IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1

Optimal Scheduling for Quality of Monitoring inWireless Rechargeable Sensor Networks

Peng Cheng, Member, IEEE, Shibo He, Member, IEEE, Fachang Jiang, Yu Gu, Member, IEEE,and Jiming Chen, Senior Member, IEEE

Abstract—Wireless Rechargeable Sensor Network (WRSN) isan emerging technology to address the energy constraint in sensornetworks. The protocol design in WRSN is extremely challengingdue to the complicated interactions between rechargeable sensornodes and readers, capable of mobility and functioning as energydistributors and data collectors. In this paper, we for the firsttime investigate the optimal scheduling problem in WRSN forstochastic event capture, i.e., how to jointly mobilize the readersfor energy distribution and schedule sensor nodes for optimalquality of monitoring (QoM). We analyze the QoM for threeapplication scenarios: i) the reader travels at a fixed speed torecharge sensor nodes and sensor nodes consume the collectedenergy in an aggressive way, ii) the reader stops to rechargesensor nodes for a predefined time during its periodic travelingand sensor nodes deplete energy aggressively, iii) the reader stopsto recharge sensor nodes but sensor nodes can adopt optimalduty cycle scheduling for maximal QoM. We provide analyticalresults for achieving the optimal QoM under arbitrary parametersettings. Extensive simulation results are offered to demonstratethe correctness and effectiveness of our results.

Index Terms—Optimal scheduling; wireless rechargeable sen-sor networks; stochastic event capture;

I. INTRODUCTION

LAST decade has witnessed the widespread adoption ofWireless Sensor Networks (WSNs) in a variety of fields,

including environment monitoring, ecosystem surveillance,physical hazards prevention, and daily activity recognition[2]–[4]. Albeit their great success, most of existing WSNsare confined to short-term applications related to scientificresearches or industrial monitoring, as sensor nodes are typi-cally battery powered and replacement of battery or sensornodes themselves will incur huge management cost. Theenergy constraint has become the major challenge that holdsback the further popularity and maturity of WSNs, whilethe rapid advances in Micro Electronic Mechanical System(MEMS) and communication technologies have been leadingto reduction in manufacture cost during the last several years.

Energy harvesting technologies have been adopted to ad-dress the energy constraint in the WSNs. An energy-harvesting

Manuscript submitted November 1, 2012; revised January 19 and March26, 2013; accepted April 2, 2013. The associate editor coordinating the reviewof this paper and approving it for publication was D. Tarchi.

Part of this paper has been presented at the 8th IEEE InternationalConference on Mobile Ad-hoc and Sensor Systems (MASS) [1].

P. Cheng, S. He, F. Jiang, and J. Chen (corresponding author) are withthe State Key Lab. of Industrial Control Technology, Zhejiang University,China, 310027 (e-mail: {pcheng, jmchen}@iipc.zju.edu.cn; {shibohe.cn,fachang.jiang}@gmail.com).

J. Gu is with Singapore University of Technology and Design, Singapore.Digital Object Identifier 10.1109/TWC.2013.13.121691

Fig. 1. WISP 4.1DL platform.

WSN can harvest energy from its surroundings and thushave a great potential to yield a perpetual network operationtime. Roughly speaking, energy harvesting technologies can becategorized into two folds: i) energy scavenging, and ii) energydistribution. Well known examples of energy scavenging in-clude rechargeable sensor nodes that can be powered by solar,wind, vibration, and biochemical processes [5]–[8]. Theseenergy scavenging sensor nodes are generally expensive andof large size since additional hardware devices are integratedfor energy collection, which limits their applications of largescale. Another drawback of energy scavenging sensor nodesis the energy reliance on the surroundings. For example,solar powered sensor nodes will fail to collect energy atnight or when sunshine is not available. The most popularapproach of energy distribution is harvesting energy fromRF signal. The great success in prevailing applications ofRadio Frequency Identification (RFID) has demonstrated thestupendous advantages and the bright future of applying RFsignal for energy distribution.

Recently, harvesting energy from RF signal has been ap-plied beyond identification to pervasive sensing and com-puting. Intel Research Center and University of Washingtonhave collaborated to develop a Wireless Identification andSensing Platform (WISP), where energy from RF signal canbe harvested, stored, and used for powering the operationof MCU (Micro Control Unit), directing the computing andsensing. Based on their model, we fabricate a prototype ofWISP 4.1DL, which is shown in Fig. 1. With the excellentthin shape and the advantage of getting rid of battery, WISPcan be widely applied in fields ranging from individual activityrecognition to large-scale urban sensing [9]–[11], opening up anew research area referred to as Wireless Rechargeable SensorNetwork (WRSN).

In a WRSN, there are mainly two components: i) a bunchof WISP tags for sensing and computing, and ii) severalreaders for data collection and energy distribution. WISPtags can be static (e.g., embedded in the infrastructures) ormobile (e.g., carried by humans or attached to vehicles), and

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2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

readers travel around to meet WISP tags in a deterministic oropportunistic way. To guarantee a desired performance, WISPtags have to determine a good on-off sampling scheduling,and readers need path planning to wirelessly recharge WISPtags and retrieve data. Solving the on-off scheduling andpath planning problems independently is very challenging.Moreover, these problems are usually coupled as the WISP’ssampling scheduling and data forwarding delay depend on thefrequency of readers’ arrival.

In this paper, we make the first attempt to address thescheduling problem in a WRSN consisting of a collection ofWISP tags embedded in the infrastructures (e.g., bridges) andone reader deterministically recharging WISP tags. Stochasticevents are assumed to occur at these infrastructures, and weaim at jointly scheduling the WISP tags and the reader tomaximize the quality of monitoring (QoM), i.e., the ratio ofcaptured events to all occurring events. We consider threeapplication scenarios: i) the reader travels at a fixed speedto recharge WISP tags and WISP tags deplete the collectedenergy without any duty cycle scheduling, ii) the readerstops to recharge WISP tags for a predefined time duringits periodic traveling and WISP tags still use energy withoutany scheduling, iii) the reader stops to recharge WISP tagsas scenario (ii) but WISP tags can adopt optimal duty cyclescheduling to maximize QoM. Note that such three scenarioscover different system complexities in practice. For example,the first scenario does not require exact localization of WISPtags or complicated motion control of reader, while boththe first and second scenarios do not require the schedulingability of WISP tags. We give analytical results of QoMunder arbitrary parameter settings for these three scenarios.Our analysis shows that the scheduling of the reader and theWISP tags has a significant impact on the QoM. We also showhow to decide the optimal scheduling by jointly addressingthe scheduling of both the reader and WISP tags. Extensivesimulation results are also provided to validate our results.

The remainder of this paper is organized as follows. Wediscuss the existing works on stochastic events capture inWSNs and WISP tags in WRSN in Section II. We givesome preliminary knowledge about the wireless rechargingand formulate our problem in the Section III. The QoMfor different application scenarios and optimal schedulingschemes are investigated in the Section IV, followed by thevalidation via extensive simulation results in the Section V.We conclude the paper in Section VI.

