Energy Efficient Resource Allocation in Machine-to-Machine Communications with Multiple Access and Energy Harvesting for IoT

Transcription

1 1 Energy Effcent Resource Allocaton n Machne-to-Machne Communcatons wth Multple Access and Energy Harvestng for IoT Zhaohu Yang, We Xu, Senor Member, IEEE, Yjn Pan, Cunhua Pan, and Mng Chen arxv: v1 [cs.it] 29 Nov 2017 Abstract Ths paper studes energy effcent resource allocaton for a machne-to-machne M2M enabled cellular network wth non-lnear energy harvestng, especally focusng on two dfferent multple access strateges, namely non-orthogonal multple access NOMA and tme dvson multple access TDMA. Our goal s to mnmze the total energy consumpton of the network va jont power control and tme allocaton whle takng nto account crcut power consumpton. For both NOMA and TDMA strateges, we show thas optmal for each machne type communcaton devce MTCD to transmt wth the mnmum throughput, and the energy consumpton of each MTCD s a convex functon wth respect to the allocated transmsson tme. Based on the derved optmal condtons for the transmsson power of MTCDs, we transform the orgnal optmzaton problem for NOMA to an equvalent problem whch can be solved suboptmally va an teratve power control and tme allocaton algorthm. Through an approprate varable transformaton, we also transform the orgnal optmzaton problem for TDMA to an equvalent tractable problem, whch can be teratvely solved. Numercal results verfy the theoretcal fndngs and demonstrate that NOMA consumes less total energy than TDMA at low crcut power regme of MTCDs, whle at hgh crcut power regme of MTCDs TDMA acheves better network energy effcency than NOMA. Index Terms Internet of Thngs IoT, machne-to-machne M2M, non-orthogonal multple access NOMA, energy harvestng, resource allocaton. I. INTRODUCTION Machne-to-machne M2M communcatons have been consdered as one of the promsng technologes to realze the Internet of Thngs IoT n the future 5th generaton network. M2M communcatons can be appled to many IoT applcatons, whch manly nvolve new busness models and opportuntes, such as smart grds, envronmental montorng and ntellgent transport systems [2]. Dfferent from conventonal human type communcatons, M2M communcatons have many unque features [3]. The unque features nclude Ths work was supported n part by the Natonal Nature Scence Foundaton of Chna under grants , & , n part by the Sx Talent Peaks projecn Jangsu Provnce under grant GDZB-005, n part by the UK Engneerng and Physcal Scences Research Councl under Grant EP/N029666/1, and n part by the Scentfc Research Foundaton of Graduate School of Southeast Unversty under Grant YBJJ1650. Ths paper was presented at the IEEE Infocom Workshops 2017 n Atlanta, GA, USA [1]. Correspondng authors: We Xu; Cunhua Pan. Z. Yang, W. Xu, Y. Pan, and M. Chen are wth the Natonal Moble Communcatons Research Laboratory, Southeast Unversty, Nanjng , Chna, Emal: {yangzhaohu, wxu, panyjn, chenmng}@seu.edu.cn. C. Pan s wth the School of Electronc Engneerng and Computer Scence, Queen Mary, Unversty of London, London E1 4NS, U.K., Emal: c.pan@qmul.ac.uk. massve transmssons from a large number of machne type communcaton devces MTCDs, small bursty natured traffc perodcally generated, extra low power consumpton of MTCDs, hgh requrements of energy effcency and securty. A key challenge for M2M communcatons s access control, whch manages the engagement of massve MTCDs to the core network. To tackle ths challenge, varous solutons have been proposed, e.g., by usng wred access cable, DSL [4], wreless short dstance technques WLAN, Bluetooth, and wde area cellular network nfrastructure Long Term Evoluton-Advanced LTE-A, WMAX [5]. Among all these solutons, an effectve approach s to deploy machne type communcaton gateways MTCGs to act as relays of MTCDs [3]. Wth the help of MTCGs, all MTCDs can be successfully connected to the base staton BS at the addtonal cost of energy consumpton [6] [9]. To enable multple MTCDs to transmt data to the same MTCG, tme dvson multple access TDMA was adopted n [10]. However, snce there are a vast number of MTCDs to be served, TDMA leads to large transmsson delay and synchronzaton overhead. By splttng users n the power doman, non-orthogonal multple access NOMA can smultaneously serve multple users at the same frequency or tme resource [11]. Consequently, NOMA based access scheme yelds a sgnfcant gan n spectral effcency over the conventonal orthogonal TDMA [11] [16]. Ths favorable characterstc makes NOMA an attractve soluton for supportng massve MTCDs n M2M networks. Consderng NOMA, [17] nvestgated an M2M enabled cellular network, where multple MTCDs smultaneously transmt data to the same MTCG and multple MTCGs smultaneously transmt the gathered data to the BS. Besdes, another key challenge s the energy consumpton of MTCDs [18] [20]. Accordng to [3], the total system throughput of an M2M network s manly lmted by the energy budget of the MTCDs. To mprove the system performance, energy harvestng EH has been appled to wreless communcaton networks [21] [24]. In partcular, drect and non-drect energy transfer based schemes for EH were nvestgated n [23], whle n [24], the optmzaton of green-energy-powered cogntve rado networks was surveyed. Recently, the downlnk resource allocaton for EH n small cells was studed n [25] [27]. By usng EH, MTCDs are able to harvest wreless energy from rado frequency RF sgnals [28] [31], and the system energy can be sgnfcantly mproved. Consequently, mplementng EH s promsng n M2M communcatons especally wth MTCDs confgured wth low power consumpton. In prevous

