Msson-Aware Placement of RF-based Power Transmtters n Wreless Sensor Networks Melke Erol-Kantarc, Member, IEEE and Hussen T. Mouftah, Fellow, IEEE School of Electrcal Engneerng and Computer Scence Unversty of Ottawa Ottawa, ON, Canada e-mals:{melke.erolkantarc, mouftah}@uottawa.ca Abstract Wreless Sensor Networks (WSNs) provde wde reach and coverage at low-cost whch enable them to be utlzed n varous felds such as health, smart grd, ndustral facltes and defense. One of the fundamental lmtatons of WSNs n long-lastng applcatons s the network lfetme. To overcome the battery constrant of sensor nodes, duty cyclng, energyeffcent protocols and energy harvestng have been consdered wdely n the lterature. A recently emergng energy harvestng technque, namely Rado Frequency (RF)-based wreless energy transfer promses to extend the lfetme of Wreless Rechargeable Sensor Networks (WRSN) wth no dependency on ntermttent ambent energy resources. In RF-based wreless energy transfer, deployng power transmtters to fxed locatons s costly due to range lmtatons of wreless power. For ths reason, moble power transmtters that vst a few selected locatons;.e. landmarks are employed. Furthermore, n WSNs sensors are expected to perform certan tasks or mssons durng ther lfetme. The achevement of each msson provdes certan profts. In ths paper, we am to optmally select the landmarks for sensor nodes that partcpate n proft maxmzng mssons. We propose an Integer Lnear Programmng (ILP) model, namely Msson- Aware Placement of Wreless Power Transmtters (MAPIT) that optmzes the placement of RF-based chargers n the WRSN by maxmzng the number of nodes recevng power from a landmark and those that contrbute the maxmum proft by achevng a msson. We show that the proft ncreases for low landmark lmt snce the number of nodes recevng power from a landmark ncreases under less landmarks. On the other hand, proft reduces by ncreased number of mssons snce the nodes partcpatng to mssons become spatally dverse. Index Terms Energy harvestng, msson-awareness, RFbased wreless power, sensor selecton, wreless sensor networks. I. INTRODUCTION Wreless Sensor Networks (WSNs) have ganed wde recognton as establshed montorng tools and they are beng used n a wde varety of applcatons ncludng mltary, health, envronment montorng, ndustry and nfrastructure montorng and so on. The fundamental challenges of WSNs have been the low data rate communcatons and lmted lfetme of the sensor nodes. Advanced data compresson and n-network aggregaton technques, and alternatve wreless communcaton technologes such as the low-power W-F offer promsng solutons for the former problem. To solve the lmted battery ssue, energy-effcency has been wdely studed n the lterature where duty-cyclng and varous energy-effcent medum access and routng protocols have been proposed. Moreover, by harvestng energy from the ambent resources, t has been possble to extend the lfetme of the sensor nodes. However, duty-cyclng and energy-effcent protocols can provde lmted lfetme whereas energy-harvestng s generally uncontrolled and may not be possble at every nstant due to unavalablty of the ambent energy. For ths reason, the recent advances n the Rado Frequency (RF)-based wreless energy transfer technology can be adopted to ncrease the lfetme of Wreless Rechargeable Sensor Networks (WRSN) n a convenent manner. Energy transfer from one devce to another devce over several meters usng electromagnetc (EM) waves s known as RF-based wreless chargng. RF-based wreless energy transmsson can be employed n WRSNs as well as the RFID technology and mplant sensors. For RF-based wreless energy transmsson, any source of RF can be utlzed. For nstance, wreless energy from the wreless nterfaces of computers can be used. Meanwhle, dedcated power transmtters can be placed at certan locatons nstead of relyng on opportunstc avalablty of the wreless power. However, deployment of power transmtters to fxed locatons ncreases the cost of the WRSN, partcularly due to the lmted range of wreless power transmsson. Hence, energy replenshment by moble wreless power transmtters has been consdered n several studes [1], []. In [1], the authors have consdered the optmal path of a moble power transmtter devce when the charger vsts each and every node for power transfer. In [], the authors have consdered selectng landmarks whch are traversed by the moble charger usng the shortest path. A transmtter located at a landmark s deally able to charge multple sensors. Sustanablty of WRSNs s a hghly desred property. Meanwhle, WRSNs are expected to perform certan tasks or mssons such as localzaton, target trackng, ambent measurement and so on. Each of these mssons can be performed by a subset of sensor nodes gven that the sensor nodes are dentcal and capable of performng all mssons. Furthermore, each accomplshed msson may be assumed to provde a certan amount of proft to WRSN. In ths context, assgnng sensor resources to mssons n an optmal manner rases as a challenge. The problem of sensor-msson assgnment have been studed n several works [3], []. Proft maxmzaton 97-1-73-713-/1/$31. 1 IEEE 1
