JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 CHASE: Charging and Scheduling Scheme for Sochasic Even Capure in Wireless Rechargeable Sensor Newors Haipeng 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 Absrac In his paper, we consider he scenario in which a mobile charger (MC) periodically ravels wihin a sensor newor o recharge he sensors wirelessly. We design join charging and scheduling schemes o maximize he Qualiy of Monioring (QoM) for sochasic evens, which arrive and depar according o nown probabiliy disribuions of ime. Informaion is considered capured if i is sensed by a leas one sensor. We focus on wo closely relaed research issues, i.e., how o choose he sensors for charging and decide he charging ime for each of hem, and how o schedule he sensors acivaion schedules according o heir received energy. We formulae our problem as he maximum QoM CHArging and SchEduling problem (CHASE). We firs ignore he MC s ravel ime and sudy he resuling relaxed version of he problem, which we call CHASE-R. We show ha boh CHASE and CHASE-R are NP-hard. For CHASE-R, we prove ha i can be formulaed as a submodular funcion maximizaion problem, which allows wo algorihms o achieve /6- and /( + ɛ)-approximaion raios. Then, for CHASE, we propose approximaion algorihms o solve i by exending he CHASE-R resuls. We conduc simulaions o validae our algorihm design. Index Terms mobile charging, scheduling, wireless rechargeable sensor newor, sochasic even capure, submodular opimizaion, approximaion algorihm. INTRODUCTION TRADITIONAL wireless sensor newors (WSNs) are consrained by limied baery energy ha powers he sensors. Their limied newor lifeime is considered a major deploymen barrier. Besides, in many applicaions sensors are locaed in hazardous or inaccessible areas such as volcanoes [], inside concree walls [], or a he boom of bridges [], which maes baeryswapping schemes [] [6] unsafe, infeasible, labor-inensive, or cosly. To exend he newor lifeime, many approaches have been proposed o harves environmenal energy such as solar [7], vibraion [8], and wind [9]. However, a limiaion of exising energy-harvesing echniques is ha i is highly dependen on he ambien environmen, which maes he harvesing rae highly unpredicable. The problem can be overcome by recen breahroughs in wireless power charging echnologies [], which allow energy o be ransferred from one sorage device o anoher H. Dai, Q. Ma, G. Chen, C. Tian are wih he Sae Key Laboraory for Novel Sofware Technology, Nanjing Universiy, Nanjing, China E-mail: haipengdai@nju.edu.cn, mg65@smail.nju.edu.cn, gchen@cs.sju.edu.cn, ianchen@nju.edu.cn X. Wu is wih Wireless Research Cenre, Universiy of Canerbury, New Zealand. E-mail: barry.wu@canerbury.ac.nz D.K.Y. Yau is wih he Informaion Technology Sysems and Design pillar, Singapore Universiy of Technology and Design, Singapore. E-mail: david yau@sud.edu.sg S. Tang is wih Naveen Jindal School of Managemen, Universiy of Texas a Dallas, 8 W. Campbell Road, Richardson, Texas, USA. E-mail: shaojie.ang@udallas.edu X. Li is wih School of Compuer Science and Technology, Universiy of Science and Technology of China, Hefei 6, China. E-mail: xiangyangli@usc.edu.cn Corresponding auhor: G. Chen (E-mail: gchen@cs.sju.edu.cn). Manuscrip received April 9, 5; revised Augus 6, 5. wirelessly wih reasonable efficiency. For example, magneic resonance coupling is shown o ransfer 6 was [] a an efficiency of 9% o abou % when he disance varies from.75 m o.5 m. Since wireless recharging can guaranee a required level of power supply, independen of he ambien environmen, and i is conacless, i has found many applicaions including smar grids [], body sensor newors [], and civil srucure monioring []. Because power chargers are expensive, i is generally no coseffecive o deploy a large number of hem saically for energy provisioning. Insead, exising pracical approaches focus on using a mobile charger (MC) o move around he sensors and charge hem in urn during a ravel schedule, for ass such as rouing [] [5] and gahering daa [6]. None of hese prior effors have solved he problem of sochasic even capure, however, in ey aspecs such as scheduling sensors duy cycles o maximize heir abiliy o capure ineresing evens of a probabilisic naure. Bu he problem is fundamenal in wireless sensor newor design, and i has received aenion for boh cases of radiional WSNs [7] [9] and wireless ambien-energy harvesing sensor newors [], []. In his paper, we opimize even capure in a newor of sensors wirelessly recharged by an MC. We assume ha sochasic evens arrive and depar according o nown ime disribuions. An even is said o be capured if i is sensed by a leas one sensor. Noe ha here are exising pracical sysem plaforms ha can enhance he performance of even monioring by wireless recharging. For example, he Wireless Idenificaion and Sensing Plaform (WISP) has been applied in individual aciviy recogniion, large-scale urban sensing [], [], and srucural healh monioring (SHM) []. In he SHM applicaion, he civil srucure Corresponding Auhor: G. Chen
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 is insrumened wih sensor nodes capable of being powered solely by energy ransmied wirelessly o hem by a mobile helicoper. Jiang e al. [] are he firs o exploi wireless power charging by MCs for efficien sochasic even capure. Their objecive is o joinly deermine he MCs movemen schedule and he sensors acivaion schedule o maximize he Qualiy of Monioring (QoM) [8], [], defined as he average informaion gained per even by he newor. They mae several simplifying assumpions: each sensor can only monior one Poin of Ineres (PoI), he charging ime for each sensor is idenical, he even saying ime follows an Exponenial disribuion, and all he sensors follow a simple (q, p) periodic schedule (i.e., he sensors monior he PoIs for q ime every p ime). We relax hese assumpions in his paper. In his paper, we consider he scenario in which an MC periodically ravels wihin a sensor newor and recharges he sensors wirelessly o enable hem o perform ass of sochasic even capure. We assume ha he MC repeas is recharge schedule every period of ime τ, and ha he schedule (couning boh he charging ime and ravel ime of he MC) mus complee wihin ime ( < τ). For example, he MC is carried by an UAV whose daily shif is from 9am o am only (here, = hour and τ = hour; ypically, τ can be a few wees or even longer), or he MC mus be wihdrawn for mainenance for some amoun of ime beween recharge schedules. We address wo closely relaed issues in he wireless recharging and even monioring. The firs is how o choose he sensors for recharging and furher decide he charging ime for each of hem, consrained by he MC s woring ime. The second is how o bes schedule he sensors acivaions based on heir received energy, considering ha nearby sensors may cover overlapping PoIs. Our goal is o joinly design a charging scheme for he MC and he sensors acivaion schedules o maximize he QoM of he sochasic even capure. We define our problem formally as he maximum QoM CHArging and SchEduling problem (CHASE). The coupling beween he MC s ravel ime and he sensor charging ime maes he problem highly challenging. Hence, we firs ignore he ravel ime and sudy he resuling relaxed version of CHASE, which we call CHASE-R. Then, based on he CHASE-R resuls, we develop soluions for he general CHASE problem. The main conribuions of his paper are as follows: We analyze he QoM of sochasic even capure, which admis he possibiliy ha he same PoI is moniored by muliple sensors. We formulae he CHASE and CHASE- R problems, and show ha boh are NP-hard. We reformulae CHASE-R as a monoone submodular funcion maximizaion problem under a special sufficien condiion. This reformulaion of CHASE-R allows wo algorihms o achieve /6-approximaion and /( + ɛ)- approximaion, respecively. Based on he CHASE-R resuls, we propose wo approximaion algorihms for CHASE, when he MC s ravel ime is considered. Imporanly, all of he proposed CHASE-R and CHASE soluions are sufficienly general o accommodae diverse acivaion schedules, even uiliy funcions, and probabiliy disribuions of he even saying imes. We conduc exensive simulaions o verify our analyical findings. Simulaion resuls show ha our schemes ouperform he sae of he ar. The remainder of he paper is organized as follows. Secion reviews relaed wor. In