Data retrieval time for energy harvesting wireless sensor networks
Mihaela Mitici, Jasper Goseling, Maurits de Graaf, Richard J. Boucherie
aa r X i v : . [ c s . S Y ] O c t Data retrieval time for energy harvesting wireless sensornetworks
Mihaela Mitici · Jasper Goseling · Maurits de Graaf · Richard J. BoucherieAbstract
We consider the problem of retrieving a reliable estimate of anattribute monitored by a wireless sensor network, where the sensors harvestenergy from the environment independently, at random. Each sensor storesthe harvested energy in batteries of limited capacity. Moreover, provided theyhave sufficient energy, the sensors broadcast their measurements in a decen-tralized fashion. Clients arrive at the sensor network according to a Poissonprocess and are interested in retrieving a fixed number of sensor measurements,based on which a reliable estimate is computed. We show that the time untilan arbitrary sensor broadcasts has a phase-type distribution. Based on thisresult and the theory of order statistics of phase-type distributions, we deter-mine the probability distribution of the time needed for a client to retrieve areliable estimate of an attribute monitored by the sensor network. We also pro-vide closed-form expression for the retrieval time of a reliable estimate whenthe capacity of the sensor battery or the rate at which energy is harvested isasymptotically large. In addition, we analyze numerically the retrieval time ofa reliable estimate for various sizes of the sensor network, maximum capacityof the sensor batteries and rate at which energy is harvested. These resultsshow that the energy harvesting rate and the broadcasting rate are the mainparameters that influence the retrieval time of a reliable estimate, while de-ploying sensors with large batteries does not significantly reduce the retrievaltime.
Keywords
Wireless sensor networks · energy harvesting · data retrievaltime · phase-type distribution · order statistics M. Mitici · J. Goseling · M. de Graaf · R.J. BoucherieStochastic Operations Research Group, Department of Applied Mathematics,University of Twente, P.O. Box 217, 7500 AE, Enschede, The Netherlands.E-mail: [email protected] Mihaela Mitici et al.
Managing energy consumption is an important component of wireless sensornetwork design, as it can lead to increased throughput and network lifetime.Recent technological advances have enabled sensor to harvest energy fromthe environment and to use this energy to recharge their batteries. Severaltechnologies have been shown to be possible for energy harvesting such assolar energy, radio frequency, vibrations, thermoelectric (see, for instance, [12]for various examples of energy harvesting technologies).In the case of battery-limited sensor networks, one of the main design goalsis to minimize the sensor energy consumption so that the lifetime of the sensornetwork is extended. In contrast, energy harvesting techniques enable sensorsto recharge their batteries over time, having the potential to extend the lifetimeof the sensor network and to improve the overall performance of the network.However, it is often the case that the harvested energy availability varies withtime in a non-deterministic manner. For instance, solar energy varies through-out the day and the intensity of the direct sunlight cannot be controlled. Toenhance energy availability, sensor are often equipped with batteries where en-ergy is stored for later use. The storage capacity of the batteries, however, isoften limited. It is important, thus, to develop mechanisms to match the energygeneration profile of the harvesting sensors with the energy consumption of thesensors. In particular, since sensors consume most of their energy to transmittheir measurements [3], it is important to jointly consider sensor transmissionsand the process of energy harvesting. Energy harvesting for wireless commu-nications has resulted in a significant number of research publications in thelast decade [14]. Considerable attention has been given to the coordinationbetween sensor transmission scheduling, which is an energy-dependent pro-cess, and energy harvesting, which depends on the availability of the energysource. Various metrics for data transmission in energy harvesting communi-cation systems have been considered. In [4,17,18] the minimization of the timeto transmit a fixed number of bits using a AWGN broadcast channel, wherethe transmitter harvests energy and has a finite-capacity rechargeable battery,is considered. In contrast to the model proposed in this paper, the energy ar-rival process is assumed to be