Exploring Device-to-Device Communication for Mobile Cloud Computing

Transcription

1 Explorng Devce-to-Devce Communcaton for Moble Cloud Computng YujnL,LeSun,andWenyeWang Department of Electrcal and Computer Engneerng North Carolna State Unversty, Ralegh, NC, USA. Abstract Wth the popularty of smartphones and exploson of moble applcatons, moble devces become the prevalent computng platform for convenent communcaton and rch entertanment. Moble cloud computng(mcc) s proposed to overcome the lmted resources of moble systems. However, when users access MCC through wreless networks, cellular network s lkely to be overloaded and W-F connectvty s ntermttent. Therefore, devce-to-devce(d2d) communcaton s exploted as an alternatve for MCC. An mportant ssue n explorng D2D communcaton for MCC s how users can detect and utlze the computng resources on other moble devces. In ths paper, we propose two moble cloud access schemes: optmal and perodc access schemes, and study the correspondng performance of moble cloud computng(.e., moble cloud sze, node s servceable tme percentage, and task success rate). We fnd that optmally, node s servceable tme percentage and task success rate approach 1. Usng more practcal perodc access scheme, node s servceable tme percentage and task success rate are determned by the rato of contact and nter-contact tme between two nodes. I. INTRODUCTION Moble devces(such as smartphones and tablets) are becomng an nseparable part of our lves for convenent communcaton and entertanment. Wth the popularty of moble devces, there s also an exploson of moble applcatons n varous categores, such as terrestral navgaton, emal and web browsng, moble games, moble healthcare, moble commerce, and socal networkng. Ths ndcates that moble devces are quckly becomng the domnant computng platform, whch enables seamless work or entertanment for users regardless of user moblty. Nonetheless, moble systems are stll lmted n ther resources(e.g., processor power, storage sze, and battery lfe) and communcatons(e.g., bandwdth, connectvty, and securty)[1]. Such resource scarceness sgnfcantly hnders the development of moble applcatons and the mprovement of moble servce qualtes. Recently, ths problem has been addressed by researchers through moble cloud computng(mcc). MCC provdes servces for resource constraned moble devces to partton and offload ther computatonally ntensve and storage demandng jobs to the cloud wth vast computatonal resources[2]. In moble cloud computng, computng-ntensve moble applcatons, such as vdeo decodng, speech recognton, and augmented realty, can to be offloaded to the cloud for processng. Ths work s supported by the NSF Award CNS Computaton offloadng can save energy and mprove performance of moble applcatons thereby overcomng lmted resource capactes of moble devces. MCC can also enable moble users to store/access large data on the cloud through wreless networks, whch can save data storage capacty and processng power on moble devces. In MCC, moble users need access computng servces on the cloud through hgh-speed and ubqutous wreless connecton. The computatonal resources n the cloud are feasble only f the nformaton exchange between the cloud and moble devces through wreless networks s fast, relable, and secure. The most wdely used network access technologes are cellular and W-F networks. Cellular network provdes the nearubqutous coverage. But cellular network s under sgnfcant pressure and lkely to be overloaded due to the ncreasng moble data traffc[3], whch may ncur long latences and slow data transfers. Although W-F has hgh data rate, W-F connectons are ntermttent. Hence, the drawback of moble cloud computng s that the performance of cloud servces depends strongly on wreless communcaton networks. In order to overcome the drawbacks of accessng MCC through cellular and W-F networks, devce-to-devce(d2d) communcaton s