Secion, we presen preliminaries and a formal definiion of he CHASE problem, as well as is relaxed version CHASE-R. In Secion, we analyze he complexiy of he problems and reformulae a special case of he relaxed problem as a monoone submodular funcion opimizaion problem. Then, we presen approximaion algorihms for he relaxed problem and he original problem. Secion 5 presens exensive simulaions o verify our heoreical resuls. Secion 6 concludes. RELATED WORK In his secion, we mainly review relaed wor on mobile charging, which generally can be classified ino hree caegories. Firs, prior wor has invesigaed he energy efficiency of mobile chargers (MCs) [5] []. For example, Wang e al. [5] proposed o coordinae muliple MCs o minimize heir aggregae ravel disance while guaraneeing coninuous operaion of each sensor, such ha he overall energy efficiency is opimized. In a laer wor [6], heir goal is o maximize he difference beween he energy harvesed by all he sensors and he ravel energy expended by all he MCs. Zhang e al. [7] presened an opimal scheme for muliple energy-consrained MCs o charge a linear WSN, where he raio of energy received by all he sensors o he ravel energy expended by all he MCs is maximized. Dai e al. [], [] sudied he problem of using minimum number of MCs o eep heir joinly charged WSN running forever. Wu e al. [] oo sensor placemen ino consideraion while deermining charging plans of muliple MCs. Second, he service delay of MCs has been considered [] [6]. Fu e al. [] minimized he overall charging delay of a single MC by planning is charging roue and sraegy. He e al. [], [5] considered he charging problem when he charging requess of sensors arrive in a dynamic fashion. Their wor has been exended o scenarios where he sensors are also mobile [6]. Third, research has addressed newor performance issues under mobile charging, from perspecives such as daa rouing, daa collecion, sensing coverage, and even monioring [] [6], [], [7] []. For daa rouing, Tong e al. [] examined he impac of mobile charging on daa rouing and WSN deploymen. Their wor has been expanded [5] o address several realisic issues (e.g., he communicaion environmen is dynamic and unreliable, he charging capaciy of an MC is limied, and he sensors are heerogeneous) by joinly considering daa rouing for he sensors and he charging scheme for an MC. Use of mobile charging o improve daa collecion in WSNs has also received significan aenion. Shi e al. [], [7] applied a single MC o improve he daa collecion, while reducing he woring ime wihin a charging ime period. In [6], [8] [], MCs are used no only as energy providers bu also as daa collecors. Zhou e al. [] solved he problem of scheduling an MC o charge sensors o mainain - coverage in he newor a low cos for he MC. Jiang e al. [] are firs o invesigae he impac of mobile charging on efficien sochasic even capure. However, hey mae several simplifying assumpions, which limi he pracicaliy of heir resuls. We relax hese assumpions in our conference version of his paper []. Wu e al. [] proposed a scheme for scheduling muliple MCs o serve collaboraive ass of sensors. Besides, here also exis a large body of relaed wor on scheduling issues in wireless chargers newors where chargers are saic, such as [] [57]. We omi deailed descripion for hem as hey are quie differen from ours.
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 Symbol o i v i O i V i L τ s S i Ŝ i w i τ τ i P c p i E i η i c i l i ν MC Meaning TABLE : Noaions Targe i Sensor i Subse of PoIs covered by sensor v i Subse of sensors covering PoI o i Lengh of sensor schedule Time duraion of a single ime slo Acivaion schedule of sensor v i Equivalen monioring schedule for PoI o i Weigh of PoI o i Maximum woring ime in one charging period Time period of charging process Charging ime allocaed o sensor v i in one charging period Woring power of MC Woring power of sensor v i Baery capaciy of sensor v i MC s charging efficiency for sensor v i Charging ime facor for sensor v i Acive ime slo budge of sensor v i MC s speed PROBLEM STATEMENT. Newor Model We assume ha here are m sensors V = {v, v,..., v m } disribued over a wo-dimensional region, which cover n Poin of Ineress (PoIs) denoed by O = {o, o,..., o n }. Le O i represen he se of PoIs covered by sensor v i. In a dense sensor newor, ypically close-by sensors may cover common PoIs. Hence, we assume ha a arge, say o i, is covered by a subse of he sensors V i. A summary of he noaions in his paper is given in Table. To prolong he lifeime of sensors, a mobile charger (MC) periodically sars from a base saion (BS) and visis each of a seleced subse of he saic sensors V s V exacly once, in order o charge he sensors wirelessly. A he end of he charging schedule, he MC reurns o he BS. The oal woring ime of he charging schedule, including he ravel overhead and he charging ime for all he seleced sensors, mus no exceed. Furhermore, we assume ha he charging schedule is repeaed every fixed period of ime τ. Hence, he off-duy ime for he MC a he BS is a leas τ. Noe ha under such a charging scheme, sensors no chosen for charging will die evenually. This is accepable since our primary concern is o maximize he overall QoM under he consrain of limied charging ime, raher han fairness among charging all he sensors. We denoe he pah of he MC by P = (π, π,..., π Vs, π Vs +) where π = π Vs + = BS and {π i } Vs i= = V s. Denoe by ij he ime required for he MC o move beween sensor v i and v j. Suppose he movemen of he MC beween sensors is dicaed by a moion planning scheme predeermined by he BS and/or he MC. The moion planning is assumed o respec physical consrains, such as following accessible pahways, avoiding obsacles, obeying mechanical limis on speeds and urns, ec, bu is deails are ou of scope of his paper. We only require he moion planning o be predeermined and eep sable, such ha ij is nown a priori and i is fixed. To use as much ime for charging as possible, he MC should ravel on a shores pah P, given Vs + by arg min P i= πiπ i+, ha complees a circui of he sensors. For simpliciy, we assume ha such a pah always exiss. Finding he pah can hen be formulaed as a Traveling Salesman 5 o o o v v 5 5 v v o Mobile Charger (MC) Base Saion (BS) o v o 9 o v o 7 o 8 Fig. : Newor model Problem (TSP), which is NP-hard. We assume ha some good approximaion algorihm is used, and he approximae soluion is given by τ T SP (V s ). Moreover, we assume ha he MC spends τ i ime for recharging he baery of v i. Then, we have: τ T SP (V s) + v i V s τ i. () The above inequaliy gives he MC s woring ime consrain. In his paper, we assume a discree ime model for a sensor s schedule, where he duraion of a ime slo is fixed and given. Specifically, every sensor follows a periodic schedule of idenical lengh L (in ime slos). In each ime slo, a sensor can schedule iself o be acive or inacive. Hence, we can express he acivaion schedule of sensor v i by a vecor S i = (a i, a i,..., a il ), where a ij = indicaes ha he sensor is acive in slo j and a ij = indicaes he opposie. We assume ha he duraion of a ime slo, say τ s, is long enough such ha he energy cos of urning he sensor on/off can be ignored. The BS is responsible for deermining he overall charging scheme, including he MC s ravel pah, he charging ime allocaed for each sensor, and he acivaion schedules of he sensors. I disseminaes he acivaion schedules o he respecive sensors eiher by a pre-esablished muli-hop communicaion mechanism, such as he Collecion Tree Proocol (CTP) [58], or hrough he MC when he MC comes near each sensor for charging. We assume ha he energy cos of disseminaing he schedules can be ignored. This is because a schedule will no change excep for excepional evens such as node breadowns. Fig. shows an example of our newor model. In his example, 5 sensors alogeher cover PoIs. The MC chooses a subse of he sensors V s = {v, v, v, v } o charge, and is ravel pah is P = {BS, v, v, v, v, BS}. The acivaion schedule of sensor v in V s is S = (a, a, a, a ) = (,,, ), and ha for v, v, and v are S = (,,, ), S = (,,, ), and S = (,,, ), respecively.. Energy Consumpion Model Le P c denoe he woring power of he MC, and p i he woring power of sensor v i. Le η i denoe he MC s charging efficiency for sensor v i, i.e., he raio of he amoun of energy received by v i o he amoun of energy consumed by he MC. The charging efficiency can vary from sensor o sensor, and i depends on facors such as he disance beween he corresponding sensor and he MC and he effeciveness of he sensor s anenna. We assume ha he leaage power of each sensor is negligible, and each sensor will have used up is energy by he ime of is nex recharge (which o 5 o6