known in advance, in an off-line manner. Theauthors optimize the transmission rate or the transmission power based onthe energy available at the transmitter and on the amount of data that needsto be transmitted such that the transmission time is minimized. Structuralproperties of the transmission policies are derived. Similar to the model pro-posed in this paper, in [13, 15] the process of energy harvesting is stochastic.Transmission policies that maximize the rate of data transmission to minimizethe mean delay of data transmission are derived. In [15], the average delay ofdata packets arriving according to a Poisson process at a transmitter whichharvests energy, is derived. In comparison, in this paper we compute the ex-pected delay of a set of sensor measurements to be transmitted by distinctsensors, each harvesting energy according to a Poisson process. ata retrieval time for energy harvesting wireless sensor networks 3
The problem of maximizing the amount of data transmitted up to a certainpoint is considered in [6, 7, 11, 16]. In [6] a general framework is providedto maximize the amount of transmitted data by a given deadline when thearrival process of energy is known in an off-line manner, at the transmitterand the battery of the transmitter is limited or suffers from energy leakage.In [7,11,16] dynamic programming is employed to determine an optimal energyallocation policy over a finite horizon so that the number of transmitted bits ismaximized. In [9] optimal transmission policies are derived to specify whetherto transmit incoming data packets or to drop them based on a value attachedto each packet and on the energy available at the transmitter.Optimal transmission policies with energy harvesting nodes that transmitusing fading wireless channels are considered in [7,8,11]. In [8], the probabilityof successful reception of data packets and the energy cost per transmittedpacket are determined for energy harvesting devices that broadcast using non-perfect transmission channels. The authors propose an erasure-based broadcastscheme to guarantee reliable transmissions.This paper considers the problem of retrieving a fixed number of sensormeasurements over an attribute from distinct wireless sensors that harvestenergy from the environment. Energy is harvested by each sensor accordingto a Poisson process, independently of the other sensors. The fact that energyis harvested at random points in time reflects the stochastic nature of theavailability of harvested energy. We further assume that the batteries havelimited storage capacity. When the battery of a sensor reaches the maximalstorage capacity, additional harvested energy is discarded. Provided that theyhave energy, the sensors broadcast measurements at an exponential rate andindependently of each other. Clients arrive at the network according to somePoisson process and are interested in retrieving several measurements over anattribute monitored by the sensors. Based on the retrieved sensor measure-ments, each client computes an estimate of the attribute. We impose thatmeasurements are retrieved from distinct sensors to avoid the situation wherethe same measurement, from the same sensor, is retrieved multiple times,which could lead to a biased estimate. We determine the probability distribu-tion of the retrieval time of a reliable estimate of an attribute monitored bythe sensor network. Consequently, we provide a closed-form expression for theexpected retrieval time of a reliable estimate. We also analyze the retrievaltime of a reliable estimate when the capacity of the battery or the rate atwhich energy is harvested is asymptotically large. These results show the im-pact of the energy availability at the sensors, as well as the energy storagecapabilities of the sensors, on the time required to retrieve a reliable estimateof an attribute from the sensor network.The remainder of this paper is organized as follows. In Section 2 we for-mulate the model and the problem statement. In Section 3 we determine thedistribution of the time for a client to retrieve a reliable estimate of an at-tribute from the sensor network. We also determine the retrieval time of areliable estimate when the rate at which energy is harvested and the maxi-mum capacity of the sensor batteries is asymptotically large. In Section 4 we
Mihaela Mitici et al. compute numerically the retrieval time of a reliable estimate under variousassumptions regarding the size of the sensor network, the energy harvestingrate and the maximum capacity of the sensor batteries. In Section 5 we discussthe results and provide conclusions.
