Energy-Efficient Mobile-Edge Computation Offloading for Applications with Shared Data

Transcription

1 Energy-Efficient Mobile-Edge Comptation Offloading for Applications with Shared Data Xiangy He, Hong Xing, Ye Chen, Armgam Nallanathan School of Electronic Engineering and Compter Science Qeen Mary University of London, London, UK xiangy.he, h.xing, ye.chen, arxiv: v1 [cs.it] 4 Sep 2018 Abstract Mobile-edge comptation offloading (MECO) has been recognized as a promising soltion to alleviate the brden of resorce-limited Internet of Thing (IoT) devices by offloading comptation tasks to the edge of celllar networks (also known as clodlet). Specifically, latency-critical applications sch as virtal reality (VR) and agmented reality (AR) have inherent collaborative properties since part of the inpt/otpt data are shared by different sers in proximity. In this paper, we consider a mlti-ser fog compting system, in which mltiple single-antenna mobile sers rnning applications featring shared data can choose between (partially) offloading their individal tasks to a nearby single-antenna clodlet for remote exection and performing pre local comptation. The mobile sers energy minimization is formlated as a convex problem, sbject to the total compting latency constraint, the total energy constraints for individal data downloading, and the compting freqency constraints for local compting, for which classical Lagrangian dality can be applied to find the optimal soltion. Based pon the semi-closed form soltion, the shared data proves to be transmitted by only one of the mobile sers instead of mltiple ones. Besides, compared to those baseline algorithms withot considering the shared data property or the mobile sers local compting capabilities, the proposed joint comptation offloading and commnications resorce allocation provides significant energy saving. I. INTRODUCTION With the advent of the era of Internet of Things (IoT), the nprecedented growth of latency-critical applications are nevertheless hardly satisfied by mobile clod compting (MCC) alone. To cater for the low-latency reqirements while alleviating the brden over backhal networks, mobileedge compting (MEC), also interchangeably known as fog compting has arosed a paradigm shift by extending clod capabilities to the very edge within the radio access network (RAN) (see [1] and the references therein). Both indstry and academia have devoted constant effort to providing the next generation mobile networks with ltra-reliable low latency commnications (RLLC). Among pioneering indstrialization on fog compting, Cisco has proposed fog compting as a promising candidate for IoT architectre [2]. In academics, [3] [6] focsed on one-toone offloading scheme where there is one mobile ser and one corresponding clodlet, [7] [8] presented mltiple-ser cases where there are mltiple edge servers, while [9] related to mltiple-to-one scenarios where mltiple mobile sers offload compting to one edge server. Recently, the intrinsic collaborative properties of the inpt data for comptation offloading was investigated for agmented reality (AR) in [10]. In fact, in many mobile applications sch as agmented reality (AR) and virtal reality (VR), mltiple mobile devices share parts of compting inpt/otpt in common, ths making it possible for frther redcing compting latency at the edge. In [11], some important insights on the interplay among the social interactions in the VR mobile social network was revealed, and a significant redce on the end-to-end latency was achieved throgh stochastic optimization techniqe. [12] investigated potential spatial data correlation for VR applications to minimize the delay of accomplishing comptation. On another front, joint optimization of comptation offloading with commnications resorces (sch as power, bandwidth, and rate) proves to improve the performance of fog compting by explicitly taking channel conditions and commnications constraints into accont. In an early research [13], the offloading decision making was examined throgh the estimation of bandwidth data withot considering the allocation of commnication resorces and channel conditions. For commnications-aware comptation offloading, [14] minimized the local ser s comptation latency in a mlti-ser cooperative scenario, while [15] minimized the energy consmption of remote fog compting nodes. However, these line of work have not taken the shared data featre aforementioned into accont, ths failing to flly reap the advantage of fog compting. In this paper, we consider a mlti-ser fog compting system, in which mltiple single-antenna mobile sers rnning applications featring shared data can choose between (partially) offloading their compting tasks to a nearby single-antenna clodlet and execting them locally, and then download the reslts from the clodlet. Mobile sers overall energy consmption is minimized via joint optimization of comptation offloading and commnications resorce allocation. Compared with existing literatre, e.g., [10], althogh it investigated the energy minimization problem of shareddata featred offloading, it did not find the optimal soltion. Moreover, it did not draw explicit conclsion regarding the channel condition s inflence in the comptation offloading. From this point of view, or work provides in-depth nderstanding of the shared-data featred offloading in MEC systems.