2 2 wreless powered communcaton networks usng relays [32] [34], t was assumed that an energy constraned relay node harvests energy from RF sgnals and the relay uses that harvested energy to forward source nformaton to destnaton. Due to the extra low power budget of MTCDs n M2M communcatons, s reasonable to let the MTCD transmt data to an MTCG, and then the MTCG relays the nformaton whle the MTCD smultaneously harvests energy from the MTCG, whch s dfferent from exstng works, e.g., [32] [34]. Enablng the source node to harvest energy from the relay node, a power-allocaton scheme for a decode-and-forward relayng-enhanced wreless system was proposed n [35] wth one source node, one relay node and one destnaton node. The above-mentoned energy consumpton models, consdered n [1], [8] [10], [17] [19], [28] [31], are only concerned wth the RF transmsson power and gnore the crcut power consumpton of MTCDs and MTCGs. However, as stated n [36], the crcut power consumpton s non-neglgble compared to RF transmsson power. Wthout consderng the crcut power consumpton, energy savng can always beneft from longer transmsson tme [37], [38]. Whle consderng the crcut power consumpton whch defntely exsts n practce, the results vary sgnfcantly n that can not be always energy effcent for long transmsson tme due to the fact that the total energy consumpton becomes nfnty as transmsson tme goes nfnty. Hence, s of mportance to nvestgate the optmal transmsson tme when takng nto account the crcut power consumpton n applcatons. Gven access control and energy consumpton challenges n M2M communcatons, both TDMA and NOMA based M2M networks wth EH are proposed n ths work. The MTCDs frst transmt data to the correspondng MTCGs, and then MTCGs transmt wreless nformaton to the BS and wreless energy to the MTCDs. To prolong the lfetme of the consdered network, the harvested energy for each MTCD s set to be no less than the consumed energy n nformaton transmsson IT stage. The man contrbutons of ths paper are summarzed as follows: We formulate the total energy mnmzaton problem for the M2M enabled cellular network wth non-lnear energy harvestng EH model va jont power control and tme allocaton. In the non-lnear EH model, we consder the recever senstvty, on whch energy converson starts beyond a threshold. Besdes, we explctly take nto account the crcut energy consumpton of both MTCDs and MTCGs. All theses factors are crtcal n practcal applcatons whch nevtably affect the system performance. Specfcally, the non-lnear EH model leads to a nonsmooth objectve functon and non-smooth constrants, and the crcut energy consumpton affects the optmal transmsson tme of the system. For the NOMA strategy, we observe that: 1 s optmal for each MTCD to transmt wth mnmal throughput; 2 s further revealed that the energy consumpton of each MTCD s a convex functon wth respect to the allocated transmsson tme. Gven these observatons, ndcates that a globally optmal transmsson tme always exsts that the optmal transmsson tme equals the maxmally allowed transmsson tme f t does not exceed a quantfed threshold derved n closed form. To solve the orgnal total energy mnmzaton problem for the NOMA strategy, we devse a low-complexty teratve power control and tme allocaton algorthm. Specfcally, to deal wth the non-smooth EH functon, we ntroduce new sets durng whch MTCDs can effectvely harvest energy. Gven new sets, the EH functon of MTCDs can be presented as a contnuous one. Moreover, to deal wth nonconvex objectve functon, nonconvex mnmal throughput constrants, and nonconvex energy causalty constrants, we transform these nonconvex ones nto convex ones by manpulatons wth the optmal condtons. The convergence of the teratve algorthm s strctly proved. For the TDMA strategy, we verfy that the two observatons for NOMA are also vald. Although the orgnal total energy mnmzaton problem for the TDMA strategy s nonconvex, the problem can be transformed nto an equvalent tractable one, whch can be teratvely solved to ts suboptmalty. For the total energy mnmzaton, numercal results dentfy that NOMA s superor over TDMA at small crcut power regme of MTCDs, whle TDMA outperforms NOMA at large crcut power regme of MTCDs. Ths paper s organzed as follows. In Secton II, we ntroduce the system and power consumpton model. Secton III and Secton IV provde the energy effcent resource allocaton for NOMA and TDMA, respectvely. Numercal results are dsplayed n Secton V and conclusons are drawn n Secton VI. II. SYSTEM AND POWER CONSUMPTION MODEL A. System Model Consder an uplnk M2M enabled cellular network wth N MTCGs and M MTCDs, as shown n Fg. 1. Denote the sets of MTCGs and MTCDs by N {1,,N} and M {1,,M}, respectvely. Each MTCG serves as a relay for some MTCDs. Assume that the decode-andforward protocol [39] s adopted at each MTCG. Denote J {J 1 +1,,J } as the specfc set of MTCDs served by MTCG N, where J 0 0, J N M, J l1 J l, and s the cardnalty of a set. 1 To reduce the recever complexty at the MTCG, the maxmal number of MTCDs assocated to one MTCG s set as four. Obvously, we have N J M. B. NOMA Strategy In tme constrant T, each MTCD has some payloads to transmt to the BS. By usng superposton codng at the transmtter and successve nterference cancellaton SIC at 1 In ths paper, we assume that MTCDs are already assocated to MTCGs by usng the cluster formaton methods for M2M communcatons, e.g., n [40] [43]. Jont optmzaton of cluster formaton and resource allocaton n M2M communcatons wth NOMA/TDMA and EH can certanly further mprove the performance, but we leave n future work n order to focus on the power control and tme allocaton n the current submsson.

3 3 MTCG1 BS Informaton transmssonit Energy harvestngeh MTCGN smultaneously MTCDs harvest energy from all MTCGs. In the -th N phase wth allocated tme, all MTCDs n J smultaneously transmt data to MTCG accordng to the NOMA prncple and MTCG detects the sgnal. In the N + k-th k K phase wth allocated tme t N+k, all MTCGs n smultaneously transmt the decoded data from the served MTCDs to the BS by usng the NOMA strategy. As a result, we have the followng transmsson tme constrant MTCD2 MTCD1 MTCD M-1 MTCDM Fg. 1. The consdered uplnk M2M enabled cellular network. T. 1 In the -th N phase, all MTCDs n J smultaneously transmt data to MTCG followng the NOMA prncple. The receved sgnal of MTCG s MTCD uplnk IT MTCGuplnkIT&MTCDEH y jj 1+1 h j pj s j +n, 2 t 1 MTCDs t 1 t N MTCDs T t N+1 MTCGs t MTCGs Fg. 2. Tme sharng scheme for NOMA strategy durng one uplnk transmsson perod. the recever, multple MTCDs or MTCGs can smultaneously transmt sgnals to the correspondng recever usng NOMA. To reduce the recever complexty and error propagaton due to SIC, s reasonable for the same resource to be multplexed by a small number usually two to four of devces [44]. Consderng the recever complexty at the BS, the sets of MTCGs are further classfed nto multple small clusters. For MTCGs, the set N s classfed nto K clusters. Let K {1,2,,K} be the set of clusters. Denote {1 + 1,, } as the specfc set of MTCGs n cluster k K, where I 0 0, I K N, k l1 I l, and 4. 2 Note that our major motvaton of usng NOMA s to enhance the ablty of servng more termnals smultaneously [11]. However, the number of termnals occupyng the same resource cannot be arbtrarly large n order to make NOMA effectve n practce. Therefore, f there would be an even hgher number of IoT termnals, we beleve that a number of ways ncludng NOMA should be further ncorporated to better address the access problem for hgher number of IoT termnals [47]. As depcted n Fg. 2, tme T conssts of N + K uplnk transmsson phases for MTCDs and MTCGs. NOMA s adopted for MTCDs to transmt data to MTCGs n the frst N phases. Both NOMA and EH are n operaton n the last K phases where MTCGs transmt data to the BS and 2 A scheme for cluster formaton for uplnk NOMA s n [45]. Accordng to [45] and [46], one common scheme wth two devces n each cluster s the strong-weak scheme,.e., the devce wth the strongest channel condton s pared wth the devce wth the weakest, and the devce wth the second strongess pared wth one wth the second weakest, and so on. where h j s the channel between MTCD j and MTCG, p j denotes the transmsson power of MTCD j, s j s the transmtted message of MTCD j, and n represents the addtve zero-mean Gaussan nose wth varance σ 2. Wthout loss of generalty, the channels are sorted as h J1+1 2 h J 2. By applyng SIC to decode the sgnals [14] [16], the achevable throughput of MTCD j J s h j r j log p j J h, 3 l 2 p l +σ 2 where B s the avalable bandwdth for transmsson. Note that we consder the case where MTCDs assocated to dfferent MTCGs are allocated wth orthogonal tme resource. Therefore, the nterference from other MTCDs assocated to dfferent MTCDs s gnored. In the N +k-th k K phase wth allocated tme t N+k, after havng successfully decoded the messages n the last N phases, MTCGs n smultaneously transmt the gathered data to the BS based on the NOMA prncple. Denote the channel between MTCG and the BS by h. Wthout loss of generalty, the channels are sorted as h Ik h Ik 2, k K. Hence, the achevable throughput of MTCG can be expressed as [14] [16] h r Bt N+ log q Ik, 4 n+1 h n 2 q n +σ 2 where q s the transmsson power of MTCG. Accordng to [3], MTCDs are always equpped wth fnte batteres, whch lmt the lfetme of the M2M enabled cellular network. To further prolong the lfetme, EH technology s adopted for MTCDs to harvest energy remotely from RF sgnals radated by MTCGs [22]. Specfcally, each MTCD harvests energy when MTCGs transmt data to the BS. Snce the nose power s much smaller than the receved power of MTCGs n practce [48] [50], the energy harvested from the channel nose s neglgble. Assume that uplnk channel and downlnk channel follow the channel recprocty [51]. The