We consider a network of N wireless sensors, each having a noisy measurementon an attribute θ . The measurements are subject to independent and identi-cally distributed additive Gaussian noise with variance σ , i.e. X i ∼ N ( θ, σ ).Clients arrive at the sensor network according to a Poisson process withrate λ a . Each client is interested in acquiring from the network a sufficientlylarge set of sensor measurements based on which they can compute a reliableestimate X of θ .We consider an estimate to be reliable if the variance of the estimate X ateach client is below a targeted threshold H . SinceVar( X ) = Var( 1 s s X i =1 X i ) = 1 s s X i =1 Var( X i ) = σ s < H, we consider s = ⌈ σ H ⌉ . Thus, any set of s measurements from distinct sensorsis sufficient for the client to retrieve a reliable estimate of θ . As there are N sensors which can provide N distinct measurements, we also assume that s ≤ N .Sensor i , 1 ≤ i ≤ N , has b i units of energy, with 0 ≤ b i ≤ B . Each sensorharvests one unit of energy at an exponential rate λ e , independently of theother sensors. The energy is harvested from the environment. If b i = B andnew energy is harvested, then this harvested energy is discarded.We further assume that the sensor network broadcasts a measurement atan exponential rate µ . This rate is shared uniformly by the N sensors in thenetwork. Thus, at an exponential rate µ/N , independently of the other sensors,a random sensor broadcasts its measurement, provided it has energy.Upon a broadcast, the energy of the broadcasting sensor decreases by 1unit (see also Figure 2.1). . . . B − Bµ/Nλ e λ e µ/N λ e µ/Nλ e µ/N Fig. 2.1: Birth-and-Death model for a single sensor that broadcasts, providedit has energy, at an exponential rate µ/N and harvests energy from the envi-ronment at an exponential rate λ e . ata retrieval time for energy harvesting wireless sensor networks 5 A measurement is retrieved by a client only if it is the first time this sensorbroadcasts its measurement to this client. We call such a measurement to be innovative for this client. Thus, a client does not retrieve a measurement fromthe same sensor multiple times. Moreover, we assume that clients do not dropinnovative measurements.We are interested in the time W s for a client to retrieve a reliable estimatefrom the sensor network.We end this section with some notation that will be useful for working withphase-type distributions. Let e k be a k × I k denote the k × k identity matrix. For n × n matrix M and m × m matrix M , let M ⊗ M denote the Kronecker product of matrices M and M andlet M ⊕ M denote the Kronecker sum, i.e. M ⊕ M = M ⊗ I m + I n ⊗ M . Finally, let M ⊗ n and M ⊕ n denote the n -fold Kronecker product and the n -foldKronecker sum with itself, respectively. In this section we first determine the distribution of the time for a singlesensor to broadcast, given that the system is in steady-state. We show thatthis is a phase-type distribution. Using these results, we next determine thedistribution of the time for a random client to retrieve s measurements fromdistinct sensors. Lastly, we compute the retrieval time of a reliable estimate X of θ for asymptotically large B , the maximum capacity of the sensor batteries, λ e , the rate at which sensors harvest energy from the environment, and N ,the size of the sensor network.3.1 A Single SensorFirst, we consider the steady-state probability that a random sensor has i unitsof energy, 0 ≤ i ≤ B , denoted by ν ( i ),The evolution of the units of energy at a sensor follows a Birth-and-Deathmodel and a finite state space { , , . . . , B } with births at rate λ e and deathsat rate µ/N ( see Figure 2.1). The steady-state distribution of such a model iswell known in literature (see, for instance, [5]) and is, therefore, stated withoutproof below. Lemma 3.1
The steady-state probability for an arbitrary sensor to have i units of energy, ≤ i ≤ B , is: ν ( i ) = ν (cid:18) λ e Nµ (cid:19) i , (3.1) where ν = ( λ e N/µ − / (( λ e N/µ ) B +1 − , if λ e = µ/N and ν = 1 / ( B + 1) otherwise. Mihaela Mitici et al.