2 II. SYSTEM MODEL We consider a mobile-edge system that consists of U mobile sers rnning AR applications, denoted as U = 1,...,U, and one base station (BS) eqipped with compting facilities working as a clodlet. All of the mobile sers and the BS are assmed to be eqipped with single antenna. The inpt data size for ser is denoted by D I, U, in which one fraction data size of DS I bits are the shared data that is the same across all U mobile sers and the other fraction of D L bits are the data exected locally by ser. The shared data can be transmitted from each ser by part, denoted by D I, U, sch that U =1 DI = DI S. The amont of inpt data that is exclsively transmitted by is ths given by D I = D I DS I DL, U. transmission power given by p l, the achievable individal data rate for offloading the shared data is expressed as: R l = W l log 2 (1+ pl h 2 ), U, (1) where W l = Wl U with W l denoting the overall bandwidth available for the plink transmission, and is the additive white Gassian noise (AWGN) power. Accordingly, = DI /Rl, and the energy consmed by the -th ser in the shared data offloading sb-phase is given as E l = tl pl = tl h 2f(DI ), U, (2) where the fnctionf(x) is defined asf(x) = (2W l 1). Similarly, the energy consmption for the -th ser in the individal data offloading sb-phase is expressed as: x E l = tl pl = tl B. Comptation Model DI S DL h 2f(DI ), U. (3) Based on the energy model in [9], given the local compting bits D L, the energy consmption for execting local comptation is given by: Fig. 1. Timing illstration for the considered mlti-ser MEC system. It can be seen from Fig. 1 that there are two consective sb-phases for both inpt data offloading and reslts downloading phases: the shared and the individal data transmission. The transmission dration for offloading the shared inpt data is denoted by, U; the offloading dration for the individal data is denoted as, U; and the drations for downloading the shared and the individal otpt data are and tdl, U respectively. The remote comptation time are also illstrated in Fig. 1, where t C S and tc, U, denote that for the shared and the individal data transmitted to the clodlet, respectively. Similarly, F and f, U, denote the comptational freqency (in cycles/s) allocated to the shared and the individal tasks, respectively, by the clodlet. In addition, the local comptation time is denoted by t C,L, U. A. Uplink Transmission As observed from Fig. 1, there are two consective plink transmission sb-phases: the shared data and the individal data offloading [10]. Each mobile ser offloads its comptation task to the clodlet server via freqency division mltiple access (FDMA). The channel coefficient from ser is given by h, U, which is assmed to remain nchanged dring the plink transmission dration. With the E C = κ (λ 0 D L)3 0 2, U, (4) t C,L where λ 0 (in cycles/bit) denotes the nmber of CPU cycles needed for processing one bit of inpt data, and κ 0 is the energy consmption capacitance coefficient. C. Downlink Transmission Similar to the plink transmission, the downlink transmission phase also has two separate sb-phases: the shared and the individal reslts downloading. The shared otpt data are mlticasted to the mobile sers by the clodlet at its maximm transmitting power P max. The achievable individal rate for the shared data downloading is ths given by R dl = Wdl log 2(1+ P max g 2 ), U, (5) where W dl = Wdl U with Wdl denotes the overall bandwidth available for downlink transmission. The downlink channel coefficient is given by g, U. The relation between the shared otpt data and the inpt data is given by DS O = a 0DS I, where a 0 is the factor representing the nmber of otpt bits for execting one bit of inpt data. Accordingly, = DO S /Rdl, U, and ths the latency for transmitting the shared otpt data to all mobile sers is given by S = max tdl. (6) This is becase the individal reslts downloading cannot be initiated ntil the shared data has finished transmission. After the mlticasting transmission, the individal otpt data is sent to each mobile ser via FDMA. Denoting the