Uiliy JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 can be conrolled by properly allocaing he recharging ime of he MC). Therefore, in one charging period of duraion τ, he required woring energy for sensor v i under is acivaion schedule is p i Si L τ, and i should be equal o he aggregaed charged p energy for v i, i.e., P c τ i η i. Thus, we have i η τ S ip cl i = τ i. For convenience, we define he charging ime facor c i = p i η ip cl τ, which is consan; i can be inerpreed as he charging ime required for he MC o provide sufficien energy for v i o be acive for one ime slo. Hence, we have: c i S i = τ i. () Furher, denoe by E i he baery capaciy of sensor v i, and l i he maximum number of acive ime slos sensor v i can susain using is limied baery capaciy, which we call he acive ime slo budge. I is clear ha l i = Ei p iτ L. Because he oal acivaed ime slos in he acivaion schedule should no exceed he acive ime slo budge, we have: S i l i, () which we call he acive ime slo consrain. If l i L for any sensor v i, we can ignore he acive ime slo consrain. This siuaion occurs when he baery capaciy is much larger compared o he woring power of he sensor (such as ulra-capaciors [59]) or when he charging process is applied frequenly. In general, however, he acive ime slo consrain should be considered; e.g., when he baeries are of low cos and limied capaciy.. Even Model and QoM Compuaion In his secion, we firs presen assumpions on he even dynamics and he properies of he sensors. Then, we propose a general paradigm o compue a PoI s QoM when i is moniored by one or more sensors... Even Model For even dynamics, we assume ha evens a a PoI occur one afer anoher, and evens a he same PoI or differen PoIs are spaially and emporally independen [7] [8] []. Afer is occurrence, an even says for some random ime before i disappears. We denoe by X he even saying ime. Similarly, he ime duraion before he nex even occurs, which we call he even iner-arrival ime, is random and denoed by Y. Hence he sequence of even arrivals and deparures forms a sochasic process. By renewable heory [6], he expeced number of even arrivals in a ime inerval d equals µ i d, where µ i = /E(Y ) and E( ) denoes expecaion. As for he even saying ime X, we denoe he probabiliy densiy funcion of X by f(x). We use a binary sensing model for he sensors [6]. An even is said o be capured if i is sensed by a leas one sensor. Assume ha he j-h occurring even a PoI i is denoed as e i j, which is wihin range of a sensor for a oal (bu no necessarily coniguous) amoun of ime i j (i j ). We assume ha he sensor will, as a resul, gain an amoun of informaion Uj i(i j ) abou ei j, where Uj i(x) is he uiliy funcion of ei j. There are differen ypes of even uiliy funcions [7]. Exising QoM analysis ypically considers simple cases of he funcion only (e.g., he Sep uiliy funcion) [8] []. Our analysis in his paper is more general, and covers oher ypes of funcions as well. Wihou loss of generaliy, we assume Uj i (x) = U(x) for all he evens a all he PoIs; i.e., he uiliy funcion is idenical for all he evens. Furhermore, we assume ha he evens are idenifiable [7], i.e., if more han one.9.7.5.... v is awae v is awae even sars.5.5.5.5 Time (s) even ends v & v are awae Fig. : An insance of uiliy compuaion wih muliple sensors S i (x) S i = (,,,) x τ s τ s τ s τ s 5τ s 6τ s 7τ s 8τ s 9τ s Fig. : An example of periodic exension funcion sensors deec he same even simulaneously, hey will now i is he same even and learn exacly he same informaion. To help undersand he uiliy compuaion for a single even moniored by muliple sensors, we use a simple example for illusraion. As can be seen in Fig., PoI o is covered by sensor v, v and v, whose schedules are S = (,,, ), S = (,,, ) and S = (,,, ), respecively. Le he ime duraion of a single slo be τ s = s. Suppose an even arrives a PoI o a ime =. s and leaves a =.8 s, and is uiliy funcion is defined as U(x) =.5x. Then, he aggregae capured uiliy of his even by he hree sensors increases monoonically wih ime as he blue solid line shows in Fig.. Specifically, he uiliy of he even increases linearly during ime period [., ] as he even is moniored by sensor v, and eeps increasing during [, ] when v is acive in ha ime duraion. Afer ha, he uiliy remains unchanged in [, ] since no sensors are acivaed. During ime period [,.8], he uiliy resumes is increase a he same speed as before, bu no double he speed, alhough boh v and v are acive. This is due o he idenifiabiliy of evens. We also plo he curve when he uiliy funcion is given by U(x) = e x, as illusraed by he green dashed line in Fig.. I can be observed ha he curve shows a similar rend... QoM Compuaion To sar, we express he acive saus of sensors over ime as a funcion of heir individual schedules, by he periodic exension funcion given below. Noe ha each sensor is assumed o sar is schedule a ime. Definiion.. (Periodic Exension Funcion) Given a schedule S i of sensor v i, he periodic exension funcion S i (x) (S i : [, + ] {, }) of S i is defined as: {, (x [(L + j )τ s, (L + j)τ s], N, S i(j) = ) S i(x) =., oherwise () Noe ha τ s denoes he ime duraion of a single ime slo. We use a simple example for illusraion. As shown in Fig., he solid line denoes he value of he periodic exension funcion
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 5 of he schedule S i = (,,, ). The funcion aes value when x [τ s, ( + )τ s ] or x [( + )τ s, ( + )τ s ] for N, and oherwise. Then, we can derive he mahemaical expression of he QoM of a PoI covered by a single sensor in he following lemma. Lemma.. The QoM of a PoI, say o i, covered by a single sensor v j (v j V i, V i = ) under schedule S j, whose periodic exension funcion is S j (x), is given by: Q(i S j) = Lτ s Lτs + U( S j(x)dx)f(y )dyd. (5) Proof: Since he schedule of each sensor is fixed and periodic, we only need o consider he uiliy of evens saring a ime ( [, Lτ s ]). Specifically, for an even ha sars a ime ( [, Lτ s ]) and ends a ime y (y [, + )), is uiliy is U( S j(x)dx). As he even saying ime follows he probabiliy densiy funcion f(x), he expeced achieved uiliy for an even is + U( S j(x)dx)f(y )dy. By considering all possible evens ha occur in [, Lτ s ], we ge Eq. (5). Suppose S i = (a i,..., a il ) and S j = (a j,..., a jl ) are wo differen vecors. We define he OR operaion of he vecors as S i S j = (a i a j,..., a il a jl ). The following lemma shows he QoM expression for a PoI covered by muliple sensors. Lemma.. The QoM of PoI o i covered by a se of sensors V i = {v, v,..., v m }, each of which having schedule S j (j =,,..., m ), is given by: Q(i) = Q(i S, S,..., S m ) = Q(i S j). (6) v j V In oher words, he QoM achieved by he muliple i sensors can be equivalenly viewed as ha by one single sensor wih schedule v j V i S j. Proof: Referring bac o he definiion of QoM, we only have o show ha he overall uiliy available for any paricular even e i j gained by he collecion of sensors V i, namely U( i j ), is exacly equal o he uiliy U a ( i j ) gained by a virual sensor v a wih schedule v j V i S j. This can be derived by he idenifiable and idenical assumpions abou he evens. We omi he deails o save space. For simpliciy of exposiion, we call Ŝi = v j V i S j he equivalen monioring schedule for PoI o i. We sress ha our analysis can compue he QoM of a PoI in he presence of boh single and muliple monioring sensors. I can also accommodae general acivaion schedules, even uiliy funcions, and probabiliy disribuions of he even saying imes f(x).. Problem Formulaion To sum up, we formally formulae our problem CHASE as P shown below. n (P) max w iq(i) S i i= s.. (), (), (). Noe ha w i is a normalized weigh associaed wih he PoI o i, which can be inerpreed as he frequency of even occurrences of o i or he imporance of o i. The decision variables are he acivaion schedules S i s of all he sensors. Noe ha he charging ime for each sensor τ i can be deermined by S i using Eq. (), and he subse of sensors V s seleced for charging exacly conains he