II. RELATED WORK

Recently, wireless sensor networks are widely applied tomonitoring applications [12]–[17], where their primary con-cerns are to capture interesting events occurring in the regionof interest. As most of time there is no prior knowledgeabout when the events occur, it is typical to model eventsas stochastic processes. Poisson process is one of the mostpopular ones that are adopted in the existing work to charac-terize the dynamics of event occurrences [12], [15]. In general,existing works on stochastic event capture can be classifiedinto two folds, identified by whether the deployed sensor nodesare static or mobile. In a static sensor network, an energy-efficient scheduling is proposed in [16], where He et al. for

the first time investigate how the periodic sensor schedulingimproves the quality of capture (QoC) by exploiting the eventdynamics. It is shown that an asynchronous network with thecoordinated sleep protocol (CSP) is the best choice in termsof QoC and energy efficiency. In some applications wherethere are not sufficient sensor nodes, it is an alternative to usemobile sensor nodes, referred to as mobile sensor networks.The mobile sensor nodes patrol in the region of interest, gatherinformation at the points of interest (PoIs), and report it to adata collection center (sink node). Bisnik et al. investigatehow the parameters, such as sensor speed, event dynamicsand the number of mobile sensors, affect the quality of eventcapture [13]. They also study how to coordinate the movementof multiple mobile sensor nodes to satisfy the applicationrequirements as well as to achieve energy efficiency. Yau et al.[15] study quality of monitoring of stochastic event capture byusing periodic schedule (q, p), where mobile sensors monitorPoIs for q time of every p time. They also consider the hetero-geneity of PoIs and adopt utility functions, e.g., Step function,Linear function, to describe the event capture process. He etal. further consider the tradeoff between energy efficiency andQoC in a mobile sensor network [18]. They propose a metric:expected information captured per unit of energy consumption(IPE), to evaluate the overall performance of a mobile sensornetwork, and systematically analyze the optimal schedulingunder different scenarios.

Energy harvesting technologies have emerged in the lastseveral years to address the challenge of energy constraint ofbattery-powered sensor networks. One approach to design abattery-free system is to scavenge energy from surroundingenergy sources. Known examples of energy sources includesolar [7], vibration [8], temperature variation [19], wind [20],etc. Another approach of battery-free system design is todistribute energy from energy-rich sources to energy-hungrynodes, which mainly involves two methods: i) through stronglycoupled magnetic resonances [21], and ii) through radio fre-quency (RF) signals [22]. In this paper, we introduce wirelessrechargeable sensors for perpetual sensing and computing, byharvesting energy through RF signal. A rechargeable sensor,also referred to as a WISP tag, is a battery-free platformwith capability of sensing and computing [23]. Buettner etal. propose a list of potential applications for WISP tags in[24], such as monitoring blood temperature during storage ortransportation, recognizing the daily activities of the elderly,etc. Yeager et al. add capacitive sensor into the WISP tagsfor monitoring the fill percentage of milk carton [22]. In [25],Holleman et al. develop a Neural WISP, which can monitorthe neural signal and periodically transmit the spike density tothe reader. How to deploy readers to provision enough energyfor every WISP tag in the region of interest is investigatedrecently in [26].

While considerable research efforts have been invested intoenergy-efficient scheduling for event capture in the traditionalbattery-powered sensor networks, few works focus on energy-efficient protocol in the wireless rechargeable sensor networks.We for the first time investigate the optimal scheduling prob-lem in WRSN for stochastic event capture, i.e., how to jointlymobilize the reader for energy distribution and to schedulesensor node for efficient event capture. We extensively study

CHENG et al.: OPTIMAL SCHEDULING FOR QUALITY OF MONITORING IN WIRELESS RECHARGEABLE SENSOR NETWORKS 3

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.5

1

1.5

2

2.5x 10

−3

Distance (m)

Har

vest

ed p

ower

(W

)

FittingExperimental

Fig. 2. Experimental and theoretical receiving power of WISP tag proposedin [26].

the problem and analyze the quality of monitoring for differentapplication scenarios.

III. PRELIMINARIES AND PROBLEM STATEMENT

Wireless Identification and Sensing Platform (WISP) ismainly composed of four components: i) energy harvestingcomponent, ii) micro-controller component, iii) sensors and iv)communication component [22], [23], [27]. MSP430 micro-controller component with several low-power modes (LPM)is in charge of all the activities in the WISP tag. The energyharvesting component can harvest the RF radio energy andstore it in the energy storage capacitor (ESCap). A WISPtag can be integrated with certain lower-power sensors, e.g.,accelerometer, light sensor and temperature sensor, and thecommunication component can upload the ID informationand sensory data to an adjacent reader through backscattermodulation.

A. Wireless Recharging Model

Before formulating our problem, we first present a wire-less recharging model based on experimental data. In Friistransmission equation, the power Prx at the receiver is Prx =GtxGrx(λ/4dπ)

2Ptx, where Ptx is the transmission power oftransmitter, d is the distance between the transmitter and thereceiver, Gtx is the antenna gain of transmitter, Grx is theantenna gain of receiver, and λ is the wavelength of the RFwave. In this paper, we adopt the following empirical wirelessrecharging model, which was proposed and verified in [26],

Prx =GtxGrxη

Lp(

λ

4π(d+ β))2Ptx, (1)

where η is referred to as rectifier efficiency, Lp is the polariza-tion loss and β is an adjustable parameter to adapt our equationto room environment. Through extensive experimental tests,[26] has obtained η = 0.125 and β = 0.2316 by employingthe least square technique to fit the experimental data.

Note that when WISP tag is too far away from the reader,the wireless charging power would be too low to be harvested.Therefore, we assume there exists a threshold of distance

Fig. 3. N WISPs placed at PoIs along a curve Ω.

denoted by r, beyond which the WISP tag cannot be wirelesslycharged. Then the empirical recharging model is expressed as

Prx(d) =

⎧⎨⎩

τ(d+β)2 , 0 ≤ d < r

0, otherwise(2)

where τ = GtxGrxηLp

( λ4π )

2Ptx is a constant, d (d ≥0) is thedistance between the reader and WISP tag.

B. Problem Statement

We consider N WISP tags, indexed from 1 to N , placed atN points of interest (PoIs) in a 2D plane as shown in Fig. 3. Areader that is capable of mobility can travel to visit the WISPtags at PoIs one by one along a curve (Ω) with a cycle time,where Ω denotes the traveling path along which the readerwill visit all tags subsequently. One typical example may beone shortest path connected all WISP tags in space. Since thereader keeps moving along Ω periodically, we can focus on thescheduling design within one cycle. Note that the results in thispaper does not rely on any specific shape of Ω. The distancebetween tag i and tag i+1 along curve Ω is li, i ∈ (1, N − 1),and lN denotes the distance between WISP tag N and 1. Notethat in order to simplify the presentation, here we assume thedistance between each two PoIs is not less than 2r, and foreach PoI we employ only one WISP tag. Thus the reader canrecharge only one tag at a time. Since the common chargingthreshold r is around 20m [26], such a setting is feasiblefor many applications where PoIs are not too intensive. Notethat different from traditional RFID tags, each WISP tag canbe equipped with various sensing components for differentsystem requirements [27]. In addition, for many applications, itis beneficial to reduce the overlapping sensing area of differenttags for reducing the system costs. Extensive works have beendevoted to developing techniques for placing sensor nodes sothat the system hardware costs can be reduced with guaranteedsensing performance [28].

Once collecting enough energy, each WISP tag turns intoactive state and gathers the physical information at each PoI.We assume stochastic events occur at each PoI sequently.When an event occurs, it stays for a while (called event stayingtime, denoted by X), and then disappears. There is a timeduration (called event absent time, denoted by Y ) before newevent occurs. We assume X and Y at each PoI i follow theExponential distributions with means 1

αiand 1

μi, respectively.

