Joint Subcarrier and CPU Time Allocation for Mobile Edge Computing

Transcription

1 Jont Subcarrer and CPU Tme Allocaton for Moble Edge Computng Ynghao Yu, Jun Zhang, and Khaled B. Letaef, Fellow, IEEE Dept. of ECE, The Hong Kong Unversty of Scence and Technology Hamad Bn Khalfa Unversty, Doha, Qatar Emal: {yyuau, eejzhang, arxv: v [cs.it] Aug 06 Abstract In moble edge computng systems, moble devces can offload compute-ntensve tasks to a nearby cloudlet, so as to save energy and extend battery lfe. Unlke a fully-fledged cloud, a cloudlet s a small-scale datacenter deployed at a wreless access pont, and thus s hghly constraned by both rado and compute resources. We show n ths paper that separately optmzng the allocaton of ether compute or rado resource as most exstng works dd s hghly suboptmal: the congeston of compute resource leads to the waste of rado resource, and vce versa. To address ths problem, we propose a jont schedulng algorthm that allocates both rado and compute resources coordnately. Specfcally, we consder a cloudlet n an Orthogonal Frequency- Dvson Multplexng Access (OFDMA) system wth multple moble devces, where we study subcarrer allocaton for task offloadng and CPU tme allocaton for task executon n the cloudlet. Smulaton results show that the proposed algorthm sgnfcantly outperforms per-resource optmzaton, accommodatng more offloadng requests whle achevng salent energy savng. I. INTRODUCTION Lmted battery lfe contnuously shows up as the top concern of smartphone users []. The problem s becomng even more severe n the predctable future, gven the evergrowng demands for compute-ntensve apps and the stallng battery capacty of smartphones. Moble edge computng recently comes up as a promsng soluton [], [], []. By deployng small-scale datacenters at wreless access ponts known as cloudlets the system allows smartphone users to offload compute-ntensve tasks to a nearby cloudlet, so as to extend ther battery lfe by tradng off heavy CPU cycles for lghtweght communcaton. The performance of offloadng crtcally depends on the allocaton of both rado and compute resources: the former determnes the data transmsson speed and the communcaton energy consumpton; the latter determnes the compute tme of tasks offloaded to a cloudlet. In general, the more resources are allocated, the better an offloadng request s served. However, both rado and compute resources are hghly constraned n a cloudlet. In partcular, cloudlets are deployed at wreless access ponts where only a lmted number of rado channels are avalable. Meanwhle, an economc, scalable deployment forces cloudlets to be no more than small-scale datacenters Ths work s supported by the Hong Kong Research Grants Councl under Grant No wth lmted compute capabltes. Therefore, to develop effectve computaton offloadng strateges, t s crtcal to take both rado and compute resources nto account. However, a large body of exstng works smply assumed an nfnte amount of compute resources avalable n a cloudlet, where the offloaded tasks were computed wth neglgble processng tme. The problem of offloadng schedulng was then reduced to rado resource allocaton. For example, Chen et al. [5] modeled the competton for rado resources as a congeston game of selfsh moble users. Kaewpuang et al. [6] studed the cooperaton game of offloadng servce provders, where the rado and compute resources were assumed to be managed by dfferent enttes separately. As we shall show n ths paper, coordnately managng both resources mproves the overall utlzaton sgnfcantly. Sardelltt et al. [7], on the other hand, smply gnored the congeston of compute resources n a cloudlet by throttlng the CPU cycles allocated to each offloaded task. Juan et al. [8] also assumed an nfntely powerful cloudlet such that the executon tme for each offloaded task was guaranteed to be a constant value. Recently, a few researchers started to jontly consder the lmtatons n rado and compute resources. Nonetheless, some of ther assumptons ether are neffcent n regards to energy reducton or wll weaken the applcablty of the result. In [9], CPU resources were allocated as percentages of the total CPU frequency, meanng that jobs are runnng n parallel. Such parallel executon mantans farness but prolongs the average executon tme. Rado resource was allocated n nonpreemptve tme slots n [0]. However, all the slots are of a fxed length, and an unnecessarly long slot-length results n waste of rado resource. Furthermore, the effcency of the proposed schedulng polcy s senstve to some parameters that need to be searched emprcally under dfferent system settngs. Motvated by the above lmtatons n exstng works, n ths paper, we propose algorthms that fully utlze the lmted rado and compute resources to reduce the energy consumpton of moble devces. Gven the small coverage of the cloudlet, we consder an OFDMA system so that nterference among users could be gnored, wth subcarrers as the rado resource. In terms of compute resource, we allocate CPU tme slots of the cloudlet non-preemptvely wth vared slot-length. We frst propose near-optmal algorthms that separately schedule subcarrers and CPU tme slots. We show that nave combnaton of per-resource allocatons greatly degrades the system

2 performance. The reason s that congeston of one resource wll cause sgnfcant waste of the other. To address ths problem, we propose a jont schedulng algorthm to coordnately manage subcarrers and CPU of the cloudlet. Smulaton results show that a noteworthy amount of energy s saved through jont schedulng compared to separate allocaton. Moreover, the coordnate management of dfferent resources s of greater advantage when more promnent performance gans could be acheved through computaton offloadng. Fg.. A moble edge computng system wth M moble devces and an nfrastructure-based cloudlet. II. SYSTEM MODEL We consder centralzed resource allocaton for moble edge computng wth OFDMA as the multple access scheme. A cloudlet wth certan computaton capablty s deployed at the wreless access pont to provde job-executon servces. Our objectve s to mnmze the total energy consumpton of moble users. In ths secton, we wll model both the remote resources n the cloudlet and the local resources of moble devces. We wll then analyze the requred energy and tme for offloadng, and formulate the energy-mnmzaton problem. A. Model of the Cloudlet and Moble Users As shown n Fg., we consder a snapshot when the CPU of the cloudlet s dle, and there are N avalable subcarrers to serve M moble users. The CPU frequency of the cloudlet s f c. Let C = {,,..., N} denote the avalable subcarrers to be allocated. The bandwdth of each subcarrer s B N. Further denote G as the channel-gan-to-nose rato matrx. We assume G remans constant durng the schedulng process. Let U = {,,..., M} denote the M users, each wth a job to execute ether locally or remotely. In the followng, we may call user and job nterchangeably. Each job J s descrbed by ts nput data sze D and deadlne T. For user U, the maxmal frequency of local CPU s F. Maxmal transmsson power and statc crcut power are denoted by p m and p c. B. Energy Consumpton ) Local executon: Accordng to [], at frequency f, the energy consumpton of each CPU cycle s κf, and the requred CPU cycles for completng a job s gven by XD, where D s the nput data sze, whle κ and X are known constants. In order to mnmze the local executon energy consumpton, the CPU frequency of U should be set to f = XD T such that ts deadlne s exactly met snce the energy consumpton of each CPU cycle ncreases wth ts frequency. Thus, the local energy cost for U s gven by ( ) El = κf XD XD = κ XD = κx D T T. () We assume that F s always larger than f, so that local executon s always feasble for all users. ) Remote executon: In the case of offloadng, t consumes energy to send the nput data D to the cloudlet. The energy consumpton for recevng the computaton results s gnored as the amount of output data s much less the nput data [9], [0], []. Therefore, the energy cost for offloadng s: E r = (p + p c ) T t, () where p and T t are the transmsson power and transmsson tme, respectvely. We now show that the optmal value of p