Joint Computation and Communication Cooperation for Mobile Edge Computing

Transcription

1 Joint Computation and Communiation Cooperation for Mobile Edge Computing Xiaowen Cao, Feng Wang, Jie Xu, Rui Zang, and Suguang Cui Sool of Information Engineering, Guangdong University of Tenology, Guangzou, Cina Department of Eletrial and Computer Engineering, National University of Singapore, Singapore Department of Eletrial and Computer Engineering, University of California, Davis, USA {fengwang3, arxiv: v [s.it] 9 Ot 7 Abstrat Tis paper proposes a novel joint omputation and ommuniation ooperation approa in mobile edge omputing MEC systems, wi enables user ooperation in bot omputation and ommuniation for improving te MEC performane. In partiular, we onsider a basi tree-node MEC system tat onsists of a user node, a elper node, and an aess point AP node attaed wit an MEC server. We fous on te user s lateny-onstrained omputation over a finite blok, and develop a four-slot protool for implementing te joint omputation and ommuniation ooperation. Under tis setup, we jointly optimize te omputation and ommuniation resoure alloation at bot te user and te elper, so as to imize teir total energy onsumption subjet to te user s omputation lateny onstraint. We provide te optimal solution to tis problem. Numerial results sow tat te proposed joint ooperation approa signifiantly improves te omputation apaity and te energy effiieny at te user and elper nodes, as ompared to oter benmark semes witout su a joint design. Index Terms Mobile edge omputing MEC, omputation offloading, joint omputation and ommuniation ooperation, resoure alloation, optimization. I. INTRODUCTION Reent tenologial advanements ave enabled various emerging appliations e.g., augmented reality and autonomous driving tat require intensive and low-lateny omputation at massive wireless devies. As tese devies are generally of small size and tus ave limited power supply, ow to provide tem wit enaned omputation apability and low omputation lateny is one ruial but allenging task to be takled. Mobile edge omputing MEC as been reognized as a promising tenique to provide loud-like omputing at te edge of radio aess networks su as aess points APs and base stations BSs. By deploying MEC servers terein, wireless devies an offload part or all of teir omputation-eavy and lateny-sensitive tasks to APs and/or BSs for remote exeution [] [3]. Depending on weter te omputation tasks are partitionable or not, te omputation offloading an be generally ategorized into two lasses, namely binary and partial offloading, respetively []. In binary offloading, te omputation task is not partitionable, and tus sould be exeuted as a wole via eiter loal omputing at te devie itself or offloading to te MEC server. In partial offloading, te task an be partitioned into two or more independent parts, wi an be exeuted by loal omputing and offloading, respetively, in parallel. Based on te binary/partial offloading models, tere ave been a andful of prior works see, e.g., [3] [] and te referenes terein investigating te joint omputation and ommuniation optimization to improve te performane of MEC. For example, [4] and [5] onsidered power-onstrained omputation lateny imization problems in a single-user MEC system wit dynami task arrivals and annel fading. [6] [8] aimed to imize te system energy onsumption wile meeting te users omputation lateny requirements in multiuser MEC systems. Furtermore, [9], [] proposed interesting wireless powered MEC systems by integrating te emerging wireless power transfer WPT tenique into MEC, in order to aieve self-sustainable mobile omputing. Fully reaping te benefit of MEC, owever, faes several design allenges. For instane, te omputation apability at te MEC server and te ommuniation apability at te AP are generally finite; terefore, wen te number of supported users inreases, te resoures alloated to ea user would be limited. Furtermore, te omputation offloading in MEC systems ritially depends on te wireless annel onditions between te users and te AP; ene, wen te devies are loated