ALONG with the maturity of mobile cloud computing,

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1 An Optiml Offloding Prtitioning Algorithm in Moile Cloud Computing Huming Wu, Dniel Seidenstüker, Yi Sun, Crlos Mrtín Nieto, Willim Knottenelt, nd Ktink Wolter system, nd their min gol is to keep the whole ost s smll s possile. The min osts for moile offloding systems re the omputtionl ost for lol nd remote exeution, respetively, nd the ommunition ost due to the extr ommunition etween the moile devie nd the remote loud. Clultions n nturlly e desried s grph in whih verties represent omputtionl osts nd edges reflet ommunition osts []. By prtitioning the verties of grph, the lultion n e divided mong proessors of lol moile devies nd remote loud servers. Trditionl grph prtitioning lgorithms (e.g., [], [], [6] nd [7]) nnot e pplied diretly to the moile offloding systems, euse they only onsider the weights on the edges of the grph, negleting the weight of eh node. Our reserh is situted in the ontext of resoureonstrined moile devies, in whih there re often multiojetive prtitioning ost funtions, suh s minimizing the totl response time or energy onsumption on moile devies y offloding prtil worklods to loud server. In this pper, we explore the methods of how to deploy suh n offlodle pplition in more optiml wy, y dynmilly nd utomtilly determining whih prts of the pplition should e omputed on the loud server nd whih prts should e left on the moile devie to hieve prtiulr performne trget (low lteny, minimiztion of energy onsumption, low response time, et.) []. We study how to disintegrte nd distriute modules of pplition etween moile devies nd loud server, nd effetively utilize the loud resoures. The prolem of whether or not to offlod ertin prts of n pplition to the loud depends on the following ftors: CPU speed of moile devie, network ndwidth, trnsmission dt size, nd the speed of the loud server [9]. With onsidering these ftors, we onstrut weighted onsumption grph (WCG) ording to the estimted omputtionl nd ommunition ost, nd further derive new min-ost offloding prtitioning (MCOP) lrxiv:0.0796v [s.dc] 7 Ot 0 Astrt Moile offloding is n effetive wy tht migrtes omputtion-intensive prts of pplitions from resoure-onstrined moile devies onto remote resoure-rih servers. Applition prtitioning plys ritil role in high-performne offloding systems, whih involves splitting the exeution of pplitions etween the moile side nd loud side so tht the totl exeution ost is minimized. Through prtitioning, the moile devie n hve the most enefit from offloding the pplition to remote loud. In this pper, we study how to effetively nd dynmilly prtition given pplition into lol nd remote prts while keeping the totl ost s smll s possile. For generl tsks (i.e., ritrry topologil onsumption grphs), we propose new min-ost offloding prtitioning (MCOP) lgorithm tht ims t finding the optiml pplition prtitioning (determining whih portions of the pplition to run on moile devies nd whih portions on loud servers) under different prtitioning ost models nd moile environments. The simultion results show tht the proposed lgorithm provides stly low time omplexity method nd n signifintly redue exeution time nd energy onsumption y optimlly distriuting tsks etween moile devies nd loud servers, nd in the mentime, it n well dpt to environment hnges. Index Terms Moile devie, moile loud omputing, ommunition networks, offloding, ost grph, prtitioning lgorithm. INTRODUCTION ALONG with the mturity of moile loud omputing, moile loud offloding is eoming promising method to redue exeution time nd prolong ttery life of moile devies. Its min ide is to ugment exeution through migrting hevy omputtion from moile devies to resoureful loud servers nd then reeive the results from them vi wireless networks. Offloding is n effetive wy to overome the resoures nd funtionlities onstrints of the moile devies sine it n relese them from intensive proessing nd inrese performne of the moile pplitions. Offloding ll omputtion omponents of n pplition to the remote loud is not lwys neessry or effetive. Espeilly, for some omplex pplitions tht n e divided into set of dependle prts, moile devie should judiiously determine whether to offlod omputtion nd whih portion of the pplition should e offloded to the loud. We need to mke offloding deisions for ll the prts, nd the deision mde for one prt depends on the other prts. As moile omputing inresingly interts with the loud, numer of pprohes hve een proposed, e.g., MAUI [] nd CloneCloud [], iming t offloding some prts of the moile pplition exeution to the loud. To hieve good performne, they prtiulrly fous on n pplition prtitioning prolem, i.e., to deide whih prts of n pplition should e offloded to powerful servers in remote loud nd whih prts should e exeuted lolly on moile devies suh tht the totl exeution ost is minimized. Therefore, prtitioning lgorithms ply ritil role in high-performne offloding H. Wu, D. Seidenstüker, Y. Sun, C. M. Nieto nd K. Wolter re with the Institut für Informtik, Freie Universität Berlin, Germny, 9. Emil: {huming.wu, seided, yi.sun, rlosmn, ktink.wolter}@fu-erlin.de. W. Knottenelt is with the Deprtment of Computing, Imperil College London, UK. Emil: wjk@do.i..uk.

