Optimal Pricing and Load Sharing for Energy Saving with Cooperative Communications

Transcription

1 Optmal Prcng and Load Sharng for Energy Savng wth Cooperatve Communcatons Ynghao Guo, Lnge Duan, and Ru Zhang arxv: v4 [cs.it] 4 Aug 2015 Abstract Cooperatve communcatons has long been proposed as an effectve method for reducng the energy consumpton of the moble termnals MTs) n wreless cellular networks. However, t s hard to be mplemented due to the lack of ncentves for the MTs to cooperate. In ths paper, we propose a prcng mechansm to ncentvze the uplnk cooperatve communcatons for the energy savng of MTs. We frst consder the deal case of MTs full cooperaton under complete nformaton. For ths scenaro as the benchmark case, where the prvate nformaton of the helpng MTs such as the channel and battery condtons s completely known by the source MT, the problem s formulated as a relay selecton problem. Then, for the practcal case of partal cooperaton wth ncomplete nformaton, the MTs need to cooperate under the uncertantes of the helpng MTs channel and battery condtons. For ths scenaro, we propose a partal cooperaton scheme wth prcng where a source MT n low battery level or bad channel condton s allowed to select and pay another MT n proxmty to help forward ts data to the base staton BS). We formulate the source MT s prcng and load sharng problem as an optmzaton problem. Effcent algorthms based on dchotomous search and alternatve optmzaton are proposed to solve the problem for the cases of splttable and non-splttable data at the source MT, respectvely. Fnally, extensve numercal results are provded to show that our proposed cooperatve communcatons scheme wth prcng can sgnfcantly decrease both the communcatons and battery outages for the MTs, and can also ncrease the average battery level durng the MTs operaton. Index Terms cooperatve communcatons, energy savng, prcng mechansm, load sharng I. INTRODUCTION WITH the recent developments n the smart phones and the multmeda applcatons, wreless cellular network s now experencng an exponental ncrease n the wreless data traffc and today s moble termnals MTs) consume a lot more energy than before. Consderng ther lmted battery capactes, MTs need to be charged more frequently and ths has become the bggest customer complant for smart phones [1]. As such, reducng the energy consumpton for the MT s of crtcal mportance for resolvng the energy shortage of the MTs and mprovng the connectvty of the wreless networks. Furthermore, t has been shown that the communcatons modules consttute a large proporton of the MTs energy consumpton, for ether the MTs from the earler Y. Guo s wth the Department of Electrcal and Computer Engneerng, Natonal Unversty of Sngapore e-mal: ynghao.guo@nus.edu.sg). L. Duan s wth the Engneerng Systems and Desgn Pllar, Sngapore Unversty of Technology and Desgn e-mal: lnge duan@sutd.edu.sg). R. Zhang s wth the Department of Electrcal and Computer Engneerng, Natonal Unversty of Sngapore e-mal:elezhang@nus.edu.sg). He s also wth the Insttute for Infocomm Research, A*STAR, Sngapore. 2G and 3G era [2] or the more modern 4G moble phones [3]. Therefore, ths gves us a good motvaton to nvestgate the energy savng for the MTs n data communcatons. Cooperatve communcatons [4] s an effectve approach for energy savng n wreless cellular networks and wreless sensor networks. However, the battery levels and ther heterogenety among MTs/sensors have not been rgorously consdered before. For the MTs wthn a cellular network, some MTs are low n battery level and others are hgh. If the battery level s gnored, t s possble that some MTs low n battery level stll help the other MTs for data relayng. Ths s clearly undesrable snce the battery of the MT can be easly depleted. Under ths crcumstance, t would be helpful f the MTs n low battery level can get help from those hgh n battery level such that ther operaton tme can be prolonged. Hence, ths motvates ths work to consder cooperatve communcatons for energy savng wth the consderaton of the battery levels of the MTs. Furthermore, another unsolved ssue n cooperatve communcatons s that the MTs may lack the proper ncentves to cooperate. For most of the exstng studes n the lterature, t s assumed that the sensors n the wreless sensor networks or MTs n the wreless cellular networks cooperate wth each other wthout self-nterests. In realty, ths mght be true for the case of wreless sensor networks, snce the sensors wthn a target area usually belong to the same entty. Whle, ths can hardly be true for the MTs n the wreless cellular networks, snce the MTs belong to dfferent ndvduals wth self-nterests. Therefore, n order to enable a practcal mplementaton of the energy-savng cooperatve communcatons n cellular networks, ncentve desgn must be consdered for the MTs. A. Related Work It s noted that there are already pror works nvestgatng the MT-sde energy savng n the lterature [5] [7]. In partcular, [5] studed the optmal modulaton scheme to mnmze the total energy consumpton for transmttng a data package of a gven sze. Both uncoded and coded systems are consdered for the modulaton optmzaton. [6] studes the optmal power control problem for the mnmzaton of the average MT energy consumpton n the mult-cell TDMA system. In [7], the authors study the energy savng of the MTs by leveragng the spare capacty at the base statons BSs) n cellular networks. The optmal desgn s obtaned by solvng the optmzaton problems for the scenaros of real-tme data traffc and data fles transmsson, respectvely. Recently, [8], [9] showed that

2 2 there s n general a trade-off between mnmzng the energy consumpton at the BSs and that at the MTs for meetng gven qualty of servce QoS) requrements of the MTs. Moreover, cooperatve communcatons for the energy savng of the MTs has been nvestgated n the lterature of wreless sensor or cellular networks [10] [12]. In partcular, [10] studes the optmal tmer-based relay selecton scheme for the mnmzaton of the sum energy consumpton and maxmzaton of the network lfetme. [11] proposed a spacetme codng scheme for the MTs to cooperatvely transmt to the BS under gven outage and capacty requrements such that total transmt energy s mnmzed. [12] consdered the mnmzaton of energy consumpton under qualty of servce QoS) constrant wth cooperatve spectrum sharng n the cogntve rado network. [13] consdered extendng the lfetme of the machne-to-machne M2M) communcatons network by consderng the cooperatve Medum Access Control MAC) protocol. Although cooperatve communcatons has long been proposed for energy savng, t s hard to be realzed n realty due to the lack of ncentves that motvate the MTs to cooperate [14]. The dea of vrtual currency for ncentvzng the cooperaton between self-organzed enttes was frst proposed n [15] under the setup of wreless sensor network. From ths perspectve, pror works [16] [19] have also proposed varous ncentve mechansms to motvate cooperatve communcatons n wreless communcatons systems. Specfcally, [16] proposed a dstrbuted game-theoretcal framework over multuser cooperatve communcaton networks to acheve