Mobile to Mobile Computational Offloading in Multi-hop Cooperative Networks

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1 Mobe to Mobe Computatona Offoadng n Mut-hop Cooperatve Networks Con Funa, Crstano Tappareo, Wend Henzeman Department of Eectrca and Computer Engneerng, Unversty of Rochester, Rochester, NY, USA Ema: {frstname.astname}@rochester.edu Abstract As the number of mobe devces that natvey support ad hoc communcaton protocos ncrease, arge ad hoc networks can be created not ony to factate communcaton among the mobe devces, but aso to assst devces that are executng computatonay ntensve appcatons. Pror work has deveoped computaton offoadng systems for mobe devces, but ths work has focused excusvey on offoadng to snge hop neghbors, due n part to the practca chaenges of settng up mut-hop networks usng exstng ad hoc communcaton protocos. However, mtng the offoadng of computaton to one-hop neghbors nherenty restrcts the number of devces that can partcpate n the dstrbuted computaton. In ths paper, we propose and evauate the performance of computatona offoadng wthn a mut-hop cooperatve network, where mobe devces are abe to share the computatona oad wth a other nodes n the network. Addtonay, we present an teratve task assgnment agorthm that can optmze the assgnment of computatona tasks to devces n such a mut-hop cooperatve network, takng nto account the communcaton overhead of the mut-hop network. Expermenta resuts, obtaned from an mpementaton on Androd devces, are ntegrated wth an anaytca mode that enabes the evauaton of system performance under a varety of condtons. These expermenta and anaytc resuts demonstrate the beneft of enabng computaton offoadng to a devces n a mut-hop cooperatve network. I. INTRODUCTION Recent advancements n VLSI and MEMs have aowed for ncreasngy compcated processors to become cheaper and more accessbe to a varety of appcatons. For nstance, these processors have become so accessbe that manufacturers are even offerng ruggedzed devces that use these processors 1, whe st hodng certfcaton for certan envronmenta condtons, such as ow pressure or extreme temperature. Wth devces that can wthstand such harsh condtons, t s not hard to magne that when encounterng hazardous stuatons, a frst responder woud beneft greaty from havng the abty to take n data from the fed, process t, and make an nformed decson based on the resut. For exampe, frst responders mght be equpped wth ruggedzed smart devces and depoyed to the ste of an earthquake to ade n reef efforts, or a mtary patro mght be abe to perform a Sgna Integence (SIGINT) appcaton for fndng mprovsed exposve devces (IEDs) 3. Typcay, these knds of appcatons requre mmense processng capabtes and Ths work was supported n part by Harrs Corporaton, RF Communcatons Dvson and n part by CEIS, an Empre State Deveopment desgnated Center for Advanced Technoogy. consume arge amounts of the devce s battery, f the devce s even abe to perform such computatons. One potenta souton s for the mobe devce to offoad these computatonay ntensve tasks to a remote server 4,, ether by settng up reay ponts throughout the area of nterest or usng satete communcaton 6. System ke MAUI 4 and ConeCoud are based on a remote executon scheme, where devces offoad computatonay ntensve portons of an appcaton to a remote server that returns the computed resut. Athough the aforementoned remote executon systems exhbt potenta for massve performance gans, oftentmes mtary or dsaster reef scenaros do not aow for the uxury of beng abe to permt hgh atences assocated wth offoadng computaton to remote servers, ether due to bandwdth or envronmenta constrants. Addtonay, n these scenaros, the devces may not be connected to a remote server at any gven tme. The concept of cyber foragng, where devces use nearby shared computng resources, was presented to provde an aternatve to the hgh atences assocated wth offoadng computaton to remote servers 7. Whe cyber foragng s presented n the context of pubc, untrustworthy and unmanaged computng