[ieee 2014 13th annual mediterranean ad hoc networking workshop (med-hoc-net) - slovenia...

Robust Probabilistic Information Dissemination inEnergy Harvesting Wireless Sensor Networks

Eleni KavvadiaDepartment of Informatics

Ionian UniversityCorfu, Greece

Email: [email protected]

George KoufoudakisDepartment of Informatics



Konstantinos OikonomouDepartment of Informatics



Abstract—Modern network environments, like energy harvest-ing wireless sensor networks in which network lifetime can beprolonged due to ambient energy collection, necessitate the revisitof classical networking problems like information dissemination.However, as observed in this paper, flooding-based informationdissemination mechanisms suffer certain limitations due to theidiosyncrasies of nodes’ operational states in energy harvestingnetwork environments. Certain observations motivate the intro-duction of Robust Probabilistic Flooding, i.e., a robust versionof Probabilistic Flooding, capable of dealing with non-operatingnodes due to exhausted batteries that later resume their operationdue to successful ambient energy collection.

A Markov chain model is also introduced to capture thequalitative aspects of such environments in which nodes may ormay not operate. This Markov chain is simplified in the sequel,based on certain observations and assumptions presented here,and subsequently used for evaluating the proposed Robust Prob-abilistic Flooding through simulations. In particular, simulationresults demonstrate the inefficiency of Probabilistic Flooding toachieve full coverage in energy harvesting environments. On theother hand, it is shown that Robust Probabilistic Flooding iscapable of fully covering the network on the expense of increasedtermination time. Furthermore, no extra overhead is introducedwith respect to the number of messages, thus avoiding extratransmissions and therefore, no additional energy is consumed.

I. INTRODUCTION

Energyvorous as a wireless sensor network (WSN) can be,following the newly introduced principles of energy harvestingmay result in increasing lifetime thus, extending the usageand multiplying the applications of such networks. WSNs haveseen an enormous growth for more than a decade [1] and theycontinuously apply to an increasing number of applicationsranging from the bottom of the sea, to historical buildingsand extra-terrestrial monitoring. The success of WSNs isattributed to their wireless nature allowing for easy to installdeployments in places where traditional wiring would be eithercostly (e.g., large distances), difficult (e.g., underwater) or evenimpossible (e.g., prohibitive legislation regarding historicalbuildings, extra-terrestrial environments). On the other hand,their wireless nature implies absence of any infrastructure suchas power supplies and therefore, the sensor devices/nodes haveto relay on their batteries, thus limiting the lifetime of thenetwork and consequently the range of possible applications[2].

However, parallel to the growth of WSNs, new advantagesin technology have been seen resulting in modern small scale

computer systems capable not only for saving energy but alsofor compensating for some of the consumed energy by exploit-ing their ambient environment, i.e., energy harvesting [3]. Forexample, miniature solar panels could be attached to sensorsdeployed in sunny open fields, air turbines at windy places,fluid turbines to exploit water currents etc. These conjoinedactivities forge nowadays a new paradigm regarding WSNs andopen new perspectives with respect to new applications anddeployments both in old and new application environments.

The previous optimism about the future of WNSs underthe light of energy harvesting is further extended by lookinginto the particulars of the conjunction of energy harvestingand WSNs. For example, when a battery is the only source ofenergy for a WSN, then extending the network’s lifetime is aone-dimensional problem from this perspective, whereas whena node is capable of recharging its battery as under energyharvesting, more dimensions are introduced. The problembecomes more complicated considering that not all nodes havethe same capabilities of recharging, since, for example, someof those equipped with solar panels may be located in areasof less sunshine than others during certain hours of the day.As usual, some nodes may eventually run out of energy andstop operating. In the traditional approach this behavior wouldbe treated as node failure (and in some cases it will signal aneminent network failure), whereas in this case the node mayrecharge itself and after a certain time period be functionalagain.

The previously mentioned idiosyncrasies of an energyharvesting WSN, impose the need to revisit traditional mech-anisms and protocols (e.g., routing, medium access control,information dissemination) from a new perspective. The focusin this paper is on information dissemination and in particular,how a sensor node can efficiently disseminate information tothe entire network, i.e., full coverage. Note that informationdissemination is key for delivering control information tonodes in order to ensure the correct operation of the network(e.g., alert messages, routing messages). Naturally, the noderesponsible for initiating such a process is the sink node, i.e.,the particular node responsible for acquiring all data sensedby the rest of the nodes.