exploted for moble cloud computng whle avodng global network bottlenecks[4]. The ncreasng densty of moble devces produces an abundance of contact opportuntes[5]. When offloadng to remote clouds fals n low connectvty scenaros, moble devces can employ local resources on moble devces n the vcnty for computng a shared task. By explotng D2D communcaton, users could mprove the performance of moble cloud computng n terms of computng speedup and money savng on smartphone dataroamng charges[4, 6, 7]. An mportant ssue n explorng D2D communcaton for MCCshowuserscanaccessthecomputngresourceson other moble devces. Because of user moblty, D2D connecton s ntermttent. Under such ntermttent connectvty, access scheme needs to be carefully developed such that users can utlze computng resources of nearby moble devces as muchaspossblewhlenotwastngtoomuchenergyondevce dscovery. In ths paper, we propose two access schemes,.e., optmal and perodc access algorthms, n whch the ntator optmally or perodcally performs node dscovery, subtask dstrbuton and retreval wth or wthout knowledge of other nodes moblty, respectvely. We study the followng MCC performance metrcs under

2 both access schemes: moble cloud sze, servceable tme percentage, and task success rate. More specfcally, moble cloud sze s the number of nodes that an ntator dscovers and utlzes for computng; servceable tme percentage s the percentage of tme that a devce computes tasks for an ntator; tasksuccessratestheprobabltythatatasktransmttedtoa devce s executed and successfully retreved by the ntator. Optmal access scheme provdes the optmal performance of MCC based on D2D communcaton. Usng perodc access scheme, performance of moble cloud s greatly affected by contact and nter-contact tme between two nodes. The more frequent a node meets the ntator and the longer ther contacts are, the hgher the node s servceable tme percentage and the task success rate are. The remander of ths paper s organzed as follows. We present a succnct summary on the exstng work n Secton II. We gve the network model, contact process, and access schemes n Secton III. The performance of MCC s analyzed nsectoniv.weconcludensectonv. II. RELATED WORK Wth the support of cloud computng of varous servces for moble users, moble cloud computng(mcc) s ntroduced to facltate moble users to take full advantages of cloud computng. Moble users can access cloud servces through wreless networks, ncludng cellular and W-F networks. However, the ncreasng moble data traffc has put a sgnfcant stran on cellular network[3]. Overloaded cellular network may ncur long latences and slow data transfers, whch make the data uploadng and downloadng expensve. At the same tme, W- F connecton coverage s ntermttent. In order to access cloud computng seamlessly, D2D communcaton s proposed to assst exstng wreless communcaton systems. Marnell [4] ponts out that n many cases, processng moble data(such as sensor logs and multmeda data) n-place and transferrng t drectly between smartphones would be more effcent and less susceptble to network lmtatons than offloadng data and processng to remote servers. Therefore, Marnell develops Hyrax, a platform derved from Hadoop that supports cloud computng on Androd smartphones. A central server wth access to each moble devce coordnates data and jobs and smartphones communcate wth each other on an solated 82.11g network. Although the performance of HyraxspoorforCPU-boundtasks,tsshowntotolerate node-departure and offer reasonable performance n data sharng. A dstrbuted multmeda search and sharng applcaton s mplemented to qualtatvely evaluate Hyrax. Smlarly, Huerta-Canepa and Lee[8] observe that moble devces can be a vrtual cloud computng provder because ther pervasveness means the ncreasng avalablty of nearby devces; they are more powerful over the tme; they nclude dfferent network nterfaces allowng devces to communcate