For simplicity, we further assume that αi = α and μi = μ,for i = 1, ..., N .

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

Suppose the reader patrols along curve Ω at a fixed speeddenoted by v (v > 0). We refer to the duration for thereader to complete traveling the curve Ω as reader period.Note that r denotes the maximal distance for recharging aWISP tag. Once the reader is within the range r, the WISPtag will get recharged. All the harvested energy is storedin capacitor ESCap. We assume the capability of ESCap islarge enough and the leakage current is negligible. WISPtags cannot work unless the voltage of ESCap is higher thanV reg. For each WISP, there are two major states: activeand sleep with the power consumption of Pact and Pslp,respectively. Another major energy cost comes from transitionfrom different states, e.g., from sleep state to active state,and we call such energy consumption switch cost, denoted byEwak. To facilitate the analysis in the following, we denotethe time for transition by Twak, and the average power costby Pwak, thus Pwak = Ewak/Twak. We assume Pact, Pslp,Pwak, Twak and Ewak are constants as they all only dependon the hardware design.

We assume each WISP tag works on a periodic scheduling(q, p), where the WISP tag will be active for q time of everyp time. Assume during time T , there are mi stochastic eventsoccurring at PoI i, and WISP tag i captures ci events. Thequality of monitoring of the stochastic event is defined as

QoM = limT→∞

N∑i=1

ci

N∑i=1

mi

(3)

Then our problem can be formulated as how to schedule themovement of the reader and the operation of WISP tags tomaximize QoM, i.e.,

maxQoM

s.t. Ei ≤ Ei, i = 1, 2, · · · , N (4)

where Ei is the energy consumption of WISP tag i, whichdepends on the schedule (q, p), and Ei is the average collectedenergy of node i within one reader period.

In this paper, in order to simplify the statements, only oneWISP tag is placed at one PoI for monitoring the events.However, we can extend the obtained results to handle thecase where multiple WISP tags are deployed at each PoI.Specifically, since multiple tags can be wirelessly rechargedsimultaneously, we can first consider the multiple tags at thesame place as one big virtual tag whose charging power is thesame as the accumulative charging power of multiple tags.Then we apply the design results in later sections in orderto obtain the scheduling scheme for the big virtual tag. Afterthat, we may evenly divide the working period of virtual taginto several parts based on the number of tags, and let eachtag take charge of one part under a given order.

IV. QOM ANALYSIS

As each WISP tag is scheduled to be active periodically,there are two cases for an event to be captured: i) theevent happens in WISP’s active period [0, q], and is capturedinstantaneously, and ii) the event happens during the WISPtag’s inactive state [q, p] but lasts long enough till the next

TABLE INOTATIONS

Symbol Definition1α

expected event staying timeL length of the curveN sensor numbern sensor scheduling times in one cycler threshold of charging distance

v, t reader speed, and reader staying timePact, Pslp , Pwak activating, sleeping, and wakeup power of sensorTiaw , Tjaw , Tjpw cycle periods for three scenariosEiaw , Ejaw , Ejpw energy collected in one cycle period for three

scenariosT jpwslot one scheduling period for JPW

T iawact , T iaw

slp activating and sleeping time in one cycle period forIAW

T jawact , T jaw

slp activating and sleeping time in one cycle period forJAW

T �act, T �

slp activating and sleeping time in one schedulingperiod for JPW

active period [p, p+ q]. From [15], we get the probability ofcapturing event for a general (q, p) scheduling as follows.

QoM =

∫ q

0 Pr(X ≥ 0)dt+∫ p

q Pr(X + t ≥ p)dt∫ p

0 Pr(X ≥ 0)dt(5)

When the event staying time X follows the Exponentialdistribution with given parameter α, the QoM can be reducedto [15]

QoM =q

p+

1− e−α(p−q)

αp. (6)

The on-off scheduling (q, p) of each WISP tag depends onthe available harvested energy and the scheduling strategies,which is largely related to both the movement of the reader,the way of recharging, and scheduling of WISP tags. It is ofgreat interest to jointly investigate how reader plans its pathto provision sufficient energy and how the WISP tags shouldduty cycle themselves to maximize the QoM. There are mainlythree typical cases: i) Independent Aggressive Wake-up (IAW),which means the WISP tags collect energy independently anddrain out all the energy at one active state, ii) Joint AggressiveWake-up (JAW), i.e., the reader would stop at each PoI torecharge WISP tags for a predetermined time, then WISP tagswake up aggressively to capture events, iii) Joint PeriodicalWake-up (JPW), which means the reader would stop at eachPoI to recharge WISP tags, and WISP tags will rationallymanage the use of energy it collects, and duty cycle to captureevents. Note that for both JAW and JPW cases, Joint meansthe reader will jointly optimize both its patrolling speed andits stationary recharging time at each PoI. We will study theoptimal scheduling problem in the following for these threecases. Table I gives a summary of notations commonly usedin this paper.

A. QoM in Independent Aggressive Wake-up

In this application scenario, the reader can travel alongthe curve Ω periodically, and recharge each WISP tag whenthe relative distance is less than r. Denote the total energycollected by each WISP tag in each traveling period by Eiaw.

CHENG et al.: OPTIMAL SCHEDULING FOR QUALITY OF MONITORING IN WIRELESS RECHARGEABLE SENSOR NETWORKS 5

We can get the expression for Eiaw as follows,

Eiaw = 2

∫ rv

0

τ

(vt+ β)2dt =

2τr

vβ(r + β). (7)

Recall that the capability of capacitor ESCap is assumedto be large enough, and thus we do not consider the energyoverflow in the capacitor during recharging process. The WISPtag cannot work unless the voltage of capacitor ESCap is largerthan the voltage threshold V reg (in WISP tag, V reg is set tobe 1.8 V in default, however, we can also change such settingas [29]).

One scheme is that a WISP tag will turn into active stateas soon as voltage threshold is satisfied. However, in thiscase the WISP tag can only work in an instant time, and theswitches from active to sleep state or sleep to active stateare too frequent. To yield continuous operation of WISP, wedesign firmware to let WISP tag wake up to work only if therecharging process in one visit of the reader completes. Denotethe period for reader to travel all the PoIs for one round, i.e.,reader period, by Tiaw. It is easy to obtain that Tiaw = L/vwhere L is the length of the curve Ω. In each reader periodTiaw, WISP spends Twak time to wake up from sleep, andworks for T iaw

act time.Remark 1: For an extreme case, where the recharge rate is

high enough to support constant work, which means Eiaw ≥PactTiaw i.e., 2τr

Lβ(r+β) ≥ Pact. Since L ≥ 2Nr, we can obtainthat the reader would be able to support all nodes constantlyworking if N ≤ τ

Pactβ(r+β) . And it is interesting to find thatsuch a bound is irrelevant to the speed of reader. However, inmany practical settings, such a critical value would be quitesmall. The reason is that the average wireless charging poweris only comparable with the WISP activation power. Therefore,the reader can only support a small number of WISP tags forconstant working. For example, according to the experimentalmeasurements, the inequality above can hold for only onenode. Thus later in this subsection, we will focus on the multi-tag case where the nodes are not able to constantly work.