In an energy harvesting network, nodes are in operationalor non-operational modes based on the performance of theenergy collection process. If batteries are adequately suppliedthen nodes operate. If not, then they stop operate until energycollection successfully recharges them. However, in such an

2014 13th Annual Mediterranean Ad Hoc Networking Workshop (MED-HOC-NET)

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environment any flooding technique is expected to have limi-tations depending on the topology characteristics. For example,any nodes located at the bottleneck links of the network andnot operating due to insufficient energy in their batteries,will not allow for the information dissemination process toproceed in the corresponding network areas. However, thesenodes are expected, after some time, to recharge their batteriesand be able to operate, thus missing the chance for furthercovering. These observations motivate flooding revisiting andparticularly its probabilistic generalization, i.e., ProbabilisticFlooding [4], [5].

In this paper, a robust version of Probabilistic Flooding,called Robust Probabilistic Flooding, is proposed taking intoaccount the particular characteristics of the energy harvest-ing environments. State information is kept to allow for theresumption of the dissemination process as soon as nodesbecome operational again. For this reason, a four state Markovchain is also proposed here and analyzed to represent theoperating (plus not operating) and charging (plus discharging)states of the network nodes. This Markov chain model is laterused for simulation evaluation purposes. Extensive simulationresults demonstrate the inefficiencies of Probabilistic Floodingover energy harvesting environments and particularly its failureto guarantee full coverage. On the other hand, it is shownthat the proposed Robust Probabilistic Flooding is capableof achieving full coverage even though termination time in-creases. In addition, the number of messages is the same asunder Probabilistic Flooding, thus no extra transmissions arerequired avoiding to consume extra energy.

In the following Section II, past related work is presentedregarding energy harvesting and probabilistic information dis-semination. In Section III the Markov chain is presented alongwith the assumptions corresponding to the energy harvestingenvironment. In Section IV robust probabilistic informationdissemination mechanism is proposed and the analysis followsin Section V. The simulation results are shown in Section VIand future work ideas and the conclusions are drawn inSection VII.

II. PAST RELATED WORK

Harvesting or collecting energy from natural resources hasbeen considered as an alternative to supplement or completelyreplace battery supplies, e.g., [6], [7], and therefore enhancethe lifetime of a remote system. Such a system may be placedin a hostile environment and it may not be possible to other-wise recharge its energy [8]. A significant advantage is thatambient energy resources can be free and, many times, eveninexhaustible, leading to significant reduction of maintainancecosts. However, this introduces an uncertainty of the energyavailability and many works in the area deal with this issue,e.g., [8]–[15]. In addition, in [9], a technique that adjuststhe rate of the duty cycle of a node is presented. In [10], amethod is presented that achieves maximum energy harvestingfrom natural resources by accomplishing fast determinationof the optimal operating condition and in [16], an adaptiveenergy harvesting management framework that adjusts theapplication quality based on the energy harvesting conditions.A model that forecasts the source availability and estimatesthe expected energy intake is studied in [17] in order tobe able to take critical decisions regarding the utilization of

the available energy. There are additional works in the areafocusing on performance and routing issues, e.g., [6], [18]–[22]. For example, in [23] it is shown that preferring the nodeswith high energy harvesting rates (e.g., those that are exposedin sunlight) can lead to network’s workload maximization.

In [4] Probabilistic Flooding is compared against a pro-posed heuristic flooding that changes the (otherwise fixed)forwarding probability and more recently in [24] stochastictopologies were studied to derive suitable values for thethreshold probability, while [25] considers generalized randomgraphs. In [5], algebraic graph theory tools were used tofurther study the performance of Probabilistic Flooding. Otherexamples include peer-to-peer networks for service discovery[26]–[28], vehicular networks for information hovering [29],and energy savings by reducing transmissions [30]. A variationof Probabilistic Flooding for reliable dissemination is alsoproposed in [31] and an adaptive version for multi-path routingpurposes is presented in [32].

III. THE ENERGY HARVESTING NETWORK SYSTEM

A network topology is common to be represented as agraph consisting of vertices (nodes) and edges (links). Giventhat the focus in this paper is on WSNs, it is assumed, as it iscommon, that the topology is a geometric random graph [33],where the n nodes of the network are randomly scattered in the[0, 1]× [0, 1] square area and a link among any pair of nodesexists if and only if their euclidean distance is smaller than orequal to the connectivity radius rc. The number of links in thisnetwork will be denoted by � and the set of neighbor nodesof node u by Su.