optmal relay selecton and power allocaton. A two-stage Stackelberg game s formulated to consder the nterests of the source and relay, where the source node s modeled as a buyer and the relay nodes are modeled as sellers for provdng relay for the source. The dfference of ths work from our s that the battery level of the MTs are not consdered. [17] studed the dynamc barganng-based cooperatve spectrum sharng between a prmary user PU) and a secondary user SU), where the PU shares spectrum to the SU and the SU helps relay the sgnal of the PU n return. Dfferent from these works, we study the energy savng of the MTs wth the new consderaton of the battery levels of the MTs n the cooperatve communcatons. [18] proposed a so-called reputaton system based on a reputaton aucton framework to provde ndrect recprocty for stmulatng node cooperaton n green wreless networks. The dfference from our work s that t does not consder the ssue of battery level n the cooperatve communcatons and the approach for motvatng the cooperaton s dfferent. [19] consdered the busness model for cooperatve networkng problem wth the aucton theory. However, t dd not gve an exact modellng for the battery level and the proposed cooperatve communcatons scheme s not n the setup of wreless cellular network. B. Man Contrbutons and Organzaton The man contrbutons of ths paper are summarzed as follows: Prcng mechansm for ncentvzng cooperaton: In ths paper, we consder that the MTs n the network are selfsh and only wllng to cooperate when they can beneft from the cooperaton. Dfferent from the prevous works on cooperatve communcatons, we take the battery level of the MT nto consderaton and explot the heterogenetes of the battery levels and channel condtons between the MTs for cooperaton. Under the uncertantes of the helpng MTs battery levels and channel condtons, we propose a new prcng mechansm to ncentvze the cooperatve communcatons between the MTs that can lead to a wn-wn stuaton. Full cooperaton under complete nformaton: Frst, for the deal case of full cooperaton under complete nformaton, the problem s formulated as a determnstc relay selecton problem among all the helpng MTs for the cases of splttable or non-splttable data at the source MT. It s further shown that n the case of splttable data, the optmal rate allocaton follows a smple threshold structure and can be mplemented effcently. Partal cooperaton under ncomplete nformaton: Then, for the practcal case of partal cooperaton under ncomplete nformaton, the MTs belong to enttes of ndvdual nterests and cannot share prvate nformaton to the other MTs. Under the uncertantes on the battery levels and channel condtons of the helpng MTs, we formulate the MT s prcng and load sharng problem as an optmzaton problem for the two cases of splttable and non-splttable data of the source MT, respectvely. Effcent algorthms based on dchotomous search and alternatve optmzaton are proposed for the solutons of the problem. The rest of ths paper s organzed as follows. Secton II ntroduces the system model wth the cooperatve communcatons for MTs energy savng, and the resultng cost and utlty functons. Secton III dscusses our proposed protocol under complete nformaton as the performance benchmark and Secton IV studes the general case of cooperatve communcatons under ncomplete nformaton. Secton V presents numercal examples to valdate the results n ths paper. Fnally, Secton VI concludes ths paper and dscusses future work. II. SYSTEM MODEL AND ENERGY-SAVING COOPERATION A. System Model All the notatons used n ths paper are summarzed and explaned n Table I for the ease of readng. As shown n Fg. 1, we consder the uplnk data transmsson wthn one sngle cell of a cellular network. 1 Dfferent roles of the MTs wll be ntroduced later n the paper. Wthn the cell, there s one sngle-antenna BS servng K sngle-antenna MTs denoted by the setk = {1,2,,K}. We assume that the locatons of the MTs follow a two-dmensonal Homogeneous Posson Pont Process HPPP) wth spatal densty λ. 2 [20] We consder that the MTs wthn the cell ntate ther data traffc ndependently 1 Our results can be extended to the case of multple cells by applyng our results to each cell ndependently. 2 For the spatal user densty λ, t can be readly obtaned by dvdng the total number of MTs wthn the cell over the total area of the cell. The number of the MTs can be estmated by the hstory data of the cell or by real-tme montorng.

3 3 BS Source MT 3 Source MT Helpng MT Idle MT out of SRC Relay MT Drect Transmsson Cooperatve Transmsson follows a smplfed channel model ncorporatng the largescale power attenuaton wth loss exponent α > 2 and the small-scale Raylegh fadng. More specfcally, we denote r k as the dstance between MT k K and the BS, and r 0 as a reference dstance, respectvely. Then, the channel coeffcent h k s expressed as ) h h k = k G rk α, 0 r0 rk > r 0, k K, 1) h k G0, otherwse Source MT 1 Source MT 2 Fg. 1: System model for drect and cooperatve data transmsson. TABLE I: Lst of notatons and ther physcal meanngs. Symbols Physcal Meanngs K Set of all the MTs K S Set of the source MTs K I Set of the dle MTs H Set of helpng MTs for source MT K S ρ Probablty that a certan MT ntates data traffc µ N Average number of helpng MTs for source MT K S h k Channel coeffcent of MT k K g k Channel gan of MT k K r 0 Reference dstance r k Dstance from MT k K to the BS α Exponent of the large-scale power attenuaton G 0 Pathloss at reference dstance r 0 G k Pathloss for MT k K D Data rate of source MT K S D S) Data rate of the source MT K S n the CT mode D R) Data rate of the relay MT H n the CT mode σ 2 Power of the nose at the recever of the BS B k Battery level of MT k K E D,S) Energy consumpton of source MT K S wth DT mode E C,S) Energy consumpton of source MT K S wth CT mode E C,R) Energy consumpton of helpng MT H wth CT mode ζ k Unt energy cost for MT k K π Payment from the source MT K S to ts helpng MTs U Utlty for the helpng MT H C Cost for the source MT K S wth CT mode ǫ Utlty margn for the relay MT η k Exponental dstrbuted Raylegh fadng power for MT k K γ Cost reducton threshold of the source MT K S C, Source MT s cost assocatng wth helpng MT H wth probablty ρ. Then, accordng to the Markng Theorem [21], these source MTs.e. MTs ntatng data traffc) also form an HPPP wth densty ρλ and the remanng dle MTs form another HPPP wth densty 1 ρ)λ. We denote these sets of source MTs and dle MTs as K S and K I, respectvely, such that K S K I = K and K S K I =. We consder the uplnk data transmsson of all MTs and assume the narrow-band block fadng channel model. To support multple MTs, orthogonal data transmsson s assumed, e.g., by applyng orthogonal frequency-dvson multple access OFDMA). We denote the complex baseband channel coeffcent from MT k K to the BS as h k, whch where h k CN0,1), k K s an ndependent and dentcally dstrbuted..d.) crcularly symmetrc complex Gaussan CSCG) random varable wth zero mean and unt varance modelng the small-scale Raylegh fadng, and G 0 s the constant path-loss between the MT and the BS at the reference dstance r 0. Therefore, the