CHASE: Charging and Scheduling Schemefor Stochastic Event Capture in Wireless

Rechargeable Sensor NetworksHaipeng Dai ,Member, IEEE, Qiufang Ma , Xiaobing Wu, Guihai Chen,Member, IEEE,

David K. Y. Yau, Senior Member, IEEE, Shaojie Tang,Member, IEEE,

Xiang-Yang Li , Fellow, IEEE, and Chen Tian ,Member, IEEE

Abstract—In this paper, we consider the scenario in which a mobile charger (MC) periodically travels within a sensor network to

recharge the sensors wirelessly. We design joint charging and scheduling schemes to maximize the Quality of Monitoring (QoM) for

stochastic events, which arrive and depart according to known probability distributions of time. Information is considered captured if it is

sensed by at least one sensor. We focus on two closely related research issues, i.e., how to choose the sensors for charging and

decide the charging time for each of them, and how to schedule the sensors’ activation schedules according to their received energy.

We formulate our problem as the maximumQoM CHArging and SchEduling problem (CHASE). We first ignore the MC’s travel time and

study the resulting relaxed version of the problem, which we call CHASE-R. We show that both CHASE and CHASE-R are NP-hard.

For CHASE-R, we prove that it can be formulated as a submodular function maximization problem, which allows two algorithms to

achieve 1=6- and 1=ð4þ �Þ-approximation ratios. Then, for CHASE, we propose approximation algorithms to solve it by extending the

CHASE-R results. We conduct simulations to validate our algorithm design.

Index Terms—Mobile charging, scheduling, wireless rechargeable sensor network, stochastic event capture, submodular optimization,

approximation algorithm

Ç

1 INTRODUCTION

TRADITIONAL wireless sensor networks (WSNs) are con-strained by limited battery energy that powers the sen-

sors. Their limited network lifetime is considered a majordeployment barrier. Besides, in many applications sensorsare located in hazardous or inaccessible areas such as volca-noes [1], inside concrete walls [2], or at the bottom ofbridges [3], which makes battery-swapping schemes [4], [5],[6] unsafe, infeasible, labor-intensive, or costly. To extendthe network lifetime, many approaches have been proposedto harvest environmental energy such as solar [7], vibra-tion [8], and wind [9]. However, a limitation of existing

energy-harvesting techniques is that it is highly dependenton the ambient environment, which makes the harvestingrate highly unpredictable. The problem can be overcome byrecent breakthroughs in wireless power charging technolo-gies [10], which allow energy to be transferred from onestorage device to another wirelessly with reasonable effi-ciency. For example, magnetic resonance coupling is shownto transfer 60 watts [10] at an efficiency of 90 percent toabout 30 percent when the distance varies from 0:75 m to2:25 m. Since wireless recharging can guarantee a requiredlevel of power supply, independent of the ambient environ-ment, and it is contactless, it has found many applicationsincluding smart grids [11], body sensor networks [12], andcivil structure monitoring [3].

Because power chargers are expensive, it is generally notcost-effective to deploy a large number of them statically forenergy provisioning. Instead, existing practical approachesfocus on using a mobile charger (MC) to move around thesensors and charge them in turn during a travel schedule, fortasks such as routing [13], [14], [15] and gathering data [16].None of these prior efforts have solved the problem of sto-chastic event capture, however, in key aspects such as sched-uling sensors’ duty cycles to maximize their ability tocapture interesting events of a probabilistic nature. But theproblem is fundamental in wireless sensor network design,and it has received attention for both cases of traditionalWSNs [17], [18], [19] andwireless ambient-energy harvestingsensor networks [20], [21]. In this paper, we optimize event

� H. Dai, Q. Ma, G. Chen, and C. Tian are with the State Key Laboratory forNovel Software Technology, Nanjing University, Nanjing 210023, China.E-mail: {haipengdai, tianchen}@nju.edu.cn, mg1633053@smail.nju.edu.cn, gchen@cs.sjtu.edu.cn.

� X. Wu is with the Wireless Research Centre, University of Canterbury,Christchurch 8041, New Zealand. E-mail: barry.wu@canterbury.ac.nz.

� D.K.Y. Yau is with the Information Technology Systems and DesignPillar, Singapore University of Technology and Design, 487372, Singa-pore. E-mail: david_yau@sutd.edu.sg.

� S. Tang is with the Naveen Jindal School of Management, University ofTexas at Dallas, 800 W. Campbell Road, Richardson, TX 75080.E-mail: shaojie.tang@utdallas.edu.

� X. Li is with the School of Computer Science and Technology, Universityof Science and Technology of China, Hefei 230026, China.E-mail: xiangyangli@ustc.edu.cn.

Manuscript received 13 Sept. 2017; revised 18 Nov. 2018; accepted 30 Nov.2018. Date of publication 21 Dec. 2018; date of current version 3 Dec. 2019.(Corresponding author: Guihai Chen.)Digital Object Identifier no. 10.1109/TMC.2018.2887381

44 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 19, NO. 1, JANUARY 2020

1536-1233� 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See ht _tp://www.ieee.org/publications_standards/publications/rights/index.html for more information.

capture in a network of sensors wirelessly recharged by anMC. We assume that stochastic events arrive and departaccording to known time distributions. An event is said to becaptured if it is sensed by at least one sensor.

Note that there are existing practical system platformsthat can enhance the performance of event monitoring bywireless recharging. For example, the Wireless Identifica-tion and Sensing Platform (WISP) has been applied in indi-vidual activity recognition, large-scale urban sensing [22],[23], and structural health monitoring (SHM) [3]. In theSHM application, the civil structure is instrumented withsensor nodes capable of being powered solely by energytransmitted wirelessly to them by a mobile helicopter. Jianget al. [24] are the first to exploit wireless power charging byMCs for efficient stochastic event capture. Their objective isto jointly determine the MCs’ movement schedule and thesensors’ activation schedule to maximize the Quality ofMonitoring (QoM) [18], [20], defined as the average infor-mation gained per event by the network. They make severalsimplifying assumptions: each sensor can only monitor onePoint of Interest (PoI), the charging time for each sensor isidentical, the event staying time follows an Exponential dis-tribution, and all the sensors follow a simple ðq; pÞ periodicschedule (i.e., the sensors monitor the PoIs for q time everyp time). We relax these assumptions in this paper.

In this paper, we consider the scenario in which an MCperiodically travels within a sensor network and rechargesthe sensors wirelessly to enable them to perform tasks ofstochastic event capture. We assume that the MC repeats itsrecharge schedule every period of time t, and that theschedule (counting both the charging time and travel timeof the MC) must complete within time tw (tw < t). Forexample, the MC is carried by an UAV whose daily shift isfrom 9am to 11am only (here, tw ¼ 2 hour and t ¼ 24 hour;typically, t can be a few weeks or even longer), or the MCmust be withdrawn for maintenance for some amount oftime between recharge schedules.

We address two closely related issues in the wirelessrecharging and event monitoring. The first is how to choosethe sensors for recharging and further decide the chargingtime for each of them, constrained by the MC’s workingtime tw. The second is how to best schedule the sensors’activations based on their received energy, considering thatnearby sensors may cover overlapping PoIs. Our goal is tojointly design a charging scheme for the MC and thesensors’ activation schedules to maximize the QoM of thestochastic event capture. We define our problem formally asthe maximum QoM CHArging and SchEduling problem(CHASE). The coupling between the MC’s travel time andthe sensor charging time makes the problem highly chal-lenging. Hence, we first ignore the travel time and study theresulting relaxed version of CHASE, which we call CHASE-R. Then, based on the CHASE-R results, we develop solu-tions for the general CHASE problem.

The main contributions of this paper are as follows:

� We analyze the QoM of stochastic event capture,which admits the possibility that the same PoI ismonitored by multiple sensors. We formulate theCHASE and CHASE-R problems, and show thatboth are NP-hard.

� We reformulate CHASE-R as a monotone submodu-lar function maximization problem under a specialsufficient condition. This reformulation of CHASE-Rallows two algorithms to achieve 1=6-approximationand 1=ð4þ �Þ-approximation, respectively.

� Based on the CHASE-R results, we propose twoapproximation algorithms for CHASE, when theMC’s travel time is considered.

� Importantly, all of the proposed CHASE-R andCHASE solutions are sufficiently general to accom-modate diverse activation schedules, event utilityfunctions, and probability distributions of the eventstaying times.

� We conduct extensive simulations to verify our ana-lytical findings. Simulation results show that ourschemes outperform the state of the art.

The remainder of the paper is organized as follows.Section 2 reviews related work. In Section 3, we present pre-liminaries and a formal definition of the CHASE problem,as well as its relaxed version CHASE-R. In Section 4, weanalyze the complexity of the problems and reformulate aspecial case of the relaxed problem as a monotone sub-modular function optimization problem. Then, we presentapproximation algorithms for the relaxed problem and theoriginal problem. Section 5 presents extensive simulationsto verify our theoretical results. Section 6 concludes.

2 RELATED WORK

In this section, we briefly review related work on mobilecharging, which generally can be classified into threecategories.