Since the nodes cannot work constantly, after the nodedepletes most of the energy, it goes to sleep for T iaw

slp .Therefore, we have

Twak + T iawact + T iaw

slp = Tiaw, (8)

Ewak + PactTiawact + PslpT

iawslp = Eiaw, (9)

where T iawact and T iaw

slp represent the active time and sleep timeof WISP in period Tiaw, respectively, and Ewak = PwakTwak

represents the energy for waking up from sleep. After simplecalculation, we can get

T iawact =

Eiaw − Pslp(Tiaw − Twak)− Ewak

Pact − Pslp,

T iawslp = Tiaw − Twak − T iaw

act .

According to the general formula of QoM in Eq. (6), we canget the QoM in IAW scenario as follows

QoM =T iawact

L/v+

1− e−α(L/v−T iawact )

αL/v. (10)

where T iawact is also a function of v,

T iawact =

1

Pact − Pslp

[(A− PslpL)

1

v−ΔE

]

with A = 2τrβ(r+β) and ΔE = Ewak − PslpTwak.

Since both T iawact and T iaw

slp are nonnegative, we must haveTwak ≤ Tiaw and Ewak ≤ Eiaw , therefore the speed v inIAW has to satisfy the following condition

min{PwakL/v,2τr

vβ(r + β)} ≥ Ewak. (11)

The underlying assumption in this paper is that the he WISPtag will wake up in every cycle. Note that such a constraint isnot necessary but only for simplifying the statements. If thereader speed is larger than the maximal allowable speed, thereader has to patrol several times for supporting one wakeup.Such situation can be transformed into the case in this paperby utilizing the relationship between the charged energy andthe traveling speed of reader.

Now we are ready to present the result about how the readerspeed affects the QoM for IAW policy.

Theorem 1: For Independent Aggressive Wake-up, themaximum value of QoM can only be achieved at the maximumallowable reader speed.

Proof: The detailed proof is given in Appendix.From Theorem 1, it can be observed that if the nodes wake

up in an independent aggressive way, it is always better forthe reader to move faster unless the constraint (11) is violated.In fact, such result also provides some insights about how theevents would be captured. Specifically, the expression of QoM(10) can be divided into two parts, where the first part equalsto

T iawact

L/v=

1

L(A− PslpL

Pact − Pslp− ΔE

Pact − Pslpv)

corresponding to the events captured when they just happen.Note that this part is a decreasing function of v due to theexistence of wakeup time Twak. Meanwhile, the second partof QoM equals to

1− e−α(L/v−T iawact )

αL/v=

1− e−α(

PactL−APact−Pslp

1v− ΔE

Pact−Pslp)

αL/v

which corresponds to the events which are captured when theylast until the nodes activate in the next cycle. This part is anincreasing function of v due to that the activation frequency isincreasing with the reader speed v. And the second part wouldplay an increasingly significant role when v grows.

B. QoM in Joint Aggressive Wake-up

In the JAW scenario, the reader would stop at each PoI torecharge WISP tag for a period of t before it moves to the nextPoI, and we call such additional charging process as stationaryrecharging. Note that based on the wireless charging model(2), the charging power during stationary recharging can bemaximized if the reader stays at d = 0. In order to shortenthe statements, we only consider the case when d = 0 in therest of this paper, which means that the reader will stay at eachWISP tag for stationary recharging. The results below wouldalso be able to extended to handle the case where 0 < d < r.

Besides, we denote the period for reader to patrol the curveΩ in JAW by Tjaw , and the amount of energy each WISP tagcollects in the period Tjaw by Ejaw . In JAW, WISP tag uses

6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

energy aggressively, and we denote its active time by T jawact ,

and the sleep time by T jawslp . Then Tjaw can be calculated by

Tjaw = L/v +Nt, where t is the stationary recharging timefor each WISP tag.

Different from the previous IAW case, the energy collectedby each WISP tag during Tjaw for JAW is composed of twoparts: i) energy harvested when the reader stops, and ii) energyharvested when the reader is traveling towards or away fromthe WISP tag. Thus we can calculate Ejaw , i.e., the totalenergy that each tag collects during every Tjaw as follows

Ejaw =τt

β2+

2τr

vβ(r + β). (12)

1) Constant Activation: We have shown that for IAW thereexists a critical sensor number above which it is impossibleto keep the tag constant working. But such a critical valueis usually too small to be applicable. Now we prove that byapplying JAW policies such a critical value can be enlarged.

Theorem 2: For any given reader patrolling speed v, allthe tags can constantly work if the following conditions aresatisfied simultaneously

1) The number of tags N satisfies N ≤ τβ2Pact

,

2) The stopping time t satisfies t ≥ (PactL−2Ar)β2

(τ−NPactβ2)v withA = 2τr

β(r+β) .Proof: The proof can be obtained from the fact that in

order to support constant working, we must have Ejaw ≥PactTjaw i.e., τt

β2 + 2τrvβ(r+β) ≥ Pact(L/v +Nt).

Note that the threshold of tag numbers for JAW can be en-larged r+β

β times than that for IAW. Meanwhile, the minimalrequired stopping time t is decreasing along with the speed v.However, it should be noted that such a critical tag number isstill quite small compared with the usual large number of tagsin different applications. For example, in our setting, the readerwill be able to support the constant working of 5 tags by JAW.Therefore, in the rest of this section, we would investigate thedesign for the non-constant activation case, i.e., N > τ

β2Pact.

2) Non-constant Activation: Denote T jawact as the active

time of WISP and T jawslp as its sleep time for JAW scenario.

We can establish the their relationship in a similar way as (8)and (9). The scheduling of each WISP tag can be expressed as(T jaw

act , Tjaw). Hence we can get the QoM in JAW scenarioas follows,

QoM =T jawact

L/v +Nt+

1− e−α(L/v+Nt−T jawact )

α(L/v +Nt), (13)

where T jawact is also a function of t and v, i.e.,

T jawact =

1

Pact − Pslp

[(τ

β2− PslpN)t+ (A− PslpL)

1

v

−Ewak + PslpTwak

](14)

Since T jawact and T jaw

slp in JAW are also nonnegative, we havethe following constraints for v and t

min{ τt

(d+ β)2+

2τr

vβ(r + β),(L+Nvt)Pwak

v} ≥ Ewak.

(15)Remark 2: From the expression of T jaw

act , it should benoticed that stationary charging would be likely to contribute

only when N < τβ2Pslp

, which means there exists an exactupper bound for the number of WISP tags above which thecontribution of reader stopping is always negative. Note thatthis value is usually considerably large as the sleeping powerPslp is very small in practice, e.g., around 1700 for oursystem parameters. Therefore, we would only focus on theperformance of JAW with τ

β2Pact< N < τ

β2Pslp.

It is natural to ask how to design the moving speed ofreader when the stationary charging time t is given, whichis answered by the following theorem.

Theorem 3: For Joint Aggressive Wake-up with τβ2Pact

<N < τ

β2Pslp, for any arbitrary t , the maximal value of QoM

can only be achieved at vmax.

Proof: The detailed proof is given in Appendix.

By Theorem 3, for an arbitrary t, it would be preferable toincrease the patrolling speed of reader to its maximum value.On the other hand, it is also of great interest to investigate howthe stationary charging time t would affect the monitoringperformance when the reader speed is given. In order tofacilitate the description, we define notions as follows

ΔE = Ewak − PslpTwak,

S =1

Pact − Pslp(PactN − τ

β2) > 0

U =PactL−A

Pact − Pslp

1

v+

ΔE

Pact − Pslp,

G(t) = e−α(St+U) +αNvU − αLS −Nv

αNvSt+ αLS +Nv.