Each node is aware about its neighbor nodes, i.e., thosenodes that a direct transmission can take place, and particularlyif these nodes are operating or not. A node is said to be in anoperational (ON) state if the available energy resources are ableto handle its workload (sensing, processing, participating in thenetwork etc.), while in a different situation it said to be on anon-operational (OFF) state. Depending on the environmentalconditions (e.g., sunny weather in the case of solar panels), anode’s battery may be charging (C) or discharging (D).

It is now possible to define four states depending onwhether a node is in an operational or a non-operationalstate and charging its battery or not. Let ON-C (State 1)be the particular state where the node is in an operationalstate and at the same time its battery is charging. If itsbattery is full, it remains full. Obviously, at this state theenergy consumed by the node’s operation (sensing, processing,receiving, transmitting etc.) is compensated by the energycollected by the environment. This is the most desired statesince it allows for prolonging the network’s operation as faras energy consumption is considered.

When the energy supplemented by the environment doesnot compensate for the node’s operation, obviously its batteryis discharging. The node will continue to be in an operationalstate as long as there is energy left in its battery. This stateis referred to hereafter as ON-D (State 2). If the node is ina non-operational state due to insufficient energy left in itsbattery, two similar cases can also be defined. It can either beat state OFF-C (State 3) in which its battery is charging, or at


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state OFF-D (State 4) in which its battery (even slowly) keepsdischarging.

These four states along with the corresponded transitionprobabilities are depicted in Fig. 1 as a four state Markov chain[34]. The transition probability between State i and State j isdenoted by αi→j . Fig. 1 depicts all possible combinations withrespect to transition probabilities. Later, in Section V theseprobabilities are simplified based on certain observations andassumptions.

ON-C ON-D OFF-C OFF-Da1→2

a1→3

a1→4

a1→1

State 1

a2→1

a2→3

a2→4

a2→2

State 2

a3→1

a3→2

a3→4

a3→3

State 3

a4→1

a4→2

a4→3

a4→4

State 4

Fig. 1. The node energy harvesting Markov chain.

This Markov chain model is suitable to shed some lightinto the particulars of the energy harvesting environment thatunderlies the topology of the network. The validation of thisMarkov chain and the derivation of the conditions under whichit sufficiently represents an energy harvesting environment isleft for future work. For the purposes of this paper, it sufficesfor the qualitative study of the proposed, in the sequel, robustversion of Probabilistic Flooding.

IV. PROBABILISTIC INFORMATION DISSEMINATION

It is assumed that the piece of information to be dissemi-nated is initially placed at some node in the network, called theinitiator, that is usually the sink node s. Under ProbabilisticFlooding, this information is probabilistically forwarded inthe form of messages to all neighbor nodes of the initiatoraccording to a forwarding probability q, common for all nodesin the network. Each node that receives this message for thefirst time, it forwards it accordingly to its neighbor nodes apartfrom the one it arrived from. If a node receives more than oncethe same message, it discards it.

Let C(t) denote the fraction of the network nodes thathave received the particular piece of information at time t. Attime t = 0, only the initiator node has the particular piece ofinformation and therefore, C(0) = 1/n. Obviously, the aimis to achieve C(T ) = 1, i.e., full coverage, where T , alsoreferred to as termination time, is the time when the floodingprocess terminates.

Note that due to the probabilistic nature of ProbabilisticFlooding, a handful of nodes may be left uncovered at theend of the information dissemination process. Therefore, fullcoverage is achieved with high probability (w.h.p.) in the sensethat limn→+∞ C(T ) = 1 and the network does not contain anybottlenecks.

In a usual environment (nodes stop operating due to energyexhaustion and do not harvest any energy), the focus is on

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Fig. 2. Example bottleneck link (u, v). The area within the dotted line willnot be covered under Probabilistic Flooding if either node u or node v are atnon-operational state.

the particular value of q that full coverage is achieved and atthe same time the number of messages is minimized. Notethat for q = 1, all nodes eventually receive the message andtermination time is upper bounded by the network diameter D,or T ∈ O(D). In this case, the number of messages M sentin the network is at least equal to the number of network links� and less than twice this number, since some links may beutilized twice or, M ∈ Θ(�). As q decreases, termination timeincreases but the number of messages is significantly reduced.However, full coverage is now probabilistically guaranteed(i.e., w.h.p.) and depending on the topology, there is a thresholdvalue q0 for the forwarding probability q, such that for anyq < q0, full coverage cannot be achieved. Several works focuson q0, e.g., [4], [5], [24]. However, the focus in this work isabout the performance of Probabilistic Flooding consideringthe energy harvesting environment.