channel power gan between the MT k and the BS s g k = h k 2 = η k G k, k K. 2) Here, we denote η k exp1) as an exponental random varable wth unt mean modelng the power envelope of the Raylegh fadng and ) G rk α, G k = 0 r0 rk > r 0, k K 3) G 0, otherwse as the power attenuaton between the BS and the MT k at the dstance of r k. For smplcty, we consder a tme-slotted system, where symbols for the message are transmtted n each tme slot. For convenence, the number of symbols transmtted per tme slot are normalzed to unty. If MT k K ntates ts data traffc, a message from the set {1,2,,2 D k } s sent, where D k s the transmtted rate n bts per symbol. Wthout loss of generalty, we also normalze the duraton of one symbol tme to unty such that the two terms energy and power can be used nter-changeably n the paper. Then, f the achevable data rate D k s normalzed by the avalable bandwdth at the MT, for gven transmsson energy per symbol E k, the normalzed) achevable data rate for MT k K n bts/sec/hz bps/hz) s D k = log 2 1+ g ) ke k σ 2, 4) where σ 2 denotes the power of the nose at the recever of BS. For source MT K S, n order to accomplsh the uplnk transmsson at normalzed) data rate D, t can choose between the followng two transmsson modes. 1) Drect Transmsson Mode DT Mode): In ths mode, the source MT transmts to the BS drectly wth normalzed data rate D. Hence, accordng to 4) the requred energy per symbol for transmttng wth data rate D s E D,S) = σ2 g 2 D 1 ), K S. 5)

4 4 2) Cooperatve Transmsson Mode CT Mode): In ths mode, for a certan source MT K S, as shown n Fg. 1, t can assocate wth one dle MT f any) wthn the dstance d as ts relay MT that can help relay the data to the BS, where d s the range of the short range communcatons SRC) such as WF-Drect [22], Bluetooth [23], etc. 3 We denote ths set of dle MTs wthn the dstance d from the source MT K S as ts set of helpng MTs H K I, where H = N s the number of MTs wthn the set. Then, t follows that N s a Posson random varable wth mean µ N = 1 ρ)λπd 2, K S and ts probablty mass functon PMF) s gven by PrN = n) = µn N n! e µn, n = 0,1,, KS. 6) From 6), we observe that the PMF of N s proportonal to the range of the SRC d, an MT s probablty of remanng dle 1 ρ and the spatal densty λ. Note that f N = 0 or H =, source MT K S wll operate n DT mode,.e., transmt drectly to the BS; whle f N 1, source MT K S can operate n CT mode by selectng one from ts helpng MTs n H to relay the data. For the CT mode, the source MT K S n general splts for the source MT to transmt drectly to the BS and D R) for ts relay MT to transmt. For transmttng the data D S), smlar to 5), the requred energy for the source MT K S s 4 data D nto two parts wth D = D S) E C,S) + D R) : D S) ) = σ2 2 DS) 1, K S. 7) g Then, for the other part of data D R), as shown by the dashed lne n Fg. 1, the source MT frst transmts t to the selected relay MT and then the relay MT decodes and forwards the sgnal to the BS. In practce, SRC technologes e.g. WF- Drect [22], Bluetooth [23], etc.) offer hgh communcatons data rate wth low transmt power. The energy consumpton and the transmsson tme s also small compared to that n the wreless cellular network. Hence, we gnore them for ths short range data transmsson. 5 Also due to the small range, the source MT K S and ts helpng MT H have roughly the same dstance to the BS.e. r = r ) and can be assumed to have the same path-loss. Hence, the channel 3 In ths paper, we assume sngle relay selecton to keep the overhead low. Smlar approach has been used n [10]. 4 In ths paper, we do not drectly consder the maxmum power constrant of the MT n order to obtan tractable problem formulaton and nsghts. However, as wll be shown later, we have already mplctly consdered the ssue of large transmt power. When the transmt power s very large, t ncurs a large cost on the MT and the MT wll try to get help from the other MTs. Hence, large energy consumpton can be avoded. The smulaton results n Secton V-B wll corroborate the effectveness of our scheme. 5 For example, the maxmum transmsson power of WF-Drect s 30 mw and the data rate can be as hgh as 250 Mbps [22]. Whle for the LTE moble termnal n wreless cellular network, the typcal transmt power s 200 mw and the peak data rate s 75 Mbps [24]. Hence, the transmt power or the duraton of the SRC between MTs s much lower compared to that of the cellular communcatons n the uplnk and can be gnored. Furthermore, the analyss can be easly extended to the case that the energy consumpton and transmsson tme of SRC are constants and there wll not be maor changes n the results. power gan between the helpng MT H of the source MT K S and the BS s g = η G, 8) where the short-term Raylegh fadng of the channel power η s stll ndependently dstrbuted among the MTs. Hence, f helpng MT H s selected as the relay MT, the energy consumpton for ths data transmsson s E C,R) ) = σ2 2 DR) 1 g B. Defnton of Costs and Utltes, H, K S. 9) At dfferent battery levels, an MT has dfferent valuatons of the remanng energy n ts battery. The energy stored n the battery s generally more valuable when the battery level s low. Hence, we defne the unt energy cost ζ k for each MT k K as a functon of ts battery level B k,.e., ζ k = fb k ), 10) where B k [0,B max ] s the battery level of MT k wth ts range from zero to the maxmum storage B 6 max, and f : [0,B max ] [0,ζ max ] s a monotoncally decreasng functon of B k whose range s from zero to the maxmum energy cost ζ max > 0. 7 In order to motvate the helpng MT s partcpaton n the cooperaton, f a helpng MT H s selected by the source MT K S as the relay MT, t wll receve a prce π for transmttng wth data rate D R). The payment can be n the form of currency or credts n a multmeda applcaton. Hence, the utlty of helpng MT H by partcpatng n the cooperaton s π ζ E C,R), where E C,R) s the energy consumpton for transmttng wth data rate D R) as defned n 9). Furthermore, the helpng MT has a reservaton utlty of ǫ 0 for acceptng the request. That s, helpng MT H wll only accept the relay request from source MT K S f ǫ. Therefore, the utlty of the helpng MT H for source MT K S s the dfference between the prce and the energy cost f the dfference s larger than ǫ and zero otherwse, whch s defned as 8 π ζ E C,R) U = { π ζ E C,R), f π ζ E C,R) ǫ, 0, otherwse.. 11) For the source MT K S, f there s at least one helpng MT acceptng the prce π, the cost of the source MT K S s the sum of the prce π and the energy cost by drect transmsson ζ E C,S). Otherwse, t needs to drectly transmt to the BS 6 For analytcal tractablty, n ths paper, we assume that all the MTs have the same battery capacty B max. 7 Ths desgn of functon f s reasonable as a user wll value energy more when facng low battery, and we assume the mnmum energy cost equal to zero when B k = B max. 8 Here, note that the utlty functon s a concave functon of D S) wth dmnshng return and the cost functon to be defned n 12) s a convex and monotoncally ncreasng functon wth respect to D S). Hence, these defntons conform to the classc defnton of cost and utlty functons n economcs [25].