n sensor-msson assgnment and partal msson satsfacton have been consdered n [3] whle sensor-msson assgnment for energy harvestng WSNs has been recently consdered n []. In ths paper, we am to address the problem of optmal placement the RF power transmtter such that the sensor nodes wth replenshed batteres partcpate n proft maxmzng mssons. The sensor batteres are replenshed by a moble wreless power transmtter where the term placement refers to landmark selecton for parkng the moble devce and performng power transmsson. We propose an Integer Lnear Programmng (ILP) model, namely Msson-Aware Placement of Wreless Power Transmtters (MAPIT) that optmzes the placement of RF-based power transmtters n the WRSN whle maxmzng the proft of acheved mssons. For ths purpose, we use a smlar proft maxmzaton noton as n [] whle adoptng the landmark selecton scheme of []. We show that lmtng the number of landmarks ncrease the proft snce power transfer s made from few condensed locatons. Meanwhle proft reduces by ncreased number of mssons snce the nodes partcpatng to mssons become spatally dverse and they requre to be charged from dfferent landmark locatons. The rest of the paper s organzed as follows. In Secton II, we present the related work. In Secton III we ntroduce the system model of the WRSN and the wreless energy transfer. In Secton IV we present the ILP formulaton of the MAPIT scheme and n Secton V we dscuss the performance of MAPIT by the help of llustratve results. Secton VI concludes the paper and provdes future drectons. II. RELATED WORK Extendng the lfetme of a WSN s a well understood and wdely studed topc. In [5], the authors present a rgorous revew of energy conservaton technques for WSNs. Whle energy conservaton may certanly extend the battery lfetme, energy harvestng provdes opportuntes for rechargng the batteres, hence promse longer operaton when combned wth energy-effcent protocols. Untl recently, energy harvestng from the sun, vbraton, body heat, foot strke and smlar ambent resources have been consdered. For an extensve revew of energy harvestng technques for WSNs, the reader s referred to []. The ntermttent nature of ambent energy have rased concerns on the avalablty of energy when needed. To address ths ssue, couplng duty cyclng wth the slow energyharvestng process has been studed n [7] where the authors schedule actve and nactve perods accordng to energy ncome. Recently, a breakthrough technology has emerged for energy harvestng n controlled manner,.e. RF-based wreless power transmsson. An RF-based wreless chargng testbed usng Powercast Co. wreless power chargers [] have been presented n [9]. The authors have proposed a chargng-aware routng protocol and an optmzaton framework n order to determne optmal chargng and transmsson cycles n a WSN. Furthermore, a WRSN wth a moble energy transmtter has been consdered n [1]. The authors have assumed that the moble vehcle vsts each and every sensor node for replenshng ther batteres. The batteres are charged such that the mnmum avalable energy s hgher than a threshold wthn one cycle of chargng. Sh et al. have shown that the optmal travelng path for the vehcle s the shortest Hamltonan cycle when the objectve s maxmzng the rato of the dockng tme over cycle tme. In [1], although the authors have consdered a dfferent wreless chargng technology,.e. energy transmsson va magnetc resonance, they have proposed combnng moble chargng ablty wth data collecton. The moble charger has been called SenCar whch houses a hgh-capacty rechargeable battery, a DC/AC converter and a resonant col. SenCar vsts a subset of sensor nodes whch requre urgent recharge, and n the meanwhle t collects data from the sensor network. In another recent work [11], the authors have studed rechargng of the RFID tags by the RFID readers usng wreless energy transfer. The authors have frst consdered