MOBILE Cloud Computing (MCC) extends the capabilities

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1 1 Resource Sharng of a Computng Access Pont for Mult-user Moble Cloud Offloadng wth Delay Constrants Meng-Hs Chen, Student Member, IEEE, Mn Dong, Senor Member, IEEE, Ben Lang, Fellow, IEEE arxv: v2 [cs.it] 16 Mar 2018 Abstract We consder a moble cloud computng system wth multple users, a remote cloud server, a computng access pont (CAP). The CAP serves both as the network access gateway a computaton servce provder to the moble users. It can ether process the receved tasks from moble users or offload them to the cloud. We jontly optmze the offloadng decsons of all users, together wth the allocaton of computaton communcaton resources, to mnmze the overall cost of energy consumpton, computaton, maxmum delay among users. The jont optmzaton problem s formulated as a mxed-nteger program. We show that the problem can be reformulated transformed nto a non-convex quadratcally constraned quadratc program, whch s NP-hard n general. We then propose an effcent soluton to ths problem by semdefnte relaxaton a novel romzaton mappng method. Furthermore, when there s a strct delay constrant for processng each user s task, we further propose a three-step algorthm to guarantee the feasblty local optmalty of the obtaned soluton. Our numercal results show that the proposed solutons gve nearly optmal performance under a wde range of parameter settngs, the addton of a CAP can sgnfcantly reduce the cost of mult-user task offloadng compared wth conventonal moble cloud computng where only the remote cloud server s avalable. Index Terms Moble cloud computng, computng access pont, offloadng decson, resource allocaton, energy cost, computaton cost, delay cost. 1 INTRODUCTION MOBILE Cloud Computng (MCC) extends the capabltes of moble devces to mprove user experence [2] [3] [4]. Moble users can offload tasks to the cloud, usng abundant cloud resources to help them gather, store, process data. However, the nteracton between moble devces the cloud ntroduces some key challenges n the system desgn. For example, the decson on whether to offload tasks to the cloud needs to balance the tradeoff between energy consumpton computng performance. Furthermore, the communcaton delay between moble users the cloud needs to be taken nto consderaton [2]. Wth an am to reduce the communcaton delay n task offloadng, Moble Edge Computng (MEC), as defned by the European Telecommuncatons Stards Insttute (ETSI), s a dstrbuted MCC system where computng resources are nstalled locally at or near the base staton of a cellular network [5] [6] [7]. MEC shares smlartes wth mcro cloud centers [8], cloudlets [9], cyber-foragng [10], fog computng [11], except that the MEC computng servers are managed by a moble servce provder, whch Meng-Hs Chen Ben Lang are wth the Department of Electrcal Computer Engneerng, Unversty of Toronto, Toronto, Canada (e-mal: {mchen, lang}@ece.utoronto.ca). Mn Dong s wth the Department of Electrcal, Computer Software Engneerng, Unversty of Ontaro Insttute of Technology, Oshawa, Canada (e-mal: mn.dong@uot.ca). Ths work has been funded n part by grants from the Natural Scences Engneerng Research Councl (NSERC) of Canada n part by the Ontaro Mnstry of Research Innovaton under an Early Researcher Award. A prelmnary verson of ths work has appeared n [1]. allows more drect control resource management. Smlar to the concept of MEC, n ths work, we use the general term computng access pont (CAP), whch refers to a wreless access pont or a cellular base staton wth bult-n computaton capablty. For example, CAPs may be provded by Internet servce provders as a value-added servce. Moble devces that wsh to offload a task frst sends t to the CAP. The CAP may serve ts conventonal networkng functon forward the task to the remote cloud server, or drectly process the task by tself. The addtonal opton of computaton by the CAP reduces the need for access to the remote cloud server, hence can potentally decrease the communcaton delay also the overall energy computaton cost. However, the avalablty of CAP adds an extra dmenson of varablty for offloadng decsons. Each task may be processed locally at the moble devce, at the CAP, or at the remote cloud server. Furthermore, both computaton communcaton resources need to be consdered n dfferent offloadng choces. Ths makes optmzng the moble task offloadng decson even more challengng. In ths work, we study the nteracton among multple users, the CAP, the cloud. In a mult-user scenaro, to offload tasks, we need to allocate communcaton computaton resources among competng users. We jontly consder both the offloadng decson resource allocaton among all users, wth an am to conserve energy mantan servce qualty for all of them. For ths jont optmzaton problem, an optmal offloadng decson must take nto consderaton the computaton communcaton energes, computaton costs, communcaton processng delays at all local user devces, as well as the

2 2 resource constrants capabltes of the CAP the remote cloud. The contrbutons of ths work are summarzed as follows: We focus on jontly optmzng the offloadng decsons as well as the computaton communcaton resource allocaton for multple moble users wth one CAP one remote cloud server. We formulate the jont optmzaton problem to mnmze a weghted sum of costs of energy, computaton, the maxmum delay among all users. Ths results n a mxed nteger programmng problem. To solve ths challengng problem, we frst reformulate transform the problem nto a non-convex quadratcally constraned quadratc program (QCQP) [12], whch s stll NP-hard n general. To obtan a soluton to ths problem, we then propose an effcent heurstc algorthm, termed sharecap, based on semdefnte relaxaton (SDR) [13] a novel romzaton mappng method. We further study the scenaro where there s a strct processng deadlne for each user s task. Wth these addtonal delay constrants, the proposed sharecap method can no longer be drectly appled to fnd a soluton due to the absence of a feasblty guarantee. To solve ths more complcated optmzaton problem, we further propose a three-step algorthm named sharecap-d, consstng of SDR, adaptve adjustment, sequental tunng, to teratvely fnd a soluton. We show that sharecap-d guarantees a locally optmal soluton. Through numercal study, by comparng wth an optmal offloadng polcy obtaned by exhaustve search, we demonstrate that the proposed sharecap sharecap-d methods gve nearly optmal performance under a wde range of parameter settngs. Furthermore, we observe that the addton of a CAP can sgnfcantly reduce the energy computatonal costs of the system, as compared wth the conventonal MCC where only the remote cloud server s avalable for task offloadng. The rest of ths paper s organzed as follows. Related works are revewed n Secton 2. In Secton 3, we descrbe the system model for moble cloud computng wth a CAP formulate the optmzaton problem. In Secton 4, we transform our problem to a QCQP problem solve t through the SDR approach. In Secton 5, we further study the scenaro wth strct delay constrants. In Secton 6, we extend our work to sum delays optmzaton. Numercal results are presented n Secton 7, followed by concluson n Secton 8. Notatons: We denote by a T A T the transpose of vector a matrx A, respectvely. The notaton dag(a) denotes the dagonal matrx wth dagonal elements beng elements of vector a. The trace functon of matrx A s denoted by Tr(A). We use A(,j) to denote the (,j)th entry of matrx A. We use A 0 to ndcate that A s a postve sem-defnte matrx. 