5 5 OneMThasdatato transmt CTmodehaslower costthandtmode? Yes Submt payment and sze ofdatatohelpngmts Helpng MT checks ts condton. Accept f satsfed. Any helpng MT accepts request? Yes Cooperatve Transmsson No No Drect Transmsson Fg. 2: Cooperatve communcatons protocol. wth rate D at the cost of ζ E D,S). Thus, the energy cost of source MT K S s { π +ζ E C,S), f H,π ζ E C,R) ǫ, C = ζ E D,S)., otherwse. 12) To ensure the mutual benefts of the source MT K S and relay MT H n the cooperaton, the prceπ should satsfy the followng nequalty ǫ a) b) π ζ E D,S) ζ E C,S), 13) where nequalty a) ensures the utlty ncrease of the helpng MTs n the cooperaton and nequalty b) ensures cost reducton for the source MT. Note that ζ E D,S) ǫ must hold for the feasblty of the CT mode. That s, the value of the energy consumpton by drect transmsson at the source MT must be larger than the reservaton utlty of the helpng MT. C. Cooperatve Transmsson Protocol Next, n order for the MTs n the cellular network to cooperate wth mutual benefts, we propose the followng cooperatve data transmsson protocol, whch s also shown by the flow chart n Fg. 2. 1) When an MT has data to transmt, t chooses between the CT mode and DT mode accordng to the crteron to be specfed later n 16) and 22), for the cases of complete and ncomplete nformaton, respectvely. 2) If the DT mode s selected, the source MT transmts drectly to the BS. If the CT mode s selected, t broadcasts the proposed payment and the relay data rate to all ts helpng MTs. 3) The helpng MT f any) accepts the request and sends an acceptance notfcaton to the source MT f the condton for cooperaton s satsfed or reects the request otherwse. 4) If multple helpng MTs accept the relay request, the source MT randomly chooses one MT as the relay MT and transmts the data wth the CT mode. 9 Otherwse, the source MT transmts wth the DT mode. In the above proposed cooperatve transmsson protocol, the key challenge s the mechansm desgn for ncentvzng the cooperaton of the MTs such that the MTs can mutually beneft. In the followng sectons, we propose prcng-based ncentve mechansm desgn for the cooperaton under dfferent nformaton sharng scenaros. III. BENCHMARK CASE: FULL COOPERATION UNDER COMPLETE INFORMATION In ths secton, we consder the deal case of full cooperaton under complete nformaton, where the prvate nformaton of the helpng MTs H, ncludng the number of helpng MTs N, ther battery levels B s and channel condtons g s, s known by each source MT K S. Ths case can happen when the MTs belong to a fully cooperatve group e.g., frends) that they are wllng to help each other wthout the requrement on the reservaton utlty ǫ and share ther prvate nformaton truthfully. Ths case wll also provde the performance benchmark upper bound) for the partal cooperaton under ncomplete nformaton n the next secton. Due to the full cooperaton nature among the MTs, the reservaton utlty of the helpng MT ǫ reduces to zero. Hence, source MT K S only needs to gve a payment to ts helpng for transmttng D R) such that the helpng MT s utlty n 11) s non-negatve. Hence, the requred amount of payment to MT H that s ust enough to cover the cost ζ E C,R) helpng MT H from source MT K S s ζ E C,R). Then, source MT K S needs to optmze the relay data rate D R) for each helpng MT H to mnmze the sum energy cost,.e., C, = mn. D R) 0 ζ E C,R) s.t. D R) +ζ E C,S) +D S) = D. 14) Problem 14) can be consdered as a weghted sum energy mnmzaton problem for the source and helpng MTs, where the weght s the unt energy cost of the ndvdual MT. It s evdent that when the weghts.e. unt energy cost) of the source and helpng MTs are equal, ths problem reduces to the sum energy mnmzaton problem. When the weght of one MT s larger than the other, the problem s more favorable for the MT wth lower energy and the optmzaton s more smlar to the max-mn optmzaton of the battery levels. 9 Because the relay data rate and prce are already determned by the source MT and each relay canddate provdes the same help to the source MT, the source MT does not care about whch helpng MT among those acceptng the request s chosen.