Then we have the following theorem.

Theorem 4: For Joint Aggressive Wake-up, if v is fixed,and τ

β2Pact< N < τ

β2Pslp, then the maximal QoM can only

be achieved:

1) at t → ∞, if a) 1α ≤ ΔE

Pact−Pslp; or b) 1

α > ΔEPact−Pslp

with v ≤ 1(Pact−Pslp−αΔE)N (Lτ

β2 − 2Nτrβ(β+r)),

2) in the set {0, {t|G(t) = 0}}, where QoM owns thelargest value.

Proof: The detailed proof is given in the Appendix.

From Theorem 4, it is interesting that the optimal stayingtime depends on both the event dynamics α and the movingspeed. Specifically, there exists a critical value for α, abovewhich, the staying time should be chosen as large as possible.Additionally, if such a critical value is not satisfied, it wouldbe still better to enlarge the staying time when the speed ofreader is less than a constant value (determined by the numberof nodes as well as the charging and recharging parameters).Otherwise, the optimal staying time should be chosen from asmall set of points including t = 0.

Theorem 4 can also be explained from the expression ofQoM under JAW policy. By enlarging staying time t, it can beexpected that the ratio of activation time over total cycle periodis increasing to a limitation, which will definitely increase thefirst part of QoM. Meanwhile, the cycle period will increaselinearly with the staying time t which may reduce the secondpart of QoM. In fact, if the maximal QoM would be achieved

CHENG et al.: OPTIMAL SCHEDULING FOR QUALITY OF MONITORING IN WIRELESS RECHARGEABLE SENSOR NETWORKS 7

when t approaches ∞, we have

QoMu = limt→∞

[T jawact

L/v +Nt+

1− e−α(L/v+Nt−T jawact )

α(L/v +Nt))

]

=

τβ2 − PslpN

N(Pact − Pslp), (16)

where the second part would approaches zero quickly whent approaches ∞. Note that in this paper, t approaches ∞does not cover the case where the reader stays at one WISPtag leaving other tags not charged1. Instead, t approaches ∞means that the staying time becomes large enough that thestationary charging plays a major role in the expression ofQoM.

Thus, it becomes understandable that it would always bebetter to enlarge the staying time t if the second part ofQoM is considerably small compared with the first part asguaranteed by the conditions provided by Theorem 4. Notethat the capacitor of WISP in practice always owns a limitedenergy capacity. Therefore, if Theorem 4 proves that undera given setting the optimal reader staying time should be aslarge as possible, we can design such an optimal t according tothe actual capacitor size. On the other hand, when the stayingtime becomes larger, the effect of further enlarging t wouldbecome smaller and smaller and the QoM will approach thelimitation quickly, which will also be shown in the simulationpart. Therefore, we may be able to achieve a better trade-off by takeing account of both the theoretical results and thepractical costs.

C. QoM in Joint Periodic Wake-up

In the previous two subsections, we investigate how to planthe moving pattern of mobile reader so that the QoM at thesensor side can be maximized. On the other hand, since therandom event may stay for a random period of time once ithappened, the sensor itself may be able to achieve better QoMby properly scheduling its activation other than aggressivelyworking [16]. However, due to the additional energy andtime cost of waking up, it is of great interest to investigatewhether sensor scheduling is beneficial and how to designthe schedule. In this subsection, we will provide the conditionunder which the periodical activation would enhance the QoMperformance. We will also show how to design the optimalperiodical schedule, i.e., Joint Periodic Wake-up (JPW), whichmaximizes the QoM under practical constraints.

Denote the time for the reader to complete a round oftraveling Ω by Tjpw and the corresponding collected energy ofWISP tag by Ejpw . Then we have Tjpw = Tjaw = L/v+Nt.Besides, the energy that a WISP tag collects in period Tjpw

is the same as that in JAW too, i.e.,

Ejpw = Ejaw =τt

(d+ β)2+

2τr

vβ(r + β). (17)

Each WISP tag depletes an amount of energy Ewak toswitch from inactive state to active state which will cost Twak

1Note that once the reader stays at one WISP tag forever, the QoM forthis tag can indeed be recovered from (16) by letting N = 1 (notice that

τβ2Pact

< 1), however, it is usually not preferred as such a strategy maywaste the mobile charging ability of reader and leave N−1 PoIs unattended.

time. And we assume that each WISP tag will divide Tjpw inton individual slots for scheduling, i.e., T jpw

slot = Tjpw/n. Thenthe energy and time costs in the switch process also increaselinearly with the slot number n. And the total time spent inswitch process during one reader period is upper bounded byTjpw , and the total energy consumption is upper bounded byEjpw , i.e.,

nEwak ≤ τt

(d+ β)2+

2τr

vβ(r + β), nTwak ≤ L/v +Nt.

Thus the v, t and n in this scenario must satisfy the followinginequality.

min{ τt

n(d+ β)2+

2τr

nvβ(r + β),(L+Nvt)Pwak

nv} ≥ Ewak.

(18)We denote the active time in each slot by T �

act, the inactivetime by T �

slp. Since all the energy is assumed to be allocatedto each slot equally, we have

T �act =

Ejpw/n− Pslp(Tjpwslot − Twak)− Ewak

Pact − Pslp,

T �slp = T jpw

slot − Twak − T �act.

Since all the slots are equally divided, we can treat each ofthem as a scheduling period T jpw

slot in which there is a constantactive duration denoted by T �

act. Then the scheduling can beexpressed as (T �

act, Tjpwslot ). According to the Eq. (6), QoM

under JPW can be expressed as

QoM =nT �

act

(L/v +Nt)+

n− ne−α[(L/v+Nt)/n−T�act]

α(L/v +Nt), (19)

where

T �act =

1

nT jawact − n− 1

n

ΔE

Pact − Pslp> 0 (20)

with ΔE = Ewak−PslpTwak. Then the effect of slot numbern is provided by the following theorem.

Theorem 5: For Joint Periodic Wake-up with given feasiblev and t,

1) if 1α ≤ ΔE

Pact−Pslp, the maximal QoM can be achieved at

n = 1.2) if 1

α > ΔEPact−Pslp

, the maximal QoM can be achieved atthe maximum n satisfying (18) and (20).

Proof: The proof is given in the Appendix.From Theorem 5, it can be observed that when the expected

event staying time is less than a critical value, it would bebetter not to periodically schedule the activation due to thecost of additional energy for waking up. Otherwise it wouldbe better to schedule the activation as frequently as possible.

The results of Theorem 5 can also be explained from theexpression of QoM under JPW, we can also see that, thefirst part of QoM will decrease linearly due to the additionalwakeup energy ΔE. However, the second part is an increasingfunction for n, and it will play a more significant role if theexpected event staying time 1

α becomes large. Note that inpractical setting the sleeping power Pslp is much smaller thanPact while the additional energy for waking up, i.e., Ewak,is usually much larger than PslpTwak. Therefore the criticalvalue for expected event staying time ΔE

Pact−Pslp can be wellapproximated by Ewak

Pact.

8 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

With given reader staying time t and moving speed v,Theorem 5 tells how to design the duty cycle of WISP tag sothat the QoM can be maximized. On the other hand, if the slotnumber n has been given in advance, we can also determinethe v or t in a similar way as the JAW scenario. Since theperiodic schedule only add another constant term (when n isgiven), the effects of parameters v, t on QoM with given n aresimilar to those in IAW and JAW. The only difference wouldbe some changes of the values in Theorem 4 which can beobtained after direct algebraic manipulations.