far away from te AP or te wireless annels suffer from deep fading, te benefit of offloading would be ompromised. To overome tese issues, we notie tat future wireless networks will onsist of massive devies e.g., smartpones, wearable omputing devies, and smart sensors, ea of wi is equipped wit ertain loal omputation and ommuniation resoures; furtermore, at any time instane, it is igly likely tat some devies are in te idle status due to te burst nature of bot te omputation and ommuniation traffis. As a result, enabling user ooperation among tese devies in bot omputation and ommuniation is an effiient and viable solution to improve te MEC performane, were nearby idle devies an sare teir omputation and ommuniation resoures to elp enane ative omputing users performane. In tis paper, we investigate a new paradigm of user ooperation in bot omputation and ommuniation for MEC systems. For te purpose of exposition, we onsider a basi tree-node system, wi onsists of a user node, a elper node, and an AP node attaed wit an MEC server. Suppose tat te omputation tasks need to be exeuted witin a time blok, and partial offloading is implemented for omputation

2 tasks. To implement te joint omputation and ommuniation ooperation, we divide te blok into four slots. In te first slot, te user offloads part of its tasks to te elper, su tat te elper an ooperatively ompute tem on bealf of te user in te remaining time. In te seond and tird slots, te elper works as a deode-and-forward DF relay for ooperative ommuniation, in order to elp te user offload some oter omputation tasks to te AP for remote exeution in te fourt slot. Under tis setup, we pursue an energy-effiient design to imize te total energy onsumption at bot te user and te elper, subjet to te user s omputation lateny onstraint. Towards tis end, we jointly optimize te alloation of time slots, te partition of te user s omputation bits for its loal omputing, te elper s ooperative omputing, and te AP s remote exeution, respetively, te entral proess unit CPU frequenies at bot te user and te elper, as well as teir transmission powers for offloading. Toug tis problem is non-onvex in general, we transform it into a onvex form and use te Lagrange duality metod to obtain te optimal solution in a semi-losed form. Numerial results sow tat te proposed joint omputation and ommuniation ooperation approa outperforms alternative benmark semes witout su a joint design. Note tat tere ave been some prior works tat independently studied te ommuniation ooperation see, e.g., [] [4] and te omputation ooperation [5], [6], respetively. In wireless ommuniation systems, te ooperative ommuniations or relaying teniques ave been extensively investigated to inrease te data rate and/or improve te transmission reliability [] [4]. In MEC systems, te soalled devie-to-devie DD fogging [5] and peer-to-peer PP ooperative omputing [6] ave been proposed to enable te omputation ooperation between end users, in wi one atively-omputing user an employ DD or PP ommuniation to offload its omputation tasks to te nearby idle user for remote exeution. Different from tese prior works wit sole ommuniation or omputation ooperation, tis work is te first attempt to pursue te joint omputation and ommuniation ooperation by unifying bot for maximizing te MEC performane. II. SYSTEM MODEL AND PROBLEM FORMULATION As sown in Fig., we onsider a basi tree-node MEC system tat onsists of one user node, one elper node, and one AP node wit an MEC server integrated. All te tree nodes are equipped wit one single antenna. We fous on a time blok wit duration T >, were te wireless annels are assumed to remain unanged over tis blok, and te user needs to suessfully exeute omputation tasks wit L > input bits before te end of tis blok. It is assumed tat te tree nodes perfetly know te global annel state information CSI and te omputation-related information; Note tat te joint omputation and ommuniation ooperation in tis paper an be extended into te senario wit multiple users and multiple elpers, by effiiently pairing one elper wit ea user for ooperation. Te