Note that in the above theorem ν is the probability that the battery of asensor is depleted.Next, we consider W , the time until a sensor broadcasts, given that thesystem is in steady-state. Based on W , we compute the time for a client toretrieve a reliable estimate by assuming that, upon arrival, this client observesthe energy level of the sensors in steady-state. This is valid since the sensorsoperate independently of the arrivals of the clients. Moreover, since clientsarrive according to a Poisson process, they see the system in steady-state(PASTA).Since the evolution of the energy at an arbitrary sensor follows a continuous-time Markov process, the distribution of W can be modeled as a phase-typedistribution as follows. Consider a continuous-time Markov chain with B + 2states. State 0 ≤ i ≤ B are transient states and correspond to a sensor having i units of energy. The ( B + 2) − th state is an absorbing state. This state isreached when the sensor broadcasts a measurement. At an exponential rate λ e , a jump occurs from state i to state i + 1, 0 ≤ i < B . This correspondsto the sensor harvesting an additional unit of energy. At an exponential rate µ/N , a transition occurs from state 1 ≤ i ≤ B to the absorbing state. This cor-responds to a sensor broadcast. Let the initial distribution over the transientstates be ν . Then, the time until absorption is W , as desired.Before giving a more formal description of the phase-type representationof W , we make a simplification by observing that in the above descriptionthe states 1 to B can be aggregated into a single transient state, which wewill denote by 1. There is a transition from state 0 to this aggregated state1 at rate λ e and there is a single outgoing transition from this aggregatedstate 1 to the absorbing state at rate µ/N . Below, we will give the formalrepresentation of this phase-type distribution as ( a, T ) and specify the vector a and the matrix T . Given this representation as a phase-type distribution,we immediately obtain P ( W ≤ t ) = 1 − ae T x e . In this case, however, since T has such a simple structure we can also obtain the distribution function in anexplicit form. This yields the following result. Lemma 3.2
The distribution of W is phase-type ( a , T ) , where a = (cid:20) ν − ν (cid:21) , T = (cid:20) − λ λ − µ/N (cid:21) . (3.2) The distribution function of W can be expressed as P ( W ≤ t ) = 1 − e − µN t + µNµN − λ e ν (0) (cid:16) e − µN t − e − λ e t (cid:17) . (3.3) Proof
The phase-type characterization follows from the discussion above thelemma. The expression of the distribution function is obtained by observingthat, given that we are in state 0, which happens with probability ν (0), thedistribution of W is given by the sum of two exponentially distributed randomvariables with parameters µN and λ e (see, for instance, [2]). Given that weare in the aggregated state 1, which happens with probability 1 − ν (0), the ata retrieval time for energy harvesting wireless sensor networks 7 distribution of W is given by an exponentially distributed random variablewith parameter µN . Therefore, P ( W ≤ t ) = (cid:18) − µNµN − λ e e − λ e t + λ eµN − λ e e − µN t (cid:19) ν (0) + (1 − e − µN t )(1 − ν (0)) . The result follows directly from the above expression.3.2 Retrieving a reliable estimateIn this section, we determine the distribution of the time for a random clientto retrieve s measurements from distinct sensors. Lemma 3.3
The distribution of the time until a client receives s measure-ments from distinct sensors is: P ( W s ≤ t ) = 1 − s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − j − k a ⊗ ( N − K ) exp (cid:16) tT ⊕ ( N − k ) (cid:17) e N − k ! . Proof
Recall that a client leaves the system as soon as he retrieves exactly s measurements. Thus, we need to compute the distribution of the time betweenthe moment a client arrives at the network and the moment when the s -thbroadcast occurs, all s broadcasts from distinct sensors. This can be seen asthe distribution of the s -th order statistic of N phase-type distributed randomvariables with representation ( a , T ), as introduced above. The distribution ofthe s -th order statistic is (see, for instance, [1]), for N variables, is P ( W s ≤ t ) = N X j = s (cid:18) Nj (cid:19) P ( W ≤ t ) j ( P ( W > t ) N − j . (3.4)Starting from the above expression we derive P ( W s ≤ t ) = 1 − s − X j =0 (cid:18) Nj (cid:19) P ( W ≤ t ) j ( P ( W > t ) N − j = 1 − s − X j =0 (cid:18) Nj (cid:19) (1 − P ( W > t )) j ( P ( W > t ) N − j = 1 − s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − j − k P ( W > t ) j − k + 1 ! ( P ( W > t ) N − j (3.5)= 1 − s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − j − k P ( W > t ) N − k ! , (3.6)where in (3.5) we expanded the polynomial (1 − P ( W > t )) j . Mihaela Mitici et al.