2 RELATED WORK Many exstng works study task offloadng from moble users to the local (or remote) processor n two-ter cloud systems. For a sngle moble user offloadng ts entre applcaton to the cloud, the authors of [14], [15], [16] presented dfferent energy models to analyze whether or not to offload applcaton to the cloud, the tradeoff between energy consumpton computng performance was studed n [17], [18]. Furthermore, many studes have consdered parttonng an applcaton nto multple tasks. Among them, MAUI [19], Clonecloud [20], Thnkar [21] are systems proposed to enable a moble devce to offload tasks to the cloud. These works focus on the mplementaton of offloadng mechansms from the moble devce to the cloud, the dscusson on optmzng the offloadng decsons was lmted. In [22] [23], heurstc offloadng polces were proposed for a moble user wth sequental tasks. In [24], [25], [26], the problem of cloud offloadng for a moble user wth dependent tasks was studed. In [27], offloadng a moble user s tasks n an ntermttently connected cloud system was consdered. The mpact of moblty was consdered n [28], where the authors proposed an opportunstc offloadng algorthm. All of the studes above focus on a sngle moble user. Task offloadng by multple moble users have been consdered n [29], [30], [31], [32], [33], [34], [35], [36], [37], where each user has a sngle applcaton or task to be offloaded to the cloud n ts entrety. In [29], [30], [31], the authors consdered optmzng offloadng decsons, amng to maxmze the revenue of the moble cloud servce provders under a fxed resource usage per user. The cooperaton among selfsh servce provders to mprove the revenue was further studed n [31]. The authors of [32], [33] studed the allocaton of rado computaton resources n the scenaro where all tasks are always offloaded. The jont optmzaton of offloadng decson communcaton computaton resources for system utlty maxmzaton was consdered n [34], where where the number of tasks that can be offloaded s lmted by the transmsson bwdth; a heurstc algorthm was proposed to obtan the resource allocaton offloadng decson sequentally. Game theoretc approaches were adopted n [35], [36], [37] to study decentralzed decson control n systems where offloadng decsons are made by moble users as selfsh players. However, these game theoretc works focus on the offloadng decsons for each user wthout consderng the allocaton of communcaton computaton resources. Furthermore, a mult-user scenaro where each user has multple ndependent tasks was consdered n [38], where the offloadng decson algorthm were proposed by mnmzng the weghted cost of energy consumpton worst-case offloadng delay. The authors of [39] consdered a moble devce cloud, whch s composed purely of proxmal moble devces, a task schedulng mechansm was proposed for concurrent applcaton management. Coordnaton of local moble devces formng a moble cloud has been studed n [40]. All of the studes above focus on a two-ter cloud network consstng of only moble users another ter of local or remote processors. The three-ter network consstng of moble users, a local computng node (e.g., cloudlet or CAP), a remote cloud

3 3 server has been studed n [41], [42], [43], [44], [45]. Wthout consderng resource allocaton, centralzed heurstc algorthms for offloadng decsons were proposed n [41], [42], [43], whle a game theoretc approach was consdered to dstrbutedly obtan the offloadng decson n [44]. Despte these works, the jont optmzaton of the offloadng decson the allocaton of computaton communcaton resources for a general three-ter system has not been nvestgated before. The jont optmzaton problem s much more complcated to solve, because the offloadng decson resource allocaton are nter-dependent. In our recent work [45], a mult-user scenaro where each user has multple ndependent tasks was consdered for jont optmzaton of offloadng allocaton of communcaton computaton resources. The dfferences of ths work [45] are as follows: 1) The problem structures are dfferent, leadng to dfferent problem formulatons soluton approaches; 2) For the sngle-task per user case studed n ths paper, we propose a low-complexty algorthm that s shown to acheve nearly optmal performance. Ths combned advantage n both the complexty performance cannot be acheved by the algorthm proposed n [45]; 3) In ths work, we further study the scenaro where a strct processng deadlne s mposed on each user s task, whch cannot be addressed by the soluton approach proposed n [45]. 3 SYSTEM MODEL AND PROBLEM FORMULATION In ths secton, we frst ntroduce the model of moble cloud computng wth a CAP, detalng the costs of processng locally, at the CAP, at the cloud. We then explan the jont offloadng decson resource allocaton optmzaton problem to mnmze a weghted sum cost. 3.1 System Model Moble Cloud wth CAP Consder a cloud access network wth N moble users, one CAP, one remote cloud server, as shown n Fg. 1. The CAP s a wreless access pont (or a cellular base staton) wth bult-n computaton capablty that may be provded by Internet servce provders as a value-added servce. Instead of just servng as a relay to always forward receved tasks from users to the cloud, the CAP also has the capablty to process user tasks subject to ts resource constrant. We denote the set of all users by N = {1,...,N}. Each moble user has one task to be ether processed locally or offloaded to the CAP, the CAP determnes whether to process each receved task by tself or further offload t to the cloud for processng. Snce there are multple tasks offloaded to the CAP some of them are processed by the CAP, we need to further allocate the communcaton computaton resources avalable at the CAP. We assume that all tasks are avalable at tme zero. Ths s smlar to many exstng studes [15], [16], [17], [18], [33], [34], [35], [36], [37]. If the tasks arrve dynamcally n tme, we may apply our model the proposed soluton n a quas-statc manner, where the system processes the tasks n batches that are collected over tme ntervals [46]. Moble User 1 1 Moble User 2 2 Moble User N Fg. 1. System model N Computng AP Offloadng Decson Remote Cloud Denote the offloadng decsons for user by x l,xa,xc {0, 1}, ndcatng whether user s task s processed locally, at the CAP, or at the cloud, respectvely. The offloadng decsons are constraned by x l +x a +x c = 1, N. (1) Notce that only one of x l, xa, xc for user can be Cost of Local Processng The nput data sze, output data sze, processng cycles of user s task are denoted by D n (), D out (), Y() 1, respectvely. Smlar to [16], [17], [18], [33], [34], [35], [36], [37], we assume that these quanttes are known, whch may be acheved by applyng a program profler [19], [20], [21]. We assume that the addtonal nstructons requred for remote processng can be downloaded drectly by the CAP or the cloud va ther access to a hgh-capacty wred network. When the task s processed locally, the processng energy s denoted by E l the processng tme s denoted by T l Cost of CAP Processng For user s task beng offloaded to the CAP, we denote the energy consumed by wreless transmsson (to the CAP) recepton (from the CAP) at user by E t E r, respectvely. We further denote the uplnk downlnk transmsson tmes between user the CAP by T t = D n ()/(η ucu ) Tr = D out ()/(η dcd ), respectvely, where c u c d are uplnk bwdth downlnk bwdth allocated to user, η u ηd are the spectral effcency of uplnk downlnk transmsson between user the CAP, respectvely 2. Furthermore, c u c d are lmted by the uplnk bwdthc UL downlnk bwdthc DL as follows c u C UL, (2) 1. The processng cycles of user s task depends on the nput data sze the applcaton type. For smplcty of llustraton, we ntally assume that a task requres the same value of Y() on dfferent CPUs so that ts processng tme s a functon of the CPU s clock speed only. We wll explan later how the proposed soluton can be trvally extended to the general case. 2. The spectral effcency can be approxmated by log(1+snr) where SNR s the lnk qualty between user the CAP.