First, prior work has investigated the energy efficiencyof mobile chargers [25], [26], [27], [28], [29]. For example,Wang et al. [25] proposed to coordinate multiple MCs tominimize their aggregate travel distance while guarantee-ing continuous operation of each sensor, such that the over-all energy efficiency is optimized. In a later work [26], theirgoal is to maximize the difference between the energy har-vested by all the sensors and the travel energy expendedby all the MCs. Zhang et al. [27] presented an optimalscheme for multiple energy-constrained MCs to charge alinear WSN, where the ratio of energy received by all thesensors to the travel energy expended by all the MCs ismaximized.

Second, the service delay of MCs has been consid-ered [30], [31], [32], [33]. Fu et al. [30] minimized the overallcharging delay of a single MC by planning its chargingroute and strategy. He et al. [31], [32] considered the charg-ing problem when the charging requests of sensors arrive ina dynamic fashion. Their work has been extended to scenar-ios where the sensors are also mobile [33].

Third, research has addressed network performanceissues under mobile charging, from perspectives such asdata routing, data collection, sensing coverage, and eventmonitoring [13], [14], [15], [16], [24], [34], [35], [36], [37],[38]. For data routing, Tong et al. [13] examined the impactof mobile charging on data routing and WSN deployment.Their work has been expanded [15] to address several real-istic issues (e.g., the communication environment isdynamic and unreliable, the charging capacity of an MC is

DAI ET AL.: CHASE: CHARGING AND SCHEDULING SCHEME FOR STOCHASTIC EVENT CAPTURE IN WIRELESS RECHARGEABLE... 45

limited, and the sensors are heterogeneous) by jointly con-sidering data routing for the sensors and the chargingscheme for an MC. Use of mobile charging to improve datacollection in WSNs has also received significant attention.Shi et al. [14], [34] applied a single MC to improve the datacollection, while reducing the working time within a charg-ing time period. In [16], [35], [36], [37], MCs are used notonly as energy providers but also as data collectors. Zhouet al. [38] solved the problem of scheduling an MC to chargesensors to maintain k-coverage in the network at low costfor the MC. Jiang et al. [24] are first to investigate the impactof mobile charging on efficient stochastic event capture.However, they make several simplifying assumptions,which limit the practicality of their results. We relax theseassumptions in this paper.

3 PROBLEM STATEMENT

3.1 Network Model

We assume that there arem sensors V ¼ fv1; v2; . . . ; vmg dis-tributed over a two-dimensional region, which cover nPoint of Interests denoted by O ¼ fo1; o2; . . . ; ong. Let Oi rep-resent the set of PoIs covered by sensor vi. In a dense sensornetwork, typically close-by sensors may cover commonPoIs. Hence, we assume that a target, say oi, is covered by asubset of the sensors Vi. A summary of the notations in thispaper is given in Table 1.

To prolong the lifetime of sensors, a mobile charger peri-odically starts from a base station (BS) and visits each of aselected subset of the static sensors Vs � V exactly once, inorder to charge the sensors wirelessly. At the end of thecharging schedule, the MC returns to the BS. The total work-ing time of the charging schedule, including the travel over-head and the charging time for all the selected sensors,must not exceed tw. Furthermore, we assume that the charg-ing schedule is repeated every fixed period of time t. Hence,the off-duty time for the MC at the BS is at least t � tw. Notethat under such a charging scheme, sensors not chosen forcharging will die eventually. This is acceptable since ourprimary concern is to maximize the overall QoM under theconstraint of limited charging time, rather than fairnessamong charging all the sensors.

We denote the path of the MC by P ¼ ðp0;p1; . . . ;

pjVsj;pjVsjþ1Þ where p0 ¼ pjVsjþ1 ¼ BS and fpigjVsji¼1 ¼ Vs.Denote by tij the time required for the MC to move between

sensor vi and vj. Suppose the movement of the MC betweensensors is dictated by a motion planning scheme predeter-mined by the BS and/or the MC. The motion planning isassumed to respect physical constraints, such as followingaccessible pathways, avoiding obstacles, obeying mechani-cal limits on speeds and turns, etc, but its details are out ofscope of this paper. We only require the motion planning tobe predetermined and keep stable, such that tij is known apriori and it is fixed. To use as much time for charging aspossible, the MC should travel on a shortest path P , given

by arg minPPjVsjþ1

i¼0 tpipiþ1 , that completes a circuit of thesensors. For simplicity, we assume that such a path alwaysexists. Finding the path can then be formulated as a Travel-ing Salesman Problem (TSP), which is NP-hard. We assumethat some good approximation algorithm is used, and theapproximate solution is given by tTSP ðVsÞ. Moreover, weassume that the MC spends ti time for recharging the bat-tery of vi. Then, we have:

tTSP ðVsÞ þXvi2Vs

ti � tw: (1)

The above inequality gives the MC’s working time constraint.In this paper, we assume a discrete time model for asensor’s schedule, where the duration of a time slot is fixedand given. Specifically, every sensor follows a periodicschedule of identical length L (in time slots). In each timeslot, a sensor can schedule itself to be active or inactive.Hence, we can express the activation schedule of sensor vi bya vector Si ¼ ðai1; ai2; . . . ; aiLÞ, where aij ¼ 1 indicates thatthe sensor is active in slot j and aij ¼ 0 indicates the oppo-site. We assume that the duration of a time slot, say tts, islong enough such that the energy cost of turning the sensoron/off can be ignored.

The BS is responsible for determining the overall charg-ing scheme, including the MC’s travel path, the chargingtime allocated for each sensor, and the activation schedulesof the sensors. It disseminates the activation schedules to

TABLE 1Notations

Symbol Meaning

oi Target ivi Sensor iOi Subset of PoIs covered by sensor viVi Subset of sensors covering PoI oiL Length of sensor scheduletts Time duration of a single time slotSi Activation schedule of sensor vibSi Equivalent monitoring schedule for PoI oiwi Weight of PoI oitw Maximum working time in one charging periodt Time period of charging processti Charging time allocated to sensor vi in one charging

periodPc Working power of MCpi Working power of sensor viEi Battery capacity of sensor vihi MC’s charging efficiency for sensor vici Charging time factor for sensor vili Active time slot budget of sensor vinMC MC’s speed

Fig. 1. Network model.

46 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 19, NO. 1, JANUARY 2020

the respective sensors either by a pre-established multi-hopcommunication mechanism, such as the Collection TreeProtocol (CTP) [39], or through the MC when the MC comesnear each sensor for charging. We assume that the energycost of disseminating the schedules can be ignored. This isbecause a schedule will not change except for exceptionalevents such as node breakdowns.

Fig. 2 shows an example of our network model. In thisexample, 5 sensors altogether cover 11 PoIs. The MC choo-ses a subset of the sensors Vs ¼ fv1; v2; v3; v4g to charge, andits travel path is P ¼ fBS; v1; v2; v3; v4; BSg. The activationschedule of sensor v1 in Vs is S1 ¼ ða11; a12; a13; a14Þ ¼ð1; 0; 0; 1Þ, and that for v2, v3, and v4 are S2 ¼ ð0; 1; 0; 0Þ,S3 ¼ ð0; 0; 0; 1Þ, and S4 ¼ ð1; 0; 0; 0Þ, respectively.

3.2 Energy Consumption Model

Let Pc denote the working power of the MC, and pi theworking power of sensor vi. Let hi denote the MC’s chargingefficiency for sensor vi, i.e., the ratio of the amount of energyreceived by vi to the amount of energy consumed by theMC. The charging efficiency can vary from sensor to sensor,and it depends on factors such as the distance between thecorresponding sensor and the MC and the effectiveness ofthe sensor’s antenna. We assume that the leakage power ofeach sensor is negligible, and each sensor will have used upits energy by the time of its next recharge (which can be con-trolled by properly allocating the recharging time of theMC). Therefore, in one charging period of duration t, therequired working energy for sensor vi under its activation

schedule is pi � jjSijjL t, and it should be equal to the aggre-

gated charged energy for vi, i.e., Pc � ti � hi. Thus, we havepi

hiPcL t � jjSijj1 ¼ ti. For convenience, we define the charging

time factor ci ¼ pihiPcL t, which is constant; it can be interpreted

as the charging time required for the MC to provide suffi-cient energy for vi to be active for one time slot. Hence, wehave:

ci � jjSijj1 ¼ ti: (2)

Further, denote by Ei the battery capacity of sensor vi,and li the maximum number of active time slots sensor vican sustain using its limited battery capacity, which we callthe active time slot budget. It is clear that li ¼ Ei

pitL. Because

the total activated time slots in the activation scheduleshould not exceed the active time slot budget, we have:

jjSijj1 �i; (3)

which we call the active time slot constraint. If li � L for anysensor vi, we can ignore the active time slot constraint. This

situation occurs when the battery capacity is much largercompared to the working power of the sensor (such asultra-capacitors [40]) or when the charging process isapplied frequently. In general, however, the active time slotconstraint should be considered; e.g., when the batteries areof low cost and limited capacity.