resources, ths practce coud be transated to mtary or dsaster reef scenaros provded that the resources are depoyed over an area of nterest a-pror. On the other hand, Hyrax 8 proved the beneft of utzng a of the nodes n a network. By portng Hadoop Apache to mobe devces, the authors were abe to create a dstrbuted computng fe system and enabe dstrbuted computng on mobe devces. Hyrax acheves ths by utzng a centra server and nfrastructure based communcatons. Startng wth the work presented n 8, severa works presented soutons that are abe to offoad computaton to other nearby mobe devces n a compete ad hoc manner Honeybee, for exampe, provdes a programmng framework to ade mobe to mobe computatona offoadng, usng Buetooth to factate the communcaton between devces 9. More recenty, the authors n 1, 11 proposed and mpemented a mobe computatona offoadng system usng Androd s mpementaton of WF Drect. Whe these works have shown the feasbty of offoadng to mobe devces n an ad hoc settng, ther mpementatons are mted to a snge hop ad hoc network. The next step n the evouton of mobe computaton offoadng s to consder a the computatona resources avaabe n a mut-hop ad hoc

2 network. In ths regard, Serendpty enabes remote computng among mobe devces over ntermttenty connected ad hoc networks 1, usng a technque smar to those used n dsrupton toerant networks 13. By utzng a modfed verson of Djkstra s routng agorthm, the authors presented a water fng based task assgnment scheme, that uses an estmate of the tme requred to transfer data between two nodes to cacuate the number of tasks to assgn to a partcuar node. Serendpty s performance has aso been verfed through an mpementaton on Androd devces. Gven the above, our work ams at overcomng the mtatons of the prevous work by expotng a the computatona resources avaabe n a mut-hop ad hoc network. In partcuar, our contrbutons n ths paper are: Usng an mpementaton on Androd devces, we demonstrate the beneft of expandng the number of computatona resources that can be utzed for task competon, and show that n a rea appcaton, a smpe task dstrbuton s abe to reduce the tota tme of the dstrbuted computaton. We deveop a provaby optma agorthm for dstrbutng tasks to nodes wthn a mut-hop network that can be used to estmate the performance of offoadng the computaton over a mut-hop ad hoc network under dfferent settngs. Hence, the overa concuson of ths paper s that not ony s t feasbe to offoad computaton to nodes that are mutpe hops away, but there s aso a measurabe performance gan that can be acheved by dong so. The rest of the paper s organzed as foows. In Secton II, we descrbe our mobe to mobe computatona offoadng system, and motvate the need for an ntegent task assgnment. In Secton III, we present an anaytca mode for cacuatng the amount of tme and energy that s requred to assgn tasks to dfferent nodes n a mut-hop network. In Secton IV, we defne an objectve functon representng the cost to compute some number of tasks, as we as a provaby optma teratve agorthm for optmzng ths objectve functon. Secton V presents measured and anaytc resuts for computaton offoadng n a mut-hop WF Drect network. Secton VI concudes the paper. II. MOTIVATION Our mobe to mobe computatona offoadng system s composed of a set of N mobe devces, organzed nto a network through ad hoc communcatons. The devces are cooperatve, meanng that we do not consder the presence of any sefsh user. Moreover, we assume the exstence of a goba routng protoco, and each node has knowedge of a other avaabe nodes wthn the network. At a certan pont n tme, one of the devces needs to run a compex appcaton, whch can be dvded nto ceary defned homogenous tasks, as s the case for a mtary SIGINT appcaton 3, or non-nvasvey determnng the structura ntegrty of a budng durng dsaster reef efforts 14. These tasks are ndependent from each other, and ther computaton Gan over offoadng to one neghbor % Group -Hops 3-Hops Group -Hops Network 3-Hops Network Unform Group Unform -Hops Network Unform 3-Hops Network Fg. 