extension is left for our future work. Offloading Loal omputing User Relaying Cooperative omputing Helper AP MEC server Fig.. A basi tree-node MEC system wit joint omputation and ommuniation ooperation. Te dased and solid lines indiate te task offloading to te elper for omputation ooperation and to te AP via te elper s ommuniation ooperation as a relay, respetively. User Helper User AP/Helper Helper AP Remote task exeution at AP Fig.. MEC protool wit joint omputation and ommuniation ooperation. aordingly, tey an ooperatively sedule teir omputation and ommuniation resoures for te MEC performane optimization. In order to implement te joint omputation and ommuniation ooperation, te L input bits sould be generally partitioned into tree parts for loal omputing, offloading to elper, and offloading to AP, respetively. Let l u denote te number of input bits for loal omputing at te user,l denote tat for offloading to te elper, and l a denote tat for offloading to te AP, respetively. We ten ave l u +l +l a = L. A. MEC Protool Wit Joint Computation and Communiation Cooperation In order to implement te joint omputation and ommuniation ooperation, te duration-t blok is generally divided into four slots as sown in Fig.. In te first slot wit duration τ, te user offloads te l task-input bits to te elper, and te elper an exeute tem in te remaining time wit duration T τ. In te seond and tird slots, te elper ats as a DF relay to elp te user offload l a task-input bits to te AP. Speifially, in te seond slot wit duration τ, te user broadasts te l a input bits to bot te AP and te elper simultaneously; after suessfully deoding, te elper forwards tem to te AP in te tird slot wit durationτ 3. After olleting te input bits from te user, te MEC server

3 an remotely exeute te offloaded tasks in te fourt time slot wit duration τ 4. It is wort noting tat we ave ignored te time for downloading te omputation results from te elper and AP to te user, due to te fat tat te omputation results are normally wit mu smaller size tan te input bits, and tus te downloading time beomes negligible. In order to ensure te omputation tasks to be suessfully exeuted before te end of tis blok, we ave te following time onstraint: τ +τ +τ 3 +τ 4 T. In te following, we first introdue te omputation offloading from te user to te elper and te AP, and ten present te omputing at te tree nodes. B. Computation Offloading Computation Offloading to Helper: In te first slot, te user offloads l task-input bits to te elper wit te transmit power P. Let > denote te annel power gain from te user to te elper, andb te system bandwidt. Ten te aievable data rate in bits/se for offloading from te user to te elper is given by r P = Blog + P Γ, 3 were represents te power of te additive wite Gaussian noise AWGN at te reeiver of te elper, and Γ is a onstant term aounting for te gap from te annel apaity due to a pratial modulation and oding seme. For simpliity, Γ = is assumed trougout tis paper. In pratie, te number of offloaded bits l from te user to te elper annot exeed τ r P. Hene, we ave l τ r P. 4 Furtermore, let P u,max denote te maximum transmit power at te user, and aordingly we ave P P u,max. We onsider te user s transmission energy as te sole energy budget for omputation offloading, and ignore te energy onsumed by iruits in te radio-frequeny RF ains, baseband signal proessing, et. Terefore, in te first slot, te energy onsumption for te user s offloading is given by E offl = τ P. 5 Computation Offloading to AP Assisted by Helper: In te seond and tird slots, te elper ats as a DF relay to elp te user offload l a task-input bits to te AP. In te seond slot wit duration τ, let P denote te user s transmit power wit P P u,max. In tis ase, te aievable data rate from te user to te elper is given by r P wit r defined in 3. Furtermore, by denoting > as te annel power gain from te user to te AP, te aievable data rate from te user to te AP is r P = Blog + P were is te AWGN power at te AP reeiver., 6 After suessfully deoding te reeived message, te elper forwards tem to te AP in te tird slot wit duration τ 3 by using te transmit power P 3. Let P,max denote te maximum transmit power from te elper, and tus it olds tat P 3 P,max. Let > denote te annel power gain from te elper to te AP. Te aievable data rate from te elper to te AP is r P 3 = Blog + P 3. 