3.3 Event Model and QoM Computation

In this section, we first present assumptions on the eventdynamics and the properties of the sensors. Then, we pro-pose a general paradigm to compute a PoI’s QoM when it ismonitored by one or more sensors.

3.3.1 Event Model

For event dynamics, we assume that events at a PoI occurone after another, and events at the same PoI or differentPoIs are spatially and temporally independent [17], [18],[24]. After its occurrence, an event stays for some randomtime before it disappears. We denote by X the event stayingtime. Similarly, the time duration before the next eventoccurs, which we call the event inter-arrival time, is randomand denoted by Y . Hence the sequence of event arrivals anddepartures forms a stochastic process. By renewable theory[41], the expected number of event arrivals in a time intervaldt equals midt, where mi ¼ 1=EðY Þ and Eð�Þ denotes expec-tation. As for the event staying timeX, we denote the proba-bility density function ofX by fðxÞ.

We use a binary sensing model for the sensors [42]. Anevent is said to be captured if it is sensed by at least one sen-sor. Assume that the jth occurring event at PoI i is denotedas eij, which is within range of a sensor for a total (but notnecessarily contiguous) amount of time tijðtij � 0Þ. Weassume that the sensor will, as a result, gain an amount ofinformation Ui

jðtijÞ about eij, where UijðxÞ is the utility func-

tion of eij. There are different types of event utility func-

tions [17]. Existing QoM analysis typically considers simplecases of the function only (e.g., the Step utility function) [18][20]. Our analysis in this paper is more general, and coversother types of functions as well. Without loss of generality,we assume Ui

jðxÞ ¼ UðxÞ for all the events at all the PoIs;i.e., the utility function is identical for all the events. Fur-thermore, we assume that the events are identifiable [17], i.e.,if more than one sensors detect the same event simulta-neously, they will know it is the same event and learnexactly the same information.

To help understand the utility computation for a singleevent monitored by multiple sensors, we use a simple exam-ple for illustration. As can be seen in Fig. 2, PoI o4 is coveredby sensor v1, v2 and v3, whose schedules are S1 ¼ ð1; 0; 0; 1Þ,S2 ¼ ð0; 1; 0; 0Þ and S3 ¼ ð0; 0; 0; 1Þ, respectively. Let thetime duration of a single slot be tts ¼ 1 s. Suppose an eventarrives at PoI o4 at time t ¼ 0:2 s and leaves at t ¼ 3:8 s, andits utility function is defined as UðxÞ ¼ 0:25x. Then, theaggregate captured utility of this event by the three sensorsincreases monotonically with time as the blue solid lineshows in Fig. 2. Specifically, the utility of the event increaseslinearly during time period [0.2,1] as the event is monitoredby sensor v1, and keeps increasing during [1,2] when v2 isactive in that time duration. After that, the utility remainsunchanged in [2,3] since no sensors are activated. During

Fig. 2. An instance of utility computation with multiple sensors.

DAI ET AL.: CHASE: CHARGING AND SCHEDULING SCHEME FOR STOCHASTIC EVENT CAPTURE IN WIRELESS RECHARGEABLE... 47

time period [3,3.8], the utility resumes its increase at thesame speed as before, but not double the speed, althoughboth v1 and v3 are active. This is due to the identifiability ofevents. We also plot the curve when the utility function isgiven by UðxÞ ¼ 1� e�0:8x, as illustrated by the greendashed line in Fig. 2. It can be observed that the curve showsa similar trend.

3.3.2 QoM Computation

To start, we express the active status of sensors over time asa function of their individual schedules, by the periodicextension function given below. Note that each sensor isassumed to start its schedule at time 0.

Definition 3.1 (Periodic Extension Function). Given aschedule Si of sensor vi, the periodic extension functionSiðxÞ ðSi : ½0;þ1� 7! f0; 1gÞ of Si is defined as:

SiðxÞ ¼ 1; ðx 2 ½ðkL þ j� 1Þtts; ðkLþ jÞtts�; k 2 N ; SiðjÞ ¼ 1Þ0; otherwise

�:

(4)

Note that tts denotes the time duration of a single timeslot.

We use a simple example for illustration. As shown inFig. 3, the solid line denotes the value of the periodic exten-sion function of the schedule Si ¼ ð1; 0; 0; 1Þ. The functiontakes value 1 when x 2 ½4ktts; ð4kþ 1Þtts� or x 2 ½ð4kþ 3Þtts;ð4kþ 4Þtts� for k 2 N , and 0 otherwise.

Then, we can derive the mathematical expression of theQoM of a PoI covered by a single sensor in the followinglemma.

Lemma 3.1. The QoM of a PoI, say oi, covered by a single sensorvj ðvj 2 Vi; jVij ¼ 1Þ under schedule Sj, whose periodic exten-sion function is SjðxÞ, is given by:

QðijSjÞ ¼ 1

Ltts

Z Ltts0

Z þ1t

U

Z y

t

SjðxÞdx!fðy� tÞdydt:

(5)

Proof. Since the schedule of each sensor is fixed and peri-odic, we only need to consider the utility of events start-ing at time t ðt 2 ½0;Ltts�Þ. Specifically, for an event thatstarts at time t ðt 2 ½0;Ltts�Þ and ends at timey ðy 2 ½t;þ1ÞÞ, its utility is UðR yt SjðxÞdxÞ. As the eventstaying time follows the probability density functionfðxÞ, the expected achieved utility for an event isRþ1t UðR yt SjðxÞdxÞfðy� tÞdy. By considering all possible

events that occur in ½0;Ltts�, we get Eq. (5). tuSuppose Si ¼ ðai1; . . .; aiLÞ and Sj ¼ ðaj1; . . .; ajLÞ are two

different vectors. We define the “OR” operation of the

vectors as Si _ Sj ¼ ðai1 _ aj1; . . . ; aiL _ ajLÞ. The followinglemma shows the QoM expression for a PoI covered by mul-tiple sensors.

Lemma 3.2. The QoM of PoI oi covered by a set of sensorsVi ¼ fv10 ; v20 ; . . . ; vm0 g, each of which having scheduleSj ðj ¼ 10; 20; . . . ;m0Þ, is given by:

QðiÞ ¼ QðijS10 ; S20 ; . . . ; Sm0 Þ ¼ Q�ij_vj2Vi

Sj

�: (6)

In other words, the QoM achieved by the multiple sensors canbe equivalently viewed as that by one single sensor with sched-uleW

vj2Vi Sj.

Proof. Referring back to the definition of QoM, we onlyhave to show that the overall utility available for any par-ticular event eij gained by the collection of sensors Vi,namely UðtijÞ, is exactly equal to the utility UaðtijÞ gainedby a virtual sensor va with schedule

Wvj2Vi Sj. This can be

derived by the identifiable and identical assumptions

about the events. We omit the details to save space. tuFor simplicity of exposition, we call bSi ¼

Wvj2Vi Sj the

equivalent monitoring schedule for PoI oi. We stress that ouranalysis can compute the QoM of a PoI in the presence ofboth single and multiple monitoring sensors. It can alsoaccommodate general activation schedules, event utilityfunctions, and probability distributions of the event stayingtimes fðxÞ.

3.4 Problem Formulation

To sum up, we formally formulate our problem CHASE asP1 shown below.

ðP1Þ maxSi

Xni¼1

wiQðiÞ

s:t: ð1Þ; ð2Þ; ð3Þ:

Note that wi is a normalized weight associated with the PoIoi, which can be interpreted as the frequency of event occur-rences of oi or the importance of oi. The decision variablesare the activation schedules Sis of all the sensors. Note thatthe charging time for each sensor ti can be determined by Si

using Eq. (2), and the subset of sensors Vs selected for charg-ing exactly contains the sensors vis with non-zero activationschedules Sis. The quantities tw, ci, pi, t, hi, Pc, L, Ei, li, andwi are given constants.

Note that different heuristic algorithms, such as evolution-ary techniques [43], [44], simulated annealing [45], and parti-cle swarm optimization [46], can be used to solve CHASE,and they may show good performance in particular practicalscenarios. Nevertheless, they do not guarantee good perfor-mance theoretically. In contrast, our proposed techniques pro-vide provable approximation ratios, which improve upon theheuristics by bounding the loss of performance.

3.5 Roadmap of our Solution

As evidenced by the above formulation, the full CHASEproblem is complex. It involves the selection of the candi-date set of sensors Vs for charging, the coupling between theMC’s travel time and the allocation of charging time among

Fig. 3. An example of periodic extension function.

48 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 19, NO. 1, JANUARY 2020

the sensors, the active time slot constraint, and careful com-putation of the QoM. Among these factors, it is particularlyhard to account for the travel time accurately. Hence, westart by considering a relaxed version of CHASE, which wecall CHASE-R, that ignores the travel time; i.e., we assumetTSP ðV Þ ¼ 0. Besides amenable to analysis, importantlyCHASE-R is also meaningful in practice, as tw can be muchbigger than tTSP ðV Þ due to long required charging timenecessitated by typically limited charging efficiencies ofMCs. For example, the charging time for the voltage to reach1:8 V for a WISP tag equipped with a 100 uF capacitor canbe as large as 155 seconds, when the RFID reader is 10.0meters away [47]. After solving CHASE-R, we will accord-ingly develop solutions for the general CHASE problem byputting the MC’s travel time back into consideration.