could be derved through bsecton search. Denote W = {W(, j) W(, j) {0, }, U, j C} as the subcarrer allocaton matrx. For user U who has been allocated a group of subcarrers W(), transmt energy effcency (n bts per joule) s convex wth the transmt power []. Therefore, va bsecton search we can fnd the optmal p that mnmzes the transmsson energy for the nput data. In addton, as the transmt power has to be larger than a threshold p t to meet the job deadlne T, we have p = max(p, p t ). () C. Tme for Offloadng ) Transmsson: Let P = {P(, j) P(, j) [0, p m ], U, j C} be the power allocaton matrx. The optmal power allocaton matrx P s obtaned by the water-fllng algorthm []. We then have the aggregated data rate as R = B N N j= and transmsson tme as W,j log( + P(, j)g(, j)), () T t = D R. (5) ) Queung and remote executon: We assume nonpreemptve CPU allocaton, whch assgns a tme slot to one user each tme untl ts job completes. The remote executon tme n the cloudlet s then gven by T c = XD f c. (6) Denote q = {q q {,,..., M}, q q j,, j U} as the executon sequence n the cloudlet, and jobs are executed n the ascendng order of q. The queung tme n the cloudlet s then Q c = α j Tc j, (7) j,q j<q

3 where α j s the ndcator of whether job J j s offloaded. Thus, the total tme for remote executon s gven by D. Problem Formulaton T r = T t + Q c + T c. (8) We now formulate the total energy consumpton mnmzaton problem as follows: P : mnmze α,w,p,q = ( ( α ) El + α Et ), (9) subject to W(, j), j C (9a) = q q j, f j., j U (9b) N p = W(, j)p(, j) p m, U (9c) j= R = B N N j= W,j log( + P(, j)g(, j)), U (9d) El = κx D T, U (9e) Er = D (p + p c ), U (9f) R T r = D R + j,q j<q α j T j c + α T c T, U. (9g) Constrant (9a) ensures that each subcarrer s assgned exclusvely to one user. (9b) enforces non-preemptve executon n the cloudlet. (9c) and (9d) are the results of bsecton search and water-fllng wth a gven the subcarrer allocaton matrx W. (9c) places an upper bound for the total transmsson power. Fnally, (9e) and (9f) respectvely calculate the localexecuton and offloadng energy and (9g) enforces the correspondng hard deadlne on each of the offloaded task. Ths resource allocaton problem s a mxed-nteger nonlnear programmng (MINLP) problem, whch n general s NPhard. The optmal soluton to such a problem s dffcult to fnd, due to the combnatoral optmzaton varables (α, q and W). Also, handlng the non-convex functons n both the objectve and constrants brngs an addtonal challenge. In the followng, we wll propose effcent algorthms to solve ths problem wth near-optmal performance. III. CLOUDLET WITH UNLIMITED COMPUTATION CAPABILITY In ths secton, we address problem (9) under a common assumpton adopted n exstng lteratures,.e., a powerful cloudlet whose computaton capablty s far beyond the offloadng demands of users. For ths specal case, we develop an effcent algorthm to allocate the rado resources, whch could serve as a performance upper bound for the case that the cloudlet possesses lmted compute resources as wll be pursued n the next secton. From the prevous dscussons, the optmal transmt power and power allocaton are respectvely obtaned through bsecton search and the water-fllng algorthm. The problem now s to allocate the subcarrers properly. The subcarrer allocaton problem n moble edge computng system poses several new challenges. Frstly, t s dffcult to derve closed-form expressons for the outcome of bsecton search and water-fllng algorthm, whch makes t mpossble to explctly compare dfferent subcarrer allocaton results. Besdes, for the users who execute ther tasks locally, the allocated subcarrers wll be wasted. To avod such waste of rado resources, subcarrers should be allocated n groups that wll ensure benefcal offloadng. Intutvely, users wth heavy computaton workload and meanwhle n good channel condtons should have hgh prortes to offload, as relatvely few subcarrers are requred by such users to meet the deadlne requrement whle energy savngs wll be large. Thus, to fnd these users, we propose an algorthm that allocates subcarrers n the mnmum group of