3 downlink transmitting power for the -th individal data by p dl, the achievable rate for individal data downloading is ths expressed as: R dl = W dl log 2 (1+ pdl g 2 ), U. (7) Similarly, denoting the individal otpt data size by D O, U, D O = a 0 D I = a 0(D I DI S DL ), and tdl = D/R O dl. For energy consmption, the overall energy consmed for decoding the reslt sent back by the clodlet at the -th mobile ser is given by [10] E dl = (tdl +tdl )ρdl, U, (8) whereρ dl (in Joles/second) captres the energy expenditre per second. In addition, the total energy consmed by the BS for reslts transmission is given by, 0(D I g 2f(a DS I DL ) ), U, (9) which is reqired not to exceed E max by the BS operator. D. Total Latency Next, we consider the overall compting latency. As illstrated in Fig. 1, it is observed the individal data downloading in Phase II cannot start ntil the clodlet completes individal data compting, and the BS finishes the shared data transmission over the downlink. Moreover, for the individal data compting, it cannot start before either the corresponding individal data finishes offloading or the clodlet completes the shared data compting, i.e., max + tl,max tl + tc S. Frthermore, also seen from Fig. 1, for the shared data reslts, it can only start being transmitted in the downlink after the clodlet completes the shared data compting and all the individal data finishes offloading in the plink, i.e., max max tl + t C S,max tl +. Combining the above facts, the total compting latency is expressed as follows: τ = max max +,max tl +t C S+t C, max max tl +t C S,max tl + U. III. PROBLEM FORMULATION + S +, (10) The overall energy consmption at the mobile sers consists of three parts: data offloading over the plink (c.f. (2) and (3)), local compting (c.f. (4)), and reslts retrieving (c.f. (8)), which is ths given by E total = κ 0 (λ 0 D L ) 3 t C,L2 + + DI S DL h 2f(DI )+ h 2f(DI ) ( + )ρ dl. (11) The objective is to minimize the overall energy consmption given by E total, sbject to the compting latency constraints, the maximm local compting freqencies, and the total energy consmption on the individal data at the BS. Specifically, the optimization problem is formlated as below: (P1) : min E total,tl,tc,l,tdl,dl,di (12a) τ T max, U, (12b) 0(D I g 2f(a DI S DL ) ) E max, (12c) 0 t C,L T max, U, (12d) λ 0 D L tc,l f,max, (12e) 0 D L D I DS, I U, (12f) D I = DS,D I I 0, (12g) 0,tl 0,tC,L 0,tdl 0, U. (12h) Constraint (12b) and (12d) gives the latency constraints that the time taken for accomplishing compting tasks cannot excess the maximm allowed length, both for offloading and local compting. (12c) tells that the available energy for downlink transmission of remote compting node shold be lower than a maximm level. (12e) restricts the nmber of allowable local compting bits imposed by local compting capabilities. Besides, (12g) pts that adding all the shared data bits offloaded by all mobile sers respectively, the vale shold be eqal to the exact amont of shared bits existing in the same ser grop. IV. OPTIMAL SCHEME FOR JOINT OFFLOADING AND COMMUNICATION RESOURCE ALLOCATION A. Problem Reformlation Althogh the latency expression (10) looks complex in its from, (12b) is still a convex constraint. For the ease of exposition, we assme herein that the clodlet exectes the shared and the individal compting within the dration of the individal data offloading and the shared reslts downloading, respectively, i.e., t C S tl, and tc tdl, U 1. As a reslt, (12b) can be simplified as below: max +tl +tdl S +tdl T max, U. (13) 1 We assme herein that the comptation capacities at the clodlet is relatively mch higher than those at the mobile sers, and ths the compting time taken is mch shorter than the data transmission time.