Assuming the already described energy harvesting environ-ment and Probabilistic Flooding, consider a certain bottlenecklink (u, v) between node u and v that connects a certainnetwork area to the rest of the network. If node v is notoperating (being either at State 3 or 4 according to theMarkov chain depicted in Fig. 1), then any message sent underProbabilistic Flooding will fail to reach the particular areaconnected to the rest of the network through the bottlenecklink (u, v). However, due to the energy harvesting capabilities,node v may become operational after some time (e.g., at State1 or 2) and being capable of forwarding the particular messagehaving only been aware of it.

The latter observation motivates the subsequent modifi-cation of Probabilistic Flooding in order to accommodatethe idiosyncrasies of the energy harvesting environments andbecome robust in the face of node failures/revivals due toenergy exhaustion and subsequent harvesting. The main ideais to let messages still be forwarded in the network aftera node becomes operational again. There are two distinctcases: (i) a message is lost since it cannot be forwarded toa particular node that is not operating; and (ii) a messageis lost because a node that has just received it for the firsttime stops operating. In order to deal with both cases, statusinformation is proposed to be maintained at the each node andwhen a node resumes its normal operation to proceed as ifthe non-operational mode was never assumed. The proposedRobust Probabilistic Flooding mechanism is presented next inAlgorithm 1, where s is the initiator node and q the particular


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value for the forwarding probability.

Let Lu be the set of neighbor nodes of u operating at aparticular time instance. Obviously, Lu ⊆ Su. It is assumedthat each node is aware about the set of operating and notoperating neighbor nodes by utilizing “hello” messages thatare commonly exchanged at the data link control layer (DLC),thus no extra overhead is introduced. Let Ru be the set ofthe remaining to be covered neighbor nodes of u, i.e., thosethat should have been informed but this was not possible eitherbecause these nodes were not operating or because node u wasnot operating.

Algorithm 1 Robust Probabilistic Flooding(s,q)1: Var u: current node; covered: Boolean flag;2: Initialize:3: if u == s then � s is the initiator node4: covered:=TRUE;5: Ru := RANDOMSELECTION(Su, q);6: for ∀ v ∈ Ru ∩ Lu do7: send < message, u >→ v;8: Ru := Ru \ {v}; � Remove node u from Ru

9: else covered:=FALSE;

10: Operate:11: if receive u ←< message, x > && !covered then12: covered:=TRUE;13: Ru := RANDOMSELECTION(Su \ {x}, q);14: for ∀ v ∈ Ru ∩ Lu do15: send < message, u >→ v;16: Ru := Ru \ {v};

17: function RANDOMSELECTION(S, q)18: Var R := ∅;19: for ∀ v ∈ S do20: if UNIFORM([0, 1]) < q then21: R := R ∪ {v};22: return R;

During initialization, all nodes (represented by the nodevariable u in Algorithm 1) update their status information asnot covered (Line 9), apart from the initiator node (Line 4).During the initialization, set Rs contains the particular neigh-bors of the initiator node s that the message will be forwardedto. Function RANDOMSELECTION(S, q) (Line 17) returns arandomly selected (uniformly) subset of nodes in S. After-wards (Lines 6–8), the message is forwarded to the selectedoperating neighbor nodes (i.e., set Rs∩Ls) and Rs is updatedaccordingly.

After initialization, at each time step any node u (includingthe initiator node s) checks whether it has received a messageor not (Line 11). If the message is received for the first time,then Ru is initialized (Line 13). For all nodes, the message issent to those neighbor nodes (if any, i.e. if Ru ∩ Lu = ∅ atLine 14) that was not sent previously due to either their or u’snon-operating status.

Algorithm 1 is claimed to operate when nodes stop operat-ing and then resume their operation. If the latter is not the case,i.e., Lu ≡ Su, then Robust Probabilistic Flooding is reducedto Probabilistic Flooding. Another interesting observation is

that no extra messages are sent under Robust ProbabilisticFlooding, since each node probabilistically forwards the mes-sage to the same set of neighbor nodes as under ProbabilisticFlooding. However, termination time is affected as will be alsoshown later in the simulations section. Note that there is alsoa certain memory overhead due to the status information thata node needs to maintain (i.e., set Ru). However, this is notsignificant and in any case upper bounded by the number ofneighbor nodes.