2 gorithm designed espeilly for the moile offloding systems. This MCOP lgorithm ims t finding the optiml ut tht minimizes given ojetive funtion (response time, energy onsumption or the weighted sum of time nd energy) nd n e pplied to WCGs of ritrry topology. The reminder of this pper is orgnized s follows. We review relted work in Setion. Setion explores the prtitioning hllenges nd proess. Setion rings in the prtitioning models suh s topology, optimiztion nd prtitioning ost models. An optiml prtitioning lgorithm for ritrry topology is proposed nd investigted in Setion. Setion 6 desries three different profilers tht re used for informtion olleting. Setion 7 gives some evlution nd simultion results. Finlly, the pper is summrized in Setion. RELATED WORK Offloding eomes n ttrtive solution for meeting response time requirements on moile systems s pplitions eome inresingly omplex [0]. Extending ttery lifetime is lso one of the most ruil design ojetives of moile devies euse they re usully equipped with limited ttery pity. Mny reserh efforts hve een devoted to offloding omputtion to remote servers in order to shorten exeution time or sve energy onsumption. Krthik et l. et l. rgued tht offloding ould potentilly sve energy nd redue exeution time for moile users, ut not ll pplitions re energy-effiient nd time-sving when they re migrted to the loud. It depends on whether the omputtionl ost sved due to offloding outperforms the extr ommunition ost. A lrge mount of ommunition omined with smll mount of omputtion should preferly e performed lolly on the moile devie, while smll mount of ommunition with lrge mount of omputtion should preferly e exeuted remotely. The prtitioning lgorithm introdued in [] ims t reduing the response time of tsks on moile devies. It finds the offloding nd integrting points on sequene of lls y depth-first serh nd liner time serhing sheme, nd n hieve low user-pereived lteny while lrgely redue the prtitioning omputtion on loud. The offloding inferene engine proposed in [] n dptively mke deisions t runtime, dynmilly prtition n pplition nd offlod prt of the pplition exeution to powerful nery surrogte. Some pplition prtitioning solutions [], [], [] hevily depend upon progrmmers nd middlewre to prtition the pplitions, whih limits their uses. Prtitioning tehnologies were dopted to identify offloded prts for energy sving [], [6], [7]. The energy ost of eh funtion of the pplition ws profiled. Aording to the profiling result, they onstruted ost grph, in whih eh node represented funtion to e performed, nd eh edge indited the dt to e trnsmitted. Finlly, the server prts were exeuted on remote servers for reduing energy onsumption. CloneCloud [] used omintion of stti nlysis nd dynmi profiling to prtition pplitions utomtilly t fine grnulrity while optimizing exeution time nd energy usge for trget omputtion nd ommunition environment. However, this pproh only onsiders limited input/environmentl onditions in the offline pre-proessing nd needs to e ootstrpped for every new pplition uilt. This work ws motivted y the ove interesting works to investigte the prtitioning prolem in moile loud omputing environment, iming t the different ojets, inluding minimum of the response time, minimum of the energy onsumption, nd minimum of weighted sum of time nd energy. We expliitly onsidered the moile nture of oth user nd pplition ehviors, nd ddressed how dynmi prtitioning n ddress these heterogeneity prolems y tking the ndwidth s vrile. Thus, we gretly extending prior work [] y onsidering dynmi prtitioning of pplitions etween wek devies nd louds, in order to etter support pplitions running on diverse devies in different environments. PARTITIONING PROBLEMS. Chllenges Applition prtitioning is very importnt for designing n dptive, ost-effetive, nd effiient offloding system. Some ritil issues onerning the prtitioning prolem inlude: Weighting: when hoosing n pplition tsk to offlod, we need to sle the weights of eh pplition tsk regrding its resoure utiliztion, suh s memory, proessing time, nd ndwidth utiliztions []. The weights n vry for different moile devies nd in different running environments. Communition overhed is introdued y the remote ommunition etween moile devie nd loud server. Rel-Time Adptility: sine ville network ndwidths vry in wireless environments, stti prtitioning lgorithms proposed y previous works with fixed ndwidth ssumption re unsuitle for moile pltforms [9]. The prtitioning lgorithms should e dptive to network nd devie hnges. For exmple, n optiml prtition for high-ndwidth low-lteny network nd low-pity lient might not e good prtition for high-pity lient with d network onnetion. Sine the network ondition is only mesurle t run time, the prtitioning lgorithm should e rel-time online proess []. Prtitioning Effiieny: mking prtitioning deisions for simple pplitions (e.g., n lrm lok) t rel-time is not diffiult, ut for some omplex pplitions (e.g., speeh/fe reognition) tht ontin lrge numer of methods [], highly effiient lgorithm is required to perform rel-time prtitioning.. Applition Prtitioning Proess To solve the ove hllenges, the workflow of n environment-dptive pplition prtitioning proess is proposed in Fig.. It strts with profiling n pplition tht n e split into multiple tsks, through stti nlysis nd dynmi profiling tehnology [0]. We then onstrut WCG of the moile pplition s shown in Fig. (). Bsed on prtitioning ost models, n elsti prtitioning lgorithm is proposed to mke proper pplition prtitioning. By lling suh n lgorithm, we n get preliminry prtitioning results for response time or energy optimiztion. During the exeution proess of the pplition, if the moile environment hnges, nd these hnges meet or exeed ertin threshold, the pplition

3 Applition Y Profiling Strt Stti Anlysis Grph Prtitioning Prtitioning Result Offloding Environment Chnged Fig.. Flowhrt of n pplition prtitioning proess N End Prtition Cost Module Sttistil Anlysis grph will e re-prtitioned ording to the new prmeters. Therefore, it n ultimtely relize the ondition-wre nd environment-dptive elsti prtitioning. Here in the ontext of moile environment, it inludes moile omputing resoures inside the devie, ttery level, CPU, memory, et., ut lso inludes n externl moile environment, suh s the network onnetion nd the loud s speed. After prtitioning, it then utomtilly offlods the distriuted pplitions tht require remote exeution to loud server nd performs the rest lolly on the moile devie ording to the prtitioning results. Therefore, the prolem of whether or not to offlod ertin prts of n pplition to the loud depends on the following ftors: CUP speed of the moile devie, network ndwidth, trnsmission dt size, nd the speed of the loud server [9]. When onsidering suh ftors, we onstrut WCG ording to the estimted omputtionl nd ommunition ost, nd further derive new prtitioning lgorithm designed espeilly for the moile offloding systems. Offlodle Tsks: some pplition omponents re flexile tsks tht n e proessed either lolly on the proessor of the moile devie, or remotely in loud