statonary readers (chargers) and mnmzed the number of chargers n the network. Then, they have assumed that the tags are moble and they can receve power from dfferent readers as they move. Hence, the problem have turned nto selectng reader locatons that provde adequate chargng for the tags along ther path. To the best of our knowledge, msson-aware optmal selecton of landmarks for RF-based wreless power transmtters have not been addressed n the lterature. Nonetheless, sensor-msson assgnment problem has been studed n several studes. In [3], the authors have analyzed the complexty of the sensor-msson assgnment problem and t has been shown to be NP-hard. To address ths problem, we follow smlar assumptons wth [3],.e. the utlty of the sensors are consdered to be addtve. Furthermore, sensor-msson assgnment for ambent energy harvestng sensor nodes have been consdered n []. The authors have assumed that ambent energy s harvested through solar panels and the sensors are assgned to mssons based on the amount of harvested energy. In ths paper, followng the promsng results of recent RFbased wreless energy transfer studes, we propose the MAPIT scheme that ams to optmze the placement of RF-based chargers such that sensor nodes that receve energy, partcpate to proft maxmzng mssons. III. SYSTEM MODEL A. Network Model We assume N s sensors are deployed n a RxR regon. Sensors have varyng energy expendtures dependng on several factors such as beng on a busy path towards the snk, or beng close to frequently occurrng mssons or events. Therefore, batteres of some sensor nodes may deplete faster than the others. The battery capacty of a sensor node s denoted by B max. The energy replenshment demand of one sensor durng one cycle of operaton s denoted by where B max. In the WRSN, a moble RF-based wreless power transmtter recharges the batteres of sensor nodes. The amount of energy suppled by the transmtter durng one cycle of operaton 97-1-73-713-/1/$31. 1 IEEE 13
Msson Sensor Landmark 5m Fg. 1. 5m WRSN topology ncludng mssons and landmarks. s lmted by the battery of the charger, τ s. Moble power transmtter parks at certan locatons n order to transmt power to the sensor nodes n ts vcnty. These locatons are called landmarks and denoted by δ xy. The number of landmarks s lmted to N l. In Fg. 1, we present a WRSN topology wth 3 nodes, fve mssons and four landmark locatons. We assume a sensor node has the sensng hardware to perform any msson j, j =1,,..M. The utlty of a sensor to a msson s determned based on the Eucldean dstance between the sensor and the msson, and t s denoted by σ j. σ j = 1 (1) d j where d j denotes the dstance between sensor- and mssonj. σ j s normalzed wth R n the presentaton of the results. Msson-j provdes a certan proft to the sensor network whch s denoted by P j. The total sensng resources requred for a msson to be acheved s gven by s j and zxy s a bnary varable that s 1 f sensor- s recevng power from a landmark at (x, y), whle r j s a bnary varable that s 1 f sensor- s partcpatng to msson-j. Thus the proft of msson-j, P j s defned as: P j = z σ j xyr j () s x y j B. Wreless Power Transmsson Model RF-based wreless energy harvestng requres a transmtter TR s and a recever RC s to be located wthn R c dstance where R c denotes the chargng range. For smplcty, we adopt the crcular dsk model for wreless power propagaton of [11]. Accordng to the free space model, the receved power s nversely proportonal to d rt and t s assumed to be when d rt > R c, where d rt s the dstance between the transmtter and the recever. The maxmum power output of TR s s denoted by Pow max. We assume omndrectonal Fg.. Wreless power transmsson; a) moble power transmtter, b) block dagram of energy harvestng unt at the recever. power transmsson capablty whle the several commercal products offer extended range wth drected beams []. Note that, n practce, the amount of receved power wll vary dependng on the propagaton propertes of the envronment. RC s converts the receved sgnal to DC voltage usng a capactor. In Fg. a and Fg. b, we present the moble power transmtter, a recever, and the block dagram of the energy harvestng unt at the recever [], respectvely. IV. MISSION-AWARE PLACEMENT OF WIRELESS POWER TRANSMITTERS (MAPIT) Msson-Aware Placement of Wreless Power Transmtters (MAPIT) optmzes the placement of RF-based chargers n the WRSN such that sensors charged from a sngle landmark s maxmzed and they partcpate n proft maxmzng mssons. Landmarks are the