wtheachother(wthnomoneycost);moreovertheyallowus to create communtes n whch we can execute shared tasks. Huerta-Canepa and Lee propose a vrtual cloud computng platform, n whch a context manager montors the locaton andnumberofnearbydevces.akoreanocrthatreadsan mage, scans for the Korean characters, and then presents a Romanze verson of them was developed for testng purposes. Paper[6] proposes a framework that uses local resources on moble devces for computng when offloadng to remote clouds fals n low connectvty scenaros. Experments are conducted n Bluetooth transmsson and an ntal prototype s also presented. The authors also dscuss a prelmnary analytcal model to determne whether or not a speedup wll be possble n offloadng. Sh et al.[7] nvestgates the scenaro that a moble devce uses the avalable, potentally ntermttently connected, computaton resources of other moble devces to mprove ts computatonal experence, e.g., mnmzng local power consumpton and/or decreasng computaton completon tme. The authors propose and mplement Serendpty on moble devces to leverage the frequent contacts between moble devces n order to speedup computng and conserve energy. A speech-to-text applcaton s mplemented to evaluate Serendpty, showng that Serendpty reduces job completon tme comparng wth executng locally. The authors also mplement a prelmnary prototype of Serendpty on the Androd platforms wth two computatonally complex applcatons(.e., a face detecton applcaton and a speechto-text applcaton). One fundamental ssue n usng moble devces for cloud computng s that an ntator needs to frst detect moble devces n proxmty, then perform task dstrbuton and retreval. Therefore, we propose two access schemes n ths paper and study moble cloud computng performance under them. III. MODELS AND DEFINITIONS Assumethat nmobledevcesaremovngnanetwork Ω n = [, n λ ], where λ s the spatal densty of moble users. Each moble devce has a transmsson radus r. Denote by X t = {X 1 (t),...,x n (t)}thepostonsofusersattme t. Nodes are movng accordng to Moblty Process M. We assumethatthemobltyprocessofanodesstatonaryand ergodcthatanode slocaton X ( )hasunformstatonary dstrbuton n the network area [9]. Moblty processes of nodes are ndependent and dentcally dstrbuted(..d.). Wthout loss of generalty, we assume that a moble user ntates to offload computatonal tasks to nearby moble devces attme.asshownnfg.1,thentatorcanconnecttonodes n ts transmsson range through drect D2D communcaton lnks, formng a moble cloud for computng. We assume that all nodes are wllng to support moble computng because of fast computaton of a common task or ncentves offered by the ntator. A. Contact Process Dynamcs: Apparently, node moblty affects connectvty of D2D communcaton. How frequently nodesmeetandhowlongtheystayconnectedaffectthesze and stablty of a moble cloud, n turn, nfluence the computngcapactyofamoblecloud.acontacteventbetweenapar ofusersoccurswhentwousersarecloseenoughtocommuncateandexchangecontentwtheachother.let X u (t)and X v (t)denotethelocatonsofusers uand vattme t,wecall

3 executedsubtasksfrom vattme t e.user vcanperformtask computatondurngthewholetme [t h,t e ]. Inrealty,tsdffcultforamobledevcetoobtanthe moblty nformaton of other devces due to prvacy ssue. A more practcal access scheme s the followng perodc access scheme, n whch an ntator moble devce perodcally performs devce detecton, task dstrbuton and retreval. Fg. 1. Devces n the proxmty form moble cloud. thatonecontactevent T C betweenusers uand voccursdurng [t,t 1 )f X u (t ) X v(t ) > rand X u(t) X v (t) r forall t [t,t 1 ),and X u (t 1 ) X v (t 1 ) > r.thenumber ofcontacteventsbetweenaparofuserswthntme tsa countng process called the contact process. We refer to the tme between the end and the start of two consecutve contact events between the same par of users as the nter-contact tme T I. As task dssemnaton and retreval can