infrstruture. Mny tsks fll into this tegory, nd the offloding deision depends on whether the ommunition osts outweigh the differene etween lol nd remote osts [0]. We do not need to tke offloding deisions for unofflodle omponents. However, s for offlodle ones, sine offloding ll tsks of n pplition to the remote loud is not neessry or effetive under ll irumstnes, it is worth onsidering wht should e exeuted lolly on the moile devie nd wht should e offloded onto the remote loud for exeution sed on ville networks, response time or energy onsumption. The moile devie hs to tke n offloding deision sed on the result of dynmi optimiztion prolem. PARTITIONING MODELS In this setion, we will illustrte whih ssumptions re mde, how WCGs for different types of pplitions re onstruted nd how the optimiztion prolem is defined.. Different Topologies Flexile prtitioning grnulrity-sed pplitions re not limited to speifi form. Previous works onsider pplition prtitioning t different levels of grnulrity: lsses [], ojets [0], methods [], omponents [7], [], nd threds []. Without loss of generlity, we refer to pplition tsks in this pper. Applition developers n hoose the pproprite prtition grnulrity ording to different pplitions. Constrution of WCGs is ritil for the pplition prtitioning. A moile pplition n e represented s list of fine-grined tsks, formulting different topologies s depited in Fig., where eh node reflets n pplition tsk, exeuted either on the moile devie or offloded onto the loud side for further exeution. () One () Liner. Applition Tsk Clssifition Different pplitions emerge in moile devie ording to some proess nd eh onsists of severl tsks. Sine not ll the pplition tsks re suitle for remote exeution, they need to e weighed nd distinguished s: Unofflodle Tsks: some should e unonditionlly exeuted lolly on the moile devie, either euse trnsferring relevnt informtion would tke tremendous time nd energy or euse these tsks must ess lol omponents (mer, GPS, user interfes, elerometer or other sensors et.) []. Tsks tht might use seurity issues when exeuted on different ple should lso not e offloded (suh s e-ommere). Lol proessing onsumes the ttery power of the moile devie, fortuntely, there re no ommunition osts or delys. () Loop 6 (d) Tree Fig.. Tsk-flow grphs for different topologies () (e) Mesh Only one tive node: representing n entire pplition (without prtitioning). Suh topology is often dopted y previous full offloding shemes suh s [], [], [], [], whih n lso e viewed s n exmple of the softwre s servie. In this se, the whole pplition is migrted to remote server involving omplete trnsfer of ode nd progrm stte to the 6

4 () () (d) (e) server [6]. The min drwk of this solution inludes inflexiility nd orse grnulrity. Liner topology: representing sequentil list of finegrined tsks []. Eh tsk is sequentilly exeuted, with output dt generted y one tsk s the input of the next one [7]. Loop-sed topology: loop-sed pplition is one in whih most of the funtionlity is given y iterting n exeution loop, suh s ll the online soil pplitions, in whih we model their proessing with grph tht onsists of yle []. Tree-sed topology: representing tree-sed hierrhy of tsks [6]. The node t the top of the tree is the pplition entry node (i.e., the min module). Mesh-sed topology: representing lttie-sed topology of tsks, e.g., Jv exmple of fe reognition s depited in [0]. When ompred with the sheme tht offlods the whole pplition (i.e., Fig. ()) into the loud, n pplition prtitioning sheme is le to hieve fine grnulrity for omputtion offloding when prtitioning topologil onsumption grph (CG) etween lol nd remote exeution. Different prtitions n led to different osts, nd the totl ost inurred due to offloding depends on multiple ftors, suh s devie pltforms, networks, louds, nd worklods. Therefore, the pplition my hve different optiml prtitions for different moile environments nd worklods.. Constrution of Weighted Consumption Grphs There re two types of osts in the offloding systems: one is omputtionl ost of running the pplition tsks lolly or remotely (inluding memory ost, proessing time ost, nd so on) nd the other is ommunition ost for the pplition tsks intertion (ssoited with movement of dt nd requisite messges). Even the sme tsk n hve different ost on the moile devie nd the loud in term of exeution time nd energy onsumption. As loud servers usully exeute muh fster thn moile devies hving powerful onfigurtion, it n sve energy nd improve performne when offloding prt of the omputtion to remote servers [9]. However, when verties re ssigned to different sides, the intertion etween them leds to the extr ommunition ost. Therefore, we try to find the optiml ssignment of verties for grph prtitioning nd omputtion offloding y trding off the omputtionl osts with the ommunition osts. Cll grphs re widely used to desrie dt dependenies within omputtion, where eh vertex represents tsk nd eh edge represents the lling reltionship from the ller to the llee. Figure () shows CG exmple onsisting of six tsks []. The omputtionl osts re represented y verties, while the ommunition osts re expressed y edges. We denote the dependeny of n pplition s tsks nd their orresponding osts s direted yli grph G = (V, E), where the set of verties V = (v, v,, v N ) denotes N pplition tsks nd n edge e(v i, v j ) E represents the frequeny of invotion nd dt ess etween nodes v i nd v j, where verties v i nd v j re neighors. Eh tsk v i is hrterized y five prmeters: type: offlodle or unofflodle tsk. m i : the memory onsumption of v i on moile devie pltform, i : the size of the ompiled ode of v i, in ij : the dt size of input from v i to v j, out ji : the dt size of output from v j to v i. We further onstrut WCG s depited in Fig. (). Eh vertex v V is nnotted with two-ost weights vi - tuple w(v) =< w lol (v), w loud (v) >, where w lol (v) nd w loud (v) represent the omputtionl ost of exeuting the tsk v lolly on the moile devie nd remotely on the loud, respetively. The j th vertex weighted vetor mens the j th tuple. Eh vertex is ssigned with one of the vlues in the tuple depending on the prtitioning result of the pplition grph it finlly ends up in or the lel it is ssigned [0]. The edge set E V V represents the ommunition ost mongst tsks. The weight of n edge w(e(v i, v j )) is denoted s: w(e(v i, v j )) = in ij B uplod + out ij B downlod, () whih is the ommunition ost of trnsferring the input nd return sttes when the tsks v i nd v j re exeuted on different sides, nd it losely depend on the network ndwidths (uplod ndwidth B uplod nd downlod ndwidth B downlod ) nd the trnsferred dt. A ndidte offloding deision is desried y one ut in the WCG, whih seprtes the verties into two disjoint sets, one representing tsks tht re exeuted on the moile devie nd the other one implying tsks tht re offloded to the remote server []. Hene, tking the optiml offloding deision is equivlent to prtitioning the WCG suh tht n ojetive funtion is minimized []. The red dotted line in Fig. () is one possile prtitioning ut, inditing the prtitioning of omputtionl