4 THEORETICAL ANALYSIS

In this section, we show that the CHASE-R and CHASEproblems stated above are NP-hard. Then, we reformulatethe problems and present approximation algorithms foreach of them, respectively.

4.1 Hardness of Problems

We now show that both CHASE-R and CHASE are NP-hard, and that they cannot be approximated within a factorbetter than ð1� 1=eÞ. To do that, we state the followingwell-known NP-hard problem and a related lemma.

Definition 4.1 (Maximum Coverage Problem) [48]. Givena collection of subsets S ¼ fS1; S2; . . . ; Smg of the universal setU ¼ fe1; e2; . . . ; eng and a positive integer k, find a subsetS0 � S such that jS0j � k and the number of covered elementsj [Si2S0 Sij is maximized.

Lemma 4.1 [49]. For any � > 0, the Maximum Coverage Problem(MCP) cannot be approximated within a factor ð1� 1=eþ �Þunless P ¼ NP .

We have the following theorem about the complexity ofour problems.

Theorem 4.1. Both CHASE-R and CHASE are NP-hard. Forany � > 0, there are no ð1� 1=eþ �Þ approximation solutionsto them unless P ¼ NP .

Proof. We can reduce the CHASE-R problem to the MCPproblem by setting L ¼ 1, wi ¼ 1=n, and ci ¼ c, where c isa constant, and setting li � L for any sensor vi such thatthe active time slot constraint can be removed. Hence,CHASE-R is at least as hard as MCP. For CHASE, we settw=c ¼ kþ 1=2 (where k is an integer) and tTSP ðV Þ <1=2 c. We can then prove that CHASE is also at least ashard asMCP. By Lemma 4.1, the result follows. tuRemark: CHASE involves finding a shortest path to visit

all the sensors in Vs. This component TSP problem is alreadyNP-hard. The method in Theorem 4.1 is useful in that itapplies to the easier CHASE-R problem that omits the traveltime as well.

4.2 Reformulation of CHASE-R

Because CHASE-R is NP-hard, we seek approximationalgorithms to solve it efficiently. In the following, we

reformulate CHASE-R as a monotone submodular functionmaximization problem subject to constraints including apartition matroid constraint. Before detailing the reformula-tion, we present some necessary definitions.

Definition 4.2 [50]. Let S be a finite ground set. A real-valuedset function f : 2S 7! R is normalized, monotonic and sub-modular if it satisfies the following three conditions: (i)fð;Þ ¼ 0; (ii) fðA [ fegÞ � fðAÞ � 0 for any A � S ande 2 SnA; and (iii) fðA [ fegÞ � fðAÞ � fðB [ fegÞ � fðBÞfor any A � B � S and e 2 SnB.For simplicity, we use fAðeÞ ¼ fðAþ eÞ � fðAÞ to denote

the marginal value of element e with respect to A. Note thathere, we use Aþ e instead of A [ feg.Definition 4.3 [50]. Given thatS ¼ S k

i¼1S0i is the disjoint union

of k sets and l1; . . . ; lk are positive integers, a partition matroidM¼ ðS; IÞ is a matroid in which I ¼ fX 2 S : jX \ S0ij � lifor i ¼ 1; 2; . . . ; kg.Denote by aij the activating time slot aij of sensor vi. We

define the ground set S as:

S ¼ fa11; a12; . . . ; a1L; . . . ; am1; am2; . . . ; amLg: (7)

We equivalently define the sensor schedule Si as asubset of S, namely Si ¼ fai10 ; ai20 ; . . . ; aiL0g if and only ifaij0 ¼ 1 ðj0 ¼ 10; 20; . . . ;L0Þ. Furthermore, S can be partitionedinto m disjoint sets, S01; S

02; . . . ; S

0m, where S0i ¼ fai1; ai2; . . . ;

aiLg. We say that S0i is the candidate activation schedule of sen-sor vi, since any feasible schedule Si is a subset of S0i. It isclear that any scheduling policy X consisting of all the sen-sor schedules, namely X ¼ fS1; S2; . . . ; Smg, is subject tojX \ S0ij ¼ jSij � li. Thus, we write the independent sets as:

I ¼ fX � S : jX \ S0ij � li for i ¼ 1; 2; . . . ;mg: (8)

Note that it is easy to prove thatM¼ fS; Ig is a matroid.Moreover, define cij ¼ ci as the charging time factor for

time slot aij. The working time constraint can be rewrittenasP

aij2X cij � tw, which is exactly a knapsack constraint.Hence, we have the following lemma.

Lemma 4.2. The working time constraint in CHASE-R can bewritten as a knapsack constraint on the ground set S, while theactive time slot constraint can be written as a partition matroidconstraint.

Consequently, we can rewrite the optimization problemCHASE-R as RP1 shown below.

ðRP1Þ maxX

fðXÞ ¼Xni¼1

wiQðij_vj2Vi

SjÞ

s:t: X 2 I ;Si ¼ X \ S0i 8i ¼ 1; 2; . . . ;m;Xaij2X

cij � tw:

Note that the decision variable in RP1 is the scheduling pol-icy X, which consists of elements aijs denoting the activa-tion time slot aij of sensor vi. By comparing RP1 with P1,we can see that the decision variables change from the

DAI ET AL.: CHASE: CHARGING AND SCHEDULING SCHEME FOR STOCHASTIC EVENT CAPTURE IN WIRELESS RECHARGEABLE... 49

activation schedules Si in P1 to the scheduling policy X inRP1, and the two are essentially equivalent.

Next, we show that the optimization function fðXÞexhibits a desirable property as stated in the followinglemma.

Lemma 4.3. If the utility function UðxÞ is concave, then theobjective function fðXÞ in RP1 is a monotone submodularfunction.

Proof. To prove the monotonicity and submodularity of theobjective function fðXÞ, we have to verify if the three con-ditions in Def. 4.2 hold for fðXÞ.

First, it is easy to see that fð;Þ ¼ 0, which means thatthe first condition holds for fðXÞ.

Second, we check whether the monotonicity propertyholds for fðXÞ. Suppose we have a set A � S and an ele-ment e1 2 SnA and e1 ¼ aij. We can then regardfðAþ e1Þ as the resulting overall QoM obtained by acti-vating the time slot aij of sensor vi based on the originalscheduling policy as far as A is concerned. As a result,the equivalent monitoring schedule of PoI ok, which iscovered by vi (ok 2 Oi), may be changed accordingly.Specifically, suppose the original and changed equiva-

lent monitoring schedules of ok are bS<A>k and bS<Aþe1 >

k ,

respectively. Suppose the time slot akj is activated forbS<Aþe1 >k . In addition, we use the expression aij 2 bS<A>

k

to indicate that the jth time slot of bS<A>k is active,

namely akj ¼ 1 and aij 62 bS<A>k the opposite. In reality,

both aij 2 bS<A>k and aij 62 bS<A>

k are possible. However,

due to limited space, we consider only the case

aij 62 bS<A>k , since it is more complicated.

We have the following important observation:

bS<Aþe1 >k ðxÞ � bS<A>

k ðxÞ ¼1; x 2 ½ðkL þ j� 1Þtts; ðkL þ jÞtts�

ðk 2 NÞ0; otherwise

8><>: ;

(9)which immediately leads to:Z y

t

bS<Aþe1 >k ðxÞdx�

Z y

t

bS<A>k ðxÞdx � 0 (10)

for any y � t � 0.Next, we note that any utility function UðxÞ must

increase monotonically from zero to one as a functionof the total observation time, i.e., UðxÞ � 0 andUðyÞ � UðxÞ � 0 for any y � x � 0. Hence, combiningEqs. (5) and (10), we have:

QðkjbS<Aþe1 >k Þ �QðkjbS<A>

k Þ

¼ 1

Ltts

Z Ltts0

Z 1t

"U

Z y

t

bS<Aþe1 >k ðxÞdx

!� U

Z y

t

bS<A>k ðxÞdx

!#� fðy� tÞdydt�0:

Therefore,

fAðe1Þ ¼Xok2Oi

wk

hQðkjbS<Aþe1 >

k Þ �Qðkj bS<A>k Þ

i� 0;

which means that the monotonicity property holdsfor fðXÞ.

Third, we check the last condition for fðXÞ. Similar tothe monotonicity property analysis, suppose we have setA � B � S and element e1 2 SnB (e1 ¼ aij). The originalequivalent monitoring schedules for ok in A and B arebS<A>k and bS<B>

k , and they respectively change tobS<Aþe1 >k and bS<Bþe1 >

k after adding the element e1. To

save space, we consider only the case of aij 62 bS<A>k and

aij 62 bS<B>k in this paper.

