

530 D. Bruneo et al. / Computer Networks 56 (2012) 521–532

non-exponentially distributed sojourn times in the activeand sleep states.

The CPH–Kronecker algebra technique has been com-pared vs. the Markov reward approach in a simple casestudy in order to mutually validate both techniques. There-fore, by evaluating a more complex scenario consideringboth linear and non-linear battery discharges, we can ar-gue that a linear approximation of the more realistic nonlinear behavior provides wrong results. In order to furtherdemonstrate such thesis, an in depth analysis by varyingthe active-sleep rate has been performed, with the aim ofevaluating the impact of such parameter on the expectedlifetime of a node. By this, a sub-linear trend has been ob-served demonstrating how the precision in the computa-tion of the expected lifetime offered by our technique isimportant during the WSN setting phase.

The obtained results strongly encourage future work. Inparticular, since the Kronecker algebra allows to overcomethe state space expansion problem, the study can be ex-tended to a WSN composed of several sensors with a com-plex topology and in presence of redundant nodes,specifically focusing on dependability aspects. A possiblesolution in such cases could be the adoption of hierarchicalmodels: for example by considering combinatorial/high le-vel models to represent the network on top of a model rep-resenting the single node, based on the proposedtechnique. Moreover, the impact of unreliable wirelesslinks on the WSN longevity can be an interesting parame-ter to evaluate, as well as causes of failures not related tothe battery discharge process.

Acknowledgment

This work has been partially supported by MIUR throughthe ‘‘Programma di Ricerca Scientifica di Rilevante InteresseNazionale 2007’’ (PRIN 2007) under grant 2007J4SKYP.

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Dario Bruneo was born in Messina onDecember 14th 1973. He received his Degreein Computer Engineering from the Engineer-ing Faculty of the University of Palermo (Italy)in 2000 and the Ph.D. in Advanced Technolo-gies for Information Engineering at the Uni-versity of Messina (Italy) in 2005. Since thenhe has been engaged in research on distrib-uted systems. He is currently an associateresearcher at the Engineering Faculty of theUniversity of Messina. His scientific activityhas been focused on studying distributed

system, particularly with regard to programming and managementtechniques. His main research interests include distributed system pro-gramming, Grid computing, mobile agents, mobile middleware, network

management techniques and QoS management.

Salvatore Distefano was born in Catania onJune 16th 1974. He received, in October 2001,the master degree in Computer Science Engi-neering from the University of Catania, finalscore 110/110 cum summa laudae. In January2006 he got the Ph.D. degree on ‘‘AdvancedTechnologies for the Information Engineer-ing’’ from the University of Messina. FromMarch to September 2005 he spent a period inthe University of Massachusetts Dartmouth,collaborating with the Electric and ComputerEngineering Department as a Ph.D. exchange

student. His research interests include Performance Evaluation, Distrib-uted Computing, Software Engineering and Dependability Techniques. Hehas been one of the members of the Multimedia and Distributed Systems

Laboratory (MDSLab). Recently, in 2011, he joined the Dipartimento diElettronica e Informazione, Politecnico di Milano as an Assistant Profes-sor.

Francesco Longo was born in Messina onNovember 16th 1982. He received his Degreein Computer Engineering from the Universityof Messina (Italy) in november 2007, finalscore 110/110 summa cum laude. The title ofhis thesis was ‘‘Symbolic representation of thereachability graph of non-Markovian sto-chastic Petri net’’. From settember 2007 toJune 2008 he worked at the University ofMessina within the PON Project ‘‘Progetto perl’Implementazione e lo Sviluppo di una e-Infrastruttura in Sicilia basata sul paradigma

della grid (PI2S2)’’ with the aim of designing and implementing a QoSmanagement system in Grid environment. In June 2008 he received aMaster’s degree in Open Source and Computer Security. He is currently

attendig his Ph.D studies in ‘‘Advanced Technologies for InformationEngineering’’ at the University of Messina. His main research interstsinclude performance and reliability evaluation of distributed systems(mainly Grid and Cloud) with main attention to the use of non-Markovianstochastic Petri nets. Since 2010 he is teaching assistant for the subject‘‘Valutazione delle prestazioni’’ (Performance evaluation) at the Faculty ofEngineering, University of Messina. Between May 2010 and October 2010he spent a period as a visiting scholar in the United States at the Duke

University (Durham, NC) where he had the opportunity to callaboratewith Prof. Kishor S. Trivedi in the modeling of Cloud systems and in thequantitative evaluation of their performance and availability.