sensors v i s wih non-zero acivaion schedules S i s. The quaniies, c i, p i, τ, η i, P c, L, E i, l i, and w i are given consans. Noe ha differen heurisic algorihms, such as evoluionary echniques [6], [6], simulaed annealing [6], and paricle swarm opimizaion [65], can be used o solve CHASE, and hey may show good performance in paricular pracical scenarios. Neverheless, hey do no guaranee good performance heoreically. In conras, our proposed echniques provide provable approximaion raios, which improve upon he heurisics by bounding he loss of performance..5 Roadmap of Our Soluion As evidenced by he above formulaion, he full CHASE problem is complex. I involves he selecion of he candidae se of sensors V s for charging, he coupling beween he MC s ravel ime and he allocaion of charging ime among he sensors, he acive ime slo consrain, and careful compuaion of he QoM. Among hese facors, i is paricularly hard o accoun for he ravel ime accuraely. Hence, we sar by considering a relaxed version of CHASE, which we call CHASE-R, ha ignores he ravel ime; i.e., we assume τ T SP (V ) =. Besides amenable o analysis, imporanly CHASE-R is also meaningful in pracice, as can be much bigger han τ T SP (V ) due o long required charging ime necessiaed by ypically limied charging efficiencies of MCs. For example, he charging ime for he volage o reach.8 V for a WISP ag equipped wih a uf capacior can be as large as 55 seconds, when he RFID reader is. meers away [66]. Afer solving CHASE-R, we will accordingly develop soluions for he general CHASE problem by puing he MC s ravel ime bac ino consideraion. THEORETICAL ANALYSIS In his secion, we show ha he CHASE-R and CHASE problems saed above are NP-hard. Then, we reformulae he problems and presen approximaion algorihms for each of hem, respecively.. Hardness of Problems We now show ha boh CHASE-R and CHASE are NP-hard, and ha hey canno be approximaed wihin a facor beer han ( /e). To do ha, we sae he following well-nown NP-hard problem and a relaed lemma. Definiion.. (Maximum Coverage Problem) [67] Given a collecion of subses S = {S, S,..., S m } of he universal se U = {e, e,..., e n } and a posiive ineger, find a subse S S such ha S and he number of covered elemens Si S S i is maximized. Lemma.. [68] For any ɛ >, he Maximum Coverage Problem (MCP) canno be approximaed wihin a facor ( /e + ɛ) unless P = NP. We have he following heorem abou he complexiy of our problems. Theorem.. Boh CHASE-R and CHASE are NP-hard. For any ɛ >, here are no ( /e+ɛ) approximaion soluions o hem unless P = NP. Proof: We can reduce he CHASE-R problem o he MCP problem by seing L =, w i = /n, and c i = c, where c is a consan, and seing l i L for any sensor v i such ha he acive ime slo consrain can be removed. Hence, CHASE-R is a leas as hard as MCP. For CHASE, we se /c = + / (where
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 6 is an ineger) and τ T SP (V ) < / c. We can hen prove ha CHASE is also a leas as hard as MCP. By Lemma., he resul follows. Remar: CHASE involves finding a shores pah o visi all he sensors in V s. This componen TSP problem is already NPhard. The mehod in Theorem. is useful in ha i applies o he easier CHASE-R problem ha omis he ravel ime as well.. Reformulaion of CHASE-R Because CHASE-R is NP-hard, we see approximaion algorihms o solve i efficienly. In he following, we reformulae CHASE-R as a monoone submodular funcion maximizaion problem subjec o consrains including a pariion maroid consrain. Before deailing he reformulaion, we presen some necessary definiions. Definiion.. [69] Le S be a finie ground se. A real-valued se funcion f : S R is normalized, monoonic and submodular if i saisfies he following hree condiions: (i) f( ) = ; (ii) f(a {e}) f(a) for any A S and e S\A; and (iii) f(a {e}) f(a) f(b {e}) f(b) for any A B S and e S\B. For simpliciy, we use f A (e) = f(a + e) f(a) o denoe he marginal value of elemen e wih respec o A. Noe ha here, we use A + e insead of A {e}. Definiion.. [69] Given ha S = i= S i is he disjoin union of ses and l,..., l are posiive inegers, a pariion maroid M = (S, I) is a maroid in which I = {X S : X S i l i for i =,,..., }. Denoe by a ij he acivaing ime slo a ij of sensor v i. We define he ground se S as: S = {a, a,..., a L,..., a m, a m,..., a ml}. (7) We equivalenly define he sensor schedule S i as a subse of S, namely S i = {a i, a i,..., a il } if and only if a ij = (j =,,..., L ). Furhermore, S can be pariioned ino m disjoin ses, S, S,..., S m, where S i = {a i, a i,..., a il }. We say ha S i is he candidae acivaion schedule of sensor v i, since any feasible schedule S i is a subse of S i. I is clear ha any scheduling policy X consising of all he sensor schedules, namely X = {S, S,..., S m }, is subjec o X S i = S i l i. Thus, we wrie he independen ses as: I = {X S : X S i l i for i =,,..., m}. (8) Noe ha i is easy o prove ha M = {S, I} is a maroid. Moreover, define c ij = c i as he charging ime facor for ime slo a ij. The woring ime consrain can be rewrien as a c ij X ij, which is exacly a napsac consrain. Hence, we have he following lemma. Lemma.. The woring ime consrain in CHASE-R can be wrien as a napsac consrain on he ground se S, while he acive ime slo consrain can be wrien as a pariion maroid consrain. Consequenly, we can rewrie he opimizaion problem CHASE-R as RP shown below. (RP) max X n f(x) = w iq(i S j) i= v j V i s.. X I, S i = X S i i =,,..., m, c ij. a ij X Noe ha he decision variable in RP is he scheduling policy X, which consiss of elemens a ij s denoing he acivaion ime slo a ij of sensor v i. By comparing RP wih P, we can see ha he decision variables change from he acivaion schedules S i in P o he scheduling policy X in RP, and he wo are essenially equivalen. Nex, we show ha he opimizaion funcion f(x) exhibis a desirable propery as saed in he following lemma. Lemma.. If he uiliy funcion U(x) is concave, hen he objecive funcion f(x) in RP is a monoone submodular funcion. Proof: To prove he monooniciy and submodulariy of he objecive funcion f(x), we have o verify if he hree condiions in Def.. hold for f(x). Firs, i is easy o see ha f( ) =, which means ha he firs condiion holds for f(x). Second, we chec wheher he monooniciy propery holds for f(x). Suppose we have a se A S and an elemen e S\A and e = a ij. We can hen regard f(a + e ) as he resuling overall QoM obained by acivaing he ime slo a ij of sensor v i based on he original scheduling policy as far as A is concerned. As a resul, he equivalen monioring schedule of PoI o, which is covered by v i (o O i ), may be changed accordingly. Specifically, suppose he original and changed equivalen monioring schedules of o are Ŝ<A> and Ŝ <A+e>, respecively. Suppose he ime slo a j is acivaed for Ŝ <A+e>. In addiion, we use he expression a ij Ŝ<A> o indicae ha he j h ime slo of Ŝ<A> is acive, namely a j = and a ij Ŝ<A> he opposie. In realiy, boh a ij Ŝ<A> and a ij Ŝ<A> are possible. However, due o limied space, we consider only he case a ij Ŝ<A>, since i is more complicaed. We have he following imporan observaion:, x [(L + j )τ s, (L + j)τ s] Ŝ <A+e > (x) Ŝ<A> (x) = ( N ), oherwise (9) which immediaely leads o: Ŝ <A+e > (x)dx Ŝ <A> (x)dx () for any y. Nex, we noe ha any uiliy funcion U(x) mus increase monoonically from zero o one as a funcion of he oal observaion ime, i.e., U(x) and U(y) U(x) for any y x. Hence, combining Eq. (5) and Eq. (), we have: Q( Ŝ<A+e > ) Q( Ŝ<A> = Lτ s Lτs f(y )dyd. [U( ) Ŝ <A+e > (x)dx) U( Ŝ <A> (x)dx)]