4 4 c d C DL. (3) Snce some uplnk downlnk transmssons may overlap wth each other, there s also a total bwdth constrant (c u +cd ) C Total. (4) If ths task s processed by the CAP, denote ts processng tme by T a = Y()/f a, where fa s the assgned processng rate, whch s lmted by the total processng rate f A at the CAP as f a f A. (5) The usage cost assocated wth the CAP processng user s task s denoted by C a. The usage cost may depend on the data sze processng cycles of a task, as well as the hardware energy cost to mantan the CAP. Detaled modelng of the usage cost s outsde the scope of ths work. Here we smply assume that C a s gven for all Cost of Cloud Processng If a task s further offloaded to the cloud from the CAP, besdes all the costs mentoned above (except T a C a related to the task processng cost by the CAP), there s an addtonal transmsson tme between the CAP the cloud, denoted by T ac = (D n ()+D out ())/r ac, where r ac s the transmsson rate between the CAP the cloud. Also, the cloud processng tme s denoted byt c = Y()/fc, where f c s the cloud processng rate for each user. The rate r ac s assumed to be a pre-determned value regardless of the number of users. Ths s because the CAP-cloud lnk s lkely to be a hgh-capacty wred connecton as compared wth the lmted wreless lnks between the moble users the CAP, thus there s no need to consder bwdth sharng among the users. Smlarly,f c s also assumed to be a pre-determned value because of the hgh computatonal capacty dedcated servce of the remote cloud server. Thus, T ac T c only depend on task tself. Fnally, the cloud usage cost of processng user s task at the cloud s denoted by C c. The above notatons are summarzed n Table Problem Formulaton Our goal s to reduce the moble users energy consumpton mantan the servce qualty to ther tasks. To do so, we defne the total system cost as the weghted sum of total energy consumpton, the costs to offload process all tasks, the correspondng maxmum transmsson processng delays among all users. We am to mnmze the total system cost by jontly optmzng the task offloadng decson vector x = [x l,xa,xc ]T the communcaton CAP processng resource allocaton vector r = [c u, cd, fa ]T. For user s task beng offloaded to the CAP, we defne E A as the weghted transmsson energy processng cost. Smlarly, we defne E C as the weghted transmsson TABLE 1 Notaton correspondng descrpton. Notaton Descrpton D n (), D out () nput data sze output data sze of user s task Y() processng cycles of user s task E l local processng energy of user s task E t, Er uplnk transmttng energy downlnk recevng energy of user s task to from the CAP T l, Ta, Tc local processng tme, CAP processng tme, cloud processng tme of T t, Tr T ac C UL, C DL C Total c u, cd η u, ηd user s task uplnk transmsson tme downlnk transmsson tme of user s task between the moble user the CAP transmsson tme of user s task between the CAP the cloud uplnk bwdth downlnk bwdth for transmsson between moble users the CAP total transmsson bwdth between moble users the CAP uplnk bwdth downlnk bwdth assgned to user spectral effcency of uplnk downlnk transmsson between user the CAP C a, Cc CAP usage cost cloud usage cost of user s task r ac transmsson rate for each user between the CAP the cloud f c cloud processng rate for each user f A total CAP processng rate f a CAP processng rate assgned to user α weght of the CAP usage cost β weght of the cloud usage cost ρ weght of the energy consumpton of user s task energy processng cost f the task s offloaded to the cloud. They are gven by E A = (E t +Er +αca ), E C = (E t +E r +βc c ), where α β are the relatve weghts between the transmsson energy the processng cost n E A E C, respectvely. The local processng delay at user, denoted by T L, s gven by T L = T l x l. Also, defne T A T C as the transmsson processng delay at the CAP the cloud, respectvely. We have ( ) T A Dn () = x a, η u cu + D out() η d cd + Y() f a

5 5 ( T C Dn () = η u + D ) out() cu η d +T ac cd +T c x c. The values of T L, TA, TC depend on the offloadng decsons x the communcaton CAP processng resource allocaton r. The jont optmzaton of offloadng resource allocaton s formulated as follows [ N ρ (E l xl +EA xa +EC xc ) mn {x },{r } ] +max{t L +T A +T C } s.t. (1), (2), (3), (4), (5), (6) c u,c d,f a 0, N, (7) x l,xa,xc {0,1}, N, (8) whereρ s the weght on energy consumpton relatve to the delay. The proposed optmzaton problem (6) can be solved by any controller n ths network after collectng all requred nformaton. In practce, the controller could be the CAP. That s, each user provdes ts nformaton to the CAP, the CAP wll broadcast the obtaned offloadng decsons ( the correspondng resource allocatons) to all users by solvng problem (6). Notce that n problem (6), the cost of delay s consdered n the total system cost objectve. We put dfferent emphass on delay by adjustng ρ. Note that, snce processng delay s n the objectve nstead of as a constrant n problem (6), any offloadng decson resource allocaton are feasble. However, n practce, there are applcatons that requre strct processng deadlnes, some offloadng decsons may not satsfy the strct delay constrant for a task. Ths scenaro wll be further dscussed n Secton 5. 