1. Expermenta measurement of the percent gan over offoadng to ony a snge neghbor of a smpe greedy and unform task dstrbuton scheme. can be paraezed by offoadng the tasks to dfferent devces. The resuts of a the tasks executon s then merged n order to compete the nta compex appcaton. As a resut, assgnng a task to a devce requres the transmsson and recepton of the task and the resuts of the task executon over the ad hoc route. Ths requres communcaton energy consumpton on a the nodes aong the routng path, and ncurs a tme deay due to the propagaton of the data throughout the network. In order to determne the beneft of utzng mut-hop mobe to mobe computaton offoadng, we mpemented a computatona offoadng system on Androd devces, utzng the WF Drect protoco 1. We note that the functonates for creatng mut-group networks usng WF Drect are not natvey mpemented n Androd 16. Nevertheess, the work presented n 16, 17 provde dfferent soutons (e.g., tme sharng between groups, broadcastng and mutcastng data between groups and modfcatons of the Androd OS) to seamessy enabe mut-group communcaton usng stock and non-stock Androd devces. Gven the above, we deveoped a testbed by nterconnectng severa 13 Nexus 7s (N7), usng the modfed verson of Androd presented n 16. The 13 N7 has 16 GB of storage, GB of memory, and a 1.GHz quad-core Snapdragon CPU 18. For a the resuts presented n ths paper, we mted each devce to compute ony one task at a tme, and the devce coud not request the next task unt t returned the resut. Each of the presented data ponts are an average of thrty tras. A of the measurements were taken n an ndoor workspace, and the nodes were statonary durng these experments. We acknowedge that n rea scenaros, nodes are generay mobe, and may enter or eave a network. However, to accuratey determne the mpact of the task assgnment and to mt the varabty across dfferent experments, we kept the nodes statonary. Moreover, a the resuts of ths paper have been obtaned wth the screen turned off. To evauate the performance of the envsoned mut-hop mobe to mobe computaton offoadng system, n Fgure 1 we compare the expermenta performance gan to compete a set of tasks reatve to a task dstrbuton that ony utzes one addtona devce wth extendng the computaton to a arger network (see Fgure 1). For assgnng the tasks, we mpemented both a smpe task dstrbuton, where

3 each devce n the network requests a new task as soon as t has competed the prevous task, and a Unform task dstrbuton, where the tasks are unformy dstrbuted throughout the network. Whe both task assgnments provde substanta performance gan compared to offoadng to a snge neghbor when the computaton tme s much arger then the tme requred to assgn the tasks and receve the resuts, extendng the computaton to the mut-hop network can be detrmenta when the communcaton tme s greater than the computaton. As a resut, t s not straghtforward to determne how to best assgn the tasks due to the communcaton costs of dstrbutng the tasks. Moreover, whe both the Unform and the smpe task dstrbuton aready provde gans n some cases, t s not possbe to drecty determne the performance of a dstrbuton before assgnng the tasks and, at the same tme, t s not cear f further performance mprovements are even possbe. Thus, n the next sectons we frst mode the dfferent eements of the system under consderaton and then present an teratve agorthm that s guaranteed to return the optma task assgnment. III. SYSTEM MODEL In ths secton we present the detas of the dfferent components of the system descrbed n Secton II, whch are then used to derve the optma task dstrbuton pocy presented n Secton IV. A. Task Tme Mode We start by consderng the tota tme D requred to compute a gven task at a partcuar devce. We defne ths tota tme as the tme between the task assgnment and the end of the transmsson of the resut of the task executon. Thus, n order to determne the tme requred to assgn, compute and then receve the resuts of the task executon we need to consder two man components, namey the communcaton