6 6 After obtanng the mnmum sum cost C, of assocatng wth each helpng MT H, source MT K S chooses the best helpng MT wth the followng relay selecton problem: P1) : Ĉ = mn. C,, 15) where C, s obtaned n problem 14). Next, we dscuss the crteron for the mode selecton of source MT K S, whch has been ntroduced n Secton II-C. For the source MT to choose the CT mode, ts cost reducton from the drect transmsson must be larger than a threshold denoted by γ, whch accounts for the overheads n cooperatve communcatons such as sgnalng and sgnal processng. Hence, f source MT K S chooses the CT mode, the followng condton has to be satsfed: Ĉ ζ E D,S) +γ. 16) In the followng, we dscuss the soluton for the mnmum cost Ĉ of full cooperaton under complete nformaton n two cases: non-splttable data.e. D R) = D ) and splttable data.e. 0 D R) D ). A. Cooperaton wth Non-splttable Data Frst, we dscuss the case where the data s not splttable at the source MT due to reasons such as lack of necessary processng functonaltes. In ths case, all the data of the cooperatve communcatons s transmtted by the relay MT = 0 and D R) = D ). Hence, accordng to 14),.e., D S) the cost of the source MT K S by assocatng wth helpng MT H reduces to C, = ζ σ 2 g 2 D 1 ) for problem P1). Then, the mnmum cost of full cooperaton wth nonsplttable data can be obtaned by solvng the smplfed relay selecton problem n P1). Therefore, the optmal transmsson of the source MT n the case of non-splttable data follows a two-step procedure: Frst, the source MT computes and fnds the helpng MT f any) wth the least energy cost. Then, t checks the condton n 16) and chooses between the DT mode and CT mode. B. Cooperaton wth Splttable Data It can be proved that problem 14) s a convex optmzaton problem and the optmal soluton s gven by the followng proposton. Proposton 3.1: The optmal data rate transmtted by the relay MT n problem 14) s gven by θ 0, f log 2 ˆD R) θ < D 1 = 2 D θ +log θ 2 θ ), f D log 2 θ < D, θ D, f log 2 θ D 17) where θ = ζ η and θ = ζ η can be nterpreted as the effectve energy cost of the source MT K S and helpng MT H, respectvely. Proof: Please refer to Appendx A for the detals. It can be observed from Proposton 3.1 that the optmal relay data rate follows a threshold structure wth respect to the log-rato between the effectve energy costs of the source and helpng MTs. When the effectve energy cost θ of the source MT K S s much lower than that of the relay MT H to the extent that log 2 θ θ < D s satsfed, the source MT wll not ask for help from ths helpng MT and transmt all by tself. If the effectve energy cost of the source and helpng MT s comparable, then the source and helpng MT wll splt the data package D for transmsson. Fnally, f the effectve energy cost of the source s much hgher than that of the helpng MT so that log 2 θ θ D s satsfed, then the helpng MT wll transmt the whole data package. IV. GENERAL CASE: PARTIAL COOPERATION UNDER INCOMPLETE INFORMATION In the prevous secton, we have consdered the full cooperaton under complete nformaton, whch s the optmal scenaro for the source MT and can serve as the benchmark scheme. However, ths scenaro s not applcable f the MTs belong to dfferent enttes that are not fully cooperatve and are unwllng to share prvate nformaton to each other. In ths secton, we consder the general scenaro where the MTs do not know exactly the other MTs channel condton and battery level and dscuss how these MTs stll can cooperate wth mutual benefts under ths scenaro. A. Problem Formulaton For cooperaton under ncomplete nformaton between the source and helpng MTs, we formulate the problem of decson makng under uncertantes wth the expected utlty theory ǫ) as the probablty that helpng MT H reects the request gven by the source MT K S. We assume that all the channel gans g s and battery statesb s of the helpng MTs H are ndependent. Hence, gven the set of helpng MTs H, the condtonal expected cost of the source MT K S for transmttng at data rate D s [26]. We denote Prπ ζ E C,R) E[C H ] = 1 Prπ ζ E C,R) ǫ) H π +ζ E C,S) )+ Prπ ζ E C,R) ǫ) ζ E D,S) H = 1 Prπ ζ E C,R) ǫ) H π +ζ E C,S) ζ E D,S) )+ζ E D,S), 18) where the expectaton s taken over the two possble outcomes of successful and unsuccessful relay assocaton n 12). By further consderng all possbltes of helpng MT set H for source MT K S n 6), the expected cost of the source MT K S can be obtaned by applyng the law of terated expectaton,.e., E[C ] = E[E[C H ]] = PrN = n)e[c H ], K S, 19) n=0

7 7 Here, t s worthwhle to dscuss the role of reservaton utlty ǫ n the expected energy cost E[C ]. As ǫ denotes the level of mnmum beneft for the relay MT n the cooperaton, t can be observed that the expected energy cost E[C ] should be monotoncally ncreasng wth ǫ. That s, wth a hgher reservaton utlty for the relay MT, the expected energy cost of the source MT s also hgher. Then, we formulate the optmzaton problem that mnmzes the expected cost of the source MT K S over the prce π and relay data D R) as follows: P2) : mn. π,d R) 0 E[C ] s.t. ǫ π ζ E D,S) D S) ζ E C,S), 20) +D R) = D. 21) Next, we dscuss the crteron for the mode selecton between the DT mode and CT mode. Smlar to the condton for the full cooperaton case n 16), for choosng the CT mode, we requre the expected) reducton of the source MT s energy cost from that of the drect transmsson to be larger than a threshold γ. In addton, consderng the feasblty condton for cooperaton n 13), the condton for the source MT to choose the CT mode s ζ E D,S) max{γ +E[C ],ǫ}, 22) where E[C ] s the mnmum expected cost obtaned n problem P2). It should be noted that the problem P2) s hard to be proved to be convex due to ts complex obectve functon; thus, t s dffcult to obtan ts optmal soluton n general. In the followng two subsectons, smlar to the prevous secton, we dscuss the mnmum expected cost E[C ] of the source MT K S n detals dependng on whether the data s splttable or not. B. Proposed Soluton for Problem P2) In ths subsecton, we frst smplfy problem P2) under some further assumptons. Then, smlar to Secton III, we dscuss the soluton of the problem under the cases of nonsplttable and splttable data, respectvely. Wth the energy consumpton E C,R) for the helpng MT H defned n 9), the probablty of successful assocaton between source MT K S and ts helpng MT H n 18) s Prπ ζ E C,R) where w s denoted as ǫ) = Pr ) ζ G π ǫ) η σ 2 2 DR) 1) = Pr ζ η w ), 23) w = G π ǫ) σ 2 2 DR) 1). 