7 By ombining te seond and tird slots, te maximum number of data bits tat an be transmitted from te user to te AP via te DF relay te elper is given as [3] τ r P +τ 3 r P 3, τ r P, wi is te upper bound for te number of te offloaded bits l a to te AP, i.e., l a τ r P +τ 3 r P 3, τ r P. 8 Similarly as for 5, we onsider te user s and elper s transmission energy onsumption for offloading as te energy budget in te seond and tird slots, respetively, wi are expressed as follows. C. Computing at User, Helper, and AP E offl = τ P 9 E3 offl = τ 3 P 3. Loal Computing at User: Te user exeutes te omputation tasks wit l u input bits trougout te wole blok wit duration T. Let u denote te number of CPU yles for omputing one bit at te user, and f u,n te CPU frequeny for te n-t CPU yle, were n {,..., u l u }. In pratie, te CPU frequeny f u,n is upper bounded by a maximum value, denoted by f u,max, i.e., f u,n f u,max. As all te loal omputing sould be aomplised before te end of te time blok, we ave te following omputation lateny requirement: ul u n= f u,n T. Aordingly, te user s energy onsumption for loal omputing is [] ul u Eu omp = κ u fu,n, 3 n= were κ u denotes te effetive apaitane oeffiient tat depends on te ip ariteture at te user. It as been sown in [9, Lemma ] tat in order for te user to save te omputation energy onsumption wile imizing te lateny, it is optimal to set te CPU frequenies to be idential for different CPU yles. By using tis fat and letting te onstraint in be met wit strit equality, we ave f u, = f u, =... = f u,ul u = u l u /T. 4

4 Substituting 4 into 3, te user s energy onsumption for loal omputing Eu omp is re-expressed as E omp u = κ u 3 u l3 u T. 5 By ombining 4 wit te maximum CPU frequeny onstraint, we ave u l u Tf u,max. 6 Cooperative Computing at Helper: After reeiving te offloaded l task-input bits in te first time slot, te elper exeutes te tasks during te remaining time wit duration T τ. Let f,n and f,max denote te CPU frequeny for te n-t CPU yle and te maximum CPU frequeny at te elper, respetively. Similarly as for te loal omputing at te user, we set te elper s CPU frequeny for ea CPU yle n as f,n = l /T τ, n {,..., l }, were represents te number of CPU yles for omputing one bit at te elper. Aordingly, te energy onsumption for te ooperative omputation at te elper is E omp = κ 3 l3 T τ, 7 were κ is te effetive apaitane oeffiient of te elper. As in 6, we ave te following onstraint on te offloaded bits due to te maximum CPU frequeny f,max : l T τ f,max. 8 3 Remote Computing at AP: In te fourt slot wit duration τ 4, te MEC server at te AP exeutes te offloaded l a task-input bits. As te MEC server normally as a stable energy supply e.g., onneted to te grid, te MEC server an ompute tasks at its maximal CPU frequeny, denoted by f a,max, in order to imize te omputation time. Hene, te time durationτ 4 for te MEC server to exeute tel a offloaded bits is τ 4 = l a /f a,max. 9 By substituting 9 into, te time alloation onstraint is re-expressed as D. Problem Formulation τ +τ +τ 3 +l a /f a,max T. In tis work, we aim to imize te total energy onsumption at bot te user and te elper i.e., 3 i= Eoffl i + Eu omp +E omp wit Ei offl s given in 5, 9, and, Eu omp in 5, and E omp in 7, subjet to te user s omputation lateny onstraint, by jointly optimizing teir omputation and ommuniation resoure alloation. Te deision variables inlude te alloation of te time slots τ = [τ,τ,τ 3 ], te task partition l = [l u,l,l a ], and te offloading power As te AP normally as reliable power supply, its energy onsumption is not te bottlenek of tis MEC system. Terefore, we only fous on imizing te user s and elper s energy onsumption for ommuniation and omputation at te wireless devies side. alloations P = [P,P,P 3 ]. Matematially, te energyeffiient joint omputation and ommuniation