each user, whch s defned as the mnmum set of subcarrers requred to guarantee benefcal offloadng. In each teraton, we fnd the mnmum subcarrer group C for each user U and obtan the correspondng energy consumpton. The tasks of the users achevng the most energy savngs wth ther mnmum subcarrer groups should be offloaded to the cloudlet. Detals of the proposed subcarrer allocaton polcy are summarzed n Algorthm. Algorthm : Mnmum-Group Allocaton Algorthm Input : U = {,,.., M}, C = {,,..., N}, G, D, T, p m, p c, E l Output: W, P, α α {0,..., 0}. whle U > 0 and C > 0 do for U do Fnd the mnmum group C, such that Et < El, T t T, where [Et, Tt ] = subcarrer-search (G(, C ), p m, pc, T, D ) 5 end 6 m arg max{el E t}; W(m) C m, C C C m, U U {m} ; α m ; 7 end 8 for j C do 9 for U and α = do 0 [(Et), (Tt ) ] = Bsecton-Search (G(, C {j}), p m, pc, T, D ) end m arg max{et (Et) }; W(m) C m {j}; end return W, P, α; IV. CLOUDLET WITH LIMITED COMPUTATION CAPABILITY Though the assumpton of unlmted compute resource at the cloudlet smplfes the offloadng and subcarrer allocaton

4 polcy desgn, t s necessary to consder the lmted computaton capablty of the cloudlets as they are small-scale n practce. In ths case, the queung delay Q c and executon tme T c n the cloudlet are non-neglgble, and the congeston n the cloudlet may lead to the volaton of the deadlne requrements. Therefore, the compute resources should be properly scheduled to maxmze the offloadng gan. In ths secton, we wll frst develop a per-resource allocaton algorthm that combnes the subcarrer allocaton polcy developed n Secton III wth an optmal CPU tme schedulng strategy, whch serves as a baselne. We wll then propose a jont schedulng scheme to coordnately allocate subcarrers and CPU tme slots. A. Per-Resource Allocaton As a baselne, we frst consder a per-resource allocaton algorthm. In ths algorthm, subcarrers are assgned frst, and the CPU tme slots are scheduled n the second stage. The cloudlet allocates subcarrers followng Algorthm, wth the only dfference on checkng deadlne constrant, where the executon tme n the cloudlet Tc s also consdered. Nonpreemptve CPU schedulng of the cloudlet n the second stage essentally determnes the job executon order. The resulted queung tme of each job dctates whether the deadlne requrements are satsfed and fnally whether offloadng requests are accepted. In the followng, we wll develop the optmal CPU schedulng polcy. Snce non-preemptve CPU schedulng problem s NP-hard [], we propose an optmal algorthm based on dynamc programmng wth pseudo polynomal complexty. In the dynamc programmng algorthm, the non-preemptve CPU schedulng problem s decomposed nto M states. In state I, we solve the subproblem of maxmzng total energy savng wth I out of the M users, and store the maxmum amount of saved energy as well as the correspondng executon tme as ntermedate results. To avod duplcated teratons, each subproblem adopts prevously computed results as nput, as elaborated n Algorthm. Assume U s one of the subsets consdered n state I. Let Savng(U) and T me(u) be the results of ths subproblem. To solve t, we dvde all possble executon sequences of these I jobs n U nto I categores by the last executed job. Now consder the category where user s executed at last. For the frst I users, the largest energy reducton s Savng(U {}), and ther executon tme s T me(u {}). Note that Savng(U {}) and T me(u {}) are collected from the results of state I. It s unnecessary for all of the frst I users to offload successfully. For user, the ready tme before ts job could be executed n the cloudlet s the longer one of queung tme T me(u {}) and ts transmsson tme T t. When the deadlne T could be satsfed after the executon n the cloudlet, ts offloadng request s accepted. The correspondng results of ths category wll be updated as and tempsavng() = Savng(U {}) + S, tempt me() = max{t t, T me(u {})} + T c. Algorthm : CPU Schedulng (Dynamc Programmng) Input : Savng(U {}), T me(u {}), for U S, T c, T, for U Output: Savng(U), T me(u) for U do tempsavng() Savng(U {}); tempt me() T me((u {})); f