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 7 Therefore, f A(e ) = w [Q( Ŝ<A+e> ) Q( Ŝ<A> )], o O i which means ha he monooniciy propery holds for f(x). Third, we chec he las condiion for f(x). Similar o he monooniciy propery analysis, suppose we have se A B S and elemen e S\B (e = a ij ). The original equivalen monioring schedules for o in A and B are Ŝ<A> and Ŝ<B>, and hey respecively change o Ŝ<A+e> and Ŝ<B+e> afer adding he elemen e. To save space, we consider only he case of a ij Ŝ<A> and a ij Ŝ<B> in his paper. Before he deailed proof, we show an imporan propery for he uiliy funcion U(x). For any y x and δ, we have: U(x + δ) y x U(x) + ( y x )U(y + δ), () y + δ x y + δ x and U(y) δ δ U(x) + ( )U(y + δ). () y + δ x y + δ x Noe ha x x + δ y + δ and x y y + δ, and U(x) is concave. Adding up he lef sides and he righ sides of Eqs. () and (), and simplifying he inequaliy, we have: for y x and δ. Since A B, we have: U(x + δ) U(x) U(y + δ) U(y) () for x [, + ]. Moreover, i is easy o see ha: Ŝ <B> (x) Ŝ<A> (x) () Ŝ <A+e > > (x) Ŝ<A> (x) = Ŝ<B+e (x) Ŝ<B> (x) (5) as a ij Ŝ<A> and a ij Ŝ<B> for x [, + ]. Therefore, from Eqs. (), (), and (5), i is obvious ha: [Q( Ŝ<A+e > ) Q( Ŝ<A> )] [Q( Ŝ<B+e > ) Q( Ŝ<B> )] = Lτ s. and: [U( Lτs f A(e ) f B(e ) = {[U( Ŝ <A+e > Ŝ <B+e > (x)dx) U( w {[Q( Ŝ<A+e> o O i Q( Ŝ<B> )]}. (x)dx) U( Ŝ <A> (x)dx)] Ŝ <B> (x)dx)]}f(y )dyd ) Q( Ŝ<A> )] [Q( Ŝ<B+e > ) We hus conclude ha he hird condiion holds for f(x) as well, and he resul follows. In general, uiliy funcions can be concave (e.g., he Sep, Exponenial, and Linear funcions in [7]) or no (e.g., he S- shaped and Delayed Sep funcions [7]). For our purposes, we consider only uiliy funcions U(x) ha are concave hereafer. The uiliy funcions of many real-world applicaions appear o be concave [7] [8] [9] [] []. Hence, our conribuions are no diminished significanly. Algorihm Basic algorihm for CHASE-R wih acive ime slo consrain Inpu: The objecive funcion f( ), he ground se S, he pariion maroid M, he napsac consrain, he candidae acivaion schedules S,..., S m. Oupu: Soluion X and he sensor schedules S,..., S m. : Reduce napsac consrain by applying Lemma. wih < ɛ <.5; le {P } T = denoe he resuling pariion maroids; : for each [T ] do : Run he greedy algorihm from [7] under pariion maroids M and P o obain soluion X ; : end for 5: arg max T = f(x ); 6: Saring wih he rivial pariion of X ino single elemens, greedily merge pars as long as each par saisfies he napsac consrain, unil no furher merge is possible. Consequenly, X can be pariioned ino pars {X j } j=; 7: X arg max j= f(x j ), Si X S i for i =,..., m;. Approximaion Algorihms for CHASE-R Having proved ha he objecive funcion of our problem is submodular, we now aim o find approximaion algorihms wih and wihou he acive ime slo consrain for CHASE-R. We will show ha he presence of he acive ime slo consrain maes he problem significanly more complex... Approximaion Algorihm wih Acive Time Slo Consrain For his case, we ailor he approach proposed by Gupa e al. [7] o our seings, and obain an improved approximaion. Their wor arges p-sysem and q-napsac in max-min opimizaion, where a p-sysem is similar o, bu more general han, he inersecion of p maroids. A a high level, heir approach exends ideas from Cheuri and Khanna [7] ha reduce napsac consrains o pariion maroids by an enumeraion mehod. We lis he main resul of his reducion as follows. Lemma.. Given any napsac consrain n i= w i x i B and fixed < ɛ <, here is a polynomial-ime compuable collecion P,..., P T of T = n O(/ɛ) pariion maroids such ha:. For every X T = P, we have i X w i ( + ɛ) B.. {X [n] i X w i B} T = P. Noe ha we use noaions similar o [7] for consisency. We assume ha Ω is he inersecion of he pariion maroid and napsac consrains. By scaling weighs in he napsac consrain, we assume wihou loss of generaliy ha he napsac has capaciy exacly one. Le C denoe he weighs in he napsac consrain. Assume ha he opimal QoM of CHASE-R is OP T and is corresponding soluion is X OP T. We propose he algorihm specified in Algorihm, which is devised based on he algorihm proposed in [7]. Theorem.. Algorihm for CHASE-R wih acive ime slo consrain can achieve /6-approximaion, and is ime complexiy is O((mL) nt ). Proof: We omi he deails of he proof o save space. We improve he approximaion facor from (p+)(q+) = /, obained by [7] for p-sysem and q-napsac consrains, o /6. This is because we give a igher bound for he number of pariioned pars a Sep 6 in Algorihm han ha in [7]. Besides, alhough he algorihm proposed in [7] for maroid
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 8 Algorihm Enhanced algorihm for CHASE-R wih acive ime slo consrain Inpu: The objecive funcion f( ), he ground se S, he pariion maroid M = (S, I), he napsac consrain, he candidae acivaion schedules S,..., S m, and he error hreshold ɛ. Oupu: Soluion X and he sensor schedules S,..., S m. : u max e S f(e); : for ρ { u u, ( + ɛ) : ζ u ρ max{f(e) : u, ( + ɛ),..., mlu } do f(e) corresponding coefficien of elemen e; : S = ; c e ρ}; c e is he 5: while ζ ɛ ml u ρ and e S c e do 6: for each e S do 7: if S + e I, f(s + e) f(s ) ζ, and f(s +e) f(s ) c e ρ hen 8: S S + e; 9: if e S c e > hen : S ρ S, X ρ S \{e}, X ρ {e}; : coninue wih he nex value of ρ; : end if : end if : end for 5: ζ +ɛ ζ; 6: end while 7: X ρ S ρ S, X ρ ; 8: end for 9: X arg max ρ X ρ, S i X S i for i =,..., m; Algorihm Unified algorihm for CHASE-R wihou acive ime slo consrain Inpu: The objecive funcion f( ), he ground se S, he napsac consrain, he candidae acivaion schedules S,..., S m. Oupu: Soluion X and he sensor schedules S,..., S m. : X, X, X, S i for i =,..., m; : If =, hen ; else ; : X arg max f(d), D S, D, a i D c i ; : for all D S ( D = and a i D c i ) do 5: I S; 6: while I\D do f D (a i ) ; c i 7: a arg max a i I\D 8: if f D(a ) hen 9: brea; : end if : if a i D c i + c hen : : D D {a }; else : 5: I I\{a }; end if 6: end while 7: if f(x ) f(d) hen 8: X D; 9: end if : end for : X arg max{f(x ), f(x )}, S i X S i for i =,..., m; and napsac consrains can achieve an ( /e ε)- approximaion, i requires ha all he ses of a mos iems o be enumeraed o form a feasible soluion a he firs sage, which limis is pracicaliy. Moreover, we employ pruning echniques when implemening his algorihm o speed up he compuaion, since he number T = (ml) O(/ɛ) of produced pariion maroids is sill large. We omi he deails o save space... Enhanced Approximaion Algorihm wih Acive Time Slo Consrain In his secion, we propose an enhanced algorihm o address CHASE-R wih acive ime slo consrain. The enhanced algorihm is based on he algorihm proposed in [7] for maximizing a submodular funcion subjec o a p-sysem and q-napsac consrains. Compared wih Algorihm, his algorihm achieves no only beer performance guaranee, bu i is also faser. Algorihm specifies he enhanced algorihm. By he classical resuls in [7], we have he following heorem. Theorem.. Algorihm for CHASE-R wih acive ime slo consrain can achieve /( + ɛ)-approximaion where ɛ is an arbirarily small posiive value, and is ime complexiy is O(n ml ɛ log ml ɛ ). Proof: We omi deails of he proof o save space... Approximaion Algorihm wihou Acive Time lo Consrain If l i L for any sensor v i, hen he acive ime slo consrain can be safely relaxed. This siuaion occurs when he baery capaciy is large relaive o he woring power of he sensor (e.g., ulracapaciors [59]) or we apply he charging process frequenly. In his case, we can resor o a unified greedy algorihm, namely Algorihm, o find an opimized QoM. Noe ha in his algorihm, c i refers o he corresponding charging ime facor for ime slo a i. This algorihm includes wo pars. The firs par enumeraes all possible subses of S wih cardinaliy less han or equal o, so as o find he bes feasible soluion for he highes QoM. The second par sars from every feasible subse D wih cardinaliy, and searches greedily in S o find a bes possible soluion. Finally, he algorihm oupus he bes observable soluion based on he resuls of he above wo pars. We have he following heorem based on he resuls obained by [75]. Theorem.. Algorihm for CHASE-R wihou acive ime slo consrain achieves approximaion facors of /e, /e /e 7, /e / /e.558, /e for =,,,, respecively. Is ime complexiy is O((mL) + n). Proof: We omi deails of he proof o save space. By Theorem., we claim ha Algorihm for = is in fac he bes possible for any polynomial-ime approach unless P = NP.. Approximaion Algorihms for CHASE Based on he proposed consan approximaion algorihms for CHASE-R, we now consider he original problem CHASE and propose approximaion algorihms for i. As shown in Algorihm, he soluion calls Algorihm a he firs sep o obain a feasible soluion X R for CHASE-R. Subsequenly, we sor he elemens in X R in descending order by heir cos efficiency in even monioring, defined as he raio of he overall QoM enhancemen yielded by a given acive ime slo o he charging ime required for an MC o enable ha ime slo o be acive. We ieraively remove an elemen wih he leas cos efficiency in X (X is iniialized as X R ) unil τ T SP ( X S i > v i) a i X c i. Noe ha we employ he neares neighbor algorihm o solve he TSP problem. Finally, we obain a feasible soluion X for CHASE.