Now, observe that the distribution of P ( W > t ) N − k in (3.6) is: P ( W > t ) N − k = P (min { Y , Y , . . . , Y N − k } > t ) , (3.7)where the Y i , ≤ i ≤ N − k are i.i.d. phase-type distributed random vari-ables with representation ( a, T ). Therefore, P ( W > t ) N − k is the first orderstatistic of a phase-type distributed random variable for which it is well known(see, for instance, [10]) that it is phase-type distributed with representation( a ⊗ ( N − K ) , T ⊕ ( N − k ) ). The result follows directly by inserting the distributionfunction of this phase-type distribution into (3.6).The result above is general in the sense that it does not depend on the specificrepresentation of the phase-type distributed random variable W .We are next interested in determining E [ W s ], the expected time for a clientto retrieve exactly s measurements. In principle, E [ W s ] can be obtained di-rectly from Lemma 3.3. However, the moments of order statistics of phase-typedistributed random variables are known in the literature [19]. Therefore, wewill resort to the results from [19]. Let m ks denote the k -th moment of the s -thorder statistic of N phase-type distributed random variables with representa-tion ( a , T ). Theorem 3.4 [19, Thm 4.1] m ks = m ks − + s X j =1 ( − j − (cid:18) N − s + jj − (cid:19) L ( k ) N − s + j , where L ( k ) j = (cid:0) Nj (cid:1) ( − k k ! (cid:0) a ⊗ j (cid:1) (cid:0) T ⊕ j (cid:1) − k e j , ≤ j ≤ s , and m k = 0 . Taking k = 1, E [ W s ] can be computed from Theorem 3.4. Note, however, thatstraightforward computation of moments based on Theorem 3.4 involves thematrices T ⊗ j , where j takes values up to N . The dimension of T ⊗ N is 2 N × N .Therefore, the complexity of these computations is exponentially increasing in N . Since we are interested in the behaviour of the system for larger values of N , we will derive in the next result an expression for E [ W s ] that has at mostpolynomial complexity in all model parameters. Theorem 3.5
The expected time, E [ W s ] , for a client to retrieve s measure-ments from distinct sensors is: s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − j − k N − k X v =0 (cid:18) N − kv (cid:19) ω v (1 − ω ) N − k − v λ e ( N − k − v ) + µN v , where ω = 1 − ν (0) µNµN − λ e . ata retrieval time for energy harvesting wireless sensor networks 9 Proof
The expected retrieval time for s measurements from distinct sensorscan be expressed using Theorem 3.3 and Lemma 3.2 as follows. E [ W s ]= Z ∞ P ( W s > t ) dt = s − X j =0 (cid:18) Nj (cid:19) Z ∞ j X k =0 (cid:18) jk (cid:19) ( − N − k (1 − P ( W ≤ t )) N − k ! dt (3.8)= s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − N − k Z ∞ (cid:16) e − µN t ω + e − λ e t (1 − ω ) N − k (cid:17) N − k dt (3.9)= s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − N − k Z ∞ N − k X v =0 (cid:18) N − kv (cid:19)(cid:16) e − µN t ω (cid:17) v (cid:0) e − λ e t (1 − ω ) (cid:1) N − k − v dt = s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − N − k N − k X v =0 (cid:18) N − kv (cid:19) ω v (1 − ω ) N − k − v · Z ∞ (cid:16) e − µN t (cid:17) v (cid:0) e − λ e t (cid:1) N − k − v dt = s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − N − k N − k X v =0 (cid:18) N − kv (cid:19) ω v (1 − ω ) N − k − v λ e ( N − k − v ) + µN v , (3.10)where (3.8) follows from the derivations in (3.6) and (3.9) follows from Lemma3.2, where we denoted ω = 1 − ν (0) µNµN − λ e .3.3 Asymptotic Analysis of Retrieval Time of EstimateIn this section we determine E [ W s ] for asymptotically large rate of energyharvesting, battery capacity and size of the sensor network. First, we introducethe following lemma. Lemma 3.6