D. Summary

In this section, we investigate how to jointly mobilizethe reader for distribution and to schedule sensor nodes forefficient event capture. We consider three different scenarios,i.e., Independent Aggressive Wake-up (IAW), Joint AggressiveWake-up (JAW), and Joint Periodic Wake-up (JPW), whichfit for different system complexities. For IAW, we show thatthere exists a maximal sensor number under which all thesensor nodes can constantly work. If the sensor number islarger than the critical value, we prove that it is always betterto increase the reader speed. For JAW, we also show that theexistence of the maximal sensor number by which all sensornodes can constantly work. For the cases when the sensornumber is larger than the critical value, we prove that for anygiven reader staying time it is always better to increase thereader speed. Moreover, with any given reader speed, we showhow to design the optimal reader staying time. Specifically, weshow the conditions under which it is always better to prolongthe reader staying time. For JPW, which mainly focuses onscheduling the sensor nodes, we prove that if the expectedevent staying time is larger than a critical value, it is betternot to schedule the sensor. Otherwise, it is always better toschedule the sensor as frequently as possible.

When multiple readers are employed, one direct solutionis to treat all the readers as one big reader and optimize thereader patrolling speed, the reader staying time, and the WISPtag scheduling scheme. After that, each individual reader canwork independently based on the optimization results. Notethat it would also be interesting to investigate how to divide theWISP tags into sub-regions so that different individual subsetof readers will take charge of each sub-region, however, theproblem is beyond the scope of this paper and will be left asour future work.

V. EVALUATION

In this section, extensive simulations are conducted toevaluate the analytical results of QoM for different applicationscenarios. According to the hardware settings, we can get thefollowing system parameters: Pact = 1.42 × 10−3w, Pslp =4.5× 10−6w, Pwak = 1.17× 10−3w, Ewak = 2.22× 10−5J ,Twak = 1.9 × 10−2s, β = 0.2316, τ = 4.23 × 10−4.Meanwhile, if there is no otherwise statement, it will beassumed that: L = 100m, N = 10, r = 2m, α = 0.5,μ = 0.5. The distance between PoI i and i + 1 is randomlyselected as long as the distance is larger than 2r. WISP tagsare placed at each PoI to collect information of interestingevents. As defined, we measure the QoM as the ratio of the

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

v−−m/s (IAW)

QoM

1/α=10 Numerical

1/α=10 Simulation

1/α=2 Numerical

1/α=2 Simulation

1/α=1 Numerical

1/α=1 Simulation

1/α=0.2 Numerical

1/α=0.2 Simulation

Fig. 4. QoM varies over v at different α.

number of captured events to that of total occurring events.We present both the Matlab numerical results (solid lines withdiscrete markers) and simulation results (dotted lines withdiscrete markers) to validate our analysis of QoM. For theease of clarity, we report the simulation results for averageQoM over 2000 different runs.

A. QoM of IAW

We vary the expected event staying time 1/α to be 0.2, 1, 2and 10s, and plot the QoM for different reader speeds v, whichis given in Fig. 4. It can be observed that the QoM increaseswith the reader speed v, which is consistent with Theorem 1.Moreover, when the expected event staying time gets longer,the QoM also increases quickly. The reason is that a fasterreader speed will increase the switching frequency betweenactivate state and inactivate state so that the stayed eventswill be more likely to be captured. Meanwhile, longer eventstaying time will also enhance the probability that the stayedevent will be captured.

In order to show the effects of L, we depict the QoM againstthe reader speed v for different values of L in Fig. 5. It canbe observed that the QoM will increase along with the readerspeed v for each L which is consistent with the result ofTheorem 1. On the other hand, for given v, the QoM willis degraded by larger L. The main reason is that the chargedenergy keeps the same while the cycling time Tiaw increaseswhich reduces the switching frequency of activation state andinactive state.

B. QoM of JAW

For JAW scenario, assume d = 0m during the stationaryrecharging in this JAW simulation, i.e., the reader stops at eachPoI to charge WISP tag. Under the default parameters, we firstshow how the QoM changes with varying reader speed v underdifferent given staying time t. Specifically, we let the stayingtime t equals 0, 1, 10, 60, and plot the corresponding QoM forvarying reader speed v, which is provided in Fig 6. Note thatt = 0 represents the IAW policy. It can be observed that for

CHENG et al.: OPTIMAL SCHEDULING FOR QUALITY OF MONITORING IN WIRELESS RECHARGEABLE SENSOR NETWORKS 9

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

v−−m/s (IAW)

QoM

L=50m Numerical

L=50m Simulation

L=100m Numerical

L=100m Simulation

L=200m Numerical

L=200m Simulation

L=400m Numerical

L=400m Simulation

Fig. 5. QoM varies over v at different L. (α = 0.5).

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

v−−m/s (JAW)

QoM

t=60s Numerical

t=60s Simulation

t=10s Numerical

t=10s Simulation

t=1s Numerical

t=1s Simulation

t=0s (IAW) Numerical

t=0s (IAW) Simulation

Fig. 6. QoM varies over v at different t. (α = 0.5).

different staying time t, the QoM would increase along withthe reader speed, which indicates the results of Theorem 3.Meanwhile, when we compare the QoM of different stayingtime t, it can be seen that under the default settings, largerstaying time t do help to improve the QoM which is alsoproved by Theorem 4. Moreover, the QoM approaches to alimitation (a capture probability around 0.55) when t getslarge. Such a value also meets our analytical performancelimitation of JAW given by (16).

However, as shown in Theorem 4, it is not always beneficialto prolong the stationary time t as large as possible. Instead,the optimal staying time t will depend on different parameters,e.g., expected event staying time, reader speed, and etc.Specifically, consider the case when the expected event stayingtime 1

α = 5. We let the staying time t equals 0, 0.5, 10, 60,and plot the corresponding QoM for varying reader speed v,which is depicted in Fig. 7. It can be observed that for differentstaying time t, QoM increases along with the reader speed,which validates the results of Theorem 3. However, when the

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

v−−m/s (JAW)

QoM

t=60s Numeical

t=60s Simulation

t=10s Numeical

t=10s Simulation

t=0.5s Numeical

t=0.5s Simulation

t=0s (IAW) Numeical

t=0s (IAW) Simulation

Fig. 7. QoM varies over v at different t. (α = 0.2).

0 1 2 3 4 5 6 7 8 9 10

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

t−−s (JAW)

QoM

v=7 Numerical

v=7 Simulation

v=20 Numerical

v=20 Simulation

Fig. 8. QoM varies over t at different v. (α = 0.2).

reader speed becomes larger, it may not be always better toenlarge the staying time t. For example, when the reader speedv = 20, it can be observed that t = 0.5s achieves better QoMthan the other settings. In order to facilitate the understanding,for two different reader speeds, i.e., v = 7 and v = 20,we depict the QoM for varying stationary time t, which isgiven in Fig. 8. It can be seen that when the moving speed iscomparatively small, i.e., v = 7, it would be better to enlargethe stationary staying time for better QoM , and the QoMwould monotonically increase to the performance limitation ofJAW. However, when the moving speed is large, i.e., v = 20,the QoM will first increase, then decrease monotonically tothe performance limitation of JAW, which means the optimalstaying time is around t = 0.5s as proved by Theorem 4.