4 by introdcing the axiliary variable, which satisfies tdl, U, (13) redces to +tl T max S tdl, U. (14) Notice that E C s (c.f. (4)) is monotonically decreases with respect to the local compting time t C,L for each mobile ser. To obtain the minimal energy consmption, it is obvios that t C,L = T max, U. Then the optimization problem to be solved is reformlated as: (P1 ) : min E total,tl,tdl,tdl,d L,DI (12c 12h),(14). tdl, U. (15a) (15b) (15c) B. Joint offloading and commnication resorce allocation Introdcing dal variables β, ω, σ, ν, the Lagrangian of problem (P1 ) is presented as: L(β,ω,σ,ν,,,,,D,D L ) I = h 2f(DI )+ DI S DL h 2f(DI ) + (λ 0 D L κ )3 0 + ( t C,L2 +tdl )ρdl + β ( + T max + S +tdl )+ ω (λ 0 D L t C,L f,max)+ σ ( tdl ) +ν[ 0(D g 2f(a I DS I DL ) ) E max ], (16) where β = β 1,...,β U are dal variables associated with the latency constraint (14), ω = ω 1,...,ω U are associated with local compting bits constraint (12e)), σ = σ 1,...,σ U are connected with the constraint for axiliary variable, and ν catches the downlink transmission energy constraint (12c). Hence, we have the Lagrangian dal fnction expressed as: g(β,ω,σ,ν) = min L(β,ω,σ,ν,,tl,tdl,tdl,D L,DI,tl,tdl,tdl, D L,DI ), (17) (12f-12h). Conseqently, the corresponding dal problem is formlated as: max g(β,ω,σ,ν) (18) β,ω,σ,ν β 0,ω 0,σ 0,ν 0. Proposition 1. Given a determined set of dal variables β, ω, σ, ν, the optimal soltion to the Lagrangian dal problem (16) can be determined as follows. The optimal primal variables, tl, and, are given by ˆ = ˆ = ˆD I W l ln2 [W 0( 1 e (β h 2, U. (19) 1))+1] D I DI S ˆD L W l ln2 [W 0( 1 e (β h 2, U. (20) 1))+1] ˆ = a 0 (D I DS I ˆD ) L W dl a 0 ln2 [W 0( 1 e ((ρdl +σ ) g 2, U. 1))+1] ν (21) where W 0 (x) is the principle branch of the Lambert W fnction defined as the soltion for W 0 (x)e W0(x) = x [15], e is the base of the natral logarithm; the optimal axiliary variable is given by: 0, ˆ = β σ > 0, (22) T max S, otherwise; and the optimal local compting data size is given by ˆD L = min T max,d I D I S [ ˆr l a 0ˆr dl ln2 3 3κ 0 λ ( 2W l 0 W l h 2 + νa 0 2 W dl W dl g 2 ) ω 3κ 0 λ 2 0, U, where ˆr l = Wl ln2 [W 0( 1 e (β h 2 1)) + 1] and ˆr dl = W dl a [W 0ln2 0( 1 e ((ρdl +σ) g 2 ν 1))+1], U. Proof: Please refer to Appendix A. In fact, on one hand, ˆr l s and ˆr dl s can be interpreted as the optimm transmission rate for the shared/individal data offloading and the individal data downloading, respectively, given the dal variables. On the other hand, for each ser, the optimal transmission rate for the shared data is seen to be identical to that of the individal data over the plink, given that the plink channel gains remain nchanged dring the whole offloading phase. Next, to obtain the optimal offloading bits of the shared data for each ser, i.e., ˆDI, we need the following lemma. Lemma 1. The optimal offloaded shared data for ser is expressed as, D I ˆD I = S, û = arg min, 1 U (23) 0, otherwise, where = f(ˆrl ) ˆr l + β h 2 ˆr l, U. Proof: Please refer to Appendix B. ] +

5 Notable, it is easily observed from Lemma 1 that the shared data is optimally offloaded by one specific ser instead of mltiple ones. Based on Proposition 1, the dal problem can ths be iteratively solved according to ellipsoid method (with constraints), the detail of which can be referred to [16]. The algorithm for solving (P1 ) is smmarized in Table I. TABLE I ALGORITHM I FOR SOLVING(P1 ) Energy consmption ( 10-3 J) Offloading withot considering shared data Fll Offloading Only Proposed Shared data Offloading ALgorithm Offloading with eqal time length Reqire: (β (0),ω (0),σ (0),ν (0) ) 1: repeat 2: Solve (17) given (β (i),ω (i),σ (i),ν (i) ) according to Proposition 1 and obtain