JOURNAL OF LATEX CLASS FILES, VOL., NO. 8, AUGUST 5 9 MC MC BS BS MC BS BS (a) (b) (c) (d) MC Fig. : An insance of Algorihm for CHASE Algorihm Algorihm for CHASE Inpu: The sensors se V = {v,..., vm }, he PoIs se O = {o,..., on }, he objecive funcion f ( ), he ground se S, he pariion maroid M, he napsac consrain, he candidae acivaion schedules S,..., Sm. Oupu: The sensor schedules S,..., Sm. : Call Algorihm o obain he soluion XR for CHASE-R; = : Sor XR {a,..., ak } such ha a f (ai ) X : : 5: 6: 7: 8: arg maxa XR \X Rc (XR = {a,..., a }); i i R X XR, P S K; while τt SP ( X S > vi ) > τw a X ci do i i X X\a ; ; end while Si X Si for i =,..., m; Fig. illusraes an insance of Algorihm. Suppose afer employing Algorihm, he schedules for Sensor,,, and are (,, ), (,, ), (,, ), and (,, ), respecively, as demonsraed in Fig. (a). Meanwhile, he ravel pah passing by Sensors,, and ha have non-empy acive ime slos is shown by he grey dashed lines, since he available ime lef for he MC afer charging Sensors,, and canno allow he MC o coninue raveling along any edge of he pah. Afer soring he elemens in XR a Sep, Algorihm finds ha he second acive ime slo of Sensor has he minimum cos efficiency. I hus removes ha acive ime slo, and he ime saved hen allows he MC o ravel from he BS o Sensor, as illusraed by he dar dashed line in Fig. (b). Liewise, Fig. (c) shows he resul afer removing he firs acive ime slo of Sensor which has he second smalles cos efficiency, and using he saved ime for he MC s furher ravel. Finally, in Fig. (d), he hird ime slo of Sensor is de-acivaed; consequenly, he MC is able o charge all he sensors and finish he charging pah wihin he ime S P consrain, which means ha τt SP ( X Si > vi ) τw ai X ci is saisfied. Noe ha he MC no longer needs o pass by Sensor because i has no acive ime slo. By doing so, we obain a feasible soluion for he original CHASE problem. Theorem.5. Algorihm for CHASE based on Algorihm τ (V )+max{ci }m i= ) -approximaion. The ime achieves 6 ( T SP τw complexiy of his algorihm is O((mL) nt ). Proof: We omi deails of he proof o save space. Where here is no confusion, we call Algorihm, which wors based on Algorihm, he CHASE algorihm. We can also base our algorihm (for he CHASE problem) on he enhanced algorihm for CHASE-R wih acive ime slo consrain. This algorihm differs from Algorihm in ha i calls Algorihm raher han Algorihm a Sep. We call his alernaive algorihm E-CHASE. We have he following heorem. Theorem.6. Algorihm, which wors based on Algorihm, τ (V )+max{ci }m i= achieves + ( T SP ) -approximaion. Is ime τw ml complexiy is O(n ml log + m L). Proof: We omi he deails of he proof o save space. Lasly, if he acive ime slo consrain is no needed, we can modify Algorihm by replacing Algorihm (called a Sep ) wih Algorihm. The following heorem gives a performance guaranee of his revised algorihm. Theorem.7. The revised algorihm for CHASE wihou acive /e ime slo consrain achieves approximaion facors of c, /e /e /e c, / /e c, ( /e)c for =,,,, respecively, τ (V )+max{c }m i i=. Is ime complexiy is where c = T SP τw + O((mL) n + m L) for =,,,. Proof: We omi deails of he proof o save space. 5 P ERFORMANCE E VALUATION In his secion, we presen simulaion resuls ha verify our analysis and illusrae he performance of our algorihms. 5. Evaluaion Seup Unless oherwise saed, we use he following parameer seings. We se he range of received power of a sensor o [5 mw, 5 mw ], which can be inerpreed as a charging efficiency ηi beween [.5 %,.5 %]. In our experimens, we randomly disribue sensors and 5 PoIs in a m m region, where any PoI is covered by a leas one sensor. The woring power pi of sensor vi is randomly seleced from [5 µw, µw ], while ha of he MC is se o W. The baery capaciy of a sensor is randomly seleced from he range [ J, J]. Furhermore, we se he sensing radius of a sensor o m and he sensor schedule lengh L =. We assume ha he considered even ype has a Sep uiliy funcion and is even saying imes follow f (x) = λe λx where λ =. We se τ = wee, τw = 8. hour, and he MC s speed νm C =.5 m/s. Lasly, he defaul duraion of a ime slo is se o be s. 5. Baseline Seup Because here are no exising algorihms for join mobile charging and scheduling of sensors for sochasic even capure in a
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5. λ=.5. λ=.5 λ= λ= (h) λ=.5. λ=.5 λ= λ= (h) Fig. 5: QoM vs. for SepFig. 6: QoM vs. for Exponenial uiliy funcion uiliy funcion wireless rechargeable sensor newor, we develop wo algorihms for comparison, i.e., JPW algorihm and RANDOM algorihm. The firs algorihm is obained by specializing he Join Periodic Wae-up (JPW) algorihm in []. Specifically, an MC in JPW disribues is charging ime evenly o each sensor; only when i arrives a he posiion of a sensor can i charge he sensor (his assumpion is realisic in ha he effecive charging disance of chargers is ypically far less han he disances beween sensors). The energy cos for urning he sensor on/off is ignored, as before. In addiion, we se η i = % for any sensor v i, and he ime duraion of a duy cycle in JPW is exacly equal o ha of he sensor schedule. ν MC is se o. m/s. For he oher parameers, we use he same seings as hose in Secion 5... The second algorihm RANDOM differs from JPW in ha i randomly chooses for charging he same number of sensors as ha for CHASE, and hen uniformly disribues he charging ime among he seleced sensors. Noe ha every poin on he plos for RANDOM represens an average resul over randomly generaed insances. 5. Performance Evaluaion for CHASE-R Algorihms In his secion, we firs invesigae he cases wihou considering he acive ime slo consrain. In paricular, we evaluae he overall QoM under differen even ypes or differen values of he conrol parameer. Then, we sudy he relaionship beween he period of he charging process τ and he overall QoM. This relaionship shows he impac of he acive ime slo consrain. 5.. Impac of Even Types In his se of experimens, we focus on he even ypes whose even saying imes follow f(x) = λe λx (λ =.5,.5,, ) [7], [], whereas he uiliy funcion U(x) is eiher he Sep uiliy funcion or he Exponenial uiliy funcion f(x) = Ae Ax, where A = 5 [7]. Noe ha we use Algorihm wih =. I can be seen in Figs. 5 and 6 ha he overall QoM always increases as increases. However, he marginal gain of he QoM decreases as increases. The reason is ha he even capure uiliy funcion is concave and redundan coverage of he PoIs becomes more liely when he sensors have larger acive ime slo budges under a larger. Moreover, a smaller λ will lead o a larger overall QoM under he same. This is because he expeced saying ime of evens grows as λ decreases; herefore, is probabiliy of being deeced, as well as he uiliy of sensing, increases. Besides, by comparing Fig. 5 and Fig. 6, we see ha he achieved overall QoM for evens wih Sep uiliy always exceeds ha for evens wih Exponenial uiliy. This can be explained by differences in he efficiency of even capure. For evens under Sep uiliy, full informaion abou an even is obained insananeously. on deecion. In conras, i can require a lo more ime o obain mos informaion abou an even under Exponenial uiliy. 