worklod in the pplition etween the moile devie nd the loud. V l nd V re sets of verties, where V l is the lol set in whih tsks re exeuted lolly nd V is the loud set in whih tsks re diretly offloded to the loud. We hve V l V = nd V l V = V. Further, E ut is the edge set in whih the grph is ut into two prts.. Cost Models Moile pplition prtitioning ims t finding the optiml prtitioning solution tht leds to the minimum exeution ost, in order to mke the est trdeoff etween time/energy svings nd trnsmission osts/dely. The optiml prtitioning deision depends on user requirements/expettions, devie informtion, network ndwidth, nd the pplition itself. Devie informtion inludes the exeution speed of the devie nd the worklods on it when the pplition is lunhed. If the devie omputes very slowly nd the im is to redue exeution time, it is etter to offlod more omputtion to the loud []. Network ndwidth ffets dt trnsmission for remote exeution. If the ndwidth is very high, the ost in terms of dt trnsmission will e low. In this se, it is etter to offlod more omputtion to the loud. The prtitioning deision is mde sed on the ost estimtion (omputtionl nd ommunition osts) efore the

5 m,, offlodle in Grph Cut out in out m,, offlodle in out out in 6 out 6 6 in in m,, offlodle m 6, 6, offlodle m, unofflodle out out m i = memory i i = ode_size i m,, offlodle type={offlodle, unofflodle} Moile Side Cloud Side () CG in < w lol (v ),w loud (v ) > < w lol (v ),w loud (v ) > E ut w(e(v,v )) w(e(v,v )) V V l w(e(v,v )) w(e(v,v )) < w lol (v ),w loud (v ) > Moile Side () WCG w(e(v,v )) < w lol (v ),w loud (v ) > w(e(v,v )) w(e(v,v 6 )) 6 < w lol (v ),w loud (v ) > < w lol (v 6 ),w loud (v 6 ) > Cloud Side Fig.. Constrution of WG nd WCG. progrm exeution. On the sis of Fig. (), we n formulte the prtitioning prolem s: C totl = v V I v w lol (v) + v V ( I v ) w loud (v) + e(v i,v j) E I e w(e(v i, v j )), () where the totl ost is the sum of omputtionl osts (lol nd remote) nd ommunition osts of ut ffeted edges. The loud server node nd the moile devie node must elong to different prtitions. One possile solution for this prtitioning prolem will give us n ritrry tuple of prtitions from the verties set < V l, V > nd the ut of edge set E ut in the following wy: I v = {, if v Vl 0, if v V nd I e = {, if e Eut 0, if e / E ut. () We seek to find n optiml ut in the WCG suh tht some pplition tsks re exeuted on the moile side nd the remining ones on the loud side. The optiml ut mximizes or minimizes n ojetive funtion nd menwhile stisfies moile devie s resoure onstrints. The ojetive funtion expresses the generl gol of prtition, this my e, for instne, minimize the energy onsumption, minimize the mount of exhnged dt, or omplete the exeution in less thn predefined time. We only tully perform the prtitioning when it is enefiil. Not ll pplitions n enefit from prtitioning euse of pplition-speifi properties. The ost estimtion of running eh pplition tsk on the moile devie nd loud server is needed. Offloding mkes sense only if the speedup of the loud server overweigh the extr ommunition osts. The ommunition time nd energy osts for the moile devie will vry ording to the mount of dt to e trnsmitted nd the wireless network onditions. Aording to (), the dynmi exeution onfigurtion of n elsti pplition n e deided sed on some different sving ojetives with respet to response time nd energy onsumption. A tsk s offloding gols my hnge due to hnge in environmentl onditions... Minimum Response Time The ommunition ost depends on the size of dt trnsfer nd the network ndwidth, while the omputtionl ost is impted y the omputtion time. If the minimum response time is seleted s the ojetive funtion, we n lulte the totl time spent due to offloding s: T totl (I) = v V I v T l v + v V ( I v ) T v + e E I e T tr e, () where Tv l = F Tv : the omputing time of tsk v on the moile devie when it is exeuted lolly; F : the speedup ftor, the rtio of the loud server s exeution speed ompred to tht of the moile devie, sine the omputtion pity of loud infrstruture is stronger thn tht of the moile devie, we hve F > ; Tv : the omputing time of tsk v on the loud server one it is offloded, Te tr = De tr /B: the ommunition time etween the moile devie nd the loud; De tr : the mount of dt tht is trnsmitted nd reeived; B: the urrent wireless ndwidth. In this senrio, the offloding deision engine then selets the est prtitioning ndidte tht minimizes the totl response time. The im of this ost model is to find the optiml pplition prtitioning: I min = { I v, I e I v, I e {0, } }, whih stisfies I min = rg min I T totl (I). The sved response time in the prtitioning sheme ompred to the sheme without offloding is lulted s: T sve (I) = T lol T totl (I) T lol 00%, () where T lol = v V T v l is the lol time ost when ll the pplition tsks re exeuted lolly on the moile devie. Besides, for given pplition nd moile devie, the optiml prtitioning results lso hnge ording to the situtions under different wireless network ndwidths nd the speedup ftors of the loud server... Minimum Energy Consumption Similrly, if the minimum energy onsumption is hosen s the ojetive funtion, we n lulte the totl energy onsumed y the moile devie due to offloding s: E totl (I) = v V I v E l v + v V ( I v ) E i v + e E I e E tr e, (6) where Ev l = P m Tv l : the energy onsumed of tsk v on the moile devie when it is exeuted lolly, Ev i = P i Tv : the energy onsumed of tsk v on the moile devie when it is offloded to the loud, E e = P tr Te tr : the energy spent on the

6 6 ommunition etween the moile devie nd the loud. P m, P i nd P tr re the powers of the moile devie for omputing, while eing idle nd for sending or reeiving dt, respetively. In this senrio, the offloding deision engine then selets the est prtitioning pln tht minimizes the prtitioning ost of energy. The im is to find the optiml pplition prtitioning: I min = { I v, I e I v, I e {0, } }, whih stisfies: I min = rg min I E totl (I). The sved energy when ompred to the sheme without offloding is: E sve (I) = E lol E totl (I) E lol 00%, (7) where E lol = v V El v is the lol energy ost when ll the pplition tsks re exeuted on the moile devie... Minimum of the Weighted Sum of Time nd Energy If we omine oth the response time nd energy onsumption, we n design the ost model for prtitioning s follows: W totl (I) = ω Ttotl(I) T lol + ( ω) Etotl(I) E lol, () where 0 ω is weighting prmeter used to indite reltive importne etween the response time nd energy onsumption. Lrge ω fvors response time while smll ω fvors energy onsumption. In some speil ses performne n e trded for power onsumption nd vie vers [], therefore we n use the ω prmeter to express suh speil ses preferenes for different pplitions. T totl (I) nd E totl (I) re the response time nd energy onsumption with the prtitioning solution I, respetively. To eliminte the impt of different sles of time nd energy, they re divided y the lol osts. If T totl (I)/T lol is less thn, the prtitioning will inrese the pplition s power onsumption. Similrly, if E totl (I)/E lol