ooperation problem is formulated as P : P,τ,l i {,,3} τ i P i + κ u 3 ul 3 u T s.t. P j P u,max, j {,} + κ 3 l3 T τ a b P 3 P,max τ i T, i {,,3} d l u, l, l a, 4, 8, 6, 8, and. e In problem P, te onstraints and denote te task partition and time alloation onstraints; 4 and 8 ensure tat te numbers of te offloaded bits from te user to te elper and te AP are limited by te aievable data rates over te respetive wireless annels; 6 and 8 orrespond to te maximum CPU frequeny onstraints at te user and te elper, respetively. Note tat problem P is non-onvex in general due to te oupling of τ i and P i in te objetive funtion a and te onstraints 4 and 8. However, we next transform it into a onvex problem and solve it optimally in Setion III. Before solving problem P, we first exae te feasibility to ek weter te tree-node MEC system an support te task exeution witin te lateny onstraint. Towards tis end, we obtain te maximum number of supportable task-input bits, denoted by L max. If L max is no smaller tan L in P, ten P is feasible. Oterwise, P is infeasible. In partiular, L max an be obtained by letting all te tree nodes use up all teir ommuniation and omputation resoures, via setting P = P = P u,max,p 3 = P,max and setting te onstraints in 8, 6, 8, and to be met wit strit equality. As a result, we ave L max max l u +l +l a τ,l s.t. l τ r P u,max, l a τ r P u,max u l u = Tf u,max, l = T τ f,max τ +τ +τ 3 +l a /f a,max = T τ r P u,max +τ 3 r P,max = τ r P u,max d, and e. Note tat problem is a linear program LP and tus an be effiiently solved via standard onvex optimization teniques su as te interior point metod [7]. After L max obtained, te feasibility of P is effiiently eked. In te next setion, we fous on solving P wen it is feasible. III. OPTIMAL SOLUTION TO P Tis setion presents te optimal solution to problem P. Towards tis end, we first transform it into a onvex form by introduing a set of auxiliary variables E = [E,E,E 3 ] wit E i P i τ i, were i {,,3}. Aordingly, we ave P i = E i /τ i, were we define P i = if eiter E i = or

5 τ i = olds, i {,,3}. By substituting P i = E i /τ i, i {,,3}, problem P is reformulated as P. : E,τ,l i {,,3} E i + κ u 3 ul 3 u T E s.t. l τ r τ E E3 l a τ r +τ 3 r τ τ 3 E l a τ r τ + κ 3 l3 T τ 3a 3b 3 3d E j τ j P u,max, j {,} 3e E 3 τ 3 P,max 3f, 6, 8,, d, and e, were 3 and 3d follow from 8. Note tat te funtion r j x is a onave funtion wit respet to x for any j {,,}, and terefore, its perspetive funtion xr y j x is jointly onave wit respet to x > and y. As a result, te onstraints 3b, 3, and 3d beome onvex. Furtermore, te funtion l 3 /τ is jointly onvex wit respet to l and τ >, and ene te term κ 3 l 3 T τ in te objetive funtion is jointly onvex wit respet to l and τ < T. Hene, problem P. is onvex and an be effiiently solved by standard onvex optimization teniques su as te interior-point metod [7]. Alternatively, we next use te Lagrange dual metod to obtain a well-strutured solution for gaining essential engineering insigts. Let λ, λ, and λ 3 denote te Lagrange multipliers assoiated wit te onstraints in 3b, 3, and 3d, and µ and µ denote te Lagrange multipliers assoiated wit te onstraints in and, respetively. For notational onveniene, we denote λ [λ,λ,λ 3 ] and µ [µ,µ ]. Te partial Lagrangian of problem P. is LE,τ,l,λ,µ = E +µ τ + κ 3 l3 T τ +λ µ l +λ τ r E E +E λ τ r λ 3 τ r +µ τ τ τ +µ τ 3 + κ u 3 u l3 u T µ l u E3 +E 3 λ τ 3 r τ 3 +λ +λ 3 +µ /f a,max µ l a µ T +µ L. Ten te dual funtion of problem P. is E gλ,µ = LE,τ,l,λ,µ 4 E,τ,l s.t. 6, 8, d, e, 3e, and 3f. Consequently, te dual problem of P. is D. : max gλ,µ 5 λ,µ s.t. µ, λ i, i {,,3}. τ We denote X as te set of λ, µ araterized by te onstraints in 5. Sine problem P. is onvex and satisfies te Slater s ondition, strong duality olds between problems P. and D.. As a result, one an solve P. by equivalently solving its dual problem D.. In te following, we first obtain te dual funtiongλ,µ for any givenλ,µ X, and ten obtain te optimal dual variables to maximize gλ, µ. For onveniene of presentation, we denote E,τ,l as te optimal solution to 4 