4 4 Harvested power 0 P 0 Lnear EH model Non-lnear EH model Input RF power Fg. 3. Comparson between the lnear and non-lnear EH model. total energy harvested by MTCD j served by MTCG can be evaluated as K Ej H t N+k u h nj 2 q n, N,j J, k1 n where nik1 +1 h nj 2 q n s the receved RF power of MTCD j durng tme t N+k, and functonu captures the EH model whch maps nput RF power nto harvested power. Two commonly used EH models are shown n Fg. 3,.e., lnear and non-lnear EH models. Accordng to [52] and [53], lnear EH model may lead to resource allocaton msmatch. In order to capture the effects of practcal EH crcuts on the end-toend power converson, we adopt the more practcal non-lnear EH model proposed n [52]: { M1+e ab ux M e ab +e ax2b e, f x P ab 0, 6 0, elsewse where a, b, M and P 0 are postve parameters whch capture the jont effects of dfferent non-lnear phenomena caused by hardware constrants. Note that P 0 s the recever senstvty threshold of each MTCD, n whch energy converson starts. Hence, s possble that some MTCDs cannot effectvely harvest energy n some slots, snce the receved power s below the recever senstvty threshold P 0. The total energy consumpton of the M2M enabled communcaton network conssts of two parts: the energy consumed by MTCDs and MTCGs. For each part, the energy consumpton of a transmtter conssts of both RF transmsson power and crcut power due to hardware processng [54]. Accordng to [55], the energy consumpton when MTCDs or MTCGs are n dle model,.e., do not transmt RF sgnals, s neglgble. Durng the -th N phase, MTCD j J served by MTCG just transmts data to MTCG wth allocated tme and transmsson power p j. Thus, the energy E j consumed by MTCD j J can be modeled as E j pj η +P C, N,j J, 7 where η 0,1] and P C denote the power amplfer PA effcency and the crcut power consumpton of each MTCD, respectvely. Accordng to the energy causalty constrann EH networks, E j has to satsfy E j Ej H. Summng the energy consumed by all MTCDs n J, we can obtan the energy E consumed durng the -th phase as E jj 1+1 E j, N. 8 Durng the N + k-th phase, the system energy consumpton, denoted by E N+k, s modeled as q E N+k t N+k ξ +QC 1 +1 N jj 1+1 t N+k u n1 +1 h nj 2 q n,9 where ξ 0,1] and Q C are the PA effcency and the crcut power consumpton of each MTCG, respectvely. Accordng to the law of energy conservaton [56], we must have 1 +1 N t N+k q jj 1+1 t N+k u n1 +1 h nj 2 q n >0, 10 whch s the energy loss due to wreless propagaton. Based on 5-9, the total energy consumpton, E Tot, of the whole system durng tme T can be expressed as E Tot N E jj K k11 +1 K N C. TDMA Strategy k1 jj 1+1 pj η +P C q t N+k ξ +QC t N+k u n1 +1 h nj 2 q n.11 Wth the TDMA strategy, tme T conssts of M+N uplnk transmsson phases for MTCDs and MTCGs, as llustrated n Fg. 4. All MTCDs transmt data to the correspondng MTCGs n the frst M phases wth TDMA, and all MTCGs transmt the collected data to the BS n the last N phases wth TDMA. Then, we obtan the followng transmsson tme constrant M+N T. 12 In the j-th j M phase, MTCD j J transmts data to ts servng MTCG wth achevable throughput r j Bt j log 2 1+ h j 2 p j σ 2, N,j J. 13

5 5 t 1 MTCD1 MTCD uplnk IT t M MTCDM T MTCGuplnkIT&MTCDEH t M+1 MTCG1 t M+N MTCGN Fg. 4. Tme sharng scheme for TDMA strategy durng one uplnk transmsson perod. In the M +-th phase, after havng decoded all the messages of ts served MTCDs, MTCG transmts the collected data to the BS wth achevable throughput r Bt M+ log 2 1+ h 2 q σ 2, N. 14 Smlar to 5, the total energy harvested of MTCD j served by MTCG s E H j N t M+n u h nj 2 q n, N,j J. 15 n1 Accordng to 6, s possble that some MTCDs cannot effectvely harvest energy n some slots, due to the fact that the receved power s below the recever senstvty threshold P 0. As n 7 and 9, the energy consumpton of a transmtter ncludes both RF transmsson power and crcut power [54]. Wth allocated transmsson tme t j, the energy E j consumed by MTCD j J can be modeled as pj E j t j η +P C, N,j J. 16 Wth allocated transmsson tme t M+, the system energy consumpton, denoted by E M+, s modeled as q E M+ t M+ ξ +QC N J n n1jj n1+1 t M+ u h j 2 q, 17 Jn jj t n1+1 M+uq h j 2 s the energy har- where N n1 vested by all MTCDs durng the transmsson tme t M+ for MTCG to transmt data to the BS. Accordng to 15-17, the total energy consumpton, E Tot, of the whole system durng tme T can be expressed as E Tot N jj 1+1 N jj 1+1 N E j + jj 1+1n1 M+N M+1 t j pj η +P C E + N q t M+ ξ +QC N t M+n u h nj 2 q n. 18 III. ENERGY EFFICIENT RESOURCE ALLOCATION FOR NOMA In ths secton, we study the resource allocaton for an uplnk M2M enabled cellular network wth NOMA and EH. Specfcally, we am at mnmzng the total energy consumpton va jontly optmzng power control and tme allocaton for NOMA. The system energy mnmzaton problem s formulated as mn p,q,t E Tot s.t. r j D j, N,j J r jj 1+1 D j, N E j E H j, N,j J T 19a 19b 19c 19d 19e 0 p j P j,0 q Q, N,j J 19f t 0, 19g where p [p 1,,p M ] T, q [q 1,,q N ] T, t [t 1,, t ] T, D j s the payload that MTCD j has to upload wthn tme constrant T, P j s the maxmal transmsson power of MTCDj, andq s the maxmal transmsson power of MTCG. Is assumed that all payloads are postve,.e., D j > 0, for all j. The objectve functon 19a defned n 11 s the total energy consumpton of both MTCDs and MTCGs. Constrants 19b and 19c reflect that the mnmal requred payloads for MTCDs can be uploaded to the BS. The consumed energy of each MTCD should not exceed ts harvested energy n tme T, as stated n 19d. Constrants 19e reflect that the payloads for all MTCDs are transmtted n tme T. Note that problem 19 s nonconvex due to nonconvex objectve functon 19a and constrants 19b-19d. In general, there s no standard algorthm for solvng nonconvex optmzaton problems. In the followng, we frst fnd the optmal condtons for problem 19 by explotng the specal structure of the uplnk NOMA rate, and then provde an teratve power control and tme allocaton algorthm. A. Optmal Condtons By analyzng problem 19, we have the followng lemma. Lemma 1: The optmal soluton p,q,t to problem 19 satsfes r j D j, N,j J. 20 Ths observaton states that the mnmal throughput leads to more energy savng, whch s smlar to [17] and s also wdely known n the nformaton theory communty. Lemma 1 states that the optmal transmt throughput for each MTCD s requred mnmum. Note that the optmal throughput for each MTCG s not always as ts mnmum requrement,.e., constrants 19c are actve at the optmum, snce MTCGs should transmt more power to mantan that the harvested energy of each MTCD s no less than the consumed energy.