5.. Impac of Conrol Parameer We proceed o evaluae he impac of he conrol parameer on he overall QoM in Algorihm, and plo he resuls in Fig. 7. No surprisingly, i can be seen ha he larger we choose, he higher overall QoM we obain. However, he differences beween he overall QoM under differen are no obvious. This observaion suggess ha we can choose a small o reduce he ime complexiy wihou incurring much performance degradaion. In addiion, i can be observed ha he overall QoM exceeds /e in Fig. 7, which is consisen wih Theorem. as he opimal overall QoM canno exceed. 5.. Impac of Period of Charging Process To see how he period of he charging process τ impacs he overall QoM, we se E i = J and p i = µw, and le he received power of each sensor randomly flucuae wihin a relaively smaller range of [ mw, 5 mw ] o ease compuaion. Fig. 8 shows he rend ha he overall QoM decreases wih an increasing τ. This is because an increasing τ leads o higher charging ime facors c i and smaller acive ime slo budges l i, boh of which finally lead o a reduced QoM. Moreover, ha he overall QoM is larger han /6.7 is consisen wih Theorem.. Again, we can see ha he achieved overall QoM increases wih. 5. Performance Evaluaion for he CHASE Algorihms We proceed o verify he performance of he algorihms for CHASE, which consider he MC s ravel ime. The experimens use he same parameers as in Secion 5... 5.. A Soluion o an CHASE Insance Fig. 9 illusraes he seleced sensors for charging, he sensor schedules, he MC s ravel pah, and he achieved QoM for he PoIs, for an E-CHASE soluion o a CHASE problem insance. Noe ha he BS is locaed a (, ); he sensors are mared as circles and he PoIs as riangles. The filled color of a riangle indicaes he achieved QoM for he corresponding PoI, which varies from o as he color bar shows. A blue circle indicaes ha he corresponding sensor is no chosen a Sep in Algorihm, while a green circle indicaes ha he corresponding sensor is greedily removed in he while loop. 5.. Impac of MC s Speed on Woring Time Allocaion and QoM Fig. shows ha if he speed of he MC ν MC increases, he ravel ime is reduced, leading o a larger aggregae ime for charging. Noe ha boh he ravel ime and aggregae charging ime are normalized wih respec o he maximum woring ime. I can be seen ha he fracion of he aggregae ravel ime eeps below % when ν MC grows o. m/s, which is sill small. Anoher ineresing finding from Fig. is ha he sum of he ravel ime and aggregae charging ime is no necessarily equal o (he gap is up o % when ν MC =.). This siuaion happens because we require he acive ime slo budge l i for each sensor o be an ineger. The requiremen can be relaxed, and we can assign he residual woring ime o charging sensors. We expec he overall QoM o increase as a resul, bu he deails are lef
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5.9 5.75.7 5 = = = =..8..6...8 (h). τ =.6 h w τ =. h w τ =5.8 h w τ =7. h w τ =9. h w.. 5 6 τ (wee) X (m) 8 6 (,,,) (,,,) (,,,) (,,,) (,,,) (,,,) (,,,) (,,,) (,,,) (,,,) 6 8 Y (m).. Fig. 7: Impac of conrol parameer Fig. 8: QoM vs. τ Fig. 9: CCF A soluion o a CHASE problem insance.9.. Aggregae Travel Time Aggregae Charging Time...5.6.7.8.9. ν MC (m/s).. CHASE MLB....5.6.7 ν MC (m/s).7.5.. τ =. h w τ = 8. h w 6 8 The Lengh of Sensor Schedule Fig. : Impac of MC s speed on woring ime allocaion Fig. : Impac of MC s speed on overall QoM Fig. : QoM vs. lengh of sensor schedule.9.9.9.7.5. E CHASE CHASE JPW RANDOM.5.5 Lengh of Timeslo (s) Fig. : QoM vs. duraion of ime slos.7.5. E CHASE CHASE. JPW random. 5 6 7 8 9 (h). E CHASE CHASE. JPW RANDOM. 5 6 7 8 9 (h) (a) Duraion of ime slo is s (b) Duraion of ime slo is.5 s Fig. : QoM vs..7.5 for fuure wor. As expeced, he overall QoM is enhanced wih a faser MC, as he red solid line shows in Fig.. Moreover, i is always bigger han he maximum lower bound, which is indicaed as he green doed line MLB in he figure and given by 6 ( τ T SP (V )+max{c i} m i= ). This finding corroboraes Theorem.5 in Secion.. 5.. Impac of Lengh of Sensor Schedule Fig. shows he rend of he overall QoM when he lengh of he sensor schedule L increases under =. hours and = 8. hours, respecively. We can see ha, alhough he overall QoM does no increase monoonically wih L exacly, here is a general endency for i o rise wih L. 5.5 Performance Comparison wih Exising Wor 5.5. Impac of Lengh of Time Slos The defaul duraion of a ime slo is s in he experimens so far. In his subsecion, we vary he ime slo duraion, and plo he overall QoM for boh JPW and CHASE in Fig.. Noe ha he proposed E-CHASE and CHASE algorihms consisenly ouperform JPW and RANDOM, especially when he ime slo duraion is long. The performance gains of E-CHASE and CHASE over JPW and RANDOM are % and 5.5 %, and. % and 6. %, respecively. Moreover, he overall QoM increases as he ime slo duraion decreases for all he four schemes. The reason is ha informaion abou an even wih he same saying ime becomes more liely o be capured in is early phase, as he ime inerval beween successive acive ime slos shrins. This resul is consisen wih he analysis in [7] and []. 5.5. Impac of Maximum Woring Time In Fig. (a), we observe ha he overall QoM rises wih an increasing maximum woring ime for all he four schemes, and ha boh E-CHASE and CHASE ouperform JPW or RANDOM wih respec o achieved QoM. Moreover, he performance gains of E-CHASE and CHASE over JPW or RANDOM become more significan as increases. They achieve improvemens of abou 5 % and 8. %, and 9.9 % and 8. %, respecively, when = 9. hours. The ime slo duraion here is se o s. When
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 Energy Consumpion (J).5.5.5.5 x 5 E CHASE CHASE JPW RANDOM.8.6 5. 6. 7. 7.8 8.6 9. (h) Fig. 5: Comparison of energy consumpions Energy Efficiency.5.5.5 x 6 E CHASE CHASE JPW RANDOM.8.6 5. 6. 7. 7.8 8.6 9. (h) Fig. 6: Comparison of energy efficiency he duraion is reduced o.5 s, he overall QoM of all he schemes are subsanially enhanced, as illusraed in Fig. (b). On average, he performance gains by E-CHASE and CHASE over JPW and RANDOM are abou 8.8 % and 5. %,. % and. %, respecively. Nex, we compare he proposed algorihms wih JPW and RANDOM in erms of energy for he second case. As an MC following JPW needs o always visi all he sensors, he energy overhead for ravel can be large. Similarly, alhough an MC following RANDOM visis he same number of sensors as CHASE, RANDOM chooses hese sensors randomly, wihou rying o manage he expeced ravel energy, and is energy consumpion can sill be high. I can be seen from Fig. 5 ha on average he energy consumpions for JPW and RANDOM are 7. % and 7. % higher han ha of E-CHASE, and 8.9 % and 9. % higher han ha of CHASE. Furhermore, wih a large, he MC under E-CHASE or CHASE is able o include more sensors for charging. Fig. 5 shows ha he resuling energy incremen can be subsanial. Lasly, we evaluae he energy efficiency defined as he raio of he overall QoM o he oal energy consumpion. Fig. 6 demonsraes ha E-CHASE obains average gains of. % and 7. % over JPW and RANDOM, respecively, whereas he corresponding numbers for CHASE are 5.7 % and.6 %. 