For any ≤ k ≤ s , k ∈ N , (cid:18) Nk (cid:19) k X v =0 (cid:18) kv (cid:19) ( − k − v N − kN − v = 1 . (3.11) Proof
This proof follows from induction on k . It is easy to see that (3.11) holds for k = 0. We assume that (3.11) holdsfor some k >
0. We next show that (3.11) holds for k + 1. (cid:18) Nk + 1 (cid:19) k +1 X v =0 (cid:18) k + 1 v (cid:19) ( − k +1 − v N − ( k + 1) N − v = (cid:18) Nk + 1 (cid:19) k +1 X v =0 (cid:18) k + 1 v (cid:19) ( − k +1 − v N − v + v − ( k + 1) N − v = (cid:18) Nk + 1 (cid:19) k +1 X v =0 (cid:18) k + 1 v (cid:19) ( − k +1 − v (1) v + (cid:18) Nk + 1 (cid:19) k +1 X v =0 (cid:18) k + 1 v (cid:19) ( − k +1 − v v − ( k + 1) N − v = 0 + (cid:18) Nk + 1 (cid:19) k +1 X v =0 (cid:18) k + 1 v (cid:19) ( − k +1 − v − ( k + 1) − vN − v = (cid:18) Nk + 1 (cid:19) k X v =0 (cid:18) k + 1 v (cid:19) ( − k − v ( k + 1) − vN − v = (cid:18) Nk + 1 (cid:19) ( k + 1) k X v =0 k ! v !( k + 1 − v − − k − v N − v = (cid:18) Nk (cid:19) k X v =0 (cid:18) kv (cid:19) ( − k − v N − kN − v = 1 , where the last equality follows from the induction hypothesis. Theorem 3.7
For ≤ N < ∞ and < B < ∞ , lim λ e →∞ E [ W s ] = s − X j =0 µ (1 − j/N ) . Proof
Taking λ e → ∞ in Theorem 3.5, we have thatlim λ e →∞ E [ W s ] = s − X j =0 (cid:18) Nj (cid:19) j X v =0 (cid:18) jv (cid:19) ( − j − v µ (1 − v/N ) . The result now follows from Lemma 3.6.We next consider the situations when the capacity of the sensors to storeenergy in the battery is unlimited, i.e. B → ∞ .For λ e < µ/N and B → ∞ , the battery of a sensor is most of the timeempty as the rate at which this sensor receives energy is lower than the rate atwhich this sensor broadcasts, and thus, consumes energy. As a consequence, inthis case, the waiting time for a client to retrieve s measurements from distinctsensors largely depends on λ e , which supports the broadcasting process. ata retrieval time for energy harvesting wireless sensor networks 11 For λ e > µ/N and B → ∞ , a sensor has most of the time energy forbroadcasting since the rate at which it harvests new energy is higher than therate at which it broadcasts. In this case, the waiting time for a client to retrieve s measurements from distinct sensors depends on the broadcasting rate µ/N . Theorem 3.8
For ≤ N < ∞ and < λ e < ∞ , lim B →∞ E [ W s ] = s − X j =0 λ e ( N − j ) , λ e < µNs − X j =0 µN ( N − j ) , λ e ≥ µN . Proof
We first consider the case λ e < µ/N . Then, from Lemma 3.1, lim B →∞ ν (0) =1 − λ e µ/N . Using this result in Theorem 3.5 we have thatlim B →∞ E [ W s ] = s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − j − k λ e ( N − k ) . The result follows from Lemma 3.6.We next consider the case λ e ≥ µ/N . Then ν (0) →
0. Using this result inTheorem 3.5 we have thatlim B →∞ E [ W s ] = s − X j =0 (cid:18) Nj (cid:19) j X k =0 (cid:18) jk (cid:19) ( − j − k µ ( N − k ) /N . Again, the result follows from Lemma 3.6.
Theorem 3.9
For < B < ∞ and < λ e < ∞ , lim N →∞ E [ W s ] = sµ . Proof
Recall that a measurement is broadcasted at an exponential rate µ from the sensor network, for any N >
0. Moreover, given that N → ∞ , theprobability that there is always at least one sensor with energy which cantransmit, tends to 1. Also, the probability that any s consecutive broadcastsare from distinct sensors, tends to 1 as N → ∞ . From the above argumentsit follows that, for each of the s measurements, a client waits, in expectation,1 /µ . The result follows. In this section we analyze numerically the expected waiting time for a client toretrieve a reliable estimate of an attribute under various assumptions concern-ing the size of the wireless sensor network, the maximum battery capacity ofthe sensors and the rate at which sensors harvest energy from the environment.