only be performed when there s a communcaton lnk between two nodes, contact and nter-contact tme between nodes affect moble cloud computng performance. Obtanng complete knowledge of contact processe can be extremely dffcult. Also mathematcally characterzng moble cloud computng performance s ntractable for arbtrary contact process. Thus, we assume that thecontactprocessofaparofuserssapossonprocess, whchhasbeenshowntobeagoodapproxmatonandused by other exstng studes [1, 11]. In other words, contact and nter-contact tme follow exponental dstrbutons wth parameters λ C and λ I,respectvely. B. Moble Cloud Access Schemes: In order to enable moble cloud, an ntator moble devce frst needs to dscover devces n proxmty. Then, the ntator can dspatch/retreve tasks to/from moble devces for MCC. Clearly, how an ntator detects other devces and employs them for computng determne moble cloud performance. Hence, we gve two moble cloud access schemes n the followng. Ideally, f an ntator has perfect knowledge of ts future contact wth other devces, the ntator can maxmally explot other devces for computng, resultng n the optmal performance of moble cloud. Therefore, we propose an optmal access scheme to evaluate the optmal performance of MCC based on D2D communcatons. Defnton 1(Optmal Access Scheme). Intally, a moble devce uhasataskthatneedstobecomputedwthntme τ.denoteby t h ( t h τ)thefrsthttngtmewhen ntator ufrstmeetsuser vwthntmeperod [,τ],and t e ( t e τ)thelastexttmethat uand vareoutofeach other stransmssonrangedurng [t e,τ].user uparttonsthe task,sendssomesubtasksto vattme t h,andretrevesthe Defnton 2(Perodc Access Scheme). Intator u perodcallyscanstsneghborngdevces.supposeat t,thesetofts neghborss N t = {v 1,...,v k }.Node uwllsendasubtask toeachoftsneghbors.attme t+, udetectstsneghbors N t+ = {v 1,...,v k }.Ifnode v k N t, uwllretrevethe subtasksentto v k attme tandsendanewsubtaskto v k for process;otherwse, uwllsendasubtaskto v k. Note that the node dscovery nterval depends on subtask computaton tme and devce s battery level. If battery s suffcent, hgh node dscovery frequency would ensure better utlzaton of moble cloud. When, an ntator can use a moble devce for computng durng all ther contact tme.ifbatterylevelslow,large canreduceenergyconsumpton wth performance compromse. A reasonable settng s lettng equal to the computatonal tme of a subtask, whch endures lttle performance compromse wthout too much energy consumpton. In ths paper, we assume equal tothecomputatonaltmeofasubtaskforouranalyss.in addton, f an ntator has multple neghbors n proxmty, t can use multple access scheme(e.g.,[12]) to dspatch and retrevesubtaskstoallofthematthesametme. C. Moble Cloud Performance Crtera: We seek to evaluate the D2D computng performance n terms of the sze of moble cloud, and the percentage of a moble devce s servceable tme, and the success rate of a task computaton. Moble cloud sze s the number of moble devces that an ntator detects wthn task delay tolerance τ. Servceable tme percentage s defned as the percentage oftmethatamobledevcesemployedbyanntator to provde computng servces. Task success rate represents the probablty that an ntatorcansuccessfullysendatasktoamobledevcefor computng and retreve the task back before t expres. IV. MOBILE CLOUD PERFORMANCE ANALYSIS We analyze the performance of moble cloud (.e., moble cloud sze, servceable tme percentage, task success rate) under optmal and perodc access schemes, respectvely. Analyss results under optmal and perodc access schemes wll provde the optmal and achevable cloud computng performance based on D2D communcaton, respectvely. A. Optmal Access Scheme Under optmal access scheme, an ntator can employ every node that t meets for cloud computng. Thus, moble cloud szesthenumberofmobledevcesthatanntatormeets wthntaskdelaytolerance τ.denoteby N o τ themoblecloud sze,andwehavethefollowngtheoremon N o τ.