tme requred to dstrbute the task and receve the resuts, and the tme actuay spent for the task executon. In order to derve the communcaton tme, we consder that each task s characterzed by three parameters, namey the task data sze t, the reatve resut data sze r and the task compexty c. Thus, accordng to ths defnton, assgnng a task(t, c, r) to a devce requres the transmsson/recepton of t bts, whe the resut of the task executon to be reported back to the task generator entas the transmsson/recepton of r bts. The tme requred to execute a snge task depends on the CPU of the mobe devce and on the compexty c of the task to be executed, whch, wthout oss of generaty, can be consdered as a functon e (c). Therefore, the tota tme requred to compute a snge task at devce depends on the devce s computatona capabtes, and on the characterstcs of the rado technoogy used for the data exchange, as we as on the number of hops H between the devce that generated the tasks and devce. Gven the above, the tota deay D (t,c,t) experenced by a task (t,c,r) assgned to devce s gven by ( H D (t,c,r) = t + r ) H 1 + T sw +e (c), (1) wheret tx represents the transmsson throughput of the communcaton hop, represents the communcaton hop n the reverse path, and T sw accounts for addtona swtchng tme when routng the data between two subsequent communcaton nks as can be the case, for exampe, wth the WF Drect protoco (see, e.g., 16). We note that the transmsson throughput aso accounts for parae use of the same communcaton nk (by, e.g., the smutaneous assgnment of dfferent tasks) and, snce the throughput on wreess networks fuctuates due to the sgna strength and channe mparments, assgnng vaues to the transmsson throughput ony serves as an exampe of the achevabe performance of a network. B. Task Energy Mode In most cases, the system descrbed n ths paper w consst of battery operated mobe devces. As a resut, t s mportant not ony to characterze the deay experenced by the computaton of each task, but aso to determne the mpact n terms of energy consumpton of the task dstrbuton. Therefore, n what foows we defne the tota energy consumpton requred to assgn a task to a partcuar devce. Smar to the task tme mode defned n the prevous secton, dstrbutng a task entas a communcaton energy consumpton and the task executon energy consumpton. In order to defne the communcaton energy mode, we frst defnep tx andp rx to be the transmsson and recepton power consumpton of a partcuar ad hoc communcaton nk, respectvey. We note that a partcuar nk actuay represents a par of nodes (h, k), where h s the devce transmttng the data and k s the devce recevng the data. Thus, wth a tte abuse of notaton we can wrte P tx = Ph tx and P rx = Pk rx, where Ptx h represents the power consumpton of mobe devce h durng transmsson, and Pk rx represents the power consumpton of mobe devce k durng recepton. In the same way, we defne P ex to represent the power consumpton of mobe devce durng task computaton. As a resut, we defne the tota energy expendture when assgnng a task (t,c,r) to devce as H E (t,c,r) = t H 1 + (P tx +P rx )+ r E sw +P ex e (c), (P tx +P rx ) where, smar to Eq. (1),represents the communcaton hop n the reverse path (e.g., f represents the par of nodes(h,k), refers to the par (k,h)), and E sw represents the amount of energy requred for swtchng when routng the data between two subsequent communcaton nks. IV. OPTIMAL TASK DISTRIBUTION The goa of our anayss s to fnd the optma task dstrbuton pocy that mnmzes an overa cost metrc by utzng a the avaabe nodes n a mut-hop network. In the foowng sectons, we frst formuate the optma tasks dstrbuton ()