6 CONCLUSION We have solved he problem of QoM maximizaion when a sensor newor is used o monior sochasic evens, by joinly designing he sensors mobile wireless charging and acivaion schedules. The problem has a general even model ha admis differen uiliy funcions and differen probabiliy disribuions of he even arrival and saying imes. To solve he problem, we firs acled a relaxed version ha ignored he MC ravel overhead. We developed approximaion algorihms for his relaxed problem by ransforming i ino a submodular funcion maximizaion problem, under he condiion ha he even uiliy funcion was concave. Based on soluions o he relaxed problems, we hen developed approximaion algorihms o solve he original problem when he MC s ravel ime overhead was also considered. Diverse simulaion resuls verified he heoreical analysis and illusraed he performance of he proposed algorihms relaive o wo comparison benchmars. I is ineresing for fuure research o improve he approximaion facors of he soluions and accoun for fairness issues when covering he whole se of PoIs. ACKNOWLEDGMENT This wor is suppored in par by he Naional Key R&D Program of China under Gran No. 8YFB7 and 8YF- B8, in par by he Naional Naural Science Foundaion of China under Gran No. 659, 68778, 685, 66776, 6877, 69, 6567, and 675, in par by he Naural Science Foundaion of Jiangsu Province under Gran No. BK85, in par by he Fundamenal Research Funds for he Cenral Universiies under Gran 879, in par by China Naional Funds for Disinguished Young Scieniss wih No. 6655, Key Research Program of Fronier Sciences, CAS, No. QYZDY-SSW-JSC, in par by NSF CNS 5668, and in par by and S pore MOE (SUTDT7). REFERENCES [] G. Werner-Allen, K. Lorincz, M. Ruiz, O. Marcillo, J. Johnson, J. Lees, and M. Welsh, Deploying a wireless sensor newor on an acive volcano, IEEE inerne compuing, 6. [] S. Jiang, Opimum wireless power ransmission for sensors embedded in concree, Ph.D. disseraion, Florida Inernaional Universiy,. [] D. Mascareñas, E. Flynn, M. Todd, G. Par, and C. 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Haipeng Dai received he B.S. degree in he Deparmen of Elecronic Engineering from Shanghai Jiao Tong Universiy, Shanghai, China, in, and he Ph.D. degree in he Deparmen of Compuer Science and Technology in Nanjing Universiy, Nanjing, China, in. He is a research assisan professor in he Deparmen of Compuer Science and Technology in Nanjing Universiy. His research papers have been published in many presigious conferences and journals such as ACM MobiSys, ACM MobiHoc, ACM VLDB, ACM SIGMETRICS, IEEE INFOCOM, IEEE ICDCS, IEEE ICNP, IEEE JSAC, IEEE TON, IEEE TMC, IEEE TPDS, and IEEE TOSN. He is an IEEE and ACM member. He received Bes Paper Award from IEEE ICNP 5, Bes Paper Award Runner-up from IEEE SECON 8, and Bes Paper Award Candidae from IEEE INFOCOM 7.
JOURNAL OF L A T E X CLASS FILES, VOL., NO. 8, AUGUST 5 Qiufang Ma received he B.S. degree in he College of Informaion Engineering from Nanjing Universiy of Finance and Economics, Nanjing, China, in 6. She is sudying owards he masers degree in he Deparmen of Compuer Science and Technology in Nanjing Universiy. Her research ineress focus on wireless charging. Xiaobing Wu is wih Wireless Research Cenre, Universiy of Canerbury, New Zealand. He received his PhD degree in 9 from Nanjing Universiy, ME degree in and BS degree in from Wuhan Universiy, all in Compuer Science. His research ineress are in he fields of wireless neworing and communicaions, Inerne of Things and cyber physical sysems. He won Honoured Menion Award in ACM MobiCom 9 Demos and Exhibiions. He was an Associae Professor a he Deparmen of Compuer Science and Technology, Nanjing Universiy. Guihai Chen received B.S. degree in compuer sofware from Nanjing Universiy in 98, M.E. degree in compuer applicaions from Souheas Universiy in 987, and Ph.D. degree in compuer science from he Universiy of Hong Kong in 997. He is a professor and depuy chair of he Deparmen of Compuer Science, Nanjing Universiy, China. He had been invied as a visiing professor by many foreign universiies including Kyushu Insiue of Technology, Japan in 998, Universiy of Queensland, Ausralia in, and Wayne Sae Universiy, USA during Sep. o Aug.. He has a wide range of research ineress wih focus on sensor newors, peer-o-peer compuing, high-performance compuer archiecure and combinaorics. Shaojie Tang is an Assisan Professor in he Deparmen of Informaion Sysems a Universiy of Texas a Dallas. He received his Ph.D degree from Deparmen of Compuer Science a Illinois Insiue of Technology in. He received B.S. in Radio Engineering from Souheas Universiy, P.R. China in 6. He is a member of IEEE. His main research ineress focus on wireless newors (including sensor newors and cogniive radio newors), social newors, securiy and privacy, and game heory. He has served on he ediorial board of Journal of Disribued Sensor Newors. He also served as TPC member of a number of conferences such as ACM MobiHoc, IEEE ICNP and IEEE SECON. Xiang-Yang Li received he bachelor degree from Tsinghua Universiy, China, in 995, and he MS and PhD degrees from Universiy of Illinois a Urbana-Champaign in and, respecively. He was a professor wih he Illinois Insiue of Technology. He is currenly a professor and an execuive dean of he School of Compuer Science and Technology wih he Universiy of Science and Technology of China. He has auhored a monograph eniled Wireless Ad Hoc and Sensor Newors: Theory and Applicaions and co-edied several boos, including Encyclopedia of Algorihms. His research ineress include wireless neworing, mobile compuing, securiy and privacy, cyber physical sysems, and algorihms. He is a recipien of he China NSF Ousanding Overseas Young Researcher (B). He was an IEEE Fellow (5) and an ACM Disinguished Scienis (5). He holds he EMC-Endowed Visiing Chair Professorship a Tsinghua Universiy from o 6. He has received he six bes paper awards, and one bes demo award. He is an edior of several journals and has served many inernaional conferences in various capaciies. David K.Y. Yau received he B.Sc. from he Chinese Universiy of Hong Kong, and M.S. and Ph.D. from he Universiy of Texas a Ausin, all in compuer science. He has been Professor a Singapore Universiy of Technology & Design since. Since, he has been Disinguished Scienis a he Advanced Digial Sciences Cenre, Singapore. He was Associae Professor of Compuer Science a Purdue U- niversiy (Wes Lafayee). He received an NS- F CAREER award. He won Bes Paper award in 7 ACM/IEEE IPSN and IEEE MFI. His papers in 8 IEEE MASS, IEEE PerCom, IEEE CPSNA, and ACM BuildSys were Bes Paper finaliss. His research ineress include cyberphysical sysem and newor securiy/privacy, wireless sensor newors, smar grid IT, and qualiy of service. He serves as Associae Edior of IEEE Trans. Newor Science and Engineering. He was Associae Edior of IEEE Trans. Smar Grid, Special Secion on Smar Grid Cyber- Physical Securiy (7), IEEE/ACM Trans. Neworing (-9), and Springer Neworing Science (-); Vice General Chair (6), TPC co-chair (7), and TPC Area Chair () of IEEE ICNP; TPC co-chair (6) and Seering Commiee member (7-9) of IEEE IWQoS; TPC Trac co-chair of IEEE ICDCS; and Organizing Commiee member of IEEE SECON. He is a senior member of IEEE. Chen Tian is an associae professor a Sae Key Laboraory for Novel Sofware Technology, Nanjing Universiy, China. He was previously an associae professor a School of Elecronics Informaion and Communicaions, Huazhong Universiy of Science and Technology, China. Dr. Tian received he BS (), MS () and PhD (8) degrees a Deparmen of Elecronics and Informaion Engineering from Huazhong Universiy of Science and Technology, China. From o, he was a posdocoral researcher wih he Deparmen of Compuer Science, Yale Universiy. His research ineress include daa cener newors, newor funcion virualizaion, disribued sysems, Inerne sreaming and urban compuing.