Figures 4.1 and 4.2 show E [ W s ] under various N , the size of the wirelesssensor network. As expected, as N increases, E [ W s ] decreases. The reason isthat, as N increases, the probability that at least one sensor has battery tobroadcast an innovative measurements increases. Thus, it is expected thatclients wait less to retrieve s measurements to compute a reliable estimate ofan attribute. Moreover, Figures 4.1 and 4.2 show that, for a fixed λ e , if B isincreased, then E [ W s ] decreases. This is because as B increases, more energycan be collected in the sensor batteries, which enables broadcasts.Figure 4.1 considers the case when λ e ≥ µ/N , whereas Figure 4.2 considersthe case when λ e < µ/N . When λ e ≥ µ/N , it is expected that most of thetime the batteries of the sensors have energy. If λ e < µ/N , the batteries areexpected to be empty most of the time. This explains the fact that E [ W s ] takeslower values in Figure 4.1 than in Figure 4.2. E [ W s ] B=8B=4B=2B=1
Fig. 4.1: λ e = 0 . µ = 0 . , s = 2. E [ W s ] B=8B=4B=2B=1
Fig. 4.2: λ e = 0 . µ = 0 . , s = 2.Fig. 4.3: E [ W s ] under various N , the size of the sensor network.Figure 4.4 shows E [ W s ] under various B , the battery capacity of a sensor,and for various λ e , the rate at which sensors harvest energy from the environ-ment. As expected, for a fixed B , E [ W s ] decreases as λ e increases. This is thecase because the battery of the sensors are more frequently replenished and,thus, the sensors have energy to broadcasts their measurements. We note thatfor λ e ∈ { . , . } , λ e > µ/N , while for λ e ∈ { . , . } , λ e < µ/N . Figure4.4 also shows that, for a fixed λ e , if B increases, then E [ W s ] decreases. Thisdecrease becomes less significant for large values of B . This can be explainedas follows. In the case that λ e ≥ µ/N , even though sensors are able to storelarge amounts of energy, i.e. B is large, the rate at which the sensors broadcastis low and thus, E [ W s ] mostly depends on the broadcasting rate, rather than B . In the case that λ e < µ/N , even though B is large, the amount of energyin the batteries is expected to be low most of the times. Thus, in this case, thefact that B is very large does not result in a significantly decrease in E [ W s ]. ata retrieval time for energy harvesting wireless sensor networks 13 E [ W s ] λ e =0.2 λ e =0.1 λ e =0.03 λ e =0.02 Fig. 4.4: E [ W s ] under various B , the maximum battery capacity of a sensor, N = 10, µ = 0 . , s = 2. In this paper, we considered the problem of retrieving a reliable estimate ofan attribute from a wireless sensor network that harvests energy from the en-vironment. We assumed that energy is available for harvesting at each sensoraccording to a Poisson process. Moreover, the sensors store the harvested en-ergy in a limited-capacity battery. Provided there is sufficient energy stored,the sensors broadcast their measurements, independently, at random.We showed that the time until an arbitrary sensor broadcasts has a phase-type distribution. Based on this, we determined the probability distribution ofthe time to retrieve from the sensor network a reliable estimate of an attribute.We also provided a closed-form expression for the expected time to retrieve thisestimate. In addition, we determined the retrieval time of a reliable estimatewhen the energy available for harvesting, the storage capacity of the sensorbattery or the size of the sensor networks is asymptotically large.Lastly, we analyzed numerically the retrieval time of a reliable estimateunder various assumptions concerning the size of the wireless sensor network,the maximum capacity of the batteries of sensors, as well as the rate at whichsensors harvest energy from the environment. The numerical results show thatdeploying sensors with very large batteries does not result in a significantdecrease in the retrieval time of a reliable estimate. On the other hand, thenumerical results show that the rate at which sensors harvest energy and therate at which they broadcast significantly influences the retrieval time of areliable estimate.
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