24) For smplcty, we further assume that the relaton between an MT s unt energy cost ζ k and ts battery level B k n 10) follows a lnear functon 10 ζ k = ζ max 1 B ) k. 25) B max We also assume that the battery levelb of the helpng MT H s known to the source MT K S as unform dstrbuton,.e. B U[0,B max ]. 11 Then, due to the lnear functon n 25), the energy cost ζ s also unformly dstrbuted as ζ U[0,ζ max ]. Hence, the probablty of successful assocaton between the source MT K S and helpng MT H s Prπ ζ E C,R) = 1 ζ max ζmax 0 ǫ) = Prη ζ ) w e η dη ζ dζ = w 1 e ζ w max ζmax w ). 26) Wth the results n 18) and 26), the obectve of problem P2) n 19) can be smplfed as E[C ] = n=0 [ PrN = n){ 1 1 w ) ) n] 1 e ζmax w ζ max } π +ζ E C,S) ) ζ E D,S) +ζ E D,S). 27) Next, we dscuss the convexty of problem P2) by the followng proposton. Proposton 4.1: Problem P2) s margnally convex wth respect to π and D R). Proof: Please refer to Appendx B for the detals. It should be noted that problem P2) s not a convex optmzaton problem. Ths s because the obectve of the problem s not ontly convex wth respect to π and D R). In the followng, smlar to Secton III, we dscuss the optmal soluton for problem P2) to obtan E[C ] under two cases: non-splttable.e. D R) = D ) and splttable data.e. 0 D R) D ). 1) Optmal Prcng for Non-splttable Data: Frst, we dscuss the case where the data s not splttable. In ths case, all the data of the source MTs s transmtted by the relay MT.e., D S) = 0 and D R) = D ). As w n 24) s now reduced to w = Gπ ǫ), problem P2) s smplfed σ 2 2 D 1) 10 It should be noted that our analyss can be extended to the other monotoncally non-ncreasng functons, whose analyss wll be techncally more nvolved but offers essentally smlar engneerng nsghts. It should also be noted that the choce of the functon f reflects the senstvty of the MTs towards the usage of the energy n the battery. By adoptng a functon that s n-dfferent to the battery level, the desgn obectve s more smlar to mnmzng the total energy consumpton. Instead, by adoptng a functon that s senstve to the battery level, ths desgn s more favorable for the MTs wth low battery level. 11 Our proposed scheme can stll be applcable to the case of heterogeneous battery capacty. One heurstc s that, based on the statstcs of the battery capactes of the MTs, the source MT K S can obtan the average battery capactes of the MTs as B max and predct the battery level of the helpng MT H as B U[0, B max]. Then, the proposed cooperatve communcatons protocol wth prcng under uncertanty stll apples.

8 8 to the followng problem wthout load sharng: P2 ) : [ mn. PrN = n){ 1 π n=0 π ζ E D,S) ) s.t. ǫ π ζ E D,S). 1 w ) ) n] 1 e ζmax w ζ max } +ζ E D,S) Because the data transmtted by the relay MT s fxed at D R) = D, accordng to Proposton 4.1, the problem s convex wth respect to π. Therefore, for ths un-varable convex optmzaton problem, the optmal soluton can be obtaned by checkng the frst-order condton of optmalty. However, the obectve functon of problem P2 ) s stll complcated, for whch the dervatve s hard to obtan. Hence, we propose Algorthm I based on the dervatve-free dchotomous search [27] to obtan the optmal soluton numercally for problem P2 ). TABLE II: One-dmensonal dchotomous search algorthm for solvng problem P2 ) wth precson δ π and τ Intalze: π l) := ǫ, π h) 2. Repeat: 1. Set temporary parameters: Algorthm I π l) := 1 2 πl) +π h) ) τ π, π h) := ζ E D,S), π := π l) π h) ; := 1 2 πl) +π h) 2. If E[C π l) )] < E[C π h) )], set the prce as π h) )+τ π ; := π l) ; ; 3. If E[C π l) )] > E[C π h) )], set the prce as π l) := π h) 4. Otherwse, set π h) := π h) and π l) := π l) ; 5. π := π l) π h) ; 3. Untl: the condton π > δ π s volated; 4. π := πh) +π l) )/2, E[C ] := E[C π )]. 2) Jont Prcng and Load Sharng for Splttable Data: Next, we dscuss the general case where the data s splttable at the source MT n problem P2). Accordng to Proposton 4.1, the obectve functon of problem P2) s convex wth respect to π gven a fxed D R) and to D R) gven a fxed π. Hence, based on the dchotomous search algorthm n Algorthm I, we propose Algorthm II that approxmately mnmzes the expected cost of the source MT K S wth alternatve optmzaton. For Algorthm II, t starts wth the optmal soluton obtaned n Algorthm I wth D R) = D. The algorthm then proceeds by teratvely optmzng and updatng π and D R) wth the other fxed untl the stoppng condton s satsfed. It should be noted that the algorthm always converges to a certan value wthn the range ofδ C from at least a locally optmal soluton. Ths s because each teraton of the algorthm reduces the obectve value and the optmal value of problemp2) s lower bounded. Fnally, for the complexty of Algorthm I, the the maxmum number of teratons requred for the searchng of the optmal TABLE III: Alternatve optmzaton algorthm for solvng problem P2) wth precson δ C. Algorthm II 1. Intalze: n := 0, D R) := D, E[C 0) ] := ζ E D,S) ; 2. Repeat: 1. Optmze the obectve of problem P2) wth respect to π by dchotomous search wth D R) fxed ; 2. Optmze the obectve of problem P2) wth respect to D R) by dchotomous search wth π fxed ; 3. n := n+1; 3. Untl: the condton E[C n) ] E[C n 1) ] > δ C s volated. TABLE IV: General smulaton setup Smulaton Parameters Values Nose power σ 2 = 110 dbm Path-loss exponent α = 3.6 Reference dstance r 0 = 10 m Path-loss at r 0 G 0 = 70 db Relay MT reservaton utlty