locatons where wreless power transmtters park and transmt wreless power to the sensor nodes gven that d rt <R c. As we mentoned before, n RF-based wreless energy harvestng, wreless transmtters can be placed at several fxed locatons, however ths ncreases the deployment cost snce R c s lmted to a few meters. Hence for wde coverage, large number of transmtters would be requred. For ths reason, moble transmtters are preferred. In ths case, t has been shown that usng selected landmark locatons rather than vstng all of the sensor nodes reduces the path length of the moble devce and allows more tme for chargng the battery of the moble devce []. MAPIT adopts the landmark dea of [] and ncorporates msson-awareness. MAPIT assumes that sensors share the mssons that are expected to be performed by the WRSN. A sensor that s close to a msson deally partcpates to that partcular msson, whle other mssons are handled by the other sensor nodes. For nstance, a group of sensor nodes may be engaged n target trackng snce a target may be n the event sensng range of those sensors whle others may measure ambent temperature, etc. A sensor partcpatng to the target trackng msson may be able to provde more accuracy based on ts proxmty to the event than the other partcpatng nodes. As the utlty of a sensor node to msson-j reduces as the dstance between 97-1-73-713-/1/$31. 1 IEEE 1
TABLE I SYMBOLS FOR CONSTANTS AND VARIABLES OF THE ILP MODEL. M N s P j σ j s j τ S zxy r j δ xy δ xy R c N l θxy j Number of mssons Number of sensors Proft functon Utlty of sensor- to msson-j Sensng demand of msson-j Energy demand ntensty of sensor- Energy lmt of power transmtter Bnary varable: 1 f sensor- s recevng power from a landmark at (x, y) Bnary varable: 1 f sensor- s partcpatng to msson-j Bnary varable: 1 f there s a landmark located at (x, y) Bnary varable: 1 f sensor s wthn R c dstance to the landmark at (x, y) Power transmtter range Landmark lmt Bnary varable: 1 f sensor- s charged from a landmark at (x, y) and sensor- s partcpatng to msson j the sensor and the msson ncreases, t wll contrbute hgher utlty than the others, and selectng that sensor ncreases the proft acheved by the target trackng msson. MAPIT s formulated as an ILP model wth the objectve functon gven n eq. (3). The varables and symbols used n the ILP are gven n Table 1. The objectve functon ams to maxmze the number of sensors that are replenshed from the selected landmarks and maxmze the proft of the whole WRSN by the partcpaton of the sensors that are charged from those landmarks. maxmze θxy j σ j (3) s j j Here θxy j s a bnary varable that s equal to 1 f sensor- s charged from a landmark at (x, y) and sensor- s partcpatng to msson j and θxy j = zxyr j. θxy j needs to be reformulated n a lnear way n order to be used n the ILP model. We use a smple lnearzaton technque whch works as follows [1]. When a and b are bnary varables and c s a product of the two varables,.e. c = a.b, c a, c b and c b a 1 holds. In the constrant set, θxy j s gven n the lnear form by eq. ()-eq. (). x θ j xy z xy, j, x, y () θ j xy r j, j, x, y (5) θxy j zxy r j 1, j, x, y () In a WRSN, most of the energy s consumed durng packet transmsson, hence the frequency of forwardng events to the snk and relayng the packets of the neghbors determne the amount of energy requred for toppng up the battery of a sensor node n each chargng cycle. Demand ntensty refers to the energy requrement of a sensor node and t depends wether the sensor s located on a busy path towards the snk or not. In the ILP formulaton, we denote the energy replenshment y demand,.e. demand ntensty wth. On the other hand, the energy supply of the charger s lmted by the capacty of ts battery. We assume at each landmark the power transmtter s able provde τ s unts of energy. Hence, the supply of the power transmtter should be greater than or equal to the energy requrement of the sensors that are recevng power from the landmark at (x, y). Ths constrant s formulated by eq. (11). zxy τ S x, y (7) We assume that a sensor node s allowed to receve power from one and only one landmark locaton, whch s assured by eq. (). We consder that landmarks are selected such that when the transmtter parks at the landmark, t can transmt power to at least one sensor node. Equatons (9) and (1) ensure that a transmtter located at (x, y) s able to transmt power to at least one sensor where δ xy s a bnary varable that s 1 f there s a landmark located at (x, y) and δxy s 1 f sensor f the