V. MARKOV CHAIN ANALYSIS

The proposed Robust Probabilistic Flooding presented inAlgorithm 1 is capable of dealing with the casual non-operating nature of the energy harvesting nodes. However,the Markov chain presented in Fig. 1 is too complicated dueto the numerous transition probabilities. In order to simplifythe Markov chain and focus further on the particulars ofProbabilistic Flooding and Robust Probabilistic Flooding, anumber of observations and assumptions allow for simplifyingthe Markov chain and the subsequent analysis.

For example, a direct transition from state ON-C to non-operational states (i.e., states OFF-C and OFF-D) is notexpected since the battery of the node will be most likelycharged allowing for moving first to state ON-D. Therefore,a1→3 = 0 and a1→4 = 0. Similarly, when a node is at stateOFF-D, this means that it looses energy from its battery andtherefore, it is not expected to move to operational mode,even if the conditions of the environment change and itsbattery starts recharging once more. Therefore, from this statethere is a direct transition only to state OFF-C. Consequently,a4→1 = 0 and a4→2 = 0.

Let α denote the probability that a node being at the desiredstate ON-C remains at this state, then a1→1 = α and given thetransition constraints, a1→2 = 1 − α. Similarly, let β denotethe transition probability for state ON-OFF, or β = a2→2. Inthe sequel, it is assumed that remaining at a charging state(i.e., ON-C and OFF-C), has the same probability, or a1→1 =a3→3 = α. Similarly, for the discharging states (i.e., ON-Dand OFF-D), or a2→2 = a4→4 = β. Therefore, a4→3 = 1−β.

Another assumption to further simplify this Markov chainmodel, corresponds to moving form state ON-D to OFF-C andvice versa. It will be assumed that moving to non-operationalmodes requires to go through the discharging state and sim-ilarly, moving from a non-operating and charging state to anoperating one, the energy harvesting conditions should ensurecharging the node’s battery. Eventually, a2→3 = a3→2 = 0.Let γ be the transition probability from a discharging state toa charging one, or a2→1 = γ. Let then, in analogy, assume that1 − γ corresponds to moving from the charging state OFF-Cto the discharging one OFF-D, or a3→4 = 1− γ.

Based on the previous observations and assumptions, itis possible to prune certain transitions of the Markov chaindepicted in Fig. 1 and simplify the rest of the transitionprobabilities, expressed as a function of α, β and γ. Note thatall transition probabilities are assumed to be = 0, 1, otherwise,the Markov chain would have been a trivial one. Therefore,a2→4 = 1 − β − γ and a3→1 = 1 − α − (1 − γ) = γ − α.Obviously, it is meaningful for 0 < 1 − β − γ < 1 and0 < γ−α < 1. Eventually, it is required that γ < 1−β < 1+γ


66

and α < γ < 1 + α. Given that 0 < γ < 1, both are satisfiedwhen, α < γ < 1 − β. The pruned version of the Markovchain of Fig. 1 is depicted in Fig. 3.

ON-C ON-D OFF-C OFF-D1 − α

α

State 1

γ

1 − β − γ

β

State 2

γ − α

1 − γ

α

State 3

1 − β

β

State 4

Fig. 3. The pruned Markov chain.

Let p1, p2, p3 and p4 denote the state probabilities corre-sponding to states ON-C, ON-D, OFF-C, and OFF-D, respec-tively. The analytical expressions for these state probabilitiesas a function of α, β and γ are presented next. The derivationsteps are presented in the Appendix.

p1 = (1−β)2(γ−α)

(2−α−β)

((1−β)(γ−α)+(1−α)(1−β−γ)

) , (1)

p2 = (1−α)(1−β)(γ−α)

(2−α−β)

((1−β)(γ−α)+(1−α)(1−β−γ)

) , (2)

p3 = (1−α)(1−β−γ)(1−β)

(2−α−β)

((1−β)(γ−α)+(1−α)(1−β−γ)

) , (3)

p4 = (1−α)2(1−β−γ)

(2−α−β)

((1−β)(γ−α)+(1−α)(1−β−γ)

) . (4)

State probability p1 is depicted in Fig. 4 as a functionof α, for β = 0.1 and various values of γ. Obviously, forγ = α, p1 becomes zero. When γ = β, the shape of thecurve changes (there is no local maximum and the slopeincreases monotonically). This is expected since for γ = β,p1 = γ

1−α−γ, that is a monotonically increasing function with

respect to α.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

p1

α

β = 0.1γ = 0.1γ = 0.3γ = 0.5γ = 0.7γ = 0.9

Fig. 4. State probability p1 (ON-C) when β = 0.1, for various values of γas a function of α.