max{t t, T me(u {})} + T c T then tempsavng() tempsavng() + S ; tempt me() max{t t, T me(u {})} + T c; end 5 end 6 m arg max{tempsavng()}; Savng(U) tempsavng(m), T me(u) tempt me(m) Lkewse, we calculate the results of all the I categores, and choose the one wth the largest energy savng as the fnal output Savng(U). The results of the fnal state, where M users are consdered, are the optmal soluton to the orgnal CPU schedulng problem. B. Jont Allocaton Separate allocaton leads to neffcent use of the rado and compute resource because users may stll have to execute ther tasks locally even f they are assgned wth subcarrers. The reason s that n the subcarrer allocaton stage, the queung tme for each user remans unknown. The congeston n the CPU of the cloudlet may cause executon deadlne volatons, leadng to the waste of subcarrers. For nstance, there mght be two users both wth strngent deadlnes and could not wat for the completon of the other s task. Thus, only the user wth a larger energy savng gets the chance of offloadng. In ths case, subcarrers orgnally allocated to the user that has to execute ts task locally could be re-allocated to other users. To address the uncertanty of successful offloadng and to optmze the utlty of lmted resources, jont allocaton s necessary,.e., we should fnd the least amount of both rado and compute resources that ensures successful offloadng, and allocate them to the users who save energy most effcently. The proposed algorthm s summarzed n Algorthm, where for each user, we fnd the mnmum subcarrer group that supports benefcal offloadng and also calculate the amount of CPU tme needed for remote processng. The users who save the most energy wth each CPU cycle are allocated wth the correspondng subcarrers and CPU tme slot. By dentfyng the mnmum subcarrer group and energy savng per CPU cycle, the utlzaton of both the rado and compute resources s optmzed. Another advantage of jont allocaton s that the queung tme s already known before schedulng. Such a pror knowledge of congeston wll help to avod the waste of resources.

5 Number of Offloaded Users Energy Consumpton (Joule) Algorthm : Jont Allocaton Algorthm Input : U = {,,.., M}, C = {,,..., N}, G, D, T, T c, E l Output: W, P, α, q α {0,..., 0}, t 0, x ; whle U > 0 and C > 0 do for U do Fnd the mnmum group C, such that < El, max{t t, t} + Tc T, where [Et, Tt ] = Bsecton-Search (G(, C ), p m, pc, T, D ) 5 end 6 S El E t; m arg max{ S T }; c W(m) C m, C C C m, U U {m} ; α m, q m x, x x+, t max{tt, t}+tc; 7 end 8 for j C do 9 for U and α = do 0 [(Et), (Tt ) ] = Bsecton-Search (G(, C {j}), p m, pc, T, D ) end m arg max{et (Et) }; W(m) C m {j}; end return W, P, α, q; V. PERFORMANCE EVALUATIONS In ths secton, we evaluate the performance of proposed algorthms. We focus on four questons: ) How much energy could offloadng save (compared wth local executon)? ) Is jont allocaton n advantage of per-resource allocaton? ) How well does the proposed algorthms perform (compared wth optmal allocaton)? ) How does the lmted computaton capablty of the cloudlet nfluence system performance? A. System Settng Users are randomly located n a crcle centered at the cloudlet. Large scale fadng of the channels s modeled as: P L = 0 log(d km ) + 0 log(f khz ) +.5 (db). The Raylegh fadng model s adopted for small scale fadng. The frequency band of the subcarrers s from 850 to 960 khz, wth the bandwdth of each subcarrer as 8.75 khz. The moble users crcut power and maxmum transmsson power are set to be 50 mw and W, respectvely. Input data sze s unformly dstrbuted n the range of bts, the job deadlnes are unformly dstrbuted n the range of ms, and κ s set as 0 [], []. The parameter X s set to be 8000 cycles per bt. B. Smulatons In the followng smulatons, optmal energy savngs are obtaned by exhaustvely searchng the subcarrer allocaton matrx. In the case when CPU capablty s lmted, we combne exhaustve search of subcarrer allocaton and optmal CPU schedulng to fnd the optmal results Local Executon Per-Resource Allocaton Jont Allocaton Optmal wth CPU Constrant Mnmum Group