dstance between the landmark at (x, y) and the sensor s less than R c. zxy =1 () x y z xy δ xy, x, y (9) δ xy δ xy, x, y (1) The maxmum number of landmarks s lmted to N l durng one cycle of operaton snce t s preferred to have a fxed duraton for a moble power transmtter to vst the landmarks and return back to ts dockng staton to charge ts battery from the mans, as ensured by eq. (11). We neglect the tme spent for travelng from one landmark to another snce the travelng tme s expected to be sgnfcantly small than the tme spent at a landmark for powerng the sensors. δ xy N l (11) x y MAPIT allows one sensor to partcpate to one and only one msson at a tme. Ths constrant s formulated by eq. (1). Furthermore, the utlty provded by the sensor nodes should be able to satsfy the requrement of the msson as gven by eq. (13). We assume that mssons are accomplshed only when the partcpatng sensors have adequate resources for the msson. In the lterature, partal satsfacton of mssons have also been consdered []. r j 1 (1) j r j σ j s j j (13) 97-1-73-713-/1/$31. 1 IEEE 15
V. PERFORMANCE EVALUATION We evaluate the performance of MAPIT by consderng a rectangular feld of 5m X 5m where N s = 3 sensor nodes are randomly deployed. We assume that M mssons can occur smultaneously n the WRSN and we vary M from 5 to 15. Landmark lmt, N l, vares between 1 to. Mssons are assumed to requre dentcal amount of sensng resources whch s s j =1. The demand ntensty of a sensor node,.e. the amount of battery that needs to be charged, vares dependng on several factors as explaned n the prevous sectons. We assume that the maxmum battery capacty of a sensor node s 1kJ [1]. We evaluate the performance of MAPIT under varous demand ntenstes. The battery capacty of the power transmtter s assumed to be kj. Presently, the range of wreless energy transfer technology s low. Usng antennas wth hgh transmsson power or drectonal antennas, t s possble to extend the range to several tens of meters. In our performance evaluatons, we set the wreless energy transfer range as R c =m. We use CPLEX to determne the optmal landmark locatons by MAPIT and we present the averaged results. In Fg. 3, we present the proft acheved by performng the mssons n the WRSN. The number of mssons are set to M =5n ths set of results. The landmark lmt, N l vares from 1 to, and we observe the proft for two dfferent values of energy replenshment demand of the sensors. The gray bar denotes the case where each sensor have 1kJ of energy demand and the blue bar denotes 3kJ of energy demand. When, for a landmark lmt of ten, the proft s almost 7 unts of utlty. Note that proft s defned n unts of utlty. As the landmark lmt ncreases, the proft reduces slghtly. The reason for ths behavor s as follows. Increasng N l results n sensor nodes to receve power from dfferent landmarks. The proft s defned as a functon of zxy whch s number of nodes recevng power from the landmark at (x, y). Hence the proft reduces as the number of sensors charged from the same landmark reduces. When, proft s hgher than the case for. Under fxed N l, ncreasng demand ntensty ncreases the number of sensors that receve power from a certan landmark locaton. When N l =1the proft s almost unts of utlty. As N l ncreases, proft reduces, smlar to the prevous case. In Fg., we present the proft under varyng number of mssons whch are set between 5 to 15 and the landmark lmt s N l =15. Smlar to the prevous settngs we consder two dfferent demand profles. For, the proft acheved from the utlty of sensor nodes reduces as the number of mssons ncreases. When there are fve mssons the proft s below 7 unts of utlty and t reduces to less than unts of utlty for M =15. As the number of mssons ncrease the dstance between the respectve mssons and sensors reduce. Snce the proft s related wth the dstance through the defnton utlty, proft reduces as the number of mssons ncrease. For M =15and proft drops even lower Proft(utlty) Proft (utlty) 9 7 5 3 1 9 7 5 3 1 Fg. 3. Fg.. 1 15 Landmark Lmt (N l ) Proft acheved under varyng landmark lmts. 5 1 15 Number of Mssons Proft acheved under varyng number of mssons. than whch can be explaned as ncreased number of mssons and demands reduce the number of sensors that receve power from the same locaton and partcpate to the same mssons. In Fg. 5, we present the number of selected landmarks under varyng number of landmark lmts. For both demand profles, the number of landmarks ncreases as the number of landmarks ncreases. For N l = 1, both of the demand profles almost utlze all of the landmarks. Hence, N l =1s the mnmum number of landmarks requred for ths network. In Fg., we present the number of selected landmarks under varyng number of mssons when N l =15. For both demand profles, number of selected landmarks are between 97-1-73-713-/1/$31. 1 IEEE 1