ˆ,ˆ,ˆ,ˆ, ˆD, L ˆD I ; 3: pdate the sbgradient of β,ω,σ,ν respectively, i.e., +tl T max + max tdl + tdl, λ 0 D L tc,lf,max, tdl tdl, g 2 f( a 0(D I DS I DL ) ) E max in accordance with the ellipsoid method [16]; 4: ntil the predefined accracy threshold is satisfied. Ensre: The optimal dal variables to the dal problem (18) (β,ω,σ,ν ) 5: Solve (17) again with (β,ω,σ,ν ) Ensre:,tl,,,D L,D V. NUMERICAL RESULTS In this section, the nmerical reslts of the proposed algorithm together with other baseline algorithms are presented. Except for the local compting only scheme where sers execte all the data bits locally, there are three other offloading schemes presented as baseline algorithms: 1) Offloading withot considering the shared data: the collaborative properties are ignored, every ser makes the offloading decision withot coordination among other sers; 2) Fll offloading only: the shared data is taken into consideration, bt the whole chnks of inpt data of every ser are forced to be offloaded to the edge compting node, exclding the local compting capability from participating in the comptation tasks; 3) Offloading with eqal time length: taking the correlated data into consideration, the data offloading and downloading are performed for each ser with eqal time length, with optimal soltions obtained throgh CVX. In the simlation, the bandwidth avaialble is assmed to be W l = W dl =10MHz, the maximm downlink transmit power P max = 1W, and the inpt data size D I = 10kbits for all sers. The spectral density of the (AWGN) power is -169 dbm/hz. The mobile energy expenditre per second in the downlink is ρ dl =0.625 J/s [10], the maximm local compting capability f,max = 1GHz. Besides, λ 0 = cycle/bit, a 0 = 1, κ 0 = The pathloss model is PL = log 10 (d ), where d represents the distance between ser and edge compting node in kilometers. Fig.2 depicts how the energy consmption changes with different latency constraints. The energy consmption are becoming lower as the latency reqirement gets longer for all listed offloading algorithms. Only the proposed offloading scheme can give the lowest energy consming performance Latency Constraint T max (s) Fig. 2. Energy consmption verss different latency constraints The best energy saving improvement can only be achieved throgh the joint participation of local compting and shared data coordination. Besides, even thogh the eqal time length offloading has lower complexity than the proposed algorithm, it cannot compete with the proposed one in terms of energy saving. Recalling or conclsion that the best way to achieve the energy saving is to let these correlated bits transmitted by one specific ser, the reason is that forcing offloading time dration to be eqal makes the shared data to be transmitted by all sers simltaneosly. The energy consmed for compting one data bit increases exponentially as the latency constraint diminishes. Hence for the local compting only scheme, when latency constraint comes to 0.01 second the energy taken to finish the comptation tasks, which is 1000 mjoles, can reach p to nearly 100 times more than those of all the offloading algorithms. Then it drops exponentially to 10 mjoles when the latency constraint goes to 0.1 second. As a reslt, the crve representing local compting only is not added in Fig.2, otherwise the comparison of the offloading schemes will not be clear. In Fig.3, the energy consmption changes with the percentage of shared data is demonstrated. Apparently, as long as we take the shared data into consideration when making offloading decisions, the lower overall energy consmption is achieved when the proportion of shared data gets higher. More energy will be saved when the percentage of shared data