4 SHARECAP OFFLOADING SOLUTION For the scenaro wthout any delay constrant, we show n ths secton that optmzaton problem (6) has an equvalent QCQP formulaton that s NP-hard n general. We then present our proposed soluton through the SDR romzaton mappng approach. 4.1 Overvew of the Proposed Soluton Gven some offloadng decsons x, problem (6) concerns only the resource allocaton vector r as ( ) mn E +max{t L +T A +T C } (9) {r } s.t. (2), (3), (4), (5), (7), where E ρ (E l xl +EA xa +EC xc ). Note that E only depends on x, thus can be treated as a constant. The resource allocaton problem (9) s convex. It can be solved optmally usng stard convex optmzaton approaches such as the nteror-pont method. Snce there are a fnte number of offloadng decsons, a globally optmal soluton for problem (6) can be obtaned by exhaustve search among 3 N possble offloadng decsons. However, the complexty grows exponentally wth the number of users thus mpractcal. In order to fnd an effcent soluton to problem (6), we frst transform t nto a separable QCQP wth a lnear objectve, then propose a separable SDR approach a novel romzaton mappng method to recover the bnary offloadng decsons. Once we obtan the bnary offloadng decsons, we can easly solve problem (9) to fnd the correspondng optmal resource allocaton. We name our method the sharecap offloadng resource allocaton soluton. 4.2 QCQP Reformulaton Semdefnte Relaxaton We frst replace the nteger constrant (8) by x s (x s 1) = 0, N, (10) for s {l, a, c}. Then, we move the delay term from the objectve to the constrants by ntroducng addtonal auxlary varable t. Optmzaton problem (6) s equvalent to the followng problem mn {x },{r }, t ρ (Ex l l +E A x a +E C x c )+t (11) ( s.t. T l xl + Dn () η ucu ( Dn () + + D out() η u cu η d cd + D out() η d cd +T ac +T c + Y() (1), (2), (3), (4), (5), (7), (10). f a ) x a ) x c t, N, (12) We now show that the optmzaton problem (11) can be transformed nto a separable QCQP problem by the followng steps. Frst, we ntroduce addtonal auxlary varables d (D u,dd,da ). Constrant (12) can be equvalently replaced by the followng four constrants T l x l +D u +D d +D a +(T ac +T c )x c t, N, (13) D n ()(x a +xc ) η u cu D out ()(x a +xc ) η d cd Y()x a f a D u, N, (14) D d, N, (15) D a, N, (16) where constrant (13) s the overall delay constrant, constrants (14) to (16) correspond to the uplnk transmsson tme, the downlnk transmsson tme, the CAP processng tme, respectvely. Next, we vectorze the varables parameters n (11). Defne w 0 [t, ] T, (17) w [x l,x a,x c,c u,d u,c d,d d,f a,d a ] T, N, (18)

6 6 whch s the decson vector for user contanng all decson varables. Then, we can rewrte the objectve n (11) as where b T w, (19) =0 b 0 [1,0 1 8 ] T, b ρ [E l,ea,ec,0 1 6] T, for 0. In the followng, we present each constrant n problem (11) n a correspondng matrx form. For the overall delay constrant (13), t can be rewrtten as where (b c k )T w k 0, N, (20) k=0 b c 0 [ 1,0 1 8] T, b c b c k [Tl,0,(T ac +T c ),0,1,0,1,0,1] T, 0, for k 0,. The matrx forms of constrants (14) - (16) are where w T A µ w +(b µ )T w 0, µ {u,d,a}, N, (21) [ ] 0 η u A u 0.5 η u, A 0 d A u A u 0 2 4, A d A d, A a [ ] , A a [ A a b u [0,D n(),d n (),0 1 6 ] T, b d [0,D out(),d out (),0 1 6 ] T, b a [0,Y(),0,0 1 6] T. [ ] 0 η d η d, 0 ], We then replace the offloadng placement constrant (1) wth (b P ) T w = 1, N, (22) whereb P [1,1,1,0 1 6 ] T. For uplnk downlnk bwdth resource constrants (2) (3), we rewrte them as where (b U )T w C UL, (23) (b D ) T w C DL, (24) b U [0,0,0,1,0 1 5 ] T, b D [0 1 5,1,0,0,0] T. Smlarly, the total bwdth constrant (4) s as follows (b S ) T w C Total, (25) whereb S [0,0,0,1,0,1,0,0,0] T. The constrant (5) on the CAP processng resource allocaton can be rewrtten as (b A )T w f A, (26) whereb A [0 1 7,1,0] T. Constrant (7), whch ensures that all varables are nonnegatve, s replaced by w 0, N {0}. (27) Fnally, we rewrte nteger constrant (10) as w T dag(e j)w (e j ) T w = 0, j {1,2,3}, N, (28) where each e j s a 9 1 stard unt vector wth the jth entry beng 1. By further defnng z [w T 1]T, for n N {0}, together wth the above matrx form expressons, optmzaton problem (11) can now be transformed nto the followng equvalent homogeneous separable QCQP formulaton where mn {z } s.t. z T G z (29) =0 z T kg c kz k 0, N, (30) k=0 z T Gµ z 0, µ {u,d,a}, N, (31) z T G P z = 1, N, (32) z T GU z C UL, (33) z T GD z C DL, (34) z T G S z C Total, (35) z T G A z f A, (36) z T G I jz = 0, j {1,2,3}, N, (37) z 0, N {0}, (38) [ b ] 1, G 2 bt 0 [ ] G c 1 k 0 2 bc k 1, 2 (bc k )T 0 [ ] A G µ µ 1 2 bµ 1, µ {u,d,a}, 2 (bµ )T 0 [ G π bπ 1 2 (bπ )T 0 ], j {1,2,3}. G I j [ dag(ej ) 1 2 e j 1 2 et j 0 ], π {P,U,D,S,A},