6 6 Based on Lemma 1, we further have the followng lemma about the optmal transmsson power of MTCDs. Lemma 2: If p,q,t s the optmal soluton to problem 19, we have where J p j + σ2 h j 2 a l ln2d l B, b jl σ 2 h j 2 e a j e t 1 l1 a l t 1 e a j t 1 b jl t e, N,j J, 21 sj+1 ln2d s, N,j,l J. B 22 Besdes, the optmal transmsson power p j of MTCD j J s always non-negatve and decreases wth the transmsson tme t. Proof: Please refer to Appendx A. From Lemma 2, large transmsson tme results n low transmsson power. Ths s reasonable as the mnmal payload s lmted and large transmsson tme requres low achevable rate measured n bts/s. Is also revealed from Lemma 2 that the optmal transmsson power of MTCD j J served by MTCG depends only on the varable of the allocated transmsson tme. As a result, the energy E j consumed by MTCD j n J s a functon of the allocated transmsson tme. Based on 7 and 21, we have E j σ 2 t η h j 2 e a l e a j b jl t 1 e + σ2 t η h j 2 e a j + P C. 23 Theorem 1: The energy E j defned n 23 s convex wth respect to w.r.t. the transmsson tme. When P C 0, the energy E j monotoncally decreases wth. When P C > 0, the energy E j frst decreases wth when 0 Tj and then ncreases wth when > Tj zero pont of the frst-order dervatve Ej E j tt j, where T j,.e., s the unque Proof: Please refer to Appendx B. Fg. 5 exemplfes the energy E j gven n 23 versus. When P C 0,.e., the crcut energy consumpton of MTCDs s not consdered, we come to the same concluson as n [37] and [38] that the consumed energy decreases as the transmsson tme ncreases accordng to Theorem 1. Wthout consderng the crcut power consumpton, R log 2 1+SNR and the energy effcency ncreases wth the decrease of power. Consequently, when P C 0 and the mnmal throughput demand D j s gven, the consumed energy E j s a decreasng functon w.r.t.. Ths fundamentally follows the Shannon s law. When P C > 0,.e., the crcut energy consumpton of MTCDs s taken nto account, however, we fnd from Theorem 1 that the consumed energy frst decreases and then ncreases E j 0 * Tj C P >0 Fg. 5. The energy E j versus transmsson tme. C P 0 wth the transmsson tme, whch s dfferent from the prevous concluson n [37] and [38]. Ths s because that the total energy contans two parts balancng each other,.e., the RF transmsson energy part whch monotoncally decreases wth the transmsson tme and the crcut energy part whch lnearly ncreases wth the transmsson tme. In the followng of ths secton, we assume that the crcut power consumpton of MTCDs and MTCGs s n general postve,.e., P C > 0 and Q C > 0. Theorem 2: If T max N mn j J {Tj }, the optmal tme allocaton t to problem 19 satsfes t t T. 25 If T T Upp, where T Upp s defned n C.4, the optmal tme allocaton t to problem 19 satsfes t < T. 26 Proof: Please refer to Appendx C. From Theorem 2, s observed that transmttng wth the maxmal transmsson tme T s optmal when T s not large. Ths s because that the reduced energy of RF transmsson domnates the addtonal energy of crcut by ncreasng transmsson tme. When the avalable tme T becomes large enough, Theorem 2 states thas not optmal to transmt wth the maxmal transmsson tme T. Ths s due to the fact that the ncreased energy of crcut power domnates the power consumpton whle the energy reducton of RF transmsson becomes relatvely margnal. B. Jont Power Control and Tme Allocaton Algorthm Problem 19 has two dffcultes: one comes from the non-smooth EH functon defned n 6, and the other one s the non-convexty of both objectve functon 19a and constrants 19b-19d. To deal wth the frst dffculty, we ntroduce notaton S j as the set of phases durng whch