gets higher for proposed offloading scheme compared to the scheme withot considering the existence of shared data. This trend applies to the fll offloading only algorithm as well, becase it also cares abot the existence of shared data when making offloading decisions. The energy consmptions for fll offloading only do not always go nder that of offloading withot considering shared data. That is becase when given specific latency constraint, the importance of local compting capabilities diminishes in saving mobile sers energy consmption as the share of common data increases. Since most of the data will be offloaded to the edge node, few inpt bits wold remain local for compting. Then the energy consmption of the fll offloading only scenario represents that it get closer to that of the proposed algorithm when the percentage of shared data increases. Similar trend applies to the eqal time length offloading as well. t C S = λ 0D I S/F (24)

6 12 10 Offloading withot considering shared data Offloading with eqal time length Fll Offloading Only Proposed Shared Data Offloading Algorithm with DI, the partial Lagrangian is expressed as ˆr l min D I L = [ tl h 2f(DI )+β ] Energy consmption ( 10-3 J) = [ D I D ˆr l h 2f(ˆrl I )+β ˆr l = D I ] (24a) Percentage of shared data (%) Fig. 3. Energy consmption verss different percentage of shared data VI. CONCLUSIONS In this paper, a mlti-ser fog compting system was considered, in which mltiple single-antenna mobile sers rnning applications featring shared data can partially offload their individal comptation tasks to a nearby singleantenna clodlet and then download the reslts from it. The mobile sers energy consmption minimization problem sbject to the total latency, the total downlink transmission energy and the local compting constraints was formlated as a convex problem with the optimal soltion obtained by classical Lagrangian dality method. Based pon the semi-closed form soltion, it was proved that the shared data is optimally transmitted by only one of the mobile sers instead of mltiple ones collaboratively. The proposed joint comptation offloading and commnications resorce allocation was verified by simlations against other baseline algorithms that ignore the shared data property or the mobile sers own compting capabilities.. APPENDIX A In order to find the optimal soltions of the primary problem, we need to examine the related partial derivatives L L L L,,,, L, L, U. After obtaining D L D I t l these partial derivatives, the KKT conditions can be applied to find the optimal soltions. For example, let L D L and L D I eqal to 0. The inverse fnction of y = f(x) xf (x) for x > 0 is given by x = Wl [W0( y ln2 e 1 ) + 1]. Then it follows that e f(ˆr ) l ˆr f l (ˆr ) l = f(ˆr l ) ˆr l f (ˆr l ) = β h 2, and the optimal plink transmission rate of the shared data ˆr l and that of the exclsively offloaded data ˆr l are ths derived. Then the expressions of the optimal primary variables are readily obtained as shown in (19), (20), (21), (22), (23), and (23). APPENDIX B To obtain how the shared inpt data offloading ˆD I are distribted among sers, we need to examine the partial Lagrangian regarding D I and. Replacing the shared data offloading time be D I = DS,D I I 0, U, (24b) where we define = f(ˆrl ) β + as a constant given the ˆr l h 2 ˆr l dal variable β s. As a reslt, the optimal soltion to the linear programming (LP) (24) is easily obtained as shown in (23). REFERENCES [1] Y. Mao, C. Yo, J. Zhang, K. Hang, and K. B. Letaief, A srvey on mobile edge compting: The commnication perspective, Commn. Srveys Tts., vol. 19, pp , Forth Qarter [2] F. Bonomi, R. Milito, J. Zh, and S. 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