is less thn, it will redue the pplition s performne. In this senrio, the offloding deision engine then selets the est prtition pln tht minimizes the prtitioning ost of weighted sum of time nd energy. The im is to find the optiml pplition prtitioning: I min = { I v, I e I v, I e {0, } }, while stisfying: I min = rg min I W totl (I). The sved weighted sum of time nd energy in the prtitioning sheme ompred to the sheme without offloding is lulted s: T totl (I) E totl (I) W sve (I) = ω Tlol +( ω) Elol 00%. T lol E lol (9) PARTITIONING ALGORITHM FOR OFFLOADING In this setion, we introdue the min-ost offloding prtitioning (MCOP) lgorithm for WCGs of ritrry topology. The MCOP lgorithm tkes WCG s input whih represents n pplition s opertions/lultions s the nodes nd the ommunition etween them s the edges. Eh node hs two osts: the first is the ost of performing the opertion lolly (e.g., on the moile phone) nd the seond is the ost of performing it elsewhere (e.g., on the loud). The weight of the edges is the ommunition ost to the offloded omputtion. It is ssumed tht the ommunition ost etween opertions in the sme lotion re negligile. The result ontins informtion out the osts nd reports whih opertions should e performed lolly nd whih should e offloded.. Steps The MCOP lgorithm n e divided into two steps s follows: ) Unofflodle Verties Merging: An unofflodle vertex is the one tht hs speil fetures mking it unle to e migrted outside of the moile devie nd therefore is loted only in the unofflodle prtition. Aprt from this, we n hoose ny tsk to e exeuted lolly ording to our preferenes or other resons. Then ll verties tht re not going to e migrted to the loud re merged into one tht is seleted s the soure vertex. By merging, we men tht these nodes re olesed into one, whose weight is the sum of the weights of ll merged nodes. Let G represent the originl grph fter ll the unofflodle verties re merged. ) Corse Prtitioning: The trget of this step is to orsen G to the orsest grph G V. To orsen mens to merge two nodes nd redue the node ount y one. Therefore, the lgorithm hs V phses. In eh phse i (for i V ), the ut vlue, i.e., the prtitioning ost in grph G i = (V i, E i ) is lulted. G i+ rises from G i y merging suitle nodes, where G = G. The prtitioning results of using the MCOP lgorithm re the minimum ut mong ll the uts in n individul phse i nd the orresponding group lists for lol nd loud exeution. Furthermore, in eh phse i of the orse prtitioning, we still hve five steps: ) Strt with A={}, where is usully n unfflodle node in G i. ) Itertively dd the vertex to A tht is the most tightly onneted to A. ) Let s, t e the lst two verties (in order) dded to A. ) The grph ut of the phse i is etween V i \{t} nd {t}. ) G i+ rises from G i y merging verties s nd t.. Merging Definition: If s, t V (s t), then s nd t n e merged s follows: ) Nodes s nd t re hosen. ) Nodes s nd t re sustituted y new node x s,t. All edges tht were previously inident to s or t re now inident to x s,t (exept the edge etween nodes s nd t when they re onneted). ) Multiple edges re resolved y dding edge weights. The weights of the node x s,t re resolved y dding the weights of s nd t. The merging funtion is used to merge two verties into one new vertex, whih is implemented s in Algorithm. For exmple, we n merge nodes nd s shown in Fig... Algorithmi Proess The lgorithmi proess is illustrted s the MinCut funtion in Algorithm, nd in eh phse i, it lls the MinCutPhse funtion s desried in Algorithm. Sine some tsks hve to e exeuted lolly, we need to merge them into one node. The ore of this lgorithm is to mke it esy to selet the next vertex to e dded to the set A, tht is Most Tightly

7 7 <9, > <6, > 6 <, > <, > () Step <9, >, <, > () Step <6, > 6 <, > <, 7>, () Step 6 <6, > 6 <, > Fig.. Exmple of merging two nodes Algorithm The Merging funtion //This funtion tkes s nd t s verties in the given grph nd merges them into one Funtion: G =Merge(G, w, s, t) Input: G: the given grph, G = (V, E) w: the weights of edges nd verties s, t: two verties in previous grph tht re to e merged Output: G : the new grph fter merging two verties : x s,t s t : for ll nodes v V do : if v {s, t} then : w(e(x s,t, v)) = w(e(s, v)) + w(e(t, v)) : //dding [ weights of edges 6: w lol (x s,t), w loud (x ] s,t) = w lol (t), w loud (s) + w loud (t) ] 7: //dding weights of nodes : E E e(x s,t, v) //dding edges 9: end if 0: E E\{e(s, v), e(t, v)} //deleting edges : end for : V V \{s, t} x s,t : return G = (V, E ) [ w lol (s) + Conneted Vertex (MTCV), whih is defined s the vertex whose (v) into A is mximum, where (v) = w(e(a, v)) [w lol (v) w loud (v)]. Further, we hve the totl ost from prtitioning: [ ] C ut(a t,t) = C lol w lol (t) w loud (t) + v A\t w(e(t, v)), (0) where C lol = v V wlol (v) is the totl of lol osts nd the ut vluec ut(a t,t) is the prtitioning ost, w lol (t) w loud (t) is the gin of node t from offloding, nd v A\t w(e(t, v)) is the totl of extr ommunition osts due to offloding. Theorem. ut(a t, t) is lwys minimum s t ut in the urrent grph, where s nd t re the lst two verties dded in the phse, the s t ut seprtes nodes s nd t on two different sides. The run of eh MinCutPhse funtion orders the verties of the urrent grph linerly, strting with nd ending with s nd t, ording to the order of ddition into A. We wnt to show tht C ut(a t,t) C ut(h) for ny ritrry s t ut H. Lemm. We define H s n ritrry s t ut, A v s set of verties dded to A efore v, nd H v s ut of A v {v} indued y H. For ll tive verties v, we hve C ut (A v, v) C ut (H v ). Algorithm The M incut funtion //This funtion performs n optiml offloding prtition lgorithm Funtion: [mincut, M incutgroupslist] = M incut(g, w, SoureV erties) Input: G: the given grph, G = (V, E) w: the weights of edges nd verties SoureVerties: list of verties tht re fored to e kept in one side of the ut Output: mincut: the minimum sum of weights of edges nd verties mong the ut MinCutGroupsList: two lists of verties, one lol list nd one remote list : w(mincut) : for i = : length(sourev erties) do : //Merge ll the soure verties (unofflodle) into one : (G, w) = Merge(G, w, SoureV erties(), SoureV erties(i)) : end for 6: while V > do 7: [ut(a t, t), s, t] = MinCutP hse(g, w) : if w(ut(a t, t)) < w(mincut) then 9: mincut ut(a t, t) 0: end if : Merge(G, w, s, t) : //Merge the lst two verties (in order) into one : end while : return mincut nd MinCutGroupsList Proof. As shown in Fig., we use indution on the numer of tive verties, k. ) When k =, the lim is true, ) Assume the inequlity holds true up to u, tht is C ut (A u, u) C ut (H u ), ) Suppose v is the first tive vertex fter u, ording to the ssumption C ut (A u, u) C ut (H u ), then we hve: C ut (A v, v) = C ut (A u, v) + C ut (A v A u, v) C ut (A u, u) + C ut (A v A u, v) (u is MTCV) C ut (H u ) + C ut (A v A u, v) C ut (H v ). Sine t is lwys n tive vertex with respet to H, y the Lemm, we n onlude tht C ut(a t,t) C ut(h) whih sys extly tht the ost of ut(a t, t) is t most s hevy s the ost of ut(h). Therefore, Theorem is now proved.