Number of Landmarks Number of Landmarks 1 1 1 1 1 1 1 1 1 1 1 15 Landmark Lmt (N l ) Fg. 5. Number of landmarks under varyng N l. 5 1 15 Number of Mssons Fg.. Number of landmarks under varyng M. 1 and 15. When, the landmark lmt s not reached, and less number of landmarks are able to maxmze the proft. For low demand ntensty maxmum number of landmarks are reached whch s related wth the objectve functon and the supply constrant of the power transmtter. Thus, ths can be explaned by the objectve functon and eq (7)- eq(1). VI. CONCLUSIONS Low-cost Wreless Sensor Network (WSN) technology s antcpated to be adopted n a wde range of applcatons ncludng health, smart grd, mltary felds, etc. Besdes the numerous advantages of the WSN technology, lmted lfetme of the sensor nodes have been consdered as a sgnfcant performance bottleneck. Duty cyclng, energy-effcent network protocols and energy-harvestng technques offer solutons to extend the lfetme of the WSNs. However, they are stll lmted ether by the capacty of the battery or by the avalablty of ambent energy. In ths paper, we have adopted the recently emergng RF-based wreless energy transfer technque n the Wreless Rechargeable Sensor Networks (WRSN). We have proposed an Integer Lnear Programmng (ILP) model, namely Msson- Aware Placement of Wreless Transmtters (MAPIT) for optmal placement of RF-based power transmtters n the WRSN such that the sensor nodes wth replenshed batteres partcpate n proft maxmzng mssons. We have shown that lmtng the number of landmarks ncrease the proft snce power transfer s made from a less number of condensed locatons. Meanwhle, proft reduces as the number of mssons ncreases, snce the nodes that are partcpatng to mssons become geographcally dverse and they requre to be charged from dfferent landmark locatons. As a future research drecton, sleep schedulng and energyeffcent communcaton protocols can be utlzed along wth the RF-based power transfer to provde the most effcent soluton to the lmted lfetme bottleneck of the WSNs. Meanwhle combnng moble chargng and msson-awareness wth moble data collecton solutons can further reduce energy consumpton of the sensor nodes and mprove the performance of the WRSN. REFERENCES [1] L. Sh, L. Xe, Y.T. Hou, H.D. Sheral, On Renewable Sensor Networks wth Wreless Energy Transfer, n Proc. of IEEE INFOCOM, Shangha, Chna, Aprl 1-15, 11, pp. 135-135. [] M. Erol-Kantarc, H.T. Mouftah, SuReSense: Sustanable Wreless Rechargeable Sensor Networks for the Smart Grd, to appear n IEEE Wreless Communcatons Magazne, June 1. [3] H. Rowahy, M. P. Johnson, O. Lu, A. Bar-Noy, T. Brown, T. La Porta, Sensor-msson assgnment n wreless sensor networks, ACM Transactons on Sensor Networks, vol., no., July 1. [] T. La Porta, C. Petrol, D. Spenza, Sensor-msson assgnment n wreless sensor networks wth energy harvestng, n Proc. of th Annual IEEE Conference on Sensor, Mesh and Ad Hoc Communcatons and Networks (SECON), pp.13-1, 7-3 June 11. [5] G. Anastas, M. Cont, M. D Francesco, A. Passarella, Energy conservaton n wreless sensor networks: A survey, Ad Hoc Networks, vol. 7, no. 3, May 9, Pages 537-5. [] S. Sudevalayam, P. Kulkarn, Energy Harvestng Sensor Nodes: Survey and Implcatons, IEEE Communcatons Surveys & Tutorals, vol.13, no.3, pp.3-1, Thrd Quarter 11. [7] V. Pryyma, D. Turgut, L. Bolon, Actve tme schedulng for rechargeable sensor networks, Elsever Computer Networks, vol. 5, no., 1, pp.31. [] Powercast Corporaton, [Onlne] http://www.powercastco.com/ [9] R. D. Mohammady, K. Chowdhury, M. D Felce, Routng and Lnk Layer Protocol Desgn for Sensor Networks wth Wreless Energy Transfer, IEEE GLOBECOM, Mam, December 1. [1] M. Zhao, J. L, Y. Yang, Jont Moble Energy Replenshment and Data Gatherng n Wreless Rechargeable Sensor Networks n Proc. of 3rd Int. Teletraffc Congress, September -, 11, San Francsco, USA. [11] S. He, J. Chen, F. Jang, D.K.Y. Yau, G. Xng, Y. Sun, Energy Provsonng n Wreless Rechargeable Sensor Networks, n Proc. of IEEE INFOCOM, Shangha, Chna, Aprl 1-15, 11, pp. -1. [1] R. Prasad, H. Wu, Gateway Deployment optmzaton n Cellular W-F Mesh Networks, Journal of Networks, July, pp. 31-39. 97-1-73-713-/1/$31. 1 IEEE 17