7 7 Comparng the optmzaton problems (11) (29), all constrants have one-to-one correspondng matrx representatons. Specfcally, constrant (30) s the overall delay constrant, constrant (31) comes from the addtonal auxlary constrants (14)-(16), constrant (32) s the offloadng placement constrant, constrants (33) (34) correspond to uplnk downlnk bwdth resource constrants, constrant (35) s the total bwdth constrant, constrant (36) s the constrant on the CAP processng resource allocaton, constrant (37) corresponds to the nteger constrant (10). Therefore, optmzaton problem (29) s equvalent to the orgnal problem (6). Note that optmzaton problem (29) s a non-convex separable QCQP problem, whch s NP-hard n general. To solve t, we apply a separable SDR approach to relax t nto a separable semdefnte programmng (SDP) problem. Defne Z z z T. We then have z T Gz = Tr(GZ ), (39) wth rank(z ) = 1. By droppng the rank constrant rank(z ) = 1, we relax problem (29) nto the followng separable SDP problem mn {Z } s.t. Tr(G Z ) (40) =0 Tr(G c k Z k) 0, N, (41) k=0 Tr(G µ Z ) 0, µ {u,d,a}, N, (42) Tr(G P Z ) = 1, N, (43) Tr(G U Z ) C UL, (44) Tr(G D Z ) C DL, (45) Tr(G S Z ) C Total, (46) Tr(G A Z ) f A, (47) Tr(G I j Z ) = 0, j {1,2,3}, N, (48) Z (10,10) = 1, N {0}, (49) Z 0, N {0}. (50) The above problem can be solved effcently n polynomal tme usng stard SDP software, such as SeDuM [47]. Denote Z as the optmal soluton of the SDP problem (40). We need to obtan the offloadng decson x of the orgnal problem (6) from Z. In the followng, we propose a romzaton method to obtan our bnary offloadng decsons. sgn of each element n these vectors. Among the generated vectors, the one that yelds the best objectve value of the orgnal problem would be chosen as the desred soluton. However, the above romzaton procedure does not produce a feasble soluton due to the offloadng decson placement constrant (1). Instead, usng the structure of Z constrants n problem (40), we propose the followng romzaton method for a feasble soluton. Denote the offloadng soluton vector as x [x T 1,...,x T N] T, where x [x l,xa,xc ]T, for N. Snce we have removed the rank-1 constrant from problem (29) to arrve at the relaxed problem (40), the obtaned soluton Z for problem (40) does not drectly provde a feasble bnary soluton for the offloadng decsons. Our goal s to obtan approprate offloadng decsons from Z by mappng ts elements to bnary numbers. Note that only the frst three elements n z correspond to the offloadng decson varables for user (see w n (18)). Also, snce Z = z z T z (10) = 1, we know that the last row of Z satsfes Z (10,j) = z (j), for allj. Hence, we can use the values ofz (10,j) to recover the bnary offloadng decson z (j), for j = 1,2,3. Before provdng the detals of the proposed romzaton method, we frst show the property ofz (10,j), forj = 1,2,3, n the followng lemma. Lemma 1. For the optmal soluton Z of problem (40), Z (10,j) [0,1], for j = 1,2,3, N. Proof: After droppng rank-1 constrant rank(z ) = 1, n the relaxed problem (40), constrant (48) can only guarantee that Z (j,j) = Z (10,j)(= Z (j,10)) for j = 1,2,3. To show that Z (10,j) [0,1], for j = 1,2,3, we note that, frst, Z (j,j) 0 snce Z 0, whch means Z (10,j) 0, for j = 1,2,3. In addton, constrant (43) guarantees that Z (10,1)+Z (10,2)+Z (10,3) = 1. Thus, we have Z (10,j) [0,1], for j = 1,2,3. Based on Lemma 1, we consder a probablstc mappng method to fnd x. We take Z (10,j) as the probablty of z (j) = 1,.e., prob(z (j) = 1) = Z (10,j), for j = 1,2,3. Denote p [p l,pa,pc ]T [Z (10,1),Z (10,2),Z (10,3)]T. Equvalently, ths means prob(x s = 1) = ps, for s = l,a,c. We reconstruct [x l,xa,xc ] usng p as margnal probabltes, whle satsfyng constrant (1). Ths leads to our proposed probablstc romzaton method as follows. Let U l = p l (1 p a )(1 p c ), 4.3 Bnary Offloadng Decsons va Romzaton One mght consder usng a common approach [13] to obtan an nteger soluton from the relaxed SDP problem, by romly generatng vectors from the Gaussan dstrbuton wth zero mean covarance Z for L tmes, then mappng them to the nteger set {0,1} 3N by usng the U a = (1 p l )pa (1 pc ), U c = (1 p l )(1 p a )p c.