Allocaton Optmal wthout CPU Constrant Wthout Computatonal Constrant Number of Users (M) Fg.. Energy Consumpton w. r. t. Total Number of Users. N =, r = 0. km, f c = 600 MHz Wthout Computatonal Constrant Per-Resource Allocaton Jont Allocaton Optmal wth CPU Constrant Mnmum Group Allocaton Optmal wthout CPU Constrant Number of Users (M) Fg.. Number of Offloaded Users w. r. t. Total Number of Users. N =, r = 0. km, f c = 600 MHz. ) Number of users: Energy consumpton acheved by the proposed algorthms as the number of users ncreases are shown n Fg.. From the curves, both Algorthm (mnmum group allocaton) and Algorthm (jont allocaton) acheve near-optmal performance, for cases wthout and wth computatonal constrants, respectvely. Per-resource allocaton only acheves half of the energy savng compared to jont allocaton, although the subcarrer allocaton (mnmum group allocaton) s close to optmal and CPU tme allocaton (dynamc programmng) s optmal. The numbers of offloaded users are compared n Fg., where we fnd that the constrant of the CPU capabltes (red curves) leads to a reducton of the offloadng number by nearly 50%. ) Coverage of the cloudlet: We further nvestgate the mpacts of coverage radus of the cloudlet n Fg.. As the radus r ncreases, the offloadng gan shrnks. The reason s that the users are dstrbuted at a longer dstance from the cloudlet on average. Therefore, fewer users could be supported for offloadng. Ths further demonstrates that to provde satsfactory offloadng servces and acheve seamless

6 Energy Savng (Joule) Energy Consumpton (Joule) connecton, the cloudlets need to be close to the users and densely deployed. Consderng the hgh deployment cost of computatonally powerful datacenters, cloudlets wth lmted compute resource are preferred n practce Local-Executon Per-Resource Allocaton Jont Allocaton Optmal wth CPU Constrant Mmmum Group Allocaton Optmal wthout CPU Constrant Radus of the Cloudlet (km) Fg.. Energy Consumpton w. r. t. Radus of the Cloudlet. M =, N =, f c = 600 MHz Opt-I(M=7) GA(M=7) Opt-II(M=7) JA(M=7) Per(M=7) Optl-I(M=) GA(M=) Opt-II(M=) JA(M=) Per(M=) " " " CPU Frequency (Hz) #0 8 Fg. 5. Energy Savng w. r. t. CPU Frequency of the Cloudlet. M = {, 7}, N =. Legends: Opt-I optmal allocaton wthout CPU constrant, Opt-II optmal allocaton wth CPU constrant, MGA mnmum group allocaton, JA jont allocaton, Per: per-resource allocaton. ) CPU frequency of the cloudlet: Fg. 5 shows how the total energy savng vares as computaton capablty of the cloudlet,.e., the CPU frequency f c, ncreases. It can be seen that there s a saturaton pont at around 800 MHz for a - user system (green curves). However, when there are 7 users (red curves), the energy savng keeps ncreasng due to rcher user-dversty (comparng wth ). Another observaton s that jont allocaton s of greater advantage over per-resource allocaton as the CPU frequency f c ncreases (comparng wth ). When f c s larger than a threshold, the performance of jont allocaton algorthm approaches the optmal schedulng polcy wthout CPU constrant. These results demonstrate that coordnate management better utlzes the resources when a larger offloadng gan can be acheved, ether by a rcher userdversty or enhanced processng power. VI. CONCLUSIONS In ths paper, we proposed jont schedulng algorthms of both rado and compute resources for moble edge computng systems usng OFDMA, whch are effcent and near-optmal n terms of energy savng for moble devces. Through extensve smulatons, we showed that the per-resource allocaton greatly degrades the system performance, even though the allocaton polcy for each type of resource s near-optmal. Therefore, rather than smply combnng separate allocaton polces, the congeston nformaton of both types of resources should be consdered smultaneously. Furthermore, such jont schedulng s more crtcal when computaton offloadng can provde a more promnent energy savng. For future nvestgatons, we wll explore the cooperatons among cloudlets to further mprove the offloadng gan. REFERENCES [] K. Kumar and Y.-H. 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