Before the detailed proof, we show an important prop-erty for the utility function UðxÞ. For any y � x � 0 andd � 0, we have:

Uðxþ dÞ � y� x

yþ d� xUðxÞ þ ð1� y� x

yþ d� xÞUðyþ dÞ;

(11)and

UðyÞ � d

yþ d� xUðxÞ þ ð1� d

yþ d� xÞUðyþ dÞ: (12)

Note that x � xþ d � yþ d and x � y � yþ d, and UðxÞ isconcave. Adding up the left sides and the right sides ofEqs. (11) and (12), and simplifying the inequality, we have:

Uðxþ dÞ � UðxÞ � Uðyþ dÞ � UðyÞ (13)

for y � x � 0 and d � 0.Since A � B, we have:

bS<B>k ðxÞ � bS<A>

k ðxÞ � 0 (14)

for x 2 ½0;þ1�.Moreover, it is easy to see that:

bS<Aþe1 >k ðxÞ � bS<A>

k ðxÞ ¼ bS<Bþe1 >k ðxÞ � bS<B>

k ðxÞ (15)

as aij 62 bS<A>k and aij 62 bS<B>

k for x 2 ½0;þ1�.Therefore, from Eqs. (13), (14), and (15), it is obvious

that:"QðkjbS<Aþe1 >

k Þ �QðkjbS<A>k Þ

#�"QðkjbS<Bþe1 >

k Þ �QðkjbS<B>k Þ

#

¼ 1

Ltts

Z Ltts0

Z 1t

(hUðZ y

t

bS<Aþe1 >k ðxÞdxÞ � Uð

Z y

t

bS<A>k ðxÞdxÞ

i�hUðZ y

t

bS<Bþe1 >k ðxÞdxÞ � Uð

Z y

t

bS<B>k ðxÞdxÞ

i)fðy� tÞdydt

�0:

and:

fAðe1Þ � fBðe1Þ

¼Xok2Oi

wk

(hQðkjbS<Aþe1 >

k Þ �QðkjbS<A>k Þ

i

�hQðkjbS<Bþe1 >

k Þ �Qðkj bS<B>k Þ

i)� 0:

50 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 19, NO. 1, JANUARY 2020

We thus conclude that the third condition holds for fðXÞas well, and the result follows. tuIn general, utility functions can be concave (e.g., the Step,

Exponential, and Linear functions in [17]) or not (e.g., theS-shaped and Delayed Step functions [17]). For ourpurposes, we consider only utility functions UðxÞ that areconcave hereafter. The utility functions of many real-worldapplications appear to be concave [17], [18], [19], [20], [21].Hence, our contributions are not diminished significantly.

4.3 Approximation Algorithms for CHASE-R

Having proved that the objective function of our problemis submodular, we now aim to find approximation algo-rithms with and without the active time slot constraint forCHASE-R. We will show that the presence of the activetime slot constraint makes the problem significantly morecomplex.

Algorithm 1. Basic Algorithm for CHASE-R with ActiveTime Slot Constraint

Input: The objective function fð�Þ, the ground set S, the parti-tion matroidM, the knapsack constraint, the candidateactivation schedules S01; . . . ; S

0m.

Output: SolutionX and the sensor schedules S1; . . . ; Sm.1: Reduce knapsack constraint by applying Lemma 4.4

with 0 < � < 0:5; let fPtgTt¼1 denote the resulting par-tition matroids;

2: for each t 2 ½T � do3: Run the greedy algorithm from [51] under 2 partition

matroidsM and Pt to obtain solutionXt;4: end for5: t arg maxTt¼1fðXtÞ;6: Starting with the trivial partition of Xt into single ele-

ments, greedily merge parts as long as each part satis-fies the knapsack constraint, until no further merge ispossible. Consequently, Xt can be partitioned into kparts fXj

tgkj¼1;7: X arg maxkj¼1fðXj

t Þ, Si X \ S0i for i ¼ 1; . . . ; m;

4.3.1 Approximation Algorithm with Active Time Slot

Constraint

For this case, we tailor the approach proposed by Guptaet al. [52] to our settings, and obtain an improved approxi-mation. Their work targets p-system and q-knapsack inmax-min optimization, where a p-system is similar to, butmore general than, the intersection of p matroids. At a highlevel, their approach extends ideas from Chekuri andKhanna [53] that reduce knapsack constraints to partitionmatroids by an enumeration method. We list the main resultof this reduction as follows.

Lemma 4.4. Given any knapsack constraintPn

i¼1 wi � xi � Band fixed 0 < � < 1, there is a polynomial-time computable

collection P1; . . . ;PT of T ¼ nOð1=�2Þ partition matroids suchthat:

1. For every X 2 S Tt¼1Pt, we have

Pi2X wi �

ð1þ �Þ �B.

2. fX � ½n�jPi2X wi � Bg � S Tt¼1Pt.

Note that we use notations similar to [52] for consistency.We assume thatV is the intersection of the partition matroidand knapsack constraints. By scaling weights in the knap-sack constraint, we assume without loss of generality thatthe knapsack has capacity exactly one. Let C denote theweights in the knapsack constraint. Assume that the opti-mal QoM of CHASE-R is OPT and its corresponding solu-tion isXOPT .

We propose the algorithm specified in Algorithm 1,which is devised based on the algorithm proposed in [52].

Theorem 4.2. Algorithm 1 for CHASE-R with active time slotconstraint can achieve 1=6-approximation, and its time com-plexity is OððmLÞ2nT Þ.

Proof. We omit the details of the proof to save space. tuWe improve the approximation factor from 1

ðpþ2Þð3qþ1Þ ¼1=12, obtained by [52] for p-system and q-knapsack con-straints, to 1=6. This is because we give a tighter bound forthe number of partitioned parts k at Step 6 in Algorithm 1than that in [52]. Besides, although the algorithm proposedin [54] for 1 matroid and k knapsack constraints can achieve

an ð1� 1=e� "Þ-approximation, it requires that all the sets

of at most 1012 items to be enumerated to form a feasiblesolution at the first stage, which limits its practicality.

Moreover, we employ pruning techniques when imple-menting this algorithm to speed up the computation, since

the number T ¼ ðmLÞOð1=�2Þ of produced partition matroidsis still large. We omit the details to save space.

4.3.2 Enhanced Approximation Algorithm with Active

Time Slot Constraint

In this section, we propose an enhanced algorithm toaddress CHASE-R with active time slot constraint. Theenhanced algorithm is based on the algorithm proposedin [55] for maximizing a submodular function subject toa p-system and q-knapsack constraints. Compared withAlgorithm 1, this algorithm achieves not only better perfor-mance guarantee, but it is also faster. Algorithm 2 specifiesthe enhanced algorithm.

By the classical results in [55], we have the followingtheorem.

Theorem 4.3. Algorithm 2 for CHASE-R with active time slotconstraint can achieve 1=ð4þ �Þ-approximation where � is anarbitrarily small positive value, and its time complexity isOðn � mL

�2log 2 mL

� Þ.Proof. We omit details of the proof to save space. tu

4.3.3 Approximation Algorithm without Active Time Slot

Constraint

If li � L for any sensor vi, then the active time slot constraintcan be safely relaxed. This situation occurs when the batterycapacity is large relative to the working power of the sensor(e.g., ultra-capacitors [40]) or we apply the charging processfrequently. In this case, we can resort to a unified greedyalgorithm, namely Algorithm 3, to find an optimized QoM.Note that in this algorithm, c0i refers to the correspondingcharging time factor for time slot a0i.

DAI ET AL.: CHASE: CHARGING AND SCHEDULING SCHEME FOR STOCHASTIC EVENT CAPTURE IN WIRELESS RECHARGEABLE... 51

Algorithm 2. Enhanced Algorithm for CHASE-R withActive Time Slot Constraint

Input: The objective function fð�Þ, the ground set S, the parti-tion matroid M¼ ðS; IÞ, the knapsack constraint, thecandidate activation schedules S01; . . . ; S

0m, and the error

threshold �.Output: SolutionX and the sensor schedules S1; . . . ; Sm.1: u maxe2SfðeÞ;2: for r 2 fu2 ; ð1þ �Þ u2 ; ð1þ �Þ2 u

2 ; . . . ;mLug do3: z ur maxffðeÞ : fðeÞce

� rg; " ce is the corre-

sponding coefficient of element e;

4: S0 ¼ ;;5: while z � �

mLur and

Pe2S0 ce � 1 do

6: for each e 2 S do7: if S0 þ e 2 I , fðS0 þ eÞ � fðS0Þ � z, and

fðS0þeÞ�fðS0Þce

� r then

8: S0 S0 þ e;9: if

Pe2S0 ce > 1 then

10: S0r S0,Xr S0nfeg,X0r feg;11: continuewith the next value of r;12: end if13: end if14: end for15: z 1

1þ� z;

16: end while17: Xr S0r S0,X0r ;;18: end for19: X arg maxrXr, Si X \ S0i for i ¼ 1; . . . ; m;

This algorithm includes two parts. The first part enumer-ates all possible subsets of S with cardinality less than orequal to k0, so as to find the best feasible solution for thehighest QoM. The second part starts from every feasiblesubset D with cardinality k, and searches greedily in S tofind a best possible solution. Finally, the algorithm outputsthe best observable solution based on the results of theabove two parts.