4 ( Theorem1. [ Themoblecloudsze ( ) ]) Nτ o followsbnomal n 1, 1 1 πr2 e λiτ. Proof: Assume at t =, there are N nodes n an ntator s transmsson range, moble cloud sze ncrement Nτ o N sthesuperpostonof n N 1numberof-1 processes 1 { TI f τ},where T I stheresdualnter-contacttme. n N 1 n N Nτ o = N + 1 { TI f τ} = n { f TI>τ}. Itsworthnotngthat N sarandomvarabledependngon ntalnodedstrbutonnthenetwork.rgorously, P(Nτ o = k) = E(P(Nτ o = k N ))and E(Nτ o ) = E(E(Nτ o N )). Therefore, Nτ o s determned by the ntal node dstrbuton and the resdual nter-contact tme between two nodes. In homogeneous ( ) networks, N satsfes P(N = m) = m ( n 1 m.then, ( m n 1 πr ) 2 1 ) πr2 k ( ) πr P(Nτ o = k) = ( m n 1 2 m ) n 1 m ) (1 πr2 m=1 ) k m ( ( n m 1 k m (F ) fti (τ) 1 F fti (t) ( )) πr = ( n 1 2 k k ) + F TI f(τ) (1 πr2 ) n 1 k ( (1 πr2 n k 1 1 F fti (τ)), ) n k 1 whchsabnomaldstrbutonwthparameters ( ) n 1and πr 2 + F TI f(τ) 1 πr2,where F fti (τ) = P( T I τ).thus, ( ) ) πr E(Nτ o 2 ) = (n 1) (1 + πr2 F fti (τ). (1) Thedenstyfunctonof T I s λ I [1 F TI (x)],where F TI (x) sthecdfof T I and λ 1 I = x F TI (dx) = (1 F TI (x))dx.whenthenter-contacttme T I followsexponentaldstrbutonwthparameter λ I, T I sdentcallydstrbuted wth T I.Hence,wecompleteourproof. Under the optmal access scheme, an ntator can schedule tosendsubtaskstoothermobledevcesnadvancesothat theycanperformcomputngevenwhentheyarenotncontact wth the ntator. Thus, the servceable tme for a moble devcesfromthefrsthttngtme t h tothelastexttme t e. Inotherwords,theservceabletmepercentages (t e t h )/τ, for t h t e τ. Theorem 2. The servceable tme percentage of a node s shownneq.(2),whchsapproachng 1when τslarge. Proof: In a contact process between two nodes, denote by ξ(t) = 1whentwonodesarencontactattme t, ξ(t) = otherwse. ()When ξ() = ξ(τ) = 1, t h = and t e = τ.clearly,ts servceabletme ST o τ = τ. ()When ξ() = and ξ(τ) = 1, t h = T I and t e = τ. ST o τ = τ T I 1 { f TI<τ}. ()When ξ() = 1and ξ(τ) =, t h = and t e = τ T I, where T I s the backward recurrence tme of T I. ST o τ = τ T I 1 { c TI<τ}. (v) When ξ() = and ξ(τ) =, f the ntatornever encounters the node wthn tme τ, the node s servceable tme s. Otherwse, t h = T I and t e = τ T I. Thus, ST o τ = [τ ( T I + T I ) 1 { f TI+ c T I<τ} ] 1 { f T I<τ}. Denote by π j (t) the equlbrum probablty, gven that ξ() =,that ξ(t) = j(,j =,1).Let p and p 1 denote P(ξ() = )and P(ξ() = 1),respectvely.Because T I and T C areexponentalrandomvarableswthparameters λ I and λ C,respectvely, T I and T I havethesamedstrbutonas T I, and T I + T I followserlang-2dstrbutonerlang(2,λ I ).Thus, E(ST o τ )/τ = τλ I e λiτ π (τ)p (2) + [1 + (π 1 (τ)p + π 1 (τ)p 1 + π (τ)p )e λiτ ] 1 λ I τ (1 e λiτ )(π 1 (τ)p + π 1 (τ)p 1 + 2π (τ)p ), where p 1 = πr2 and p = 1 p 1, and the equlbrum probablty π j (τ)canbedervedbasedoncox srenewal Theory(Chapter7.4)[13]: π (τ) = β+γe βτ/λc, π 1 (τ) = γ γe βτ/λc, π 1 (τ) = β βe βτ/λc, and π 11 (τ) = γ + βe βτ/λc,where β = λc and γ = λi.when τ slarge, E(STτ o )/τ 1,.e.,themeanservceabletme percentage approaches 1 n the long run. Smlarly, when an ntator has full knowledge of ts contact and nter-contact events wth other devces, t can make sure dssemnatngtaskstoamobledevceonlyftheycanbe successfully retreved. Thus, the task success rate of optmal access scheme s 1. In practce, t s