8 Algorithm The MinCutPhse funtion //This funtion perform one phse of the prtitioning lgorithm Funtion: [ut(a t, t), s, t]=mincutp hse(g i, w) Input: G i: the grph in Phse i, i.e., G i = (V i, E i) w: the weights of edges nd verties SoureVerties: list of verties tht re fored to e kept in one side of the ut Output: s, t: the lsted two verties tht re dded to A ut(a t, t): the ut etween {A t} nd {t} in phse i : ritrry vertex of G i : A {} : while A V i do : mx = : v mx = null 6: for v V i do 7: if v / A then : //Performne gin through offloding the tsk v to the loud 9: (v) w(e(a, v)) [w lol (v) w loud (v)] 0: //Find the vertex tht is the most tightly onneted to A : if mx < (v) then : mx = (v) : v mx = v : end if : end if 6: end for 7: A A {v mx} : Merge(G, w,, v mx) 9: end while 0: t the lst vertex (in order) dded to A : s the lst seond vertex (in order) dded to A : return ut(a t, t) s t As the running time of the lgorithm MinCut is essentilly equl to the dded running time of the V runs of Min- CutPhse, whih is lled on grphs with deresing numer of verties nd edges, it suffies to show tht single MinCutPhse needs t most O( V log V + E ) time yielding n overll running time. The omputtionl omplexity of the MCOP lgorithm n e noted s O( V log V + V E ). As omprison, liner progrmming (LP) solvers re widely used in shemes like [] nd []. The LP solver is sed on rnh nd ound, whih is n lgorithm design prdigm for disrete nd omintoril optimiztion prolems, s well s generl rel vlued prolems []. The numer of its optionl solutions grows exponentilly with the numer of tsks, whih mens higher time omplexity O ( V ). Therefore, the MCOP lgorithm hs muh lower time omplexity when ompred to the existing lgorithms, whih is proportion to the squre of the numer of tsks nd hene n hieve n optiml offloding strtegy s quikly s possile.. Cse Study Figure 6 shows tht node is defined s the strting point in whih the orresponding tsk will lwys e omputed y the moile devie. We hve s = d nd t = f, nd the indued ordering,,, e, d, f of the verties. Node f is ut off from the grph. The first ut-of-the-phse orresponds to the prtitions {,,, e, d} nd {f}. Sine the overll lol ost is C lol = v V wlol (v) =, we n lulte the ut ost y using (0) s: C ut(a f,f) = ( ) + = 0. At the end, we merge nodes s = d nd t = f into one. From Figs. 7-0, we repet the sme proess of the Min- CutPhse funtion s the first phse in Fig. 6. There re V = phses, nd t the end, ll nodes re merged into one. Then, we ompre ll the ut vlues, the minimum vlue refers to the phse whih hs the optiml prtitioning ut. In this senrio, the minimum ut of the grph G is the fourth ut-of-the-phse. The optiml ut is etween {, } nd {, d, e, f} s depited in Fig. with the minimum ost of C ut({, }, {, d, e, f}) = ( ) + ( + ) =. Here, tsks, d, e, f re offloded to the remote loud server while tsks nd re exeuted lolly. +-(-)=- def <, > A t () The s t ut H t Fig.. The optiml ut in phse s t 6 PROFILING Fig.. The proof of Lemm () An ritrry s t ut. Computtionl Complexity How to uild the WCG is tully the ottlenek of whole tehnique, whih losely depends on profiling, i.e., the proess of gthering the informtion required to mke offloding deisions. Suh informtion my onsist of the omputtion nd ommunition osts of the exeution units (progrm profiler), the network sttus (network profiler), nd the moile devie speifi hrteristis suh s energy onsumption (energy profiler). Profilers re needed to ollet informtion out the devie nd network hrteristis, whih is ritil prt of the prtitioning lgorithm: the more urte nd lightweight they re, the more orret deisions n e mde, nd the lower overhed is introdued [6]. We will in the following introdue ll types of profilers. 6. Progrm Profiler A progrm profiler (stti or dynmi) ollets hrteristis of pplitions, e.g., the exeution time, the memory usge nd

9 9 <6, > <6, > <9, > e <9, > e -(9-)=- -(9-)=- d f d f <, > <, > <, > <, > -(-)=-7 -(-)=6 -(6-)=- <6, > <9, > e d f <, > <, > <6, > <9, > e d <, > ++-(-)=0 +-(-)=- G : A = {} G : A = {, } G : A = {,, } G : A = {,,, e} f <, > <9, > <6, > e d f <, > <, > <9, > -(-)=- G : A = {,,, e, d} G : A = {,,, e, d, f} s d e <6, > <, > <, > t f <9, > s nd t merged Fig. 6. The st phse of MinCutPhse funtion. The indued ordering of the verties is,,, e, s, t, where s = d nd t = f. The st ut-of-the-phse orresponds to the prtitions {,,, e, d} nd {f} with the ut vlue: C ut(a f,f) = ( ) + = 0. df <7, 9> e <6, > <9, > e <6, > <9, > e <6, > -(9-)=- -(9-)=- df df <7, 9> <7, 9> -(7-9)=-7 -(-)=6 G : A = {} G : A = {, } -(6-)=- <9, > e <6, > df <7, 9> +-(7-9)=- G : A = {,, } <9, > <6, > ++-(7-9)=-0 G : A = {,,, e} e df <7, 9> <9, > s t Fig. 7. The nd phse of MinCutPhse funtion. The indued ordering of the verties is,,, s, t, where s = e nd t = {df}. The nd ut-of-the-phse orresponds to the prtitions {,,, e} nd {d, f} with the ut vlue: C ut(a {d,f},{d,f}) = (7 9) + ( + + ) =. df e <7, 9> G : A = {,,, e, {df}} <6, > <9, > s nd t merged <9, > <9, > <9, > <9, > -(9-)=- -(9-)=- def def <, > def def def <, > <, > <, > <, > -(-)=- +-(-)=-6 -(-)=6 G : A = {} G : A = {, } G : A = {,, } G : A = {,,, {def}} s nd t merged s t Fig.. The rd phse of MinCutPhse funtion. The indued ordering of the verties is,, s, t, where s = nd t = {def}. The rd ut-of-the-phse orresponds to the prtitions {,, } nd {d, e, f} with the ut vlue: C ut({,,},{d,e,f}) = ( ) + ( + ) = 9. def <, > -(-)=- +-(-)=- def def def <, > <, > <, > def <, > -(-)=6 G s nd t merged : A = {} G : A = {, } G : A = {,, {def}} s t Fig. 9. The th phse of MinCutPhse funtion. The indued ordering of the verties is, s, t, where s = nd t = {def}. The th ut-of-the-phse orresponds to the prtitions {, } nd {, d, e, f} with the ut vlue: C ut({, }, {, d, e, f}) = { ( ) ( + ) } =.