4 probem, and then we present an teratve agorthm that s abe to fnd the optma task assgnment. A. Optmzaton Probem Let N be the number of mobe devces partcpatng n the dstrbuted computaton and M be the number of homogeneous tasks 1 to be dstrbuted. We defne φ to be a vector such that φ = φ,φ 1,...,φ N, where φ represents the number of tasks assgned to a partcuar devce, such that φ Z +, φ M and N =1 φ = M, and Φ to be the set of a the possbe task assgnments that satsfy these condtons. Moreover, et C = C 1,C,...,C N be a cost vector, where C wth = 1,...,N represents the cost of assgnng a task to devce. Gven the above, our objectve s to sove the foowng optmzaton probem: mnf(φ), (3) φ Φ where F(φ) = max {φ C } represents the cost of a partcuar task assgnment φ Φ. We note that F(φ) accounts for the assgnment of tasks to botteneck nodes and s sutabe for, e.g, fndng the mnmum tme requred to perform the dstrbuted computaton of the M tasks. B. Proposed Pocy The optmzaton probem defned n Eq. (3) can be seen as a varaton of the Lnear Botteneck Assgnment Probem (LBAP) 19. Dfferent from a tradtona LBAP, we consder the possbty of assgnng to a partcuar agent (.e., devce) more than one task. We note that whe dfferent agorthms for sovng the LBAP probem have been proposed n the terature, e.g.,, aowng an agent to execute more than one task adds addtona compexty to the probem. At the same tme, the assumpton of homogeneous tasks aows us to smpfy the probem, whch can be optmay soved wth an teratve agorthm. In what foows, we frst propose an agorthm to sove probem (3), and then prove the optmaty of the souton determned by our agorthm. In partcuar, we am to fnd an optma pocy φ for probem (3). To ths end, et X α be a vector of ength N, so that X α represents the number of tasks that have been assgned to devce up to the assgnment of task α, wth α M. In addton, et Y be a decson vector of ength N, so that a but one of the eements of Y are zero. Gven the above, the agorthm for fndng the optma task assgnment s defned n Agorthm 1. C. Performance Anayss To prove that Agorthm 1 s abe to determne an optma souton to the optmzaton probem defned n Eq. (3), we frst show that startng from an optma task assgnment for K tasks, t s possbe to construct an optma task assgnment for K +1 tasks. 1 The extenson of the resuts of ths paper to the case of non-homogeneous tasks s consdered as future work. Agorthm 1 Cost Optma Task Dstrbuton 1: X ={,,,...,} : for Each Task α 1,...,M do 3: Y={,,,...,} 4: = argmn((x α 1 + 1) C) : Y = 1 6: X α = X α 1 +Y 7: end for Theorem IV.1. Gven an optma K tasks assgnment Γ K, an optma K +1 tasks assgnment Γ K+1 can be obtaned as Γ K+1 = argmn ψ Ψ K+1 max {ψ C = 1,...,N}, where Ψ K+1 = {ψ }, ψ = Γ K + ǫ and ǫ s an a-zeros vector wth a one n poston, for each = 1,...,N. Proof. We w prove ths n two steps. In the frst step, we consder the case where Γ K s the ony optma task assgnment for K tasks, whe n the second step, we consder the case of mutpe K task assgnments that have the same optma cost. For the case where there s a unque optma souton to dstrbute K tasks, et s assume by contradcton that Γ K+1 determned accordng to the above defnton s not optma. Ths, n turn, means that there exsts a K+1 tasks assgnment Θ K+1, such that Θ K+1 / Ψ K+1 and F(Θ K+1 ) < F(Γ K+1 ). (4) Moreover, we can consder Θ K+1 to be derved from a K tasks assgnment Θ K, such that F(Γ K ) < F(Θ K ) and F(Θ K ) F(Θ K+1 ). () As a resut, by combnng (4) and (), we obtan F(Θ K ) < F(Γ K+1 ). (6) On the other end, assumng that F(Θ K ) = Θ K m C m (.e., devce m s the botteneck devce that determnes the overa cost of the non-optma task assgnment Θ K ), we have that Γ K m < ΘK m. Thus, ΓK m +1 ΘK m. Now, f ΓK m +1 < ΘK m, then F(Γ K + e m ) = F(Γ K ) < F(Θ K+1 ), whch contradcts the hypothess n (4). If Γ K m +1 = ΘK m, nstead, F(ΓK +e m ) = F(Θ K ) F(Θ K+1 ), whch st contradcts the hypothess n (4). As a resut, f Γ K s the unque optma K tasks assgnment, then the K + 1 task assgnment Γ K+1 obtaned by Theorem IV.1 s an optma K +1 tasks assgnment. For the case where F(Γ K ) = F(Θ K ), there exst at east one Γ K m, m = 1,...,N such that Γ K m < Θ K m snce Γ K Θ K. Thus, Γ K m +1 ΘK m whch, foowng an argument smar to the one descrbe above, resuts n F(Γ K+1 ) = F(Γ K +e m ) F(Θ K+1 ), whch contradcts the hypothess n (4). Therefore, Γ K+1 obtaned by Theorem IV.1 s an optma K +1 tasks assgnment.