7 7 MTCD j J can effectvely harvest energy,.e., S j {k nik1 +1 h nj 2 q n > P 0, k K}. Wth S j n hand, the harvested power of MTCD j J can be presented by the smooth functon ūx defned n 28. To deal wth the second dffculty, we substtute 3-7, 11 and 21 nto 19, and the orgnal problem 19 wth fxed sets S j s can be equvalently transformed nto the followng problem: mn q,t N jj 1+1 N + + jj 1+1 K k N jj 1+1 σ 2 t η h j 2 e a l e a j b jl t 1 e σ 2 t η h j 2 e a j t N+k q ξ +QC t N+k ū k S j j J D j s.t. h 2 Bt q 2 N+1 1 where N + l+1 jj 1+1 n1 +1 k K, σ 2 η h j 2 e a l e a j + σ2 η h j 2 t N+k ū k S j n1 +1 J T e a j n P C h l 2 q l +σ 2 b jl t e P C h nj 2 q n, 27a 27b h nj 2 q n, N,j J 27c h nj 2 q n P 0, N,j J,k S j 27d σ 2 h j 2 e a l e a j b jl t 1 e 27e + σ2 h j 2 e a j P j, N,j J 27f 0 q Q, N 27g t 0, 27h ūx M1+eab e ab + e M ax2b eab, x Problem 27 s stll nonconvex w.r.t. q,t due to nonconvex objectve functon 27a and constrants 27b-27c. Before solvng problem 27, we have the followng theorem. Theorem 3: Gven transmsson tme τ [t N+1,, t ] T, problem 27 s a convex problem w.r.t. q, t, where t [t 1,,t N ] T. Gven q, t, problem 27 s equvalent to a lnear problem w.r.t. τ. Proof: Please refer to Appendx D. Accordng to Theorem 3, problem 27 wth gven transmsson tme τ can be effectvely solved by usng the standard convex optmzaton method, such as nteror pont method [57]. Besdes, problem 27 wth gven q, s a lnear problem, whch can be optmally solved va the smplex method. Based on Theorem 3, we propose an teratve power control and tme allocaton for NOMA IPCTA-NOMA algorthm wth low complexty to obtan a suboptmal soluton of problem 19. The dea s to teratvely update sets S j s accordng to the power and tme varables obtaned n the prevous step. Algorthm 1: Iteratve Power Control and Tme Allocaton for NOMA IPCTA-NOMA Algorthm 1: Set S 0 j {k j n Ik J n,k K}, I, j J j, ntalze a feasble soluton q 0,t 0 to problem 27 wth S 0 j s, the tolerance θ, the teraton number v 0, and the maxmal teraton number V max. 2: repeat 3: Set τ [t v N+1,,tv ]T. 4: repeat 5: Obtan the optmal q, t of convex problem 27 wth fxed τ and sets S v j. 6: Obtan the optmal τ of lnear problem 27 wth fxed q, t and sets S v j. 7: untl the objectve value 27a wth fxed sets S v j converges. 8: Set v v +1. 9: Denote q v q, t v [ t T,τ T ] T. 10: Calculate the objectve value 27a wth fxed sets S v j as U v Obj E Tot q v,t v. 11: Update S v j {k nik1 +1 h nj 2 q n v > P 0, k K}, I,j J. 12: untlv 1 and U v /U v1 Obj < θ orv > V max. Obj Uv1 Obj C. Convergence and Complexty Analyss Theorem 4: Assumng V max, the sequence q,t generated by the IPCTA-NOMA algorthm converges. Proof: Please refer to Appendx E. Accordng to the IPCTA-NOMA algorthm, the major complexty les n solvng the convex problem 27 wth fxed τ. Consderng that the dmenson of the varables n problem 27 wth fxed τ s 2N, the complexty of solvng problem 27 wth fxedτ by usng the standard nteror pont method s ON 3 [57, Pages 487, 569]. As a result, the total complexty of the proposed IPCTA-NOMA algorthm s OL NO L IT N 3, where L NO denotes the number of outer teratons of the IPCTA-NOMA algorthm, and L IT denotes the number of nner teratons of the IPCTA-NOMA algorthm for teratvely solvng nonconvex problem 27 wth fxed sets S j s.

8 8 IV. ENERGY EFFICIENT RESOURCE ALLOCATION FOR TDMA In ths secton, we study the energy mnmzaton for the M2M enabled cellular network wth TDMA. Accordng to and 18, the energy mnmzaton problem can be formulated as mn p,q,ˆt N jj 1+1 N jj 1+1n1 pj t j η +P C + s.t. Bt j log 2 1+ h j 2 p j σ 2 N N t M+n u h nj 2 q n Bt M+ log 2 1+ h 2 q σ 2 pj t j η +P C N+1 T q t M+ ξ +QC 29a D j, N,j J 29b jj 1+1 D j, N 29c N t M+n u h nj 2 q n, N,j J n1 29d 29e 0 p j P j,0 q Q, N,j J 29f ˆt 0, 29g where ˆt [t 1,, t M+N ] T. Obvously, problem 29 s nonconvex due to nonconvex objectve functon 29a and constrants 29b-29d. In the followng, we frst provde the optmal condtons for problem 29, and then we propose a low-complexty algorthm to solve problem 29. A. Optmal Condtons Smlar to Lemma 1, s also optmal for each MTCD to transmt wth the mnmal throughput requrement. Accordngly, the followng lemma s drectly obtaned. Lemma 3: The optmal soluton p,q,ˆt to problem 29 satsfes Bt j log 2 1+ h j 2 p j σ 2 D j, N,j J. 30 Accordng to 30, the optmal transmsson power of MTCD j can be presented as p j 1 D j Bt h j 2 2 j 1, N,j J. 31 Substtutng 31 nto 16 yelds E j t D j j Bt h j 2 2 j 1 +t j P C, N,j J. 32 η By analyzng 32, we can obtan the followng theorem. Theorem 5: The energy E j defned n 32 s convex w.r.t. t j. When P C 0, the energy E j monotoncally decreases wth the transmsson tme t j. When P C > 0, the energy E j frst decreases wth t j when 0 t j Tj and then ncreases wth t j when t j > Tj, where T j s the unque zero pont of the frst-order dervatve Ej t j,.e., E j t j tjt j Snce Theorem 5 can be proved by checkng the frst-order dervatve Ej t j as n Appendx B, the proof of Theorem 4 s omtted. Smlar to Theorem 2 for NOMA, we come to the smlar concluson for TDMA that transmttng wth the maxmal transmsson tme T s optmal when T s not large, whle for larget s not optmal to transmt wth the maxmal transmsson tme T. B. Iteratve Power Control and Tme Allocaton Algorthm Smlar to problem 19, problem 29 has two dffcultes: one s the non-smooth EH functon n 6, and the other s the nonconvex objectve functon 29a and constrants 29b- 29d. To deal wth the frst dffculty, we ntroduce notaton S j as the set of MTCGs from whch MTCD j J can effectvely harvest energy,.e., S j {n h nj 2 q n > P 0, n I}. Wth S j n hand, the harvested power of MTCD j J from MTCG n S j can be presented by the smooth functon ūx defned n 28. To tackle the second dffculty, we show that problem 29 wth fxed sets S j s can be transformed nto an equvalent convex problem. Theorem 6: The orgnal problem n 29 wth fxed sets S j s can be equvalently transformed nto the followng convex problem as mn ˆp,ˆq,ˆt N jj 1+1 N jj 1+1 ˆpj η +t jp C + s.t. Bt j log 2 1+ h j 2ˆp j σ 2 t j N ˆq ξ +t M+Q C hnj 2ˆq n t M+n ū 34a t M+n n S j D j, N,j J 34b Bt M+ log 2 1+ h 2ˆq σ 2 t M+ jj 1+1 D j, N 34c ˆp j η +t jp C hnj 2ˆq n t M+n ū, N,j J t M+n n S j 34d h nj 2ˆq n > P 0 t M+n, N,j J,n S j 34e N+1 T 34f 0 ˆp j P j t j, N,j J 34g 0 ˆq Q t M+, N 34h ˆt 0, 34 where ˆp [ˆp 1,, ˆp M ] T and ˆq [ˆq 1,,ˆq N ] T.