ǫ = 0.2 Cost reducton threshold γ = 1 Maxmum battery level B max = 100 J Maxmum unt energy cost ζ max = 1 ζ prcng π wth precson δ π s Olog E D,S) 2 δ π ). Next, for the complexty of Algorthm II, the upper bound of each lne search for D R) and π are N D R) = log 2 D ζ N π = log E D,S) 2 δ π, respectvely, where δ R) D requrement for the lne search of D R) δ D R) ) and s precson. The upper bound for the total number of teratons n the above alternatve optmzaton s M = ζed,s) δ C. Hence, the upper bound for the complexty of Algorthm II s OMN π + N R) D )). Moreover, gven the data rate of the source MT, user densty, energy cost and channel condton, the optmal soluton can be computed off-lne and stored n a look-up table for practcal mplementaton. V. NUMERICAL RESULTS In ths secton, we frst show the convergence of the algorthm for the sngle source MT and examne ts performance under dfferent transmsson schemes. Then, the smulaton of multple source MTs s gven to show ther real-tme operaton under our proposed protocol n a sngle-cell system. The general smulaton parameters are gven n Table IV and specfc smulaton setup and parameters for the cases of sngle source MT and multple source MTs wll be elaborated later n each subsecton. A. Sngle Source MT In ths subsecton, we consder the smulaton for sngle source MT. We frst show the convergence of Algorthm II for the partal cooperaton wth splttable data rate and compare the convergent cost to that wth non-splttable data rate by 12 The typcal range of LTE n the urban envronment s 1-5 km. [28]

9 9 TABLE V: Smulaton setup for sngle source MT Smulaton Parameters Values Dstance from the source MT K S to BS 12 r = 50 m Short-term fadng of ths sngle source MT η = 0.5 Intal battery level of the source MT B = 10 J Algorthm I. Then, we show the smulaton results for the expected cost of the sngle source MT versus battery levels under dfferent schemes. The specfc smulaton parameters for ths case of sngle source MT are gven n Table V. 1) Convergence of Algorthm II for partal cooperaton: Frst, we show the convergence of Algorthm II for the partal cooperaton wth splttable data compared wth that wth nonsplttable data by Algorthm I n Secton IV-B2 for the data transmsson of a sngle source MT K S under dfferent data rates D and average number of helpng MTs µ N. Frst, exhaustve search on π and D R) wth quantzaton of 0.2 and 0.1 n the feasble regons s conducted for three cases wth dfferent pars of µ N and D and the mnmum expected costs are 11.51, 4.46 and 1.55, respectvely. Then, the result of the ont optmzaton of π and D by Algorthm II wth splttable data SD) s shown n Fg. 3 wth the sold lne. The expected cost at teraton{0} denotes the cost by the drect transmsson, whch are 19.96, and 2.22, respectvely. The procedure n Algorthm II s executed 4 teratons and each of the unvarable dchotomous search sub-routnes n Algorthm II s executed 8 tmes, wth sub-routnes {1, 3, 5, 7} for mnmzaton wth respect to π and sub-routnes {2,4,6,8} for that wth respect to D R) n Algorthm II. For comparson, the result by Algorthm I wth non-splttable data NSD) s shown by the three dash lnes. The smulaton result s shown n Fg. 3 and t can be observed that the convergence s fast. The converged expected energy costs for the three cases are 11.51, 4.46 and 1.55, respectvely, whch are the same as the results by exhaustve search. Hence, the global optmal soluton s obtaned n ths case. Furthermore, the expected cost reducton from drect transmsson for the three cases are 8.45, and 0.67, respectvely. Hence, accordng to the condton n 22), the transmsson mode selected by the source MT for the three cases wll be CT, CT and DT, respectvely. By comparng the two cases wth D = 6 bps/hz, t can be observed that a hgher densty of helpng MTs can further reduce the expected energy cost. By comparng the two cases wth splttable and non-splttable data, t can also be observed that, n addton to the optmal prcng, load sharng can ndeed further reduce the expected energy cost. Furthermore, the cost reductons wth load sharng are 1.57, 1.25 and 1.12 tmes of those wthout load sharng, respectvely. Therefore, load sharng s more cost-effectve when the sze of the data s large and the average number of helpng MTs s small. Fnally, t should be noted that the case of splttable data leads to a lower energy cost compared wth that of non-splttable data. Ths s because the energy consumpton s exponentally ncreasng wth respect to the transmsson data rate and splttng the data and further optmzng the relay data rate result n a smaller total energy Expected Energy Cost SD, µ N =1, D =3 bps/hz SD, µ N =1, D =6 bps/hz SD, µ N =4, D =6 bps/hz NSD, µ N =1, D =3 bps/hz NSD, µ N =1, D =6 bps/hz NSD, µ N =4, D =6 bps/hz Number of Sub routne Executons Fg. 3: Expected energy cost of partal cooperaton under ncomplete nformaton wth splttable and non-splttable data versus the number of sub-routne executons under dfferent µ N s and D s. Expected Energy Cost DT Part. Coop. In Comp Info NSD Part. Coop. In Comp. Info. SD Full Coop. Comp. Info. NSD Full Coop. Comp. Info. SD Battery Level J) Fg. 4: Expected cost of the sngle source MT versus ts battery level under dfferent schemes. consumpton. In summary, for the cooperatve transmsson under complete nformaton wth splttable data, the optmal soluton s obtaned by the followng three-step procedure: Frst, the source MT computes the optmal data rate for each helpng MT accordng to Proposton 3.1. Then, t searches for the one wth the lowest energy cost from all the helpng MTs by problem P1). Last, t checks the condton n 16) and chooses between the DT and CT mode. 2) Expected energy cost under dfferent battery levels and transmsson schemes: Next, we show the expected cost of dfferent schemes under dfferent battery levels. The smulaton setup s shown as follows. We consder the smulaton under the followng 5 schemes: Drect Transmsson DT) n 5).