We have the following theorem based on the resultsobtained by [56].

Theorem 4.4. Algorithm 3 for CHASE-R without active time slotconstraint achieves approximation factors of 1�1=e

2 0:3161,1�1=e2�1=e 0:3873, 1�1=e

3=2�1=e 0:5584, 1� 1=e 0:6321 for k ¼0; 1; 2; 3, respectively. Its time complexity isOððmLÞkþ2nÞ.

Proof.We omit details of the proof to save space. tuBy Theorem 4.1, we claim that Algorithm 3 for k ¼ 3 is in

fact the best possible for any polynomial-time approachunless P ¼ NP .

4.4 Approximation Algorithms for CHASE

Based on the proposed constant approximation algorithmsfor CHASE-R, we now consider the original problemCHASE and propose approximation algorithms for it.

As shown in Algorithm 4, the solution calls Algorithm 1at the first step to obtain a feasible solution XR for CHASE-R. Subsequently, we sort the elements in XR in descendingorder by their cost efficiency in event monitoring, definedas the ratio of the overall QoM enhancement yielded by agiven active time slot to the charging time required for an

MC to enable that time slot to be active.We iteratively removean element with the least cost efficiency inX (X is initializedas XR) until tTSP ð

SjX\S0

ij> 0viÞ � tw �

Pa0i2X c0i. Note that

we employ the nearest neighbor algorithm to solve the TSPproblem. Finally, we obtain a feasible solutionX for CHASE.

Algorithm 3. Unified Algorithm for CHASE-R withoutActive Time Slot Constraint

Input: The objective function fð�Þ, the ground set S, the knap-sack constraint, the candidate activation schedulesS01; . . . ; S

0m.

Output: SolutionX and the sensor schedules S1; . . . ; Sm.1: X ;,X1 ;,X2 ;, Si ; for i ¼ 1; . . . ; m;2: If k ¼ 0, then k0 1; else k0 k� 1;3: X1 arg maxfðDÞ, 8D 2 S, jDj � k0,

Pa0i2D c0i � tw;

4: for allD 2 S (jDj ¼ k andP

a0i2D c0i � tw) do

5: I S;6: while InD 6¼ ; do7: a0t arg maxa0

i2InD

fDða0iÞc0i;

8: if fDða0tÞ � 0 then9: break;10: end if11: if

Pa0i2D c0i þ c0t � tw then

12: D D [ fa0tg;13: else14: I Infa0tg;15: end if16: end while17: if fðX2Þ � fðDÞ then18: X2 D;19: end if20: end for21: X arg maxffðX1Þ; fðX2Þg, Si X \ S0i for

i ¼ 1; . . . ;m;

Algorithm 4. Algorithm for CHASE

Input: The sensors set V ¼ fv1; . . . ; vmg, the PoIs set O ¼fo1; . . . ; ong, the objective function fð�Þ, the ground setS, the partition matroid M, the knapsack constraint,the candidate activation schedules S01; . . . ; S

0m.

Output: The sensor schedules S1; . . . ; Sm.1: Call Algorithm 1 to obtain the solutionXR for CHASE-R;2: Sort XR fa01; . . . ; a0Kg such that a0t ¼ arg maxa0

i2XRnXt�1

RfXt�1Rða0

iÞ

c0i

(Xt�1R ¼ fa01; . . . ; a0t�1g);

3: X XR, t K;4: while tTSP ð

SjX\S0

ij> 0viÞ > tw �

Pa0i2X c0i do

5: X Xna0t;6: t t� 1;7: end while8: Si X \ S0i for i ¼ 1; . . . ;m;

Fig. 4 illustrates an instance of Algorithm 4. Supposeafter employing Algorithm 1, the schedules for Sensors 1,2, 3, and 4 are (0,1,1), (1,1,1), (1,0,1), and (0,0,0), respec-tively, as demonstrated in Fig. 4(a). Meanwhile, the travelpath passing by Sensors 1, 2, and 3 that have non-emptyactive time slots is shown by the grey dashed lines, sincethe available time left for the MC after charging Sensors1, 2, and 3 cannot allow the MC to continue traveling

52 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 19, NO. 1, JANUARY 2020

along any edge of the path. After sorting the elements inXR at Step 2, Algorithm 4 finds that the second activetime slot of Sensor 1 has the minimum cost efficiency. Itthus removes that active time slot, and the time savedthen allows the MC to travel from the BS to Sensor 2, asillustrated by the dark dashed line in Fig. 4b. Likewise,Fig. 4c shows the result after removing the first activetime slot of Sensor 3 which has the second smallest costefficiency, and using the saved time for the MC’s furthertravel. Finally, in Fig. 4d, the third time slot of Sensor 1 isde-activated; consequently, the MC is able to charge allthe sensors and finish the charging path within the timeconstraint, which means that tTSP ð

SjX\S0

ij> 0viÞ � tw �P

a0i2X c0i is satisfied. Note that the MC no longer needs to

pass by Sensor 1 because it has no active time slot. Bydoing so, we obtain a feasible solution for the originalCHASE problem.

Theorem 4.5. Algorithm 4 for CHASE based on Algorithm 1

achieves 16 ð1�

tTSP ðV Þþmaxfcigmi¼1tw

Þ-approximation. The time

complexity of this algorithm is OððmLÞ2nT Þ.Proof.We omit details of the proof to save space. tu

Note that the literature has recent results that considersubmodular optimization with routing constraints [57], [58].However, they cannot be used to solve our problem becausetheir solutions only address routing constraints, whereasours must address one more partition matroid constraintand one more knapsack constraint.

Where there is no confusion, we call Algorithm 4, whichworks based on Algorithm 1, the CHASE algorithm. We canalso base our algorithm (for the CHASE problem) on theenhanced algorithm for CHASE-R with active time slot con-straint. This algorithm differs from Algorithm 4 in that itcalls Algorithm 2 rather than Algorithm 1 at Step 1. We callthis alternative algorithm E-CHASE. We have the followingtheorem.

Theorem 4.6. Algorithm 4, which works based on Algorithm 2,

achieves 14þ� ð1�

tTSP ðV Þþmaxfcigmi¼1tw

Þ-approximation. Its time

complexity is Oðn � mL�2

log 2 mL� þm3LÞ.

Proof.We omit the details of the proof to save space. tuLastly, if the active time slot constraint is not needed, we

can modify Algorithm 4 by replacing Algorithm 1 (called atStep 1) with Algorithm 3. The following theorem gives aperformance guarantee of this revised algorithm.

Theorem 4.7. The revised algorithm for CHASE without activetime slot constraint achieves approximation factors of 1�1=e

2 c,1�1=e2�1=e c,

1�1=e3=2�1=e c, ð1� 1=eÞc for k ¼ 0; 1; 2; 3, respectively,

where c ¼ 1� tTSP ðV Þþmaxfcigmi¼1tw

. Its time complexity is

OððmLÞkþ2nþm3LÞ for k ¼ 0; 1; 2; 3.

Proof. We omit details of the proof to save space. tu

5 PERFORMANCE EVALUATION

In this section, we present simulation results that verify ouranalysis and illustrate the performance of our algorithms.

5.1 Evaluation Setup

Unless otherwise stated, we use the following parametersettings. We set the range of received power of a sensor to½15mW; 45mW �, which can be interpreted as a charging effi-ciency hi between ½0:5%; 1:5%�. In our experiments, we ran-domly distribute 20 sensors and 50 PoIs in a 120 m� 120 mregion, where any PoI is covered by at least one sensor. Theworking power pi of sensor vi is randomly selected from½50mW; 100mW �, while that of the MC is set to 3W . The bat-tery capacity of a sensor is randomly selected from therange ½100J; 1000J �. Furthermore, we set the sensing radiusof a sensor to 20 m and the sensor schedule length L ¼ 4.We assume that the considered event type has a Step utilityfunction and its event staying times follow fðxÞ ¼ �e��x

where � ¼ 1. We set t ¼ 2 week, tw ¼ 8:2 hour, and theMC’s speed nMC ¼ 0:05 m=s. Lastly, the default duration ofa time slot is set to be 1 s.

5.2 Baseline Setup

Because there are no existing algorithms for joint mobilecharging and scheduling of sensors for stochastic event cap-ture in a wireless rechargeable sensor network, we developtwo algorithms for comparison, i.e., Joint Periodic Wake-up(JPW) algorithm and RANDOM algorithm. The first algo-rithm is obtained by specializing the Joint PeriodicWake-up)algorithm in [24]. Specifically, an MC in JPW distributes itscharging time evenly to each sensor; only when it arrives atthe position of a sensor can it charge the sensor (this assump-tion is realistic in that the effective charging distance ofchargers is typically far less than the distances between sen-sors). The energy cost for turning the sensor on/off isignored, as before. In addition, we set hi ¼ 1% for any sensorvi, and the time duration of a duty cycle in JPW is exactlyequal to that of the sensor schedule. nMC is set to 0:1 m=s. For

Fig. 4. An instance of Algorithm 4 for CHASE.