9 9 Proof: Please refer to Appendx F. Based on Theorem 6, we propose an teratve power control and tme allocaton for TDMA IPCTA-TDMA algorthm wth low complexty to obtan a suboptmal soluton of problem 29. The dea s to teratvely update sets S j s accordng to the power and tme varables obtaned n the prevous step. Algorthm 2: Iteratve Power Control and Tme Allocaton for TDMA IPCTA-TDMA Algorthm 1: Set S 0 j {}, I, j J j, the tolerance θ, the teraton number v 0, and the maxmal teraton number V max. 2: repeat 3: Obtan the optmal ˆp v,ˆq v,ˆt v of convex problem 34 wth fxed sets S v j. 4: Calculate the objectve value 34a wth fxed sets S v j as U v Obj E Tot ˆp v,ˆq v,ˆt v. 5: Set v v +1. 6: Update S v j { n I,j J. 7: untl v 2 and U v hnj 2ˆq v1 n t v1 M+n Obj Uv1 Obj /U v1 } > P 0, n I, Obj < θ orv > V max. 8: Output p ˆp v,ˆt ˆt v,q ˆqv /t v M+, I. Energy E 11 J 2.5 x NOMA, P C 0 NOMA, P C 5 mw NOMA, P C 10 mw Transmsson tme t s 1 Fg. 6. Energy E 11 consumed by MTCD 1 versus the transmsson tme t 1 for NOMA strategy. Total energy J NOMA TDMA 2 C. Convergence and Complexty Analyss Theorem 7: Assumng V max, the sequence ˆp,ˆq,ˆt generated by the IPCTA algorthm converges. Theorem 7 can be proved by usng the same method as n Appendx E. The proof of Theorem 7 s thus omtted. Accordng to the IPCTA-TDMA algorthm, the major complexty les n solvng the convex problem 34. Consderng that the dmenson of the varables n problem 34 s 2M + N, the complexty of solvng problem 34 by usng the standard nteror pont method s OM + N 3 [57, Pages 487, 569]. As a result, the total complexty of the proposed IPCTA-TDMA algorthm s OL TD M +N 3, where L TD denotes the number of teratons of the IPCTA- TDMA algorthm. V. NUMERICAL RESULTS In ths secton, we evaluate the proposed schemes through smulatons. There are 40 MTCDs unformly dstrbuted wth a BS n the center. We adopt the data-centrc clusterng technque n [41] for cluster formaton of MTCDs, the number of MTCGs s set as 12, and the maxmal number of MTCDs assocated to one MTCG s 4. For NOMA, all MTCGs are classfed nto 6 clusters based on the strong-weak scheme [45], [46]. The path loss model s log 10 d d s n km and the standard devaton of shadow fadng s4db [58]. The nose power σ dbm, and the bandwdth of the system s B 18 KHz. For the non-lnear EH model n 6, we set M 24 mw, a 1500 and b accordng to [52], whch are obtaned by curve fttng from the measurement Number of teratons Fg. 7. Convergence behavors of IPCTA-NOMA and IPATC-TDMA. data n [59]. The recever senstvty threshold P 0 s set as 0.1 mw. The PA effcences of each MTCD and MTCG are set to η ξ 0.9, and the crcut power of each MTCG s Q C 500 mw as n [55]. We assume equal throughput demand for all MTCDs,.e., D 1 D M D, and equal maxmal transmsson power for each MTCD or MTCG,.e., P 1 P M P, and Q 1 Q N Q. Unless otherwse specfed, parameters are set as P 5 mw, P C 0.5 mw, Q 1 W, D 10 Kbts, and T 5 s. Fg. 6 depcts, for nstance,e 11 n 23 consumed by MTCD 1 served by MTCG 1 versus the transmsson tme t 1 for NOMA. Is observed that E 11 monotoncally decreases wth t 1 when P C 0. For the case wth P C 5 mw or P C 10 mw, E 11 frst decreases and then ncreases wth t 1. Both observatons valdate our theoretcal fndngs n Theorem 1. The convergence behavors of IPCTA-NOMA and IPCTA- TDMA are llustrated n Fg. 7. From ths fgure, the total energy of both algorthms monotoncally decreases, whch confrms the convergence analyss n Secton III-C and IV-C. It can be seen that both IPCTA-NOMA and IPCTA-TDMA converge rapdly. In Fg. 8, we llustrate the total energy consumpton versus the crcut power of each MTCD. Accordng to Fg. 8, the total energy of NOMA outperforms TDMA when the crcut power of each MTCD s low,.e., P C 4 mw n the test case. At low crcut power regme, the total energy consumpton of the network manly les n the RF transmsson power of MTCDs

10 10 Total energy J NOMA TDMA Total energy J NOMA, P C 0.5 mw TDMA, P C 0.5 mw NOMA, P C 5 mw TDMA, P C 5 mw Crcut power of each MTCD mw Fg. 8. Total energy versus the crcut power of each MTCD Maxmal transmsson power of each MTCG W Fg. 9. Total energy versus the maxmal transmsson power of each MTCG. and the energy consumed by MTCGs to charge the MTCDs through EH. For the NOMA strategy, MTCDs served by the same MTCG can smultaneously upload data and the MTCG decodes the messages accordng to NOMA detectons, whch requres lower RF transmsson power of MTCDs than the TDMA strategy. Thus, the total energy of NOMA s less than TDMA for low crcut power of each MTCD. From Fg. 8, we can fnd that the total energy of TDMA outperforms NOMA when the crcut power of each MTCD becomes hgh,.e., P C 5 mw n our tests. At hgh crcut power regme, the total energy consumpton of the network manly les n the crcut power of MTCDs and the energy consumed by MTCGs to charge the MTCDs through EH. For the NOMA strategy, the transmsson tme of each MTCD wth NOMA s always longer than that wth TDMA, whch leads to hgher crcut power consumpton of MTCDs than the TDMA strategy. As a result, TDMA enjoys better energy effcency than NOMA for hgh crcut power of each MTCD. Fg. 9 llustrates the total energy versus the maxmal transmsson power of each MTCG. Is observed that the total energy decreases wth the ncrease of maxmal transmsson power of each MTCG for both NOMA and TDMA. Ths s because that the ncrement of maxmal transmsson power of each MTCG allows the MTCG to transmt wth large transmsson power, whch leads to short EH tme of MTCDs to harvest enough energy and low total energy consumpton of the network. Moreover, s found that the total energy of NOMA s more senstve to the maxmal transmsson power of each MTCG than that of TDMA for hgh crcut power case as P C 5 mw for each MTCD. The reason s that MTCD wth low channel gan receves ntra-cluster nterference due to NOMA and the energy consumpton s hence especally large for low maxmal transmsson power of each MTCG and hgh crcut power of each MTCD. The total energy versus the maxmal transmsson power of each MTCD s shown n Fg. 10. Is observed that the total energy decreases wth growng maxmal transmsson power of each MTCD for both NOMA and TDMA. Ths s due to the fact that a larger maxmal transmsson power of each MTCD ensures MTCDs can transmt wth more power, and the requred payload can be uploaded n a shorter tme, whch results n low crcut power consumpton and low Total energy J NOMA, P C 0.5 mw TDMA, P C 0.5 mw NOMA, P C 5 mw TDMA, P C 5 mw Maxmal transmsson power of each MTCD mw Fg. 10. Total energy versus the maxmal transmsson power of each MTCD. energy consumpton. It can be found that the total energy of TDMA converges faster than that of NOMA as the maxmal transmsson power of each MTCD ncreases. Ths s because that the MTCDs served by the same MTCG smultaneously transmt data for NOMA, and the requred transmsson power of each MTCD for NOMA s always larger than that of MTCD for TDMA. Fnally, n Fg. 11, we llustrate the total energy versus the requred payload of each MTCD. The fgure shows that the total energy ncreases wth the requred payload of each MTCD. Ths s due to the fact that large payload of each MTCD requres large energy consumpton of MTCDs and MTCGs, whch leads to hgh energy consumpton of the network. VI. CONCLUSIONS Ths paper compares the total energy consumpton between NOMA and TDMA strateges n uplnk M2M communcatons wth EH. We formulate the total energy mnmzaton problem subject to mnmal throughput constrants, maxmal transmsson power constrants and energy causalty constrants, wth the crcut power consumpton taken nto account. By applyng the condtons thas optmal to transmt wth the mnmal throughput for each MTCD, we transform the orgnal problem for NOMA strategy nto an equvalent problem, whch s suboptmally solved through an teratve algorthm. By usng a proper varable transformaton, we transform the nonconvex