10 0 -(-)=- def def def <, > G : A = {} G : A = {, {def}} s nd t merged s t Fig. 0. The th phse of MinCutPhse funtion. The indued ordering of the verties is s, t, where s = nd t = {def}. The th ut-of-the-phse orresponds to the prtitions {}, nd {,, d, e, f} with ut vlue C ut({}, {,, d, e, f}) = ( ) + = 7. <, > <, > the size of dt. We n omine stti nlysis nd dynmi profiling to onstrut the WCG of n pplition. Stti nlysis otins the ontrol flow grph of n pplition y nlyzing the yteode with nodes representing ojets nd edges representing reltions etween ojets. We n get ll the ojets nd the reltions etween them sed on method invotions y trversing the grph. Construting ll grphs y hnd nd without the help of nlysis tools would hve ost fr more time nd resoures. Mny tools nd frmeworks hve een developed to generte the ll grph. Mny tools nd frmeworks hve een developed to generte the ll grph of given pplition, e.g., Sprk [7], Cg [], nd Soot [], nd this utomtion is huge dvntge. Dynmi profiling is dopted to otin weights of the nodes nd edges. Sine there is ertin rtio of exeution time to the totl yteode instrution ount for Jv progrms, exeution time of ojets n e evluted y the orresponding yteode instrution ount [9]. Dt trnsmission dt etween tsks inlude prmeters nd return vlues of method invotions. Comining Jv yteode rewriting with pretretment informtion like speedup ftor F nd wireless ndwidth B, we n otin the exeution time for eh tsk (node weight) nd the trnsmission time for eh invotion (edge weight). These weights n e dynmilly ssigned ording to the different proessing pilities of the loud server nd the wireless ndwidth. We tke fe reognition pplition s n exmple. By nlyzing this pplition with Soot, the ll grph ould e onstruted s tree-sed topology in Fig.. From the lol estimted exeution time, we n get the remote estimted exeution time, dividing y the speedup ftor F. When offloding tsk to the loud, the ommunition ost inurred etween the moile devie nd the loud is the dt trnsfer divided y the ndwidth. Then, we hve the weighted onsumption grph for this pplition. Finlly, with remote exeution nd trnsmission osts, we now hve ll informtion to get the WCG. 6. Network Profiler A network profiler ollets informtion out wireless onnetion sttus nd ville ndwidth. It mesures the network hrteristis t initiliztion, nd it ontinuously monitors environmentl hnges. Network throughput n e otined y mesuring the time durtion when sending ertin mount of dt s in []. Due to the moile nture, the sttus of wireless onnetion ould hnge frequently (e.g. user moves to other lotion). Fresh informtion out wireless. The fe reognition pplition is uilt upon n open soure ode whih implements the Eigenfe fe reognition lgorithm FeBundle sumitfe.9ms 0KB 600KB FeBundle ompute 7.ms lss nme method nme exeution time EigenFeCretor hekaginst 7.ms 0.9KB 0.KB EigenFeCretor redimge 0.7ms JPGFile <init> 7.ms JPGFile redimge 77.7ms TestFeReognition min.ms 67.KB 0KB Jm.Mtrix eig.ms KB 0.KB 00KB EigenFeCretor omputebundle 6.ms 906KB EigenFeCretor sumit 6.6ms 000KB Jm.Mtrix trnspose.0ms Fig.. Cll grph of fe reognition pplition EigenFeCretor redfebudles 6ms 006KB EigenFeCretor sumitset 7.ms 00KB 006KB Jm.Mtrix times 6.6ms EigenFeCretor svebundle 9ms onnetion is ritil for the optimizer to mke orret offloding deisions. The profiler trks severl prmeters for the WiFi nd G interfes, inluding the numer of pkets trnsmitted nd reeived per seond, nd reeiving nd trnsmitting dt rte [6]. These mesurements enle etter estimtion of the urrent network performne eing hieved. We n use Speedtest to mesure the moile network ndwidth. 6. Energy Profiler There re two wys to estimte the energy onsumption, nmely, softwre nd hrdwre monitors. For exmple, MAUI [] used power meter tthed to the smrtphone s ttery to uild n energy profile. Power Monitor (e.g. Monsoon monitor) is devie tht mesures energy onsumption when dt is trnsmitted from the moile devie to the loud server y supplying ertin level of power to the moile devie. We n lso use PowerTutor to mesure the power onsumption of the pplitions. Although PowerTutor doe not give very urte results s hrdwre power monitor does, the result is still resonle nd does provide some vlues euse it gives the detiled energy onsumption informtion for eh hrdwre omponent.. A free onnetion nlysis tool, whih shows rel-time downlod nd uplod grphs, stores results oth lolly nd on the Internet for shring, PowerTutor is n pplition for Android phones tht provides urte, rel-time power onsumption estimtes for power-intensive hrdwre omponents,

11 Running Time/s 7 EVALUATION 7. Setup To evlute the prtitioning lgorithm, we need to know three different kinds of vlues: Fixed Vlues: they re set y the moile pplition developer, determined sed on lrge numer of experiments. For exmple, the power onsumption vlues of P m, P i, nd P tr re prmeters speifi to the moile system. We use n HP ipaq PDA with 00-MHz Intel XSle proessor tht hs the following vlues: P m 0.9 W, P i 0. W, nd P tr. W [0]. Speifi Vlues: suh prmeters represent some stte of moile devies, e.g., the size of trnsferred dt, the vlue of urrent wireless ndwidth B (for onvenient, we ssume B uplod = B downlod ) nd the speedup ftor F tht depends on the speed of urrent loud server nd the moile devie. Clulted Vlues: these vlues nnot e determined y pplition developers. For given pplition, the omputtionl ost is ffeted y input prmeters nd devie hrteristis, whih n e mesured using progrm profiler. The ommunition ost is relted to trnsmitting odes/dt vi wireless interfes suh s WiFi or G, whih n e trked y network profiler. Performne evlution results enompss omprisons with other existing shemes, in ontrst to the energy onservtion effiieny nd exeution time. We ompre the prtitioning results with two other intuitive strtegies without prtitioning nd, for ese of referene, we list ll three kinds of offloding tehniques: No Offloding (Lol Exeution): ll omputtion tsks of n pplition re running lolly on the moile devie nd there is no ommunition ost. This my e ostly sine s ompred to the powerful omputing pility t the loud side, the moile devie is limited in proessing speed nd ttery life. Full Offloding: ll omputtion tsks of moile pplitions (exept the unofflodle tsks) re moved from the lol moile devie to the remote loud for exeution. This my signifintly redue the implementtion omplexity, whih mkes the moile devies lighter nd smller. However, full offloding is not lwys the optiml hoie sine different pplition tsks my hve different hrteristis tht mke them more or less suitle for offloding [9]. Prtil Offloding (With Prtitioning): with the help of the MCOP lgorithm, ll tsks inluding unofflodle nd offlodle ones re prtitioned into two sets, one for lol exeution on the moile devie nd the other for remote exeution on loud server node. Before tsk is exeuted, it my require ertin mount of dt from other tsks. Thus, dt migrtion vi wireless networks is needed etween tsks tht re exeuted t different sides. We define the sved ost in the prtil offloding sheme ompred to tht in the no offloding sheme s Offloding Gin, whih n e formulted s: Offloding Gin = Prtil Offloding Cost No Offloding Cost 00%. () The offloding gins in terms of time, energy nd the weighted sum of time nd energy re desried in (), (7) nd (9), respetively. 