10 10 Full Cooperaton under Complete Informaton wth Non-Splttable Data Full Coop. Comp. Info. NSD) n Secton III-A. Full Cooperaton under Complete Informaton wth Splttable Data Full Coop. Comp. Info. SD) n Secton III-B. Partal Cooperaton under Incomplete Informaton wth Non-Splttable Data Part. Coop. In-Comp. Info. NSD) n Secton IV-B1. Partal Cooperaton under Incomplete Informaton wth Splttable Data Part. Coop. In-Comp. Info. SD) n Secton IV-B2. Specfcally, for the schemes of partal cooperaton under ncomplete nformaton wth splttable and non-splttable data, the mnmum expected costs are obtaned by Algorthms I and II, respectvely. For the schemes of full cooperaton under complete nformaton wth splttable and non-splttable data, the number of helpng MTs N for source MT K S s generated accordng to the Posson dstrbuton wth µ N = 2 and the source MT transmts the data at the rate of D = 4 bps/hz. Ther battery levels B, H are unformly generated on [0,B max ] and short term Raylegh fadng η, H s generated accordng to exp1). The mnmum energy costs can be obtaned by the results n Sectons III-A and III-B, respectvely, and the results are averaged over 1000 ndependent realzatons for accurately obtanng the expected energy costs for comparson. The expected cost of the transmsson wth DT mode s also obtaned by averagng over 1000 ndependent realzatons. The smulaton result s shown n Fg. 4. It can be observed that our proposed cooperatve communcatons scheme performs sgnfcantly better than the drect transmsson benchmark. Moreover, cooperatve communcatons s more effectve when the battery level of the source MT s low. Ths s because when the battery level of the source MT s hgh and cost for drect transmsson s low, t s less lkely to seek help from the other MTs. Furthermore, t can also be observed that there are gaps n expected energy costs between the schemes wth complete nformaton n Secton III and those wth ncomplete nformaton n Secton IV, whch are explaned as follows: ) In the case of complete nformaton wth non-splttable data, the source MT can observe the set of helpng MTs as well as ther channel condtons and battery levels, and choose the most cost-effcent one as the relay. Whle for the ncomplete nformaton case, the source MT can only randomly choose one from the possble helpng MTs that accept the offer wth the rsk of endng up wth drect transmsson. ) In the case of complete nformaton wth splttable data, n addton to the reason n ), the source MT can ontly optmze the relay data rate and the payment and choose the helpng MT that leads to the mnmum sum energy cost as n Proposton 3.1). Whle n the case of ncomplete nformaton wth splttable data, the source can only optmze the payment and relay data rate wth respect to the expected energy cost, whch has the possblty that the source MT ends up wth drect Meters Source MTs Idle MTs BS Meters Fg. 5: Setup for the smulaton of multple source MTs wth K = 100 MTs. TABLE VI: Smulaton setup for multple source MTs Smulaton Parameters Values Total number of MTs K = 100 Probablty at whch MTs ntate data transmsson ρ = 0.2 Normalzed data rate 13 D = 6 bps/hz Range of the SRC for a source MT d = 7 m transmsson due to the lack of helpng MTs nformaton. ) In both the cases wth and wthout splttable data under complete nformaton, the reservaton utlty margn ǫ for the helpng MT s zero, whch further reduces the cost of the source MT from that under ncomplete nformaton and fully motvates the cooperaton. Fnally, the fgure shows that, when battery levels equal 100 J, the expected energy costs of all cases also become zero. Ths s due to our assumpton that the energy cost at full battery capacty.e. battery level equals 100 J) s zero and at ths case, there s no cooperaton between the source and relay MTs. B. Multple Source MTs In ths subsecton, we conduct a smulaton wth multple source MTs and show the real-tme operaton of our proposed cooperatve communcatons protocol wthn a sngle cell. We examne the fve schemes consdered n the prevous subsecton and show the performance mprovement n terms of battery and communcatons outage, average battery level and battery level dstrbuton, under a sngle-cell setup. The specfc smulaton parameters for multple source MTs are gven n Table VI and the smulaton setup s descrbed as follows. We consder our smulaton wthn a m 2 square area as shown n Fg. 5. The operaton of the system begns wth the battery levels of the MTs unformly generated on [24] 13 The uplnk spectrum effcency of the LTE system s bps/hz.

11 11 TABLE VII: Number of communcatons and battery outages for the fve cases after 300 tme slots. Part. Coop. In-Comp. Info. Full Coop. Comp. Info. DT NSD SD NSD SD Commun. Outage Battery Outage [0,B max ]. For the purpose of nvestgaton, at the begnnng of each tme slot, the postons of the MTs are unformly regenerated wthn the above mentoned area. In ths setup, t s possble that there s overlap between the set of helpng MTs for dfferent source MTs, where one helpng MT can possbly be assocated wth two source MTs. In order to avod ths stuaton, we re-generate the postons of the source MTs f there s overlap between the helpng MTs. Accordng to the functon µ N = 1 ρ)λπd 2, K S, the average number of helpng MTs n ths setup s µ N = 1.2. Due to the physcal constrant of the MT, we set the maxmum transmt energy of the MT ase max = 3 J for any tme slot. If the transmt energy of the MT exceeds E max, a communcatons outage wll be declared by the MT and the data package s dscarded. Durng the operaton of the system, f the battery of a certan MT s draned out, ths MT declares a battery outage and ceases any operaton from that tme on, ncludng data transmsson as a source MT or cooperatve relay for the other source MTs as relay MT. We show the total number of communcatons and battery outages for the 100 MTs after 300 tme slots for the same 5 schemes as n Secton V-A. The smulaton results are shown n Table VII. It can be observed that, compared wth the benchmark case of DT, all of the four proposed schemes wth cooperatve communcatons perform better n terms of communcatons and battery outage. The reducton of the battery outages reflects the effectveness of our protocol desgn for the energy savng of the MT, especally for those MTs that are low n battery level. In addton to the reducton of the battery outage, our proposed scheme also shows sgnfcant reducton n the number of communcatons outage. Ths s because, n the case of drect transmsson, f the channel condton of the source MT s poor, the transmsson power wll exceed the peak power constrant E max and communcatons outage wll occur. Whle, under the same crcumstances wth cooperatve communcatons, the source MT can seek help from the other helpng MTs, whose transmt power s possbly lower than the peak power constrant and the transmsson can be successful. Hence, our proposed schemes can mprove the uplnk data transmsson of the MTs n terms of both the communcatons relablty and battery sustanablty. Next, we show the average battery level k B k/ K of the MTs durng the 300 tme slots n Fg. 6. It can be observed that the average battery levels of dfferent schemes drop wth dfferent rates. Compared wth the benchmark-case of DT, our proposed protocol can effectvely ncrease the average battery level of the MTs over tme. Even though the MTs under cooperatve communcatons successfully delver more data packages as shown n Table VII, these schemes stll perform Average Battery Level J) DT Part. Coop. In Comp Info NSD 20 Part. Coop. In Comp. Info. SD Full Coop. Comp. Info. NSD Full Coop. Comp. Info. SD Tme Slots Fg. 6: Average battery level k B k/ K of the MTs over tme. Fg. 7: Dstrbuton of the battery levels of 100 MTs after 300 tme slots. better n terms of average battery level. Fnally, we show the dstrbuton of the battery levels of the 100 MTs at the end of the 300 tme slots n Fg. 7. It can be observed that, for the benchmark case of DT, a large proporton of the MTs have draned out ther batteres. Whle for the other cases wth cooperaton, ther battery levels reman on the relatvely hgher level than the drect transmsson case by the dstrbuton. It should also be noted that although a lot of MTs under cooperatve communcatonss stay n the low battery