DAI ET AL.: CHASE: CHARGING AND SCHEDULING SCHEME FOR STOCHASTIC EVENT CAPTURE IN WIRELESS RECHARGEABLE... 53

the other parameters, we use the same settings as those inSection 5.3.3. The second algorithm RANDOM differs fromJPW in that it randomly chooses for charging the same num-ber of sensors as that for CHASE, and then uniformly distrib-utes the charging time among the selected sensors. Note thatevery point on the plots for RANDOM represents an averageresult over 100 randomly generated instances.

5.3 Performance Evaluation for CHASE-RAlgorithms

In this section, we first investigate the cases without consid-ering the active time slot constraint. In particular, we evalu-ate the overall QoM under different event types or differentvalues of the control parameter k. Then, we study the rela-tionship between the period of the charging process t andthe overall QoM. This relationship shows the impact of theactive time slot constraint.

5.3.1 Impact of Event Types

In this set of experiments, we focus on the event typeswhose event staying times follow fðxÞ ¼ �e��x (� ¼0:25; 0:5; 1; 2) [17], [24], whereas the utility function UðxÞ iseither the Step utility function or the Exponential utilityfunction fðxÞ ¼ Ae�Ax, where A ¼ 5 [17]. Note that we useAlgorithm 3 with k ¼ 3. It can be seen in Figs. 5 and 6 thatthe overall QoM always increases as tw increases. However,the marginal gain of the QoM decreases as tw increases. Thereason is that the event capture utility function is concaveand redundant coverage of the PoIs becomes more likelywhen the sensors have larger active time slot budgets under

a larger tw. Moreover, a smaller � will lead to a larger over-all QoM under the same tw. This is because the expectedstaying time of events grows as � decreases; therefore, itsprobability of being detected, as well as the utility of sens-ing, increases. Besides, by comparing Figs. 5 and 6, we seethat the achieved overall QoM for events with Step utilityalways exceeds that for events with Exponential utility. Thiscan be explained by differences in the efficiency of eventcapture. For events under Step utility, full informationabout an event is obtained instantaneously on detection. Incontrast, it can require a lot more time to obtain most infor-mation about an event under Exponential utility.

5.3.2 Impact of Control Parameter k

Weproceed to evaluate the impact of the control parameter k onthe overall QoM in Algorithm 3, and plot the results in Fig. 7.Not surprisingly, it can be seen that the larger k we choose, thehigher overall QoM we obtain. However, the differencesbetween the overall QoM under different k are not obvious.This observation suggests thatwe can choose a small k to reducethe time complexity without incurring much performance deg-radation. In addition, it can be observed that the overall QoMexceeds 1� 1=e 0:63 in Fig. 7, which is consistent withTheorem4.4 as the optimal overall QoMcannot exceed 1.

5.3.3 Impact of Period of Charging Process

To see how the period of the charging process t impacts theoverall QoM, we set Ei ¼ 100J and pi ¼ 100mW , and letthe received power of each sensor randomly fluctuatewithin a relatively smaller range of ½20mW; 35mW � to easecomputation. Fig. 8 shows the trend that the overall QoMdecreases with an increasing t. This is because an increasingt leads to higher charging time factors ci and smaller activetime slot budgets li, both of which finally lead to a reducedQoM. Moreover, that the overall QoM is larger than1=6 0:17 is consistent with Theorem 4.2. Again, we cansee that the achieved overall QoM increases with tw.

5.4 Performance Evaluation for the CHASEAlgorithms

We proceed to verify the performance of the algorithms forCHASE, which consider the MC’s travel time. The experi-ments use the same parameters as in Section 5.3.3.

Fig. 5. QoM versus tw for Step utility function.

Fig. 6. QoM versus tw for Exponential utility function.

Fig. 7. Impact of control parameter k.

54 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 19, NO. 1, JANUARY 2020

5.4.1 A Solution to an CHASE Instance

Fig. 9 illustrates the selected sensors for charging, the sensorschedules, the MC’s travel path, and the achieved QoM forthe PoIs, for an E-CHASE solution to a CHASE probleminstance. Note that the BS is located at (0,0); the sensors aremarked as circles and the PoIs as triangles. The filled color ofa triangle indicates the achieved QoM for the correspondingPoI, which varies from 0 to 1 as the color bar shows. A bluecircle indicates that the corresponding sensor is not chosen atStep 1 in Algorithm 4, while a green circle indicates that thecorresponding sensor is greedily removed in thewhile loop.

5.4.2 Impact of MC’s Speed on Working Time

Allocation and QoM

Fig. 10 shows that if the speed of the MC nMC increases, thetravel time is reduced, leading to a larger aggregate time for

charging. Note that both the travel time and aggregatecharging time are normalized with respect to the maximumworking time tw. It can be seen that the fraction of the aggre-gate travel time keeps below 10 percent when nMC grows to0:1 m=s, which is still small. Another interesting findingfrom Fig. 10 is that the sum of the travel time and aggregatecharging time is not necessarily equal to tw (the gap is up to4 percent when nMC ¼ 0:02). This situation happens becausewe require the active time slot budget li for each sensor tobe an integer. The requirement can be relaxed, and we canassign the residual working time to charging sensors. Weexpect the overall QoM to increase as a result, but the detailsare left for future work. As expected, the overall QoM isenhanced with a faster MC, as the red solid line shows inFig. 11. Moreover, it is always bigger than the “maximum”lower bound, which is indicated as the green dotted line

“MLB” in the figure and given by 16 ð1�

tTSP ðV Þþmaxfcigmi¼1tw

Þ.This finding corroborates Theorem 4.5 in Section 4.4.

5.4.3 Impact of Length of Sensor Schedule

Fig. 12 shows the trend of the overall QoM when the lengthof the sensor schedule L increases under tw ¼ 4:1 hours andtw ¼ 8:2 hours, respectively. We can see that, although theoverall QoM does not increase monotonically with Lexactly, there is a general tendency for it to rise with L.

5.5 Performance Comparison with Existing Work

5.5.1 Impact of Length of Time Slots

The default duration of a time slot is 1 s in the experi-ments so far. In this subsection, we vary the time slotduration, and plot the overall QoM for both JPW andCHASE in Fig. 13. Note that the proposed E-CHASE and

Fig. 8. QoM versus t.

Fig. 9. A solution to a CHASE problem instance.

Fig. 10. Impact of MC’s speed on working time allocation.

Fig. 11. Impact of MC’s speed on overall QoM.

Fig. 12. QoM versus length of sensor schedule.

DAI ET AL.: CHASE: CHARGING AND SCHEDULING SCHEME FOR STOCHASTIC EVENT CAPTURE IN WIRELESS RECHARGEABLE... 55

CHASE algorithms consistently outperform JPW andRANDOM, especially when the time slot duration is long.The performance gains of E-CHASE and CHASE overJPW and RANDOM are 40.8 and 53.5, and 34.0 and46.1 percent, respectively. Moreover, the overall QoMincreases as the time slot duration decreases for all thefour schemes. The reason is that information about anevent with the same staying time becomes more likely tobe captured in its early phase, as the time intervalbetween successive active time slots shrinks. This result isconsistent with the analysis in [17] and [24].

5.5.2 Impact of MaximumWorking Time

In Fig. 14a, we observe that the overall QoM rises with anincreasing maximum working time tw for all the fourschemes, and that both E-CHASE and CHASE outperformJPW or RANDOM with respect to achieved QoM. More-over, the performance gains of E-CHASE and CHASE overJPW or RANDOM become more significant as tw increases.They achieve improvements of about 50.6 and 82.1, and 49.9and 81.3 percent, respectively, when tw ¼ 9:4 hours. Thetime slot duration here is set to 1 s. When the duration isreduced to 0:5 s, the overall QoM of all the schemes are sub-stantially enhanced, as illustrated in Fig. 14b. On average,the performance gains by E-CHASE and CHASE overJPW and RANDOM are about 28.8 and 35.0, 24.3 and30.1 percent, respectively.

Next, we compare the proposed algorithms with JPWand RANDOM in terms of energy for the second case. Asan MC following JPW needs to always visit all the sen-sors, the energy overhead for travel can be large. Simi-larly, although an MC following RANDOM visits the

same number of sensors as CHASE, RANDOM choosesthese sensors randomly, without trying to manage theexpected travel energy, and its energy consumption canstill be high. It can be seen from Fig. 15 that on averagethe energy consumptions for JPW and RANDOM are 37.1and 37.3 percent higher than that of E-CHASE, and 38.9and 39.4 percent higher than that of CHASE. Further-more, with a large tw, the MC under E-CHASE or CHASEis able to include more sensors for charging. Fig. 15shows that the resulting energy increment can be substan-tial. Lastly, we evaluate the energy efficiency defined asthe ratio of the overall QoM to the total energy consump-tion. Fig. 16 demonstrates that E-CHASE obtains averagegains of 30.0 and 27.2 percent over JPW and RANDOM,respectively, whereas the corresponding numbers forCHASE are 35.7 and 32.6 percent.