5 We w now use the resut of Theorem IV.1 to show that the agorthm presented n Secton IV-B returns an optma souton to the optmzaton probem (3). Theorem IV.. Agorthm 1 yeds an optma souton to the optmzaton probem defned n Eq. (3). Proof. In what foows, we w prove by nducton on the number of tasks M that the souton determned by Agorthm 1 s an optma souton to the optmzaton probem defned n Eq. (3). For the case M = 1, the set of a possbe tasks assgnment s represented by Φ = {1,,...,,,1,...,,...,,,...,1}; as a resut, we can easy see that C 1 opt = mn φǫφ max {φ C } = mn( 1 C), (7) where 1 s a vector of a ones. Thus, C 1 opt can further be smpfed to C 1 opt = C, (8) where = argmn( 1 C). Now to prove that our agorthm s capabe of yedng an optma souton for M = 1 task, et C ago be the cost of the souton produced by the agorthm, defned as C 1 ago = max { X 1 C }. (9) Snce X s ntazed to a vector of zeros, as shown n step of our agorthm, Cago 1 smpfes to Cago 1 = max {(X +1)C } {(Xj +)C j j }, (1) where X s the devce whch s chosen at steps 4 and of our agorthm, and X j s the subset of X that were not chosen at step 4 (.e., by the argmn functon). Eq. (1) can further be rewrtten as Cago 1 = max {1 C } { C j j } = C, (11) whch, compared wth Eq. (8), shows that Cago 1 =C1 opt. Let s now assume thatcago K = CK opt up tom = K, and that X K = Γ K, where Γ K s the optma souton for M = K. For the case where tasks M = K +1 we can wrte opt = mn φǫφ max {φ C }. (1) However, as shown n Theorem IV.1, we can restrct the search of the optma souton to the set Ψ K+1, whch has reduced cardnaty to N. Ths means that we can rewrte Eq. (1) as opt = mn ψǫψ K+1 max {ψ C }, (13) whch n turn can be expanded and utmatey yeds three possbe outcomes,.e., opt = mn Γ K g Cg S.T.(ΓK +1)C =Γ K g Cg (Γ K g +1)Cg (Γ h +1)C h hs.t.(γ K h +1)C h>γ K g Cg, (14) where Γ K g C g = C K opt, whch can be rewrtten as opt = max {(Γ K +1)C } {(Γ K j +)C j j }, (1) where (Γ K +1)C, = argmn((γ K +1)C). Now for the souton determned by the agorthm, we can wrte: ago = max{xk+1 C = 1,...,N}, (16) whch n turn can be smpfed to ago = max {(X K +1)C } {(X K j +)C j j } (17) whch, combned wth Eq. (1), returns ago = Copt K+1. Thus, accordng to Theorem IV.1, the K+1 tasks assgnment determned by the agorthm s guaranteed to be optma, whch concudes the proof. V. NUMERICAL RESULTS Usng the testbed descrbed n Secton II, we compared the task dstrbuton found usng the agorthm presented n Secton IV-B wth both the greedy and unform task dstrbuton agorthms descrbed prevousy. Each tra was started after the correct task dstrbuton, f appcabe, was found and was stopped when the ast resut of the computaton was receved. We note that, when optmzng for energy, the optmzaton probem presented n Secton IV can be smpfed to mnmze the tota energy consumpton of the task assgnment. Nevertheess, the cost-optma task dstrbuton presented n Secton IV mnmzes the maxmum energy consumpton requred for assgnng tasks to each devce, thus fary dstrbutng the tasks throughout the network. A. Performance Evauaton For a the resuts presented n ths secton, we consder a homogeneous network and defne the cost vector C = C 1,C,...,C N of the optmzaton probem n Eq. (3) such that C = D (t,c,r), where t = 13B, r {1KB,1KB,1MB} and c s determned so that e (c) {1ms,1ms,1s,1s}. Whe we do not ncude the energy consumpton at each devce (.e.,e n Eq. ()) n the defnton of the cost vector C, when our agorthm fnds mutpe task assgnments that resut n the same deay cost, we seect the task assgnment that mnmzes the overa network energy consumpton. Fnay, we note that dfferent defntons of the cost vector C are aso possbe. As an exampe, the cost C defned above can addtonay ncude a eve of reabty R,1, thus becomng C = D (t,c,r)/r. 1) Offoadng Resuts: In order to vadate our mode, n Fgures -4 we pot the tme requred to compute a set of tasks as a functon of dfferent computaton/communcaton tme ratos, for both an Androd mpementaton of the task dstrbuton as we as a smuaton of the task dstrbuton usng the mode descrbed n Secton III. In these fgures, the dashed nes represent the smuated performance of the task dstrbuton obtaned through the teratve agorthm descrbed,