under any given λ,µ X, E opt,τ opt,l opt as te optimal primal solution top., and λ opt,µ opt as te optimal dual solution to problem D.. Derivation of Dual Funtion gλ, µ: First, we obtain gλ,µ by solving 4 under any givenλ,µ X. Note tat 4 an be deomposed into te following five subproblems. E,τ,l E +µ τ λ τ r E τ + κ 3 l3 T τ +λ µ l s.t. 8 and E τ P u,max τ T, l. 6 E,τ E +µ τ λ τ r E τ E λ 3 τ r s.t. E τ P u,max, τ T. 7 E 3,τ 3 E 3 +µ τ 3 λ τ 3 r E3 s.t. E 3 τ 3 P,max, τ 3 T. 8 l u τ 3 κ u 3 ulu 3 µ l u T s.t. u l u Tf u,max. 9 λ +λ 3 +µ /f a,max µ l a. 3 l a L For problems 6 3, we present teir optimal solutions in te following lemmas. Due to te similar strutures of problems 6 8, we present te proof of Lemma 3. and omit te proofs of Lemmas for brevity. Lemma 3.: Under given λ,τ X, te optimal solution E,τ,l to problem 6 satisfies E = P τ, 3 l = M T τ, 3 = T, if ρ <, τ [,T], if ρ =, 33 =, if ρ >, [ ] were P λ = B ln Pu,max wit [x] a b {a, max{x, b}} and [ ] f,max µ λ M = 3κ, if µ 3 λ, 34, if µ λ <, τ

6 ρ =µ λ r P +κ M 3 + λ BP / +P / ln α P u,max + β f,max, 35, if P < P u,max, α = λ B / ln+p /, if 36 P = P u,max, β = {, if M < f,max, µ λ 3κ 3 M, if M = f,max. 37 Proof: See Appendix A. Lemma 3.: Under given λ,τ X, te optimal solution E,τ to problem 7 satisfies were P = [ v 4uw v ln B + E = P τ, 38 = T, if ρ <, τ [,T], if ρ =, 39 =, if ρ >, u λ +λ 3 ] Pu,max wit u = ln B,w = ln B λ λ 3, v =,ρ = +P ln µ λ r P+ λbp λ +P 3r P ln + λ3bp α P u,max, and, if P < P u,max, α = λ 3B + λ B, if ln ln P = P u,max. +P +P Lemma 3.3: Under given λ,τ X, te optimal solution E 3,τ 3 to problem 8 satisfies E3 = P 3 τ 3, 4 = T, if ρ 3 <, τ3 [,T], if ρ 3 =, 4 =, if ρ 3 >, [ ] were P3 = λb ln P,max and ρ 3 = µ + λbp 3 +P3 ln λ r P3 α 3P,max wit {, if P 3 < P,max, α 3 = λ B / +P 3 / ln, if P 3 = P,max. Lemma 3.4: For given λ,τ X, te optimal solution lu to problem 9 is [ ] Tfu,max lu µ u = T 3κ u 3. 4 u Lemma 3.5: For given λ,τ X, te optimal solution la to problem 3 is =, if λ +λ 3 +µ /f a,max µ >, la [,L], if λ +λ 3 +µ /f a,max µ =, 43 = L, if λ +λ 3 +µ /f a,max µ <. Proof: Note tat te objetive funtion is linear wit respet to l a wen λ +λ +µ /f a,max µ. Te proof of Lemma 3.5 is straigtforward; we ten omit it erein. Note tat in 33, 39, 4, or 43, if ρ i = for any i {,,3} or λ + λ 3 + µ /f a,max µ =, ten te optimal solutionτi orl a is non-unique in general. In tis ase, we oose τi = and la = for te purpose of evaluating te dual funtion gλ,µ. It is wort noting tat su oies may not be feasible nor optimal for te primal problem P.. To takle tis issue, we will use an additional step in Setion III-3 later to find te primal optimal τ opt i s and la opt for P.. By ombining Lemmas , te dual funtion gλ, µ is obtained for any given λ,µ X. Obtainingλ opt andµ opt to Maximizegλ,µ: Next, we sear overλ, µ X to maximize gλ, µ for solving problem D.. Sine te dual funtiongλ,µ is always onave but non-differentiable in general, we an use subgradient based metods, su as te ellipsoid metod, to obtain te optimal λ opt and µ opt for D.. Note tat for te objetive funtion in problem 4, te subgradient wit respet to λ,µ is [ l τ r E la τr τ, E E τ,la τ r E τ τ3 r 3 τ3, ] 3 τi +la/f a,max T,L lu l la. i= For te onstraints µ and λ i, te subgradients are e 4 and e i, i {,,3}, respetively, were e i R 5 is te standard unit vetor wit one in te i-t entry and zeros elsewere. 3 Optimal Solution to P: Wit λ opt and µ opt obtained, it remains to detere te optimal solution to problemp. and tus P. By replaing λ and µ in Lemmas as λ opt and µ opt, we denote te orrespondingpi s, l u, and M as P opt i s, lu opt, and Mopt, respetively. Aordingly, P opt = [P opt,p opt,p opt 3 ] orresponds to te optimal solution of P to problem P, and lu opt orresponds to te optimal solution of l u to bot problems P and P.. Neverteless, due to te non-uniqueness of τi s and l a, we need an additional step to onstrut te optimal solution of oter variables to problem P. Fortunately, wit P opt, M opt, and lu opt obtained, we know tat te optimal solution must satisfyl = M opt T τ and E i = P opt i τ i, i {,,3}. By substituting tem in P or P., we ave te following LP to obtain τ opt and l opt τ,l a 3 i= τ i P opt i a. +κ M opt 3 T τ 44 s.t. M opt T τ τ r P opt l a τ r P opt +τ 3 r P opt 3 l a τ r P opt M opt T τ +l a +lu opt = L and τ i T, i {,,3}. Te LP in 44 an be effiiently solved by te standard interior-point metod [7]. By ombining τ opt, and lopt p, lopt a,