8 To satsfy the placement constrant (1), we defne rom vectoru, whch represent the locaton that user s task wll be processed, as follows: [1,0,0] T, wth probabltyp l (local processng), u = [0,1,0] T, wth probabltyp a (CAP processng), (51) [0,0,1] T, wth probabltyp c (cloud processng), where P s = U s U l +Ua +Uc P l +P a +P c = 1., s {l,a,c}, We generate M..d. feasble offloadng solutons x (m) = [(u (m) 1 ) T...(u (m) N )T ] T usng the above procedure, for m = 1,..., M, solve the correspondng resource allocaton problem (9) for each x (m). We then choose among these feasble solutons the one that gves the mnmum objectve value of the optmzaton problem (6) to obtan the offloadng soluton x sdr the correspondng optmal resource allocaton {r sdr }. For the best decson, n practce, we should also compare x sdr wth the solutons from local processng only cloud processng only methods, select the one that gves the mnmum cost as the fnal offloadng decson x sdr the correspondng optmal resource allocaton {r sdr }. The detals of the overall sharecap offloadng resource allocaton algorthm are gven n Algorthm 1. Notce that the SDP problem (40) can be solved wthn precson ǫ by the nteror pont method n at most O( N log(1/ǫ)) teratons n whch the amount of work per teraton s O(N 6 ) [48]. Ths compares well wth the 3 N choces n exhaustve search to fnd an optmal offloadng decson. In addton, we observe from numercal results that a small number of romzaton trals (e.g., M = 10) s enough to gve system performance very close to the optmal one. Remark 1. The proposed soluton can be easly extended to the general case where the number of processng cycles for each task on dfferent CPUs are dfferent because these quanttes are constants n optmzaton problem (6). 5 OFFLOADING WITH DELAY CONSTRAINTS Tme-senstve applcatons n practce may have strct processng deadlnes, whch complcates the offloadng decsons resource allocaton. In ths secton, we further study the scenaro where each task must be completed before some gven deadlne. That s, there s a strct delay constrant for each user s task gven by T L +T A +T C T, N, (52) where we note that only one of T L, TA, TC s nonzero by ther defntons constrant (1). To ensure that at least one feasble offloadng soluton exsts, we assume that local processng tme T l T so that each user can at least process ts task locally to meet the deadlne regardless of the Algorthm 1 ShareCAP Offloadng Algorthm 1: Obtan optmal solutonz of the SDP problem (40). Extract Z (10,j), for j = 1,2,3, from Z. Set the number of romzaton trals as M. 2: Record the values of Z (10,j), for j = 1,2,3, as p. 3: for m = 1,...,M do 4: x (m) = [u (m) 1,...,u (m) N ]T wth u (m) generated as n (51); 5: Gven x (m), solve resource allocaton problem (9) record the mnmum cost value of (9) as J (m) ; 6: end for 7: Choose among x (1),...,x (M) the one that yelds the mnmum system cost: x sdr = argmn {x }J (m) (m) 8: Compare the mnmum cost of (9) underx sdr wth those under the local processng only cloud processng only solutons. Select the one that yelds the mnmum system cost as x sdr. 9: Output: the proposed offloadng soluton x sdr the correspondng optmal resource allocaton {r sdr }. avalablty of the remote processng. Wth above addtonal delay constrants, the optmzaton problem becomes [ N mn x,{r } ρ (E l xl +EA xa +EC xc ) ] +max{t L +T A +T C } s.t. (1), (2), (3), (4), (5), (7), (8), (52). 8 (53) Due to addtonal delay constrants (52), the optmzaton problem (53) s more complcated than the orgnal problem (6). In addton, dfferent from (6) where any offloadng decson s always feasble, only some offloadng decsons are feasble for problem (53). To solve ths problem, n the followng, we modfy the orgnal sharecap soluton propose a three-step algorthm, named sharedcap wth Delay Constrants (sharecap-d). Furthermore, we wll show that the newly obtaned bnary offloadng decson x computaton communcaton resource allocaton {r } by sharecap-d algorthm are locally optmal. 5.1 Step 1: QCQP Transformaton Semdefnte Relaxaton As mentoned above, optmzaton problem (53) s more complcated, snce ndvdual strct delay constrants are mposed to all users tasks. Followng the smlar procedure n Secton 4.2, we move the delay term from the objectve to the constrants by ntroducng addtonal auxlary varables t, rewrte (53) as mn {x },{r }, t ρ (E l xl +EA xa +EC xc )+t (54) ( s.t. Tx l l Dn () + η ucu ( Dn () + η u cu + D out() η d cd + D out() η d +T cd ac +T c + Y() f a ) x a (1), (2), (3), (4), (5), (7), (10), (12), ) x c T, N, (55)

9 9 where constrant (55) comes from the strct delay constrant (52). Comparng optmzaton problem (54) wth problem (11), we observe that they share a smlar structure, except that problem (54) has the addtonal delay constrant (55). Therefore, we can apply a smlar procedure to transform problem (54) nto a non-convex separable QCQP problem, solve the correspondng separable SDP relaxaton problem. Rewrtng the addtonal constrant (55) nto a matrx form, we have z T GC z T, N, (56) where z s defned as n Secton 4.2, b C [T,0,(T l ac G C +T c [ bc 1 2 (bc )T 0 ),0,1,0,1,0,1] T, ]. The optmzaton problem (54) can now be transformed nto the followng equvalent separable QCQP formulaton mn {z } z T G z (57) =0 s.t. (30) (38), (56). Smlar to (39), we have Z = z z T z T GC z = Tr(G C Z ). Therefore, we can further reformulate problem (57) nto a separable SDP problem as follows mn {Z } Tr(G Z ) (58) =0 s.t. Tr(G C Z ) T, N, (59) (41) (50), whch s SDP problem (40) wth the addtonal delay constrant (59). Note that SDP problem (58) s a relaxaton of problem (54) always feasble, so that we can obtan the optmal soluton{z }. However, the romzaton procedure ntroduced n Secton 4.3 cannot be drectly appled to fnd a feasble soluton for problem (53) due to the ndvdual delay constrant for each user s task (52). In other words, there s no feasblty guarantee for the romly generated offloadng vectorxw.r.t (52) hence ts assocated resource allocaton problem. To deal wth ths ssue, we propose a determnstc approach n whch we choose an ntal offloadng soluton that wll subsequently be mproved n Steps 2 3 below. Intal offloadng soluton: Frst, we have x = [x l,xa,xc ]T p = [p l,pa,pc ]T as defned n Secton 4.3. Applyng Lemma 1, we can guarantee that p s [0,1], s {l,a,c}. Then, we recover the offloadng decsons x sdr usng p as follows: [1,0,0] T, f max x sdr s {l,a,c} ps = pl (local processng), = [0,1,0] T, f max s {l,a,c} ps = pa (CAP processng), (60) [0,0,1] T, f max s {l,a,c} ps = pc (cloud processng), obtan the overall offloadng decson as x sdr = [(x sdr 1 )T,...,(x sdr N )T ] T. Snce an offloadng decson x sdr generated usng the above procedure may not satsfy ndvdual delay constrants (52), n the followng, we ntroduce an adaptve adjustment procedure to obtan a feasble soluton through teraton, wth x sdr as the ntal soluton. 