Fig. 13. QoM versus duration of time slots.

Fig. 14. QoM versus tw.

Fig. 15. Comparison of energy consumptions.

Fig. 16. Comparison of energy efficiency.

56 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 19, NO. 1, JANUARY 2020

6 CONCLUSION

We have solved the problem of QoM maximization when asensor network is used to monitor stochastic events, byjointly designing the sensors’ mobile wireless charging andactivation schedules. The problem has a general eventmodel that admits different utility functions and differentprobability distributions of the event arrival and stayingtimes. To solve the problem, we first tackled a relaxed ver-sion that ignored the MC travel overhead. We developedapproximation algorithms for this relaxed problem bytransforming it into a submodular function maximizationproblem, under the condition that the event utility functionwas concave. Based on solutions to the relaxed problems,we then developed approximation algorithms to solve theoriginal problem when the MC’s travel time overhead wasalso considered. Diverse simulation results verified the the-oretical analysis and illustrated the performance of the pro-posed algorithms relative to two comparison benchmarks.It is interesting for future research to improve the approxi-mation factors of the solutions and account for fairnessissues when covering the whole set of PoIs.

ACKNOWLEDGMENTS

This work is supported in part by the National Key R & DProgram of China under Grant No. 2018YFB1004704 and2018YFB0803400, in part by the National Natural ScienceFoundation of China under Grant No. 61502229, 61872178,61832005, 61672276, 61872173, 61321491, 61520106007, and61751211, in part by the Natural Science Foundation ofJiangsu Province under Grant No. BK20181251, in part bythe Fundamental Research Funds for the Central Universi-ties under Grant 021014380079, in part by China NationalFunds for Distinguished Young Scientists with No.61625205, Key Research Program of Frontier Sciences, CAS,No. QYZDY-SSW-JSC002, in part by NSF CNS 1526638, andin part by and S’pore MOE (SUTDT12017004).

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Haipeng Dai received the BS degree from theDepartment of Electronic Engineering, ShanghaiJiao Tong University, Shanghai, China, in 2010,and the PhD degree from the Department ofComputer Science and Technology, Nanjing Uni-versity, Nanjing, China, in 2014. His researchinterests include wireless charging, mobile com-puting, and data mining. He is a research assis-tant professor with the Department of ComputerScience and Technology, Nanjing University. Hisresearch papers have been published in many

prestigious conferences and journals such as ACM MobiHoc, ACMVLDB, ACM SIGMETRICS, IEEE INFOCOM, IEEE ICDCS, IEEE ICNP,ICPP, IEEE IPSN, the IEEE Journal on Selected Areas in Communica-tions, the IEEE/ACM Transactions on Networking, the IEEE Transac-tions on Mobile Computing, the IEEE Transactions on Parallel andDistributed Systems, the IEEE Transactions on Services Computing, andtheACMTransactions on Sensor Networks. He serves/ed as poster chairof the IEEE ICNP’14, TPC member of the IEEE ICNP’14, IEEE ICC’14-17, IEEE ICCCN’15-17, and the IEEE Globecom’14-17. He received theBest Paper Award from IEEE ICNP’15 and the Best Paper Award Candi-date from IEEE INFOCOM’17. He is a member of the IEEE and the ACM.

Qiufang Ma received the BS degree from theCollege of Information Engineering, NanjingUniversity of Finance and Economics, Nanjing,China, in 2016. She is working towards the mas-ter’s degree in the Department of Computer Sci-ence and Technology, Nanjing University. Herresearch interest includes wireless charging.

Xiaobing Wu received the BS and ME degreesin computer science from Wuhan University, in2000 and 2003, respectively, and the PhD degreein computer science from Nanjing University, in2009. He is with the Wireless Research Centre,University of Canterbury, New Zealand. He wasan associate professor with the Department ofComputer Science and Technology, NanjingUniversity. His research interests include wire-less networking and communications, internet ofthings, and cyber physical systems. His publica-

tions appeared in the IEEE Transactions on Parallel and Distributed Sys-tems (TPDS), the ACM Transactions on Sensor Networks (TOSN), theComputer Communications (ComCom), the Wireless Networks, IEEEINFOCOM 2015/2014, IEEE ICDCS 2014, IEEE MASS 2013, etc. Hereceived the Honored Mention Award in ACM MobiCom 2009 Demosand Exhibitions.

Guihai Chen received the BS degree in com-puter software from Nanjing University, in 1984,the ME degree in computer applications fromSoutheast University, in 1987, and the PhDdegree in computer science from the Universityof Hong Kong, in 1997. He is a professor anddeputy chair of the Department of Computer Sci-ence, Nanjing University, China. He was invitedas a visiting professor by many foreign universi-ties including the Kyushu Institute of Technology,Japan, in 1998, the University of Queensland,

Australia, in 2000, and Wayne State University, during September 2001to August 2003. He has a wide range of research interests with focus onsensor networks, peer-to-peer computing, high-performance computerarchitecture, and combinatorics. He is a member of the IEEE.

58 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 19, NO. 1, JANUARY 2020

David K. Y. Yau received the BSc degree from theChinese University of Hong Kong, and the MS andPhD degrees from the University of Texas at Austin,all in computer science.He has been a professor withthe Singapore University of Technology & Designsince 2013. Since 2010, he has been a distinguishedscientist with the Advanced Digital Sciences Centre,Singapore. He was an associate professor of com-puter science with Purdue University (West Lafay-ette). His research interests include cyber-physicalsystemandnetwork security/privacy, wireless sensor

networks, smart grid IT, and quality of service. He received an NSF CAREERaward, Best Paper award in 2017 ACM/IEEE IPSN, and 2010 IEEEMFI. Hispapers in 2008 IEEE MASS, 2013 IEEE PerCom, 2013 IEEE CPSNA, and2013ACMBuildSyswere Best Paper finalists. He serves as an associate edi-tor of the IEEE Transactions Network Science and Engineering. He was anassociate editor of the IEEE Transactions Smart Grid, Special Section onSmart Grid Cyber-Physical Security (2017), the IEEE/ACMTransactions Net-working (2004-09), and the Springer Networking Science (2012-2013); vicegeneral chair (2006), TPCco-chair (2007), andTPCarea chair (2011) of IEEEICNP; TPC co-chair (2006) and steering committee member (2007-09) ofIEEE IWQoS; TPC track co-chair of 2012 IEEE ICDCS; and organizing com-mitteemember of 2014 IEEESECON.He is a seniormember of the IEEE.

Shaojie Tang received the BS degree in radioengineering fromSoutheast University, P.R. China,in 2006 and the PhD degree from the Departmentof Computer Science, Illinois Institute of Technol-ogy, in 2012. He is an assistant professor with theDepartment of Information Systems, University ofTexas at Dallas. His main research interestsinclude wireless networks (including sensor net-works and cognitive radio networks), social net-works, security and privacy, and game theory. Hehas served on the editorial board of the Journal of

Distributed Sensor Networks. He also served as a TPC member of a num-ber of conferences such asACMMobiHoc, IEEE ICNP, and IEEESECON.He is amember of the IEEE.

Xiang-Yang Li received the bachelor’s degreefrom Tsinghua University, China, in 1995, and theMS and PhD degrees from the University of Illinoisat Urbana-Champaign, in 2000 and 2001, respec-tively. He was a professor with the Illinois Instituteof Technology. He is currently a professor and anexecutive dean of the School of Computer Scienceand Technology, University of Science and Tech-nology of China. His research interests includewireless networking, mobile computing, securityand privacy, cyber physical systems, and algo-

rithms. He has authored amonograph entitledWireless Ad Hoc andSensorNetworks: Theory and Applications and co-edited several books, includingthe Encyclopedia of Algorithms. He is a recipient of the China NSF Out-standing Overseas Young Researcher (B). He holds the EMC-Endowedvisiting chair professorship at Tsinghua University from 2014 to 2016. Hehas received the six best paper awards, and one best demo award. He isan editor of several journals and has servedmany international conferencesin various capacities. Hewas a fellow of the IEEE (2015) and anACMdistin-guished scientist (2015).

Chen Tian received the BS, MS, and PhD degreesfrom theDepartment of Electronics and InformationEngineering, Huazhong University of Science andTechnology, China, in 2000, 2003, and 2008,respectively. He is an associate professor with theState Key Laboratory for Novel Software Technol-ogy, Nanjing University, China. He was previouslyan associate professor with the School of Electron-ics Information and Communications, HuazhongUniversity of Science and Technology, China.From 2012 to 2013, he was a postdoctoral

researcher with the Department of Computer Science, Yale University. Hisresearch interests include data center networks, network function virtuali-zation, distributed systems, Internet streaming, and urban computing. Heis amember of the IEEE.

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DAI ET AL.: CHASE: CHARGING AND SCHEDULING SCHEME FOR STOCHASTIC EVENT CAPTURE IN WIRELESS RECHARGEABLE... 59