The analysis of the Markov chain is useful in the sequelwhen evaluating the proposed Robust Probabilistic Flooding.As already mentioned, the aim of the Markov chain is to

capture the basic behavior of an energy harvesting network,i.e., the nodes’ operation and not operation modes.

VI. SIMULATION RESULTS

For the simulation purposes a proprietary simulator in C++has been developed to simulate both Probabilistic Floodingand the proposed in this paper Robust Probabilistic Flooding.The purpose of this section is to show the limitations ofProbabilistic Flooding in energy harvesting environments anddemonstrate the robustness of the proposed Robust Probabilis-tic Flooding in the same environments. In order to fulfill this,the framework followed next consists of a number of steps:(i) four different geometric random graph topology categoriesof n = 1000 nodes are taken into account correspondingto different connectivity radius rc (i.e., 0.05, 0.10, 0.20 and0.30) and for each category ten topologies are created (thepresented results are averaged values for these ten cases); (ii)Probabilistic Flooding takes place and the particular thresholdvalue q0 for the forwarding probability is estimated based onthe simulation results; (iii) the energy harvesting environmentis activated by allowing all nodes to operate according to theMarkov chain presented in Fig. 3 for various values of α, β andγ; (iv) Probabilistic Flooding is shown to be ineffective in thesense that it does not fully cover the network for some range ofthese values; (v) Robust Probabilistic Flooding takes place andit is shown to be capable of achieving full coverage w. h. p. (asmentioned a handful of nodes may not receive the informationmessage due to the probabilistic nature of the disseminationapproach) in the same energy harvesting environment thatProbabilistic Flooding failed; and (vi) the overhead in termsof termination time increment is also presented.

TABLE I. AVERAGE VALUES FOR �, #NEIGHBORS, T , M AND q0(rc)FOR PROBABILISTIC FLOODING.

rc � #Neighbors T M q0

0.05 3747.5 7.5 31.1 3564.4 0.780.10 14549.2 29.1 21.6 3177.3 0.120.20 53149.3 106.3 14.0 2964.3 0.030.30 107499.7 215.0 10.0 4139.2 0.02

Table I presents average values with respect to the numberof links �, the number of neighbors for the four consideredtopology categories, termination time T and the number ofmessages M sent under Probabilistic Flooding. The (average)values of the threshold probability for each case are also given.Note that for rc > 0.30 the average number of neighbor nodesis significantly large (215.0). Therefore, more dense topologies(i.e., rc > 0.30) are not considered in this paper as not realisticfor a typical wireless sensor network.

Next, Probabilistic Flooding takes place when the energyharvesting environment is activated. Fig. 5 presents coverageC(T ) under Probabilistic flooding at termination time T , forα = 0.1 and various values of β as a function of γ forrc = 0.10. Fig. 6 presents the same scenarios for β = 0.1 andvarious values of α. For the rest, the particular values selectedfor q will be the ones of q0 corresponding to the particulartopology as given in Table I.

As it is obvious from both figures, there is a range of valuesfor α, β and γ for which Probabilistic Flooding fails to coverthe entire network. As already mentioned, the interest hereis not on the particular values of α, β, γ, but on the actual


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0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

C(T )

γ

α = 0.1β = 0.1β = 0.2β = 0.3β = 0.4β = 0.5β = 0.6

Fig. 5. Coverage at termination time C(T ) under Probabilistic Flooding forα = 0.1 and various values of β as a function of γ for rc = 0.10.

fact that the introduced Markov chain succeeds to highlightthe inefficiencies of Probabilistic Flooding by capturing theenergy harvesting environment.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

C(T )

γ

β = 0.1α = 0.1α = 0.2α = 0.3α = 0.4α = 0.5α = 0.6

Fig. 6. Coverage at termination time T under Probabilistic Flooding forβ = 0.1 and various values of α as a function of γ for rc = 0.10.