dffcult for an ntator to obtan full nformaton of ts contact wth other users. To make the optmal access scheme practcal, we can use the followng heurstc access scheme by explotng the regularty of human moblty. Human moblty shows a very hgh degree of temporal and spatal regularty[14] and can be predcted wth hgh probablty[15]. Beneftng from the predctablty of human moblty, an ntator u can estmate ts contact tme wth other users based on ther contact hstory. Heurstc Access Scheme: Intator u records the hstory of tscontactwthotherusersoveraperodoftmeandusesths nformatonfortaskcomputng.foragveperod [t,t+τ]of a day d, ntator u estmates ts contact and nter-contact tme (e.g.,frsthttngtme t h andlastexttme t e )wthauser v basedonmobltyhstoryofday d c,where csasmall nteger(usually1or2).then, uapplestheoptmalaccess scheme to utlze the computng resource of user v. Remark 1. When an ntator has pre-knowledge of node moblty or can predct other nodes moblty based on moblty hstory, an ntator can utlze computng resources on other devces through optmal or heurstc access schemes. Under the optmal access scheme, the expected number of nodes that an ntator can use wthn tme τ s (n

5 [ ( ] 1) 1 1 πr2 )e λiτ. The long-term servceable tme percentage and task success rate are approachng 1. B. Perodc Access Scheme Suppose an ntator perodcally detects devces n proxmty wth frequency 1/. Wthn tme [,τ], the ntator performs devce dscovery at tme {,,2,..., τ }. In otherwords,adevcesdetectedbythentatoronlyft s n contact wth the ntator for at least one tme pont of {,,2,..., τ }.Applyngrenewalprocesstheory,we derve the followng theorem on the sze of moble cloud. Theorem3. Themoblecloudsze N p τ followsbnomal ( n 1,1 P),where PcanbefoundnEq.(5). Proof:Ifadevcesnotdetectedbyanntatorwthn tme τ, t s n nter-contact wth the ntator at all tme ponts {,,2,..., τ }.Letusconsderthecontactprocess betweenanntatorandamobledevceandlet F TC (f TC ) and F TI (f TI )denotethedstrbuton(densty)functonsof the contact and nter-contact tme. Defne renewal functons H 2 (t) = F (n) T I F (n) T C (t)andrenewaldensty h 2 (t) = dh 2 (t)/dt,where F (n) (F (n) T C )sthe n-foldconvolutonof T I F TI (F TC )tself. Cox[13] derves the probablty that the system s at ntercontactstateattme t.condtonngonthentalstate ξ() = (.e., nter-contact state) IT (t) = 1 F TI (t) + (1 F TI ) H 2 (t). (3) Further, Baxter [16] derves the jont probablty that the system s at nter-contact state at m dstnct tme ponts {t 1,t 2,...,t m }.Let m = τ, t 1 =,...,t =,...,t m = τ, m 1 IT (m) (t 1,t 2,...,t m ) = R (t m t 1,t 1 ) + IT (t 1 ) t+1 t 1 t t 1 (4) φ (x,t 1 )IT (m ) (t +1 t 1 u,...,t m t 1 u)dx, and condtonng on ξ() = 1(.e., contact state) R (x,t) = 1 F TI (t+x)+ t φ (x,t) = 1 [ f TI (t + x) + IT (t) h 2 (u)(1 F TI (t+x u))du, t ] h 2 (u)f TI (t + x u)du. IT (m) (t 1,t 2,...,t m )canbecomputedrecursvelyfromeq. (3).Thus,theprobabltythatanodesnotdetectedbythe ntator s P = p IT (m) (t 1,t 2,...,t m ), (5) where p = 1 πr2.hence,thetotalnumberofnodesdetected by the ntator follows Bnomal dstrbuton wth parameters (n 1,1 P). ( ) Remark2. Clearly, P > 1 πr2 e λiτ.thus,themoble cloud sze under perodc access scheme s stochastcally domnated by that under optmal access scheme. Whenever an ntator senses a devce n proxmty, t transmts a subtask to the devce for computng, whch takes