C. QoM of JPW

For JPW, the WISP tags will duty cycle its activation forimproving the QoM of either IAW or JAW. rationally tomaximize the QoM performance. Note that in the default

10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

0 20 40 60 80 100 120 140 160 180 2000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n (JPW)

QoM

v=1m/s Numerical

v=1m/s Simulation

v=5m/s Numerical

v=5m/s Simulation

v=15m/s Numerical

v=15m/s Simulation

Fig. 9. QoM varies over n at different v. (α = 0.5).

setting, the critical value of expected event staying time inTheorem 5, i.e., ΔE

Pact−Pslpis around 0.015. Thus for the

default 1α = 2, it is always better to increase the switching

number n for better QoM. Let the staying time t = 5s, wecompare the QoM for different reader speeds under varyingswitching times n in Fig. 9. It can be seen that for each givenv, as predicted by Theorem 5, the increase of n improves theQoM dramatically. For example, when v is larger than 5m/s,QoM becomes quite close to 1 when n is larger than 50.Meanwhile, we can also observe that for given n, the QoM stillkeeps increasing along with the reader speed which supportsthe analysis in Section IV-C. In order to show the effectivenessof rationally scheduling the available energy, we also compareJPW with JAW and IAW under the same settings. Specifically,we let the reader staying time t equal to 5s for both JAW andJPW, and choose the switching times n = 100 for JPW. Thenwe depict the QoM for three scenarios under varying readerspeed v in Fig. 10. It can be observed that the gain of QoMfor JPW is around averagely 100% over that of JAW underthe same reader staying time. For example, when the readerspeed v = 1, the QoM of JAW is around 0.2 while the QoMof JPW is close to 0.8, which demonstrates the effectivenessof periodic scheduling at tag side. Meanwhile, it can also beseen that both JPW and JAW provides superior performanceover IAW.

VI. CONCLUSION

In this paper, we for the first time studied the QoMproblem in wireless rechargeable sensor networks. We dividedthe problem into three application scenarios: i) IndependentAggressive Wake-up, ii) Joint Aggressive Wake-up, and iii)Joint Periodic Wake-up, and derived the corresponding explicitperformance expressions. We also analyzed extensively theimpacts of varying parameters on the QoM. Our resultsprovide fundamental insights into the scheduling problem inthe wireless rechargeable sensor networks. Numerical resultsare offered to demonstrate the effectiveness and advantages ofour solutions.

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

v−−m/s

QoM

t=5s,n=100 (JPW) Numerical

t=5s,n=100 (JPW) Simulation

t=5s,n=1 (JAW) Numerical

t=5s,n=1 (JAW) Simulation

t=0s,n=1 (IAW) Numerical

t=0s,n=1 (IAW) Simulation

Fig. 10. QoM varies over v for three scenarios. (α = 0.5).

ACKNOWLEDGMENT

This work was supported in part by the NSFC underGrants 61061130563, 61004060, 61222305, the 863 High-Tech Project under Grant 2011AA040101-1, the SRFDPunder Grants 20100101110066, 20120101110139, NCET-11-0445, and the Fundamental Research Funds for the CentralUniversities under Grants 2013QNA5013 and 2013FZA5007.The work of Dr. Gu was also supported by Grants SUTD-ZJU/RES/03/2011 and NRF2012EWT-EIRP002-045. The au-thors would like to thank the editors and anonymous reviewersfor their valuable comments and suggestions to improve thequality of this paper.

APPENDIX

Proof of Theorem 1Proof: From the expression of QoM in Eq. (10), we have

QoM =T iawact

Tiaw+ 1− T iaw

act

Tiaw+

[− 1

2!α2(Tiaw − T iaw

act )2

+1

3!α3(Tiaw − T iaw

act )3 + · · ·

]1

αTiaw

= 1 +

[− 1

2!α2(

L

v− T iaw

act )2 +

1

3!α3(

L

v− T iaw

act )3

+ · · ·]

1

αTiaw

= 1 +v

αLf(v),

where f(v) = − 12!α

2(Lv − T iawact )

2 + 13!α

3(Lv − T iawact )

3 + · · ·,and f(v) < 0. Denote df(v)

dv as the derivative of f(v) on v.We have

df(v)

dv= −α2

[L

v− T iaw

act

] [L

v− T iaw

act

]′×[1

−α(Lv − T iawact )

2!+

α2(Lv − T iawact )

2

3!+ · · ·

]

= α2

[L

v− T iaw

act

]LPact −A

(Pact − Pslp)v2

CHENG et al.: OPTIMAL SCHEDULING FOR QUALITY OF MONITORING IN WIRELESS RECHARGEABLE SENSOR NETWORKS 11

×1− e−α(Lv −T iaw

act )

α(Lv − T iawact )

= αLPact −A

(Pact − Pslp)v2

[1− e−α(L

v −T iawact )

],

where A = 2τrβ(r+β) . Recall that Pact−Pslp > 0, Pact

Lv − A

v >

0, and Lv − T iaw

act > 0. We get df(v)dv > 0, indicating f(v) is

an increasing function of v. Denote dQoMdv as the derivative of

QoM on v, we have

dQoM

dv=

1

αL

[f(v) + v

df(v)

dv

]

Now we will show that dQoMdv is always positive within the

feasible set of v. Clearly, we have

L

v− T iaw

act =L

v−

Av − Pslp(

Lv − Twak)− Ewak

Pact − Pslp

which can be denoted as Mv + B with M = PactL−A

Pact−Pslpand

B =Ewak−PslpTwak

Pact−Pslp. Then we have

f(v) + vdf(v)

dv= 1− α

M

v− αB − e−αM

v −αB

+αM

v

[1− e−αM

v −αB]

= 1− αB − (1 + αM

v)e−αM

v −αB

Denote f(v) = f(v) + v f(v)dv . We have e−1+αB f(v) =

(1−αB)e−1+αB − (1 + αMv )e−1−αM

v which is a decreasingfunction along with v ≥ 0. On the other hand, dQoM

dv ispositive when v approaches ∞ which proves that f(v) ispositive when v approaches ∞. Therefore, f(v) would bealways positive for v ≥ 0, which proves that the maximumvalue of QoM can only be achieved at vmax.

Proof of Theorem 3Proof: Following similar algebraic manipulations as the

proof of Theorem 1, we have

QoM = 1 +v

αL + αNtvf(v),

where f(v) = − 12!α

2(Lv + Nt − T jawact )2 + 1

3!α3(Lv + Nt −

T jawact )3 + · · ·, Note that T jaw

act can be calculated by (14), andf(v) < 0. On the other hand, we have

df(v)

dv= α

LPact −A

(Pact − Pslp)v2(1− e−α(L

v +Nt−T jawact )),

where A = 2τr+β , and df(v)

dv > 0. Then, for the quality ofmonitoring, we have

dQoM

dv=

L

α(L+Ntv)2f(v) +

v

αL+ αNtv

df(v)

dv

=1

α(L+Ntv)

[αL

α(L +Ntv)f(v) + v

df(v)

dv

]

>1

α(L+Ntv)

[f(v) + v

df(v)

dv

]

With a similar technique as in the proof of Theorem 1, wecan prove the dQoM

dv is always positive along with feasible v.Thus the maximum QoM can only be achieved at vmax.