This inefficiency of Probabilistic Flooding is more clearlydemonstrated in Fig. 7 for the considered values of rc. Notethat if the network was a common one with no harvestingcapabilities, full coverage would be achieved for the particularvalues of q. However, the nodes’ non-operation mode (eitherthose to send the message or those to receive it) prohibitsthe information dissemination process to arrive to the entirenetwork area.

The proposed Robust Probabilistic Flooding overcomes theobserved inefficiency of Probabilistic Flooding by keepingtrack of those neighbor nodes that the message should havebeen forwarded to under Probabilistic Flooding and does sowhen nodes are operational again. Thus, it overcomes theproblem of non-operation. Note that if there all nodes are oper-ational, Robust Probabilistic Flooding reduces to ProbabilisticFlooding as observed from Algorithm 1.

Fig. 8 presents coverage C(t) as a function of time tfor both Probabilistic Flooding (PF) and Robust ProbabilisticFlooding (RPF) for α = 0.3, β = 0.2 and γ = 0.6, for thefour topology categories rc = 0.05, 0.10, 0.20 and 0.30.

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.2 0.3

C(T )

rc

α = 0.3, β = 0.2, γ = 0.6

+

+

+

+

+α = 0.1, β = 0.2, γ = 0.6

×

×

× ×

×α = 0.1, β = 0.1, γ = 0.5

∗ ∗

∗

∗

∗

Fig. 7. Coverage at termination time T under Probabilistic Flooding forthree different sets of α, β and γ for rc = 0.05, 0.10, 0.20 and 0.30.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90

C(t)

Time Step t

α = 0.3, β = 0.2, γ = 0.6rc = 0.05 RPF

PF

∗∗∗∗∗∗∗∗∗∗

∗∗∗∗∗∗

∗∗∗∗∗∗∗∗∗∗∗∗∗

∗rc = 0.10 RPF

PF

××××××××××××××

××××××××

×××××××

×××××××××

××××××××××××

×rc = 0.20 RPF

PF

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�rc = 0.30 RPF

PF◦◦◦◦

◦

◦

◦

◦

◦◦◦◦◦◦◦◦◦◦◦

◦

Fig. 8. Coverage C(t) under both Probabilistic Flooding (PF) and RobustProbabilistic Flooding (RPF) for α = 0.3, β = 0.2 and γ = 0.6 and rc =0.05, 0.10, 0.20 and 0.30 as a function of time t.

It is interesting to observe that both flooding mechanismsstart simultaneously to cover the network. However, as timepasses, Probabilistic Flooding shows signs of inefficiency (theslope stops the rapid increment) and it stops long before fullcoverage is achieved. On the other hand, Robust ProbabilisticFlooding follows the particular rapid increment until fullcoverage is achieved. Note that full coverage is achieved w.h.p.meaning that for the considered case coverage correspondsto values of C(T ) greater than 0.97, since some nodes mayremain uncovered due to the probabilistic nature of bothinformation dissemination mechanisms.

Fig. 9 depicts simulation results regarding termination timeT under Robust Probabilistic flooding for three cases of theunderlying Markov chain and one case when no energy har-vesting is employed (i.e., the usual WSN environment). Notethat the latter case corresponds to termination time identicalto that under Probabilistic Flooding since when there are nonode failures in the network, Robust Probabilistic Floodingreduces to Probabilistic Flooding. As seen, termination timeincreases when compared to the no energy harvesting case.Note that this increment can be six times over the latter onewhich may be prohibitive in some cases. However, this is dueto the particular characteristics of the environment and eventhough there can be a significant termination time increment


68

0

20

40

60

80

100

120

0.05 0.1 0.2 0.3

T

rc

α = 0.3, β = 0.2, γ = 0.6

+

++ +

+α = 0.1, β = 0.2, γ = 0.6

×

×

× ×

×α = 0.1, β = 0.1, γ = 0.5

∗

∗

∗∗

∗no energy harvesting

�

�

��

�

Fig. 9. Termination time T under Robust Probabilistic Flooding for threedifferent sets of α, β and γ for rc = 0.05, 0.10, 0.20 and 0.30. Line“no energy harvesting” corresponds to the case that no energy harvesting isactivated in the network.

under Robust Probabilistic Flooding, the network is eventuallyfully covered.