tme to fnsh. A devce computes all subtasks that t receves no matter whether the subtasks are retreved. Thus, thetotalservngtmeofadevces tmesthenumberof task transmssons. We have the followng theorem on the servceable tme percentage by studyng the number of task transmssons between an ntator and a devce. Theorem 4. A moble devce s servceable tme percentage s approachng when τslargeand ssmall. λ I Proof: Let us consder a contact event between an ntator andadevcedurng [t,t 1 ].Dvdng [t,t 1 ]ntotmeslots [t,t +],[t +,t +2],...,wehave (t 1 t )/ number ofsubntervalswtheachslotequalsto exceptthelastone. Durng each tme slot, the ntator must perform one and only one node detecton. In the best scenaro, the ntator performs nodedetectonandtransmtsthetasktoaneghboratthe begnnngofeachtmeslot,resultngn (t 1 t )/ number of task transmssons. In the worst scenaro, the ntator performs node detecton and transmts the task to a neghbor attheendofeachtmeslot,resultngn (t 1 t )/ number of task transmssons. Denoteby T C thecontacttmerandomvarableand the number of contacts between an ntator and a devce wthntmeduraton τ.therefore,thetotalnumberoftasksa devcecomputedsupperboundedby T C andlower boundedby T C.Hence,theservcetmesatsfes E T C E(STτ p ) E T C. Basedonrenewaltheory, condtonngonthental state ξ() = (.e., nter-contact state) E(N (τ)) = F TI (τ) + E(N 1 (τ)) = F TC (τ) + τ τ E(N (τ))df TI+T C (s); E(N 1 (τ))df TI+T C (s). Takng the Laplace transform of these two equatons, L E(N (τ))(s) = L TI (s)/s + L N (τ)(s)l TI+T C (s), L E(N 1 (τ))(s) = L TC (s)/s + L N 1 (τ)(s)l TI+T C (s). Because T C and T I haveexponentaldstrbutonswthparameters λ C and λ I,respectvely, L E(N (τ))(s) = λ I(s + λ C ) s 2 (s + λ I + λ C ), L E(N 1 (τ))(s) = λ C(s + λ I ) s 2 (s + λ I + λ C ). Takng the nverse Laplace transform, we then have E(N (τ)) = λ Iλ C τ λ 2 I + λ I + λ C (λ I + λ C ) 2 (1 e (λi+λc)τ ),

6 E(N 1 (τ)) = λ Iλ C τ λ 2 C + λ I + λ C (λ I + λ C ) 2 (1 e (λi+λc)τ ). Therefore, the expected number of contact between two nodes wthn tme duraton τ satsfes E() = E( I = )p + E( I = 1)p 1, (6) = λ Iλ C τ λ I + λ C + p λ 2 I + p 1λ 2 C (λ I + λ C ) 2 (1 e (λi+λc)τ ), where p 1 = πr2 and p = 1 p 1. Applyng ths result to the servce tme, we have E()[E(T C ) ] E(ST p τ ) E()[E(T C ) + ]. When τbecomeslargeand ssmall, E(STτ p ) λ I lm =. (7) τ, τ λ I + λ C Remark 3. Usng the perodc access scheme, an ntator can λ utlze a moble devce for approxmately I percentage oftme,whchsdetermnedby λc λ I.Thelongeradevceand an ntator are n contact and the more frequently they meet, the hgher utlzaton of the devce s computng resources s. After an ntator dspatches a subtask to a moble devce, t can retreve ths subtask f the ntator detects the moble devce agan before the task expres. Suppose the ntator detectsamobledevce M τ tmesdurngtmeperod τ,the ntator only fals to retreve the last dspatched subtask. Hence, the task success rate s 1 1/M τ. Followng ths methodology, we have the followng theorem on the task success rate. Theorem 5. Task success rate satsfes P p s 1 1 E()( 1 (8) λ C + 1). Proof:Todervethetasksuccessrate,weneedtofndout the total number of transmtted subtasks from an ntator to a moble devce. From our proof of servceable tme percentage, we have E T C E(M τ ) E T C. Asthenversefunctonsconvexonthenterval (,+ ),the task falure rate satsfes ( ) 1 E 1/E T C 1 M τ E()( 1 λ + 1), C where E()canbefoundnEq.