Proof of Theorem 4Proof: Since QoM = 1 + v

αL+αNvt f(t), where f(t) =

− 12! (

Lv +Nt−T jaw

act )2+ 13! (

Lv +Nt−T jaw

act )3+· · ·, and T jawact is

calculated from (14). For simplicity of expression, we denoteLv +Nt− T jaw

act = St+U ≥ U Hence we need to determinethe sign of

dQoM

dt=

v

L+Nvt

[(Nv

αL+ αNvt+ S

)(e−α(St+U)

−1

)+

αNv

αL+ αNvt(St+ U)

]

which is equivalent to investigate the sign of the followingfunction

G(t) = e−α(St+U) +αNvU − αLS −Nv

αNvSt+ αLS +Nv.

Let G1(v) = αNvU − αLS − Nv, which can be expressedas

G1(v) = (Pslp + αΔE − Pact)Nv + α(Lτ

β2− 2Nτr

β(β + r)),

where ΔE = Ewak −PslpTwak. Note that α(Lτβ2 − 2Nτr

β(β+r)) >

0. It is straightforward to see that if G1(v) > 0, it can beguaranteed that G(t) > 0. Thus it would be always better toenlarge the staying time, i.e., the maximum QoM is achievedwhen t → ∞, if either of the following conditions is satisfied

1) 1α ≤ ΔE

Pact−Pslp;

2) 1α > ΔE

Pact−Pslpand v ≤ 1

(Pact−Pslp−αΔE)N (Lτβ2 −

2Nτrβ(β+r)).

Otherwise, the maximum QoM can only be achieved at thepoints where G(t) = 0 or t = 0. It should be noticed that,from the expression of G(t), there are at most two pointswhich satisfy G(t) = 0. Thus we just need to check the valuesof at most three points2 to find the t∗ which maximizes theQoM.

Proof of Theorem 5Proof: For simplicity, we denote the QoM of JPW with

n slots as QoMn. We first investigate the difference of n-slotand 1-slot schedule, Define ΔQoM = QoMn−QoM1, whichcan be calculated as

ΔQoM = − (n− 1)ΔE

(Pact − Pslp)Tjaw+

1

αTjaw

[(n− 1)

+e−α(Tjaw−T jawact ) − ne−α[ 1

n (Tjaw−T jawact −EP )+EP ]

]

Denote ΔEP = ΔEPact−Pslp

and Tjaw = 1n (Tjaw−T jaw

act −EP ).By taking derivative of ΔQoM , we have

dΔQoM

dn

= −ΔEP

Tjaw+

1

αTjaw

[1− (1 + αTjaw)e

−α(Tjaw+ΔEP )]

=e1−αΔEP

αΔEP

[(1− αΔEP )e

−(1−αΔEP )

−(1 + αTjaw)e−1−αTjaw

]2If some points are not feasible, we just need to check the boundary points

and the remaining points satisfying G(t) = 0.

12 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, ACCEPTED FOR PUBLICATION

By utilizing the property of function xe−x similar as in theproof of Theorem 1, it can be proved that:

If 1α ≤ EP , then dΔQoM

dn < 0, which indicates that QoMis a decreasing function of n, thus it would be better not todivide the Tjaw into different slots and schedule the activation.

Otherwise, if 1α > EP , then in order to enhance QoM , n

should be no greater than n� satisfying

(1 + αT �jaw)e

−1−αT�jaw = (1− αΔEP )e

−(1−αΔEP ).

Note that the left side is a decreasing function of T ∗jaw and

the right side can only be achieved when T ∗jaw = −ΔEP

with n approaches ∞. Recall that Tjaw should be positive,thus the maximum QoM will be achieved at the maximumvalue satisfying (18) and (20).

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Peng Cheng (IEEE M’10) received the B.E. degreein Automation, and the Ph.D. degree in ControlScience and Engineering in 2004 and 2009 respec-tively, both from Zhejiang University, Hangzhou,P.R. China. Currently he is Associate Professor withDepartment of Control Science and Engineering,Zhejiang University. He serves as the publicity co-chair for IEEE MASS 2013. His research inter-ests include networked sensing and control, cyber-physical systems, and robust control.

Shibo He received Ph.D degree in Control Scienceand Engineering at Zhejiang University, Hangzhou,China, 2012. He was a visiting scholar at Universityof Waterloo from 2010 to 2011. He serves as a trackchair for ANT 2013 and EUSPN 2013, publicity co-chair for WiSARN 2010 (Fall), and Web Chair forIEEE MASS 2013. His research interests includewireless sensor networks, crowd sensing and dataanalysis in smart grid.

Fachang Jiang received the B.E. degree in Measur-ing and Control Technology and Instrumentationsin 2009 from University of Electronic Science andTechnology of China, Chengdu, and M.S degree inControl Science and Engineering in 2012 from Zhe-jiang University, Hangzhou, P.R.China. Currently heis a software engineer in Internet of Thing Instituteof China Electronics Technology Group Corpora-tion, Hangzhou.

CHENG et al.: OPTIMAL SCHEDULING FOR QUALITY OF MONITORING IN WIRELESS RECHARGEABLE SENSOR NETWORKS 13

Yu (Jason) Gu is currently an assistant professorin the Pillar of Information System Technology andDesign at the Singapore University of Technologyand Design. He received the Ph.D. degree from theUniversity of Minnesota, Twin Cities in 2010. Dr.Gu is the author and co-author of over 50 papers inpremier journals and conferences. His publicationshave been selected as graduate-level course materialsby over 20 universities in the United States and othercountries. His research includes Networked Em-bedded Systems, Wireless Sensor Networks, Cyber-

Physical Systems, Wireless Networking, Real-time and Embedded Systems,Distributed Systems, Vehicular Ad-Hoc Networks and Stream ComputingSystems. Dr. Gu is a member of ACM and IEEE.

Jiming Chen (IEEE M’08 SM’11) received B.Scdegree and Ph.D degree both in Control Science andEngineering from Zhejiang University in 2000 and2005, respectively. He was a visiting researcher atINRIA in 2006, National University of Singaporein 2007, and University of Waterloo from 2008 to2010. Currently, he is a full professor with De-partment of Control Science and Engineering, andthe coordinator of group of Networked Sensing andControl in the State Key laboratory of IndustrialControl Technology at Zhejiang University, China.

His research interests are estimation and control over sensor network, sensorand actuator network, coverage and optimization in sensor network. Hecurrently serves associate editors for several international journals includingIEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, IEEETRANSACTIONS ON INDUSTRIAL ELECTRONICS and IEEE Network. He is aguest editor of IEEE TRANSACTIONS ON AUTOMATIC CONTROL, ComputerCommunication (Elsevier), Wireless Communication and Mobile Computer(Wiley) and Journal of Network and Computer Applications (Elsevier). Healso serves as a Co-chair for Ad hoc and Sensor Network Symposium, IEEEGlobecom 2011, general symposia Co-Chair of ACM IWCMC 2009 andACM IWCMC 2010, WiCON 2010 MAC track Co-Chair, IEEE MASS 2011Publicity Co-Chair, IEEE DCOSS 2011 Publicity Co-Chair, IEEE ICDCS2012 Publicity Co-Chair and TPC member for IEEE ICDCS 2010, IEEEMASS 2010, IEEE SECON 2011, IEEE INFOCOM 2011, IEEE INFOCOM2012, IEEE ICDCS 2012 etc.