7 Average energy onsumption Joule - Loal omputing Computation ooperation Communiation ooperation - Joint ooperation Te lengt of te time blok, T se Average energy onsumption Joule Loal omputing Computation ooperation Communiation ooperation Joint ooperation Number of omputation bits, L Mbits Fig. 3. Te average energy onsumption versus te time blok lengt. togeter wit P opt and lu opt, te optimal solution to problem P is finally found. IV. NUMERICAL RESULTS In tis setion, we present numerial results to validate te performane of te proposed joint omputation and ommuniation ooperation design, as ompared to te following benmark semes witout su a joint design. Loal omputing: te user exeutes te omputation tasks loally by itself. Te imum energy onsumption an be obtained as Eu lo = κ u 3 ul 3 /T. Computation ooperation: te omputation tasks are partitioned into two parts for te user s loal omputing and offloading to te elper, respetively. Tis orresponds to solving problem P by setting l a = and τ = τ 3 =. Communiation ooperation: te omputation tasks are partitioned into two parts for te user s loal omputing and offloading to te AP, respetively. Te offloading is assisted by te elper s ommuniation ooperation as a DF relay. Tis orresponds to solving problem P by setting l = and τ +τ 3 = T. In te simulation, we onsider tat te user and te AP are loated wit a distane of 5 meters m and te elper is loated on te line between tem. Let D denote te distane between te user and te elper. Te pat-loss between any ζ, d two nodes is denoted as β d were β = 6 db is te pat loss at te referene distane of d = m, d denotes te distane from te transmitter to te reeiver, and ζ = 3 denotes te pat-loss exponent. Furtermore, we set B = MHz, = = 7 dbm, u = = 3 yles/bit, κ u = 7, κ =.3 7, P u,max = P,max = 4 dbm, f u,max = GHz, f,max = 3 GHz, and f a,max = 5 GHz. Fig. 3 sows te average energy onsumption versus te time blok lengt T, were L =. Mbits and D = m. It is observed tat te average energy onsumption by all Fig. 4. Te average energy onsumption versus te number of omputation bits. te semes dereases as T inreases. Te ommuniationooperation seme is observed to aieve lower energy onsumption tan te omputation-ooperation seme wen T is small e.g., T <.3 se; wile te reverse is true wen T beomes large. It is also observed tat te omputationooperation and te ommuniation-ooperation semes bot outperform te loal-omputing seme, due to te fat tat te two ooperation based semes additionally exploit omputation resoures at te elper and te AP, respetively. Te proposed joint-ooperation seme is observed to aieve te lowest energy onsumption. Fig. 4 depits te average energy onsumption versus te number of omputation bits L, were T =. se and D = m. In general, similar observations are as in Fig. 3. In partiular, it is observed tat at small L values e.g., L <.6 Mbits, te loal-omputing seme aieves a similar performane as te joint-ooperation seme. Wen L beomes larger, te benefit of joint omputation and ommuniation ooperation is observed. V. CONCLUSION In tis paper, we investigated a new joint omputation and ommuniation ooperation approa in a simplified treenode MEC system, were a nearby elper node is enabled to sare its omputation and ommuniation resoures to elp improve te user s performane for mobile omputation. We proposed a four-slot protool to enable tis approa and developed a new energy-effiient design framework to imize te total energy onsumption at bot te user and te elper wile meeting te omputation lateny requirements, by jointly alloating teir omputation and ommuniation resoures. It is our ope tat tis paper an open a new avenue in exploring te multi-resoure user ooperation to improve te omputation performane for MEC.