7. Evlution in Computtionl Complexity We implement the MCOP lgorithm in Jv tht n serve s omprison to the theoreti results, nd the ode n e found in []. As n exmple, we prtition the onstruted WCG in Fig. under the ondition of the speedup ftor F = nd the ndwidth B = MB/s, where the min nd hekaginst methods re ssumed s unofflodle nodes. The optiml prtitioning result is depited in Fig.. The red nodes represent the pplition tsks tht should e offloded to the remote loud nd the lue nodes re the tsks tht re supposed to e exeuted lolly on the moile devie. The prtition results will hnge s the wireless ndwidth B or the speedup ftor F vries. <.90, 7.9> 0.0 <7.0,.60> <7.0, 6.90> <0.70, 0.> 67. <7.0, 7.60> 0.0 <77.70,.> <.0,.0> <.0, 777.6> <6.00, 7.00> <7.0, 6.0> 00.0 <6.0,.> <6.60,.0> <.00, 6.0> <9.00, 96.00> <6.60,.0> Fig.. Optiml prtitioning result of the fe reognition pplition when F = nd B = MB/s The running time of the jv implementtion under different numer of pplition tsks is depited s Fig.. We ompre it with the theoreti omputtionl omplexity denoted s O( V log V + V E ) in Setion.. We find they hve good mth with eh other, whih further proofs tht our prtitioning lgorithm hs muh lower time omplexity thn the LP solver whih hs exponentil time omplexity Simultion Theory Numer of Tsks Fig.. Running time of the MCOP lgorithm under different numer of tsks

12 7. Evlution in Dynmi Conditions We uild grphil user interfe (GUI) in MATLAB s shown in Fig.. The GUI is responsile for intertion with the user: inputing prmeters ordingly nd displying the pplition prtitioning results. The GUI is responsile for user intertion suh s reeiving input prmeters nd displying the pplition prtitioning results. Fig.. The user interfe for demonstrtion The user first inputs or selets the reltive prmeters, suh s Applition Grph, Unofflodle Nodes nd Optimiztion Model. We n either use the predefined pplition grphs of liner, loop, tree nd mesh or just hoose user to input ny ritrry CG. Then, y liking the Grph utton, WCG will e onstruted sed on the ove prmeters. Further, y liking the Strt Prtitioning utton, the prtitioning proess will egin, y lling the prtitioning lgorithm of MCOP. We n get the prtitioning results suh s Prtil Offloding Cost, No offloding Cost, Full Offloding Cost nd Offloding Gin. In ddition, the optiml prtitioning grph will pper like Fig. 6, whih further proves the orretness of the prtitioning result in Fig. with the minimum ost of. We n get the different results under different prmeters of speedup ftor F nd wireless ndwidth B. :<0,0> :<9,> :<,> d:<6,> e:<,> f:<,> Fig. 6. An optiml prtitioning result of using the MCOP lgorithm As depited in Fig. 7, the speedup ftor is set s F =. Sine the low ndwidth results in muh higher osts for dt trnsmission, the full offloding sheme nnot enefit from offloding. Given reltively lrge ndwidth, the response time or energy onsumption otined y the full offloding sheme slowly pprohes to the prtil offloding sheme euse the optiml prtition inludes more nd more tsks running on the loud side until ll offlodle tsks re offloded to the loud. With the higher ndwidth, they egin to oinide with eh other nd only derese euse ll possile nodes re offloded nd the trnsmissions eome fster. Both response time nd energy onsumption hve the sme trend s the wireless ndwidth inreses. Therefore, ndwidth is ritil ondition for offloding sine the moile system ould enefit lot from offloding in high ndwidth environments, while with low ndwidths, the no offloding sheme is preferred. As shown in Fig., the ndwidth is fixed s B = MB/s. It n e seen tht offloding enefits from higher speedup ftors. When F is very smll, the full offloding sheme n redue energy onsumption of the moile devie, however it tkes muh more response time thn the no offloding sheme. The prtil offloding sheme tht dopts the MCOP lgorithm n effetively redue exeution time nd energy onsumption, while dpting to environmentl hnges. From Figs. 7- we n tell tht the full offloding sheme performs muh etter thn the no offloding sheme under ertin dequte wireless network onditions, euse the exeution ost of running methods on the loud server is signifintly lower thn on the moile devies when the speedup ftor F is lrge. The prtil offloding sheme outperforms the no offloding nd full offloding shemes nd signifintly improves the pplition performne, sine it effetively voids offloding tsks in the se of lrge trnsition osts etween onseutive tsks ompred to the full offloding sheme, nd offlods more pproprite tsks to the loud server. In word, neither running ll pplition tsks lolly on the moile terminl nor lwys offloding their exeution to remote server, n offer n effiient solution, ut rther our prtil offloding sheme n do. We then ompre the ost svings under three different ost models. For the model of minimum weighted time nd energy, the weights of response time nd energy onsumption re oth set to 0.. As shown in Fig. 9(), it n e seen tht when the ndwidth is low, the offloding gins for ll three ost models re very smll nd lmost oinide. Tht s euse more time/energy will e spent in trnsferring the sme dt due to the low network ndwidth, resulting in exeution time inreses. As the ndwidth inreses, the offloding gins firstly rise drstilly nd then the inreses eome slower. It n e onluded tht the optiml prtition inludes more nd more tsks running on the loud side until ll the tsks re offloded to the loud when the ndwidth inreses. Among the prtitioning ost models, the minimum energy onsumption model hs the lrgest offloding gin, followed y the minimum weighted sum of time nd energy, while the response time enefits the lest from the offloding. Similrly, Figure 9() demonstrtes how the prtitioning result vries s the speedup ftor F hnges. When F is smll, the offloding gins for ll three ost models re very low sine smll vlue mens very little omputtionl ost redution from remote exeution. As F inreses, the offloding gins firstly rise drstilly nd then pproh to the sme vlue. Tht s euse the enefits from offloding nnot neglet the extr ommunition ost. From Fig. 9, it n e seen tht the