Antonio Puliafito was born in Catania onAugust 24th 1965. He received a Degree inElectrical Engineering from the University ofCatania (November 18, 1988) and Ph.D. inElectrical Engineering from University ofPalermo (October 1, 1993). He received aFellowship from CNR on ‘‘Parallel I/O subsys-tems’’ (November 1993–June 1994) and aFellowship from CNR for research activities atthe Dept. of Electrical Engineering, DukeUniversity, Durham, NC (USA) (June 1994–June 1995). He was the Assistant Professor of

Computer Engineering at the University of Catania, Italy (July 1995–October 1998). He worked as the Associate Professor of Computer Engi-neering at the University of Messina, Italy (November 1998–May 2002).

He served in the capacity of Full Professor of Computer Engineering at theUniversity of Messina, Italy (June 2002). He was the Coordinator of thecourse of study in ‘‘Computer Engineering and Telecommunications’’active at the University of Messina and of the Ph.D. course in ‘‘AdvancedTechnologies for the Information Engineering’’ active at the University ofMessina. With R. Sahner and K.S. Trivedi, he has authored the book‘‘Performance and Reliability Analysis of Computer Systems: An Example-based Approach Using the SHARPE Software Package’’, edited by KluwerAcademic Publishers, November 1995. He was the Auditor for Domain 5ACTS European program (January 99–July 2000) and Evaluator IST Euro-pean program (September 2000. He was Winner of the ICT InnovationAward 2005. He is responsible for the ICT and e-learning activities insidethe University of Messina. He is Director of the Center on InformationTechnology Development and Their Applications (CIA).His scientific activity is focused on studying distributed systems. Topicsdeveloped in this context include parallel processing, problems connectedwith reliability and fault tolerance and advanced programming tech-niques based on agents technology. Specific applications regard networkand quality of service management through software agents. He is cur-rently investigating issues related with user mobility and security, inwireless and ad hoc networks. Emphasis is currently being devoted also inGrid computing and to the adoption of such paradigm to provideadvanced multimedia services in a wireless environment. Specific inter-ests involve reliability and performance analysis of parallel and distrib-uted systems, adopting either standard and advanced modelingtechniques. His research activity includes: (1) Software Agents, (2) Reli-ability and fault tolerance, (3) Multimedia systems, (4) System modelingand Performance evaluation, (5) Multimedia, (6) GRID computing, (7)Wireless systems.

Marco Scarpa received the degree in Com-puter Science Engineering from University ofCatania, in 1994 by discussing a thesis onMarkov Regenertive Process applied to PetriNets. He received Ph.D. in Computer Sciencefrom University of Torino.During 2000–2001, he was engaged by the Fac-ulty of Engineer of Catania University as assistantprofessor in ‘‘Fundamentals of Computer Sci-ence’’, and during 2001–2002, he was assistantprofessor in ‘‘Operating Systems’’ at the Faculty ofEngineer of the University of Messina.

He was assistant professor in ‘‘Laboratory of Computer Science’’ (2001/2002) and ‘‘Operating System’’ (2002/2003) at the Faculty of Engineer ofthe University of Messina.

He was engaged as contract researcher at the ‘‘Dipartimento di Matem-atica’’ of the Universtity of Messina during 2003–2006. He is actuallyenganged as associate professor in computer science at the Universuty ofMessina and his main research interest is about performance evaluationof distributed systems.Peformance evaluation of system with faults and repairings, with par-ticular interest to modeling tecniques and performability measures overdistributed systems.

532 D. Bruneo et al. / Computer Networks 56 (2012) 521–532

Analysis of non Markovian Petri nets models by using non exponentiallydistributed firing times. Particular interest is devoted to the class of Petrinets with more than one non exponentially distributed transition in amarking.Performance and reliability analysis of real time and wireless communi-cation systems (with particular interest to the Ad-Hoc networks).

He coordinates the development of the software tool WebSPN. WebSPN isa tool for the analysis and solution of stochastic Petri nets, provided witha graphical interface written in Java; the solution algorithm implementedin WebSPN is based on an expansion tecnique thanks to which it has beenpossible to relax some restrictions present in many others tool for theanalysis of Petri nets.