11 11 Total energy J NOMA, P C 0.5 mw TDMA, P C 0.5 mw NOMA, P C 5 mw TDMA, P C 5 mw Requred payload of each MTCD kbts Fg. 11. Total energy versus the requred payload of each MTCD. problem for TDMA nto an equvalent problem, whch can be effectvely solved. Through smulatons, ether NOMA strategy or TDMA strategy may be preferred dependng on dfferent crcut power regmes of MTCDs. At low crcut power regme of MTCDs, NOMA consumes less energy, whle TDMA s preferred at hgh crcut power regme of MTCDs snce the energy consumpton for NOMA ncreases sgnfcantly as the crcut power of MTCDs ncreases. APPENDIX A PROOF OF LEMMA 2 By nsertng r j D j nto 3 from Lemma 1, we have 2 D j h l 2 p l +σ 2 2 D j B lj h l 2 p l, A.1 for j J 1 +1,,J. To solve those J J 1 equatons, we frst defne u j h l 2 p l, j J. A.2 lj Applyng A.2 nto A.1 yelds u j 2 D j u j+1 +σ 2 2 D j B, j J. A.3 Denote u [u J1+1,,u J ] T, D J1 v [σ 2 +1 Bt 2 1,,σ 2 2 D J T B], A.4 and W D J1 +1 D J1 +2 Bt Equatons n A.3 can be rewrtten as D J 1 0. A.5 E W u v, A.6 where E s an dentty matrx of sze J J 1 J J 1. From A.6, we have u E W 1 v. A.7 Before obtanng the nverse matrx E W 1, we present the followng lemma. Lemma 4: When l [1,J J 1 1], the l-th power of matrx W can be expressed as 0 T l 2 W l 1 +l sj 1 +1 Ds 1 +l+1 sj 1 +2 Ds sj l Ds 0 l, A.8 where 0 l denotes a l 1 vector of zeros. When l J J 1, W l 0 JJ 1 J J 1, where 0 JJ 1 J J 1 s a J J 1 J J 1 matrx of zeros. Proof: Lemma 4 can be proved by the prncple of mathematcal nducton. Bass: It can be verfed that Lemma 4 s vald for l 1. Inducton Hypothess: For l [1,J J 1 2], assume that the l-th power of matrx W can be expressed as A.8. Inducton Step: Accordng to A.8, we can obtan W l+1 0 T l 2 W l W 1 +l sj 1 +1 Ds D J1 +1 J1+l+1 0 T l l+1 sj 1 +2 Ds D J1 +2 Bt sj 1 +1 Ds 1 +l+2 sj 1 +2 Ds D J sj l Ds 0 l 1 sj l1 Ds 0 l+1, whch verfes that the l + 1-th power of matrx W can also be expressed as A.8. Accordng to A.8 and A.5, t s verfed that W JJ1 W JJ11 W 0 JJ 1 J J 1. A.9

12 12 Therefore, Lemma 4 s proved. Now, s ready to obtan the nverse matrx E W 1. Snce E W E + we have J J 11 l1 W l E W JJ1 E, J J 11 E W 1 E + W l. l1 Substtutng A.8 and A.11 nto A.7 yelds lj A.10 A.11 l1 u j σ 2 2 D l sj Ds Bt 1 2, j J, A.12 where we defne j1 u j σ 2 2 sj 2 Ds sj Ds B 2 0. From A.12, we can obtan. Snce J lj +1 p j 0, we have u J+1 0. A.13 From A.2 and A.13, we can obtan the transmsson power of MTCD j as p j u j u j+1 h j 2, j J. A.14 Applyng A.12 and A.13 to A.14, we have p j lj σ 2 l1 h j 2 2 D l sj Ds Bt σ2 h j 2 σ 2 l1 h j 2 2 D l sj+1 Ds Bt 1 2 σ 2 h j 2 2 D l B 2 D l1 j sj+1 Ds Bt D j1 j sj Ds Bt 1 2, σ 2 h j 2 e a l e a j b jl t 1 e + σ2 h j 2 e a j, j J. A.15 where a l and b jl are defned n 22. Snce e x 1 wth x 0 s non-negatve and decreases wth, p j s also non-negatve and decreases wth from A.15. APPENDIX B PROOF OF THEOREM 1 To show that the energy E j s convex w.r.t., we frst defne functon f jl x e a lx 1e ajx 1e b jlx, x 0. B.1 Then, the second-order dervatve follows f jl x a2 l +2a lb jl e ajx 1e a l+b jl x +2a l a j e a l+a j+b jl x +a 2 j +2a j b jl a a lx 1e aj+b jlx +b 2 jle a lx 1e ajx 1e b jlx 0, B.2 whch ndcates that f jl x s convex w.r.t. x. Accordng to [57, Page 89], the perspectve of ux s the functon vx,t defned by vx,t tux/t, dom v {x,t x/t dom u,t > 0}. If ux s a convex functon, then so s ts perspectve functon vx,t [57, Page 89]. Snce f jl x s convex, the perspectve functon f jl x, f jl x B.3 s convex w.r.t. x,. Thus, functon f jl 1, s also convex w.r.t.. Defnng functon g j x e ajx 1, B.4 whch s convex w.r.t. x. By usng the property of perspectve functon [57, Page 89], we also have that functon ḡ j x, g j x B.5 s convex w.r.t. x, and functon ḡ j 1, s accordngly convex w.r.t.. Substtutng B.3 and B.5 nto 23, we can obtan E j σ 2 η h j 2 f jl 1, + σ2 η h j 2ḡj1, + P C. B.6 Due to the fact that both f jl 1, and ḡ j 1, are convex, E j s consequently convex w.r.t. from B.6. Accordng to 23, the frst-order dervatve of E w.r.t. s expressed as E j σ 2 η h j 2 e a l e a j b jl t 1 e σ 2 a l η h j 2 e a j a l +b jl t 1 e σ 2 a j η h j 2 e a a l j +b jl t 1 e σ 2 b jl η h j 2 e a l e a j b jl t 1 e + σ2 η h j 2 e a j σ2 a j η h j 2 e aj +P C. B.7 Snce E j s convex w.r.t., functon Ej ncreases wth. Because D j s postve for all j, we have a l > 0, b jl > 0 and