(6).Hence,wehavethe tasksuccessrateneq.(8). Remark 4. Usng perodc access scheme, the task success rate supperboundedbyafunctonof τ, λ I, λ C,and.Themore frequently an ntator meets a devce and the longer they stay n contact, the closer ths upper bound approaches 1. V. CONCLUSION In ths paper, we have studed the performance of moble cloud computng based on D2D communcaton, n whch an ntator dstrbutes tasks to other nearby devces and retreves tasks after executon. We propose optmal and perodc access schemes that an ntator detects devces n proxmty and dstrbutes tasks to them for computng optmally or perodcally. Optmal access scheme results n optmal moble cloud performance wth servceable tme percentage and task success rate approachng 1. Usng more practcal perodc access scheme, an ntator moble devce can employ a moble λ I devce for computng n percentageoftme,andthe tasksuccessratesupperboundedbyafunctonof λ I and λ C. In summary, moblty patterns of users have sgncant mpact on performance of moble cloud, whch can be measured by λ C λi.inourfuturework,wewllusemobleapplcatonsto evaluate the moble cloud computng performance. REFERENCES [1] A. K. Gupta, Challenges of moble computng, n 2nd Natonal Conference on Challenges& Opportuntes n Informaton Technology(COIT-28),(Mand Gobndgarh), 28. [2] N. Fernando, S. W. Loke, and W. Rahayu, Moble cloud computng: A survey, Future Generaton Computer Systems, vol.29,no.1,pp.84 16,213. [3] A.Ajaz,H.Aghvam,andM.Aman, Asurveyonmobledata offloadng: techncal and busness perspectves, IEEE Wreless Communcatons, vol. 2, no. 2, pp , 213. [4] E. E. Marnell, Hyrax: Cloud computng on moble devces usng mapreduce, Master s thess, Carnege Mellon Unversty, 29. [5] S. Lu and A. D. Stregel, Explorng the potental n practce for opportunstc networks amongst smart moble devces, n Proc. of the ACM MobCom, 213. [6] N. Fernando, S. Loke, and W. Rahayu, Dynamc moble cloud computng: Ad hoc and opportunstc job sharng, n Proc. of IEEE UCC, 211. [7] C. Sh, V. Lakafoss, M. H. Ammar, and E. W. Zegura, Serendpty: enablng remote computng among ntermttently connected moble devces, n Proc. of ACM MobHoc, 212. [8] G. Huerta-Canepa and D. Lee, A vrtual cloud computng provder for moble devces, n ACM Workshop on Moble Cloud Computng& Servces(MCS 1), 21. [9] L.SunandW.Wang, Onlatencydstrbutonandscalng:from fnte to large cogntve rado networks under general moblty, n Proc. of IEEE INFOCOM, 212. [1] Y.LandW.Wang, Theunheraldedpowerofcloudletcomputng n the vcnty of moble devces, n Proc. of IEEE GLOBECOM, 213. [11] Y. L and W. Wang, Can moble cloudlets support moble applcatons?, n Proc. of IEEE INFOCOM, 214. [12] R. Mao and H. L, A novel multple access scheme va compressed sensng wth random data traffc, Journal of Communcatons and Networks, vol. 12, no. 4, pp , 21. [13] D.Cox,RenewalTheory.Methuen&Co,1962. [14] Y. L, M. Zhao, and W. Wang, Internode moblty correlaton for group detecton and analyss n vanets, Vehcular Technology, IEEE Transactons on, vol. 62, no. 9, pp , 213. [15] M. C. Gonzalezl, C. A. Hdalgo1, and A.-L. Barabás, Understandng ndvdual human moblty patterns, Letters to Nature, 28. [16] L. A. Baxter, Avalablty measures for a two-state system, Journal of Appled Probablty, vol. 18, no. 1, 1981.