6 Tme ms Iteratve Average Throughput Unform Average Throughput Iteratve Hgh Throughput Unform Hgh Throughput Iteratve Low Throughput Unform Low Throughput Impemented Iteratve Impemented Unform Impemented Expermenta Resuts Tme ms Iteratve Average Throughput Unform Average Throughput Iteratve Hgh Throughput Unform Hgh Throughput Iteratve Low Throughput Unform Low Throughput Impemented Iteratve Impemented Unform Impemented Expermenta Resuts e (c)=1s 1 e (c)=1ms e (c)=1ms e (c)=1s e (c)=1s 4 e (c)=1ms e (c)=1ms e (c)=1s Fg.. Impementaton and Smuaton resuts for dfferent computaton/communcaton ratos. Resut sze s 1 MB. Fg. 3. Impementaton and Smuaton resuts for dfferent computaton/communcaton ratos. Resut sze s 1 KB. n Secton IV-B, whe the contnuous nes represent the smuated performance of a unform task dstrbuton for dfferent vaues of transmsson throughputs. In partcuar, we set = {7Mbps,Tr avg,.mbps} n Eq. (1), wheretr avg s the average expermenta throughput measured for the dfferent data sze r (.e., T avg 1KB =.87Mbps, Tavg 1KB = 8.4Mbps and T avg 1MB = 3Mbps). The squares, damonds and stars, nstead, represent the resuts obtaned by mpementng the Iteratve, Unform and the task dstrbuton poces n our rea network of Androd devces, respectvey. The resuts n Fgures -4 show that when the computaton takes about tmes onger than the communcaton, a unform task dstrbuton provdes the same performance as the optma task dstrbuton. Beow ths pont, there are cear benefts n usng the task dstrbuton found wth our teratve agorthm, wth gans that become more evdent when the tme spent computng s comparabe to the communcaton tme. When the communcaton takes sgnfcanty onger than the computaton, the optma task dstrbuton can compete the set of tasks about twce as qucky as the unform task dstrbuton (.e., resut sze of 1MB and computaton tme of 1ms). Moreover, these resuts vadate the mode presented n Secton III, and show that the smuated resuts can provde a good approxmaton of the mpementaton resuts so ong as the approprate average nk throughputs are used. Hence, our teratve agorthm can be used to expore the performance of computaton offoadng n mut-hop ad hoc networks. The smpe task dstrbuton descrbed n Secton II s abe to adapt to the nstantaneous varatons n computatona tme (due to, e.g., operatng system operaton unreated to the actua task executon), as we as to the channe mparments that can severey affect the actua throughput of the communcaton nks. As a resut, the task dstrbuton s abe to attan performance very cose to the smuated performance of the optma task assgnment wth the hgh WF Drect throughput. When mnmzng the tota competon tme, thanks to ts adaptabty to the nstantaneous system varatons, the task dstrbuton outperforms the mpementaton of the optma task assgnment, thus makng t the de-facto choce for rea devce mpementatons. To gan further nsght nto the dfferences between the Tme ms Iteratve Average Throughput Unform Average Throughput Iteratve Hgh Throughput Unform Hgh Throughput Iteratve Low Throughput Unform Low Throughput Impemented Iteratve Impemented Unform Impemented e (c)=1ms Expermenta Resuts e (c)=1ms e (c)=1s e (c)=1s Fg. 4. Impementaton and Smuaton resuts for dfferent computaton/communcaton ratos. Resut sze s 1 KB. task dstrbuton and our teratve agorthm, n Fgure we compare the average task assgnments that were used