13 13 c jl > 0 from 22. To calculate the lmt of Ej at 0+, we calculate the followng lmt, lm 0+ σ 2 η h j 2 e a l e a j b jl t e σ2 a l η h j 2 e a j e a l +b jl σ 2 lm lx x + η h j 2ea 1e ajx 1 e b jlx σ2 a l x η h j 2eajx 1 e a l+b jl x σ 2 1a l x lm x + η h j 2 e a l+a j+b jl x. B.8 Thus, when approaches zero n the postve drecton, we have E j lm. B.9 0+ When approaches postve nfnty, the lmt of frst-order dervatve Ej can be calculated as E j lm P C. + B.10 If P C 0, then Ej 0 for all 0. In ths case, the energy E j always decreases wth. If P C > 0, we can observe that lm Ej t + > 0 from B.10. Snce lm Ej t 0+ < 0 and Ej ncreases wth, there exsts one unque soluton Tj satsfyng 24, whch can be solved by usng the bsecton method. In partcular, E j decreases wth 0 Tj and ncreases wth j > Tj. APPENDIX C PROOF OF THEOREM 2 We frst consder the case that T max N mn j J {Tj }. Wthout loss of generalty, we denote T Tnm max N mn j J {Tj }. Assume that the optmal soluton p,q,t to problem 19 satsfes t < T. Due to that t 0 for all, we can obtan that t n T nm Tnl, l J n. Wth all other power p j, q and tme t k fxed, j M\J n, N, k N \{n}, we ncrease the tme t n to t n t n +ǫ by an arbtrary amount 0,T N, n t ]. Usng 21, the correspondng power p l strctly deceases to p l, l J n. Accordng to Theorem 1, the energy E nl decreases wth the transmsson tme 0 t n T nl, l J n. As a result, wth new power-tme par p J n1+1,,p J n,t n, the objectve functon 19a decreases wth all the constrants satsfed. By contradcton, we must have t T for the optmal soluton. Ths completes the proof of the frst half part of Theorem 2. The last half part of Theorem 2 ndcates that transmttng wth maxmal transmsson tme s not optmal when T s larger than a threshold T Upp. Ths can be proved by usng the contradcton method. Specfcally, assumng that total transmsson tme of the optmal soluton s the maxmal transmsson tme T, we can fnd a specal soluton wth total transmsson tme less than T, whch strctly outperforms the optmal soluton. To obtan the threshold T Upp, we consder a specal soluton that satsfes all the constrants of problem 19 except the maxmal transmsson tme constrant 19e. Snce E j s convex w.r.t. accordng to Theorem 1, the energy E J jj E 1+1 j consumed by all MTCDs n J served by MTCG s also convex w.r.t.. Based on the proof of Theorem 1, we drectly obtan the followng lemma. Lemma 5: The energy E consumed by all MTCDs n J frst decreases wth the transmsson tme when0 T and then ncreases wth the transmsson tme when > T, wheret s the unque zero pont of frst-order dervatve E,.e., E tt jj 1+1 E j tt 0. C.1 Snce Lemma 5 can be proved by checkng the frst-order dervatve E as n Appendx B, the proof of Lemma 5 s omtted here. Set T, and p j can be obtaned from 21 wth, N, j J. For the transmsson power of the MTCGs, we set q Q, N. Denote p [ p 1,, p M ] T, q [ q 1,, q N ] T. Wth power p, q and tme,, t N fxed for now, the energy mnmzaton problem 19 wthout constrant 19e becomes mn τ K k11 +1 K N k1 jj 1+1 Q t N+k ξ +QC t N+1 u J k nj k1 +1 h nj 2 Q n C.2a h s.t. Bt N+ log Q Ik D j, n+1 h n 2 Q n +σ 2 j J k K, pj K η +P C t N+k u τ 0, k1 N,j J n1 +1 h nj 2 Q n, C.2b C.2c C.2d where τ [t N+1,,t ] T. Problem C.2 can be obtaned by substtutng 11 nto 19a, 4 nto 19b, and 5 and 7 nto 19c. Problem C.2 s a lnear problem, whch can be optmally solved va the smplex method. Denote the optmal soluton of problem by τ [T N+1,,T ]T. Denote t N+k T N+k, k K, and t [ t 1,, t ] T. As a result, we obtan a specal soluton p, q, t that satsfes all the constrants of problem 19 except the maxmal transmsson tme constrant 19e. Wth soluton p, q, t, the total energy consumpton ẼTot obtaned from 11 can be expressed

14 14 as Ẽ Tot N Ẽ jj K k11 +1 K N k1 jj 1+1 pj η +P C q t N+k ξ +QC t N+k u n1 +1 h nj 2 q n,c.3 where Ẽ s the energy consumed by all MTCDs n J, N, and ẼN+k s the system energy consumpton durng the N +k-th phase. Denote an upper bound of maxmal transmsson tme by where T Upp max T Amp { T,T Amp }, C.4 1+ N β K k1ẽn+k αq C, C.5 wth α defned n C.8 and β defned n C.11. If T T Upp, we show that optmal soluton p,q,t to problem 19 must satsfy constrant 26,.e., 19e s nactve, by contradcton. Assume that t T. C.6 Wth p,q,t, we denote E as the energy consumed by all MTCDs n J, N, EN+k as the system energy consumpton durng the N + k-th phase, and ETot as the total energy of the whole system. Thus, we have E Tot a N+1 b c > E Ẽ + K k1 N K Ẽ + u k1 n1 +1 E N+k t N+k N Ẽ +αq C d Ẽ ẼTot, n1 +1 h nj 2 qn +Q C K t N+k k1 q n ξ N K k1 jj 1+1 t N+k 1 C.7 where nequalty a follows from the fact that E acheves the mnmum when T accordng to Lemma 5, equalty b holds from 5 and 9, and nequalty c follows from 5, 10, ξ 0,1] and α mn k K 1. C.8 To explan procedure d, we substtute 5 and 7 nto energy causalty constrants 19d to obtan p t j K η +P C t N+k u h nj 2 q n, C.9 k1 n1 +1 for all N,j J. Consderng that p j 0 n the left hand sde of C.9 and ux s a ncreasng functon as well as q Q n the rght hand sde of C.9, we have K t k1 t N+k u Ik nik1 +1 h nj 2 Q n mn j J, β K k1 where β mn j J max k K t N+k, N, C.10 u Ik Combnng C.6 and C.8 yelds K t N+k k1 P C n1 +1 h nj 2 Q n. P C C.11 T 1+ N β. C.12 Hence, nequalty d follows from C.4 and C.12. Accordng to C.2 and C.4, soluton p, q, s a feasble soluton to problem 19. From C.7, the objectve value 19a can be decreased wth soluton p, q, t, whch contradcts that p,q,t s the optmal soluton to problem 19. Hence, the optmal soluton to problem 19 must satsfy constrant 26. APPENDIX D PROOF OF THEOREM 3 We frst show that the feasble set of problem 27 wth gven τ s convex. Obvously, constrants 27b, 27d, 27e and 27g and 27h are all lnear w.r.t. q, t. Accordng to B.1 and B.3, constrants 27c and 27f can be, respectvely, reformulated as f jl 1, + P C t N+k ū k S j for all N,j J, and n1 +1 h nj 2 q n D.1 f jl 1 P j, N,j J. D.2 Based on 28, we have ū x Ma2 1+e ab e axb e ab 1+e axb 3 0, x 0, D.3