5.2 Step 2: Obtanng a Feasble Soluton va Adaptve Adjustment Smlar to (9), optmzaton problem (53) s reduced to the optmzaton of computaton communcaton resource allocaton {r } gven by mn {r } ( E +max {T L +T A +T C s.t. (2), (3), (4), (5), (7), (52), ) } (61) where E s defned below (9). We can determne whether a gven offloadng decson x s feasble by solvng problem (61) whch s convex. If t s feasble, we can obtan the correspondng optmal resource allocaton {r }. We now provde an adaptve adjustment procedure to obtan a feasble offloadng soluton teratvely. Set x aa = x sdr. At each teraton: Check the feasblty ofx aa by solvng problem (61). Defne a set N l = { : x l = 0, N}, whch contans all users wth current decsons to offload ther tasks. If x aa s nfeasble, romly pck N l modfy the decson to be local processng as x aa = [1,0,0] T. Repeat steps untl x aa s feasble for problem (61), record the correspondng resource allocaton as {r aa }. Then output the soluton of the adaptve adjustment procedure as (x aa,{r aa }). Note that the above procedure always converges to some feasble offloadng decson x aa ( correspondng optmal resource allocaton {r aa }). To see ths, we note that n the worst case, the offloadng solutons x aa converges to the no offloadng decson profle where each task s processed locally, whch s feasble snce T l T for N. We summarze ths property n the followng proposton. Proposton 1. (x aa,{r aa }) obtaned from the adaptve adjustment procedure s always a feasble soluton to the orgnal optmzaton problem (53) wth strct delay constrants. 5.3 Step 3: Obtanng a Local Optmum va Sequental Turnng Wth a feasble soluton (x aa,{r aa }) obtaned n Step 2, we now propose an teratve procedure, termed sequental tunng, to further reduce the system cost obtan a local optmum for problem (53). Set (x st,{r st }) = (x aa,{r aa }) as the ntal pont. At each teraton: Romly order the lst of all users. Go through the user lst one by one. For each examned user, check the three possble offloadng decsons for ts task, whle keep the offloadng decsons of all other users unchanged. For each offloadng decson, fnd the total system cost by

10 solvng problem (61). As soon as some user s found to admt a lower total system cost by changng ts offloadng decson, update (x st,{r st }) to the new offloadng decson resource allocaton that gve the lower cost, ext the teraton. Repeat steps untl x st converges,.e., no change for x st can be made. Then output the soluton of the sequental turnng procedure as (x st,{r st }). The above procedure s guaranteed to converge. Ths s because there s a fnte number of possble values for x st [t]. The teraton eventually wll reach some(xst,{r st }), where the total system cost cannot be further reduced by modfyng any user s offloadng decson ( correspondng resource allocaton). It s straghtforward to show that (x st,{r st }) s a local optmum of problem (53), snce t gves the lowest system cost n the jont bnary-valued neghborhood of x neghborhood of {r }. Ths result s stated n the followng proposton. Proposton 2. Gven any feasble ntal pont, (x st,{r st }) obtaned from the sequental tunng procedure s a local optmal soluton to the orgnal optmzaton problem (53) wth strct delay constrants. We summarze the above three-step sharecap-d algorthm n Algorthm 2. Note that, by desgn, the fnal soluton (x st,{r st }) obtaned by adoptng the sequental tunng procedure s better than or at least as good as (x aa,{r aa }). In Secton 7, we show that the proposed sharecap-d method provdes not only a local optmum soluton but also nearly optmal performance compared wth the optmal polcy. Remark 2. The sequental tunng procedure can also be appled as an extenson of sharecap for the case wthout the delay constrants, to obtan a locally optmal soluton wth an even lower total system cost to problem (6). 6 EXTENSION TO SUM DELAY OPTIMIZATION In prevous optmzaton problems (6) (53), we have consdered the maxmum transmsson processng delays among all users as part of the total system cost. Our approach proposed soluton can also be extended to the consderaton of the sum delay of all users tasks as part of the total system cost. Ths optmzaton problem s gven as mn {x },{r } [ ρ (E l x l +E A x a +E C x c ) ] +(T L +T A +T C ) s.t. (1), (2), (3), (4), (5), (7), (8). (62) Same as n problems (6) (53), we can adjust ρ to put dfferent emphass on the sum delay. In addton, the strctly delay constrant (52) can be ncluded when consderng tme-senstve applcatons. Usng a smlar procedure as descrbed n Secton 4.2, we can obtan the correspondng non-convex separable QCQP problem separable SDP relaxaton problem for problem (62). The only dfference between problems (62) (6) s the structure of the resultng SDR problems. Therefore, the Algorthm 2 ShareCAP-D Offloadng Algorthm Step 1: Intal offloadng soluton va SDR 1: Obtan optmal soluton {Z } of the SDR problem (58). Extract the upper-left 3 3 sub-matrces {Ẑ } from {Z }. 2: Record the values of dagonal terms n Ẑ by p = [p l,pa,pc ]T. 3: Setx sdr = [(x sdr 1 )T,...,(x sdr N )T ] T, wherex sdr s gven by (60), as the ntal offloadng soluton. Step 2: Adaptve adjustment 4: Set x aa = x sdr. 5: Set AA = False. 6: whle AA == False do 7: Check the feasblty of x aa by solvng problem (61); 8: f x aa s nfeasble then 9: Determne the set of users wth offloaded task: Set N l = { : x l = 0, N}; 10: Romly pck user N l set ts offloadng decson to local processng: x aa = [1,0,0] T ; 11: else 12: Record the correspondng resource allocaton {r aa }; 13: Set AA = True; Ext whle loop 14: end f 15: end whle Step 3: Sequental tunng 16: Set (x st,{r st }) = (x aa,{r aa }), record the correspondng total system cost as J st. 17: Set ST = False. 18: whle ST == False do 19: Romly order the lst of all users; 20: Set user ndex j = 1; 21: whle j N do 22: Keep x st j,j j, j N unchanged. For the three possble offloadng choces of x st j, fnd ther respectve total system costs by solvng problem (61); set J st as the mnmum cost among these choces, record the correspondng soluton as x st j {r st }; 23: f J st < J st then 24: Set x st j = x st j, {rst } = {r st }, J st = J st ; 25: j N +1; 26: else f j = N then 27: j N +1; ST = True; No change of x st can be found; ext 28: else 29: j j +1; 30: end f 31: end whle 32: end whle 33: Output: The offloadng decson x st the correspondng resource allocaton {r st }. same approach as sharecap or sharecap-d (when consderng the strctly delay constrant) can agan be appled to obtan the fnal offloadng decson the correspondng optmal resource allocaton for problem (62). We omt the dervaton detals to avod repetton. 10