Finally, in all experiments it was observed that the numberof messages sent under Robust Probabilistic Flooding is thesame for any network conditions and equal to the correspond-ing topology case as depicted in Table I for the case ofProbabilistic Flooding. This was expected given the descriptionin Algorithm 1 and it is one of the advantages of RobustProbabilistic Flooding since it does not require additionaltransmissions that would eventually result in further energyconsumption.

VII. CONCLUSIONS AND FUTURE WORK

The information dissemination problem using Probabilis-tic Flooding is revisited in this paper, assuming an energyharvesting wireless sensor network environment. In such anenvironment where nodes are expected to stop operating dueto discharged batteries and then resume their operation aftersuccessful ambient energy harvesting, certain messages fail tobe delivered and therefore, full network coverage cannot beachieved. This motivated the introduction of a robust versionof Probabilistic Flooding, i.e, Robust Probabilistic Flooding,that is presented in this paper to deal with the idiosyncrasiesof the considered environment.

In order to capture the latter’s behavior, a Markov chainmodel is introduced and its simplified version is used forsimulation purposes in order to highlight the inefficiencies ofProbabilistic Flooding in such environments and the efficiencyof the proposed Robust Probabilistic Flooding. In particular,it is shown, using simulations, that the latter is capable ofachieving full coverage in any scenario captured by the Markovchain model in the expense of increased termination time.Given that the number of messages remain unchanged bydefinition, Robust Probabilistic Flooding is a suitable choicefor information dissemination in such environments.

Future work will focus on the Markov chain and the deriva-tion of the particular conditions under which it sufficientlyrepresents an energy harvesting environment. This will allowfor further analysis of Robust Probabilistic Flooding given the

environment and further maximize its performance in terms ofcoverage, number of messages and termination time.

APPENDIX

The objective here is to derive the analytical expressionsregarding p1, p2, p3 and p4. It holds that p1 + p2 + p3 + p4 =1. Based on the transition probabilities of the Markov chainpresented in Fig. 3, the following system of equations needsto be solved [34],

p1 + p2 + p3 + p4 = 1αp1 + γp2 + (γ − α)p3 = p1(1− α)p1 + βp2 = p2αp3 + (1− β)p4 = p3(1− β − γ)p2 + (1 − γ)p3 + βp4 = p4

.

Taking into account the second, third and fourth equations,

γp2 + (γ − α)p3 = (1− α)p1(1− α)p1 = (1− β)p2(1− β)p4 = (1− α)p3

.

Next, p2, p3 and p4 are expressed as a function of p1,

p3 = (1−α)p1−γp2

γ−α

p2 = 1−α1−β

p1p4 = 1−α

1−βp3

⇐⇒p3 =

(1−α)p1−γ 1−α1−β

p1

γ−α

p2 = 1−α1−β

p1p4 = 1−α

1−βp3

⇐⇒

p3 = (1−α)(1−β−γ)(1−β)(γ−α) p1

p2 = 1−α1−β

p1p4 = 1−α

1−βp3

⇐⇒

p3 = (1−α)(1−β−γ)(1−β)(γ−α) p1

p2 = 1−α1−β

p1

p4 = (1−α)2(1−β−γ)(1−β)2(γ−α) p1

.

Note that the fifth equation (i.e.,(1 − β − γ)p2 + (1 −γ)p3 + βp4 = p4) is always satisfied. Based on the first one(i.e., p1 + p2 + p3 + p4 = 1), and substituting accordingly,p1 + 1−α

1−βp1 + (1−α)(1−β−γ)

(1−β)(γ−α) p1 + (1−α)2(1−β−γ)(1−β)2(γ−α) p1 = 1,

or, (1−β)2(γ−α)+(1−β)(γ−α)(1−α)+(1−β)(1−α)(1−β−γ)(1−β)2(γ−α) p1 +

(1−α)2(1−β−γ)(1−β)2(γ−α) p1 = 1, or, (1−β)(γ−α)(2−α−β)

(1−β)2(γ−α) p1 +(1−α)(1−β−γ)(2−α−β)

(1−β)2(γ−α) p1 = 1, or, (2 − α −

β) (1−β)(γ−α)+(1−α)(1−β−γ)(1−β)2(γ−α) p1 = 1. Finally,

p1 = (1−β)2(γ−α)

(2−α−β)

((1−β)(γ−α)+(1−α)(1−β−γ)

) . By substituting

this expression for p1, the analytical expressions for p2, p3and p4 are also derived.

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