n each case. As can be seen n Fgure (a), both the teratve and task dstrbuton schemes start assgnng tasks n a unform way when the computaton tme (1s) s much onger than the tme spent communcatng. When the stuaton s reversed (.e., communcaton s much onger than the 1ms computaton), nstead, n some cases no tasks are actuay assgned to the furthest node (see Fgure (d)). Overa, these resuts show that the throughput used to compute the optma task assgnment s over estmated, whch s partcuary evdent by the fact that the devce that s generatng the tasks (.e, the devce at hops), s most of the tme computng fewer tasks than when usng the approach. ) Benefts of Offoadng to Mut-hop Neghbors: The resuts presented n the prevous secton show that a task assgnment can adapt to changng computaton and communcaton envronments and hence can return a the tasks n the east amount of tme. However, our teratve agorthm provdes a task dstrbuton that s cose to that provded by the agorthm. Addtonay, the resuts n the prevous secton show that the smuated resuts match the mpementaton resuts and, as a consequence, smuatons based on the mode from Secton III can be used to provde nsght nto the performance of the system. Thus, n what foows we use the anaytca mode and the teratve agorthm descrbed n Sectons III-IV to further expore the gans that can be acheved by extendng the dstrbuted computaton to

7 hops 1 hop hops 3 hops 4 hops (a) 1s cacuaton Iteratve hops 1 hop hops 3 hops 4 hops (c) 1ms cacuaton Iteratve Iteratve hops 1 hop hops 3 hops 4 hops (b) 1s cacuaton Iteratve hops 1 hop hops 3 hops 4 hops (d) 1ms cacuaton Fg.. Task dstrbuton for the and Iteratve approach wth resut sze fxed to 1 MB. Gan over offoadng to one neghbor % Group -Hops 3-Hops 4-Hops Snge Group -Hops Network 3-Hops Network 4-Hops Network Fg. 6. Smuated measurement ndcatng the percent gan over offoadng to ony a snge neghbor. a the avaabe network resources. In partcuar, n order to hghght the benefts of offoadng the computaton to a the mobe devces n a network, n Fgure 6 we compare the gan n tme to compete a tasks reatve to a task dstrbuton that ony utzes one addtona devce (.e., snge-hop task dstrbuton as consdered n the terature 9 11) wth: a) offoadng the computaton to a of the nearest neghbors of the devce that s generatng the tasks (.e., Snge Group), and b) offoadng the computaton to a mut-hop network (.e., -hops, 3-hops and 4-hops network). As shown n Fgure 6, n ths smuaton we consder a treeke network wth the root node generatng the tasks havng four chdren nodes, and each subsequent chd servcng one addtona node. Ths network extends for, 3 or 4 hops, resutng n a tota of 4 devces n the source node s group that can be used for group task dstrbuton, and 8, 1 and 16 devces that can be used for network task dstrbuton. Fnay, Fgure 6 ceary shows the beneft of offoadng to a mut-hop network. In partcuar, offoadng to the arger network provdes up to 3% faster computaton tme than offoadng the tasks ony to the frst group, and a gan of up to 88% aganst offoadng to ony a snge devce (as s currenty supported n the terature). VI. CONCLUSIONS In ths paper we expored the benefts of enhancng mobe to mobe computatona offoadng n mut-hop cooperatve networks. By mpementng a mut-hop computatona offoadng system, we were abe to mpement dfferent task dstrbuton agorthms and verfy the accuracy of an anaytc mode and mpemented dfferent task dstrbuton strateges. Usng ths mode, we were abe to show the overa beneft of enabng offoadng to mut-hop neghbors n a network, whch can be qute arge when the tme to compute s much hgher than the tme to communcate the data. In our future work, we w expore the nter-reaton between reay nodes and nodes that are assgned tasks. 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