8 A. Proof of Lemma 3. APPENDIX As problem 6 is onvex and satisfies te Slater s ondition, strong duality olds between problem 6 and its dual problem. Terefore, one an solve tis problem by applying te Karus-Kun-Tuker KKT onditions [7]. Te Lagrangian of problem 6 is given by L =E +µ τ λ τ r E µ l +λ l + κ 3 l 3 τ T τ a E +α E τ P u,max b τ +b τ T d l +β l T τ f,max, were a,α,b,b, d, and β are te non-negative Lagrange multipliers assoiated wit E,E τ P u,max,τ,τ T, l, and l T τf,max, respetively. Based on te KKT onditions, it follows tat a E =, α E τ P u,max =, b τ T = 45a b τ =, d l =, β l T τf,max = 45b L E = λ B ln + E τ a +α = 45 L τ = κ 3 l 3 T τ 3 +µ λ Blog + E τ + λb E τ ln + E τ + βf,max b +b +α P u,max = 45d L l = 3κ 3 l T τ µ +λ d +β =, 45e were 45a and 45b denote te omplementary slakness ondition, and 45, 45d and 45e are te first-order derivative onditions of L wit respet to E, τ, and l, respetively. Based on te KKT onditions, 3 follows from 45, and 3 olds due to 45e. Furtermore, based on 45, 45d, and 45e and wit some manipulations, we ave 36 and 37. Furtermore, by substituting 3 and 3 into 45d and assug ρ = b b, we tus ave ρ in 35. Hene, te optimal τ is given in 33. Until now, tis lemma is proved. [6] C. You, K. Huang, H. Cae, and B. Kim, Energy-effiient resoure alloation for mobile-edge omputing offloading, IEEE Trans. Wireless Commun., vol. 6, no. 3, pp , Mar. 7. [7] F. Wang, J. Xu, and Z. Ding, Optimized multiuser omputation offloading wit multi-antenna NOMA, to appear in IEEE GLOBECOM, 7. [Online]. Available: ttps://arxiv.org/pdf/ pdf. [8] M. Cen, M. Dong, and B. Liang, Joint offloading deision and resoure alloation for mobile loud wit omputing aess point, in Pro. IEEE ICASSP, Sangai, Cina, May, 6, pp [9] F. Wang, J. Xu, X. Wang, and S. Cui, Joint offloading and omputing optimization in wireless powered mobile-edge omputing system, in Pro. IEEE ICC, Paris, Frane, May, 7, pp. 6. [] C. You, K. Huang, and H. Cae, Energy effiient mobile loud omputing powered by wireless energy transfer, IEEE J. Sel. Areas Commun., vol. 34, no. 5, pp , May 6. [] J. N. Laneman, D. Tse, and G. W. Wornell, Cooperative diversity in wireless networks: Effiient protools and outage beavior, IEEE Trans. Inform. Teory, vol. 5, no., pp , Nov. 4. [] A. Sendonaris, E. Erkip, and B. Aazang, User ooperation diversity: Part I. system desription, IEEE Trans. Commun., vol. 5, no., pp , Nov. 3. [3] Y. Liang and V. V. Veeravalli, Gaussian ortogonal relay annels: Optimal resoure alloation and apaity, IEEE Trans. Inform. Teory, vol. 5, no. 9, pp , Sep. 5. [4] H. Ju and R. Zang, User ooperation in wireless powered ommuniation networks, in Pro. IEEE GLOBECOM, Austin, TX, USA, De. 4, pp [5] L. Pu, X. Cen, J. Xu, and X. Fu, DD fogging: An energy-effiient and inentive-aware task offloading framework via network-assisted DD ollaboration, IEEE J. Sel. Areas Commun., vol. 34, no., pp , Nov. 6. [6] C. You and K. Huang, Exploiting non-ausal CPU-state information for energy-effiient mobile ooperative omputing, 7. [Online]. Available: ttps://arxiv.org/abs/ [7] S. Boyd and L. Vandenberge, Convex Optimization, Cambridge, U.K.: Cambridge Univ. Press, Mar. 4. REFERENCES [] P. Ma and Z. Bevar, Mobile edge omputing: A survey on ariteture and omputation offloading, IEEE Commun. Surveys Tuts., vol. 9, no. 3, pp , Mar. 7. [] Y. Mao, C. You, J. Zang, K. Huang, and K. B. Letaief, A survey on mobile edge omputing: Te ommuniation perspetive, To appear in IEEE Commun. Surveys Tuts., 7. [3] S. Barbarossa, S. Sardellitti, and P. D. Lorenzo, Communiating wile omputing: Distributed mobile loud omputing over 5G eterogeneous networks, IEEE Signal Proess. Mag., vol. 3. no. 6, pp , Nov. 4. [4] J. Liu, Y. Mao, J. Zang, and K. B. Letaief, Delay-optimal omputation task seduling for mobile-edge omputing systems, in Pro. IEEE ISIT, Spain, Jun. 6, pp [5] Y. Zang, D. Niyato, and P. Wang, Offloading in mobile loudlet systems wit intermittent onnetivity, IEEE Trans. Mobile Comput., vol. 4, no., pp , De. 5.