11 local processng cloud processng local cloud sharecap rom mappng optmal polcy local processng cloud processng local cloud sharecap rom mappng optmal polcy β ρ Fg. 2. The total system cost versus weght β (J/bt) local processng cloud processng local cloud sharecap rom mappng optmal polcy f A 10 9 Fg. 3. The total system cost versus CAP CPU rate f A (cycles/s). 7 NUMERICAL RESULTS In ths secton, we provde numercal results based on Monte Carlo repeated samplng to study the performance of both sharecap sharecap-d under dfferent parameter settngs. 7.1 Parameter Setup In the followng, the default parameter values are descrbed, unless otherwse ndcated later. We adopt the moble devce characterstcs from [49], whch s based on a Noka moble phone, set the number of users as N = 8. Accordng to Tables 1 3 n [49], the moble devce has CPU rate cycles/s processng energy consumpton J/cycle, the local computaton tme s/bt local processng energy consumpton J/bt are calculated when the x264 CBR encode applcaton (1900 cycles/byte) s consdered for Y() = 1900D n (). The nput output data szes of each task are assumed to be unformly dstrbuted from 10 to 30 MB from 1 to 3 MB, respectvely. Both uplnk bwdth C UL downlnk bwdth C DL between moble users the CAP are set to 20 MHz, wth no addtonal lmt on the total bwdth, the transmsson recevng energy consumptons of the moble user are both J/bt as ndcated n Table 2 n [49]. For smplcty, we set η u = ηd = 3.5b/s/Hz for all Fg. 4. The total system cost versus weght ρ = ρ (s/j) local processng cloud processng 200 local cloud sharecap rom mappng optmal polcy number of users Fg. 5. The total system cost versus number of users.. The CPU rates of the CAP each server at the remote cloud are cycle/s cycle/s, respectvely. When tasks are offloaded to the cloud, the transmsson rate r ac s 6 Mpbs. Also, we set the values of cost C a C c to be the same as that of the nput data sze D n (), α = J/bt β = J/bt. We further set ρ = ρ = 0.5 s/j for all. Fnally, all numercal results are obtaned by averagng over 100 realzatons of the nput output data szes of each task. To study the performance of sharecap sharecap-d, we compare them wth the followng methods: 1) the local processng only method where all tasks are processed by moble users, 2) the cloud processng only method where all tasks are offloaded to the cloud, 3) the local-cloud offloadng method where the same approxmaton procedure as the sharecap method s appled except that there s no CAP, 4) the rom mappng method where each task s processed at dfferent locatons wth equal probablty, 5) the optmal polcy where the optmal value s obtaned by exhaustve search. When compared wth sharecap-d, all of the adaptve adjustment procedure n Secton 5 s added to all of the above methods when ther offloadng decsons are not feasble. 7.2 Performance of ShareCAP In Fg. 2, we show the system cost vs. weghtβ on the cloud processng cost. When β becomes large, the total system cost puts more emphass on the cloud usage cost. As a

12 local processng cloud processng 1400 local cloud sharecap rom mappng number of users local processng cloud processng local-cloud sharecap-d rom mappng optmal polcy θ Fg. 6. The total system cost versus number of users wth scaled resources. TABLE 2 Average run tme comparson under varous number of users. Number of sharecap (sec) optmal polcy (sec) users (exhaustve search) N/A N/A N/A N/A consequence, all tasks are more lkely to be processed by ether the moble user or the CAP. The local-cloud method n ths case converges to the local processng method. On the other h, when β decreases, the cost of cloud processng becomes nsgnfcant, sharecap, local-cloud, optmal polcy all converge to cloud processng. Though the exstence of the CAP can provde addtonal computaton capacty, all tasks processed at the CAP need to share the CAP CPU rate f A by optmally allocatng the processng rate to each user s task. In Fg. 3, we plots the total system cost vs. f A. As expected, a more powerful CAP can dramatcally ncrease system performance, sharecap converges to local-cloud when the CAP CPU rate s too slow to help. In Fg. 4, we study the system cost when weght ρ (weght of energy consumpton relatve to delay) changes. In Fgs. 5, 6, we study the system cost under varous number of users N. In partcular, n Fg. 6, the amount of lmted resources (.e., uplnk downlnk bwdth the total CAP processng rate) s scaled proportonal to the number of users N. From Fgs. 4 to 6, We observe that wth the help of the CAP, sharecap outperforms all other methods. Furthermore, all of these fgures show that, over a wde range of system parameter values, sharecap provdes performance close to that of optmal polcy, where the latter, obtaned by exhaustve search, has an exponental computatonal complexty n N,.e., O(3 N ). The correspondng average run tmes for dfferent values of N are also provded n Table 2. They are obtaned on a desktop PC wth Intel Fg. 7. The total system cost versus delay factor θ wth strct delay constrants local processng cloud processng local-cloud 150 sharecap-d rom mappng optmal polcy f A 10 9 Fg. 8. The total system cost versus CAP CPU rate f A (cycles/s) wth strct delay constrants. Core GHz processor 8 GB RAM. For optmal polcy by exhaustve search, we only obtan the run tmes up to 10 users as the requred computatonal tme becomes very hgh for the cases beyond 10 users. 7.3 Performance of ShareCAP-D For numercal results wth sharecap-d, we assumet θt l for all. Fg. 7 plots the total system cost vs. the delay factor θ. We observe that sharecap-d provdes near-optmal performance unless θ s close to 1,.e., when the problem (53) s nearly nfeasble. When θ s moderately relaxed, there are more offloadng decsons can satsfy delay constrants, whch allows sharecap-d to choose a local optmum that s closer to the optmal soluton. In Fg. 8, we plots the total system cost vs. f A wth θ = 1.1. We see that sharecap-d outperforms all other methods except the optmal soluton. Furthermore, we observe that sharecap-d s the only method that can effcently utlze the ncreasng CAP processng capacty, as demonstrated by t steeply declnng cost curve as f A ncreases. In partcular, when f A , sharecap-d s nearly dentcal to an optmal polcy. Fnally, we plot the total system cost vs. number of users N wth θ = 1.1 n Fg. 9. Though the optmzaton problem