This is a repository copy of Enabling low-latency applications in LTE-A based mixed fog/cloud computing systems.

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1 This is a repository copy of Eablig low-latecy applicatios i LTE-A based mixed fog/cloud computig systems. White Rose Research Olie URL for this paper: Versio: Accepted Versio Article: Du, J., Zhao, L., Chu, X. orcid.org/ et al. 3 more authors) 18) Eablig low-latecy applicatios i LTE-A based mixed fog/cloud computig systems. IEEE Trasactios o Vehicular Techology. ISSN IEEE. Persoal use of this material is permitted. Permissio from IEEE must be obtaied for all other users, icludig repritig/ republishig this material for advertisig or promotioal purposes, creatig ew collective works for resale or redistributio to servers or lists, or reuse of ay copyrighted compoets of this work i other works. Reproduced i accordace with the publisher's self-archivig policy. Reuse Items deposited i White Rose Research Olie are protected by copyright, with all rights reserved uless idicated otherwise. They may be dowloaded ad/or prited for private study, or other acts as permitted by atioal copyright laws. The publisher or other rights holders may allow further reproductio ad re-use of the full text versio. This is idicated by the licece iformatio o the White Rose Research Olie record for the item. Takedow If you cosider cotet i White Rose Research Olie to be i breach of UK law, please otify us by ig eprits@whiterose.ac.uk icludig the URL of the record ad the reaso for the withdrawal request. eprits@whiterose.ac.uk

2 1 Eablig Low-Latecy Applicatios i LTE-A Based Mixed Fog/Cloud Computig Systems Jiabo Du, Liqiag Zhao Member, IEEE, Xiaoli Chu Seior Member, IEEE, F. Richard Yu Fellow, IEEE, Jie Feg, ad Chih-Li I Seior Member, IEEE Abstract I order to eable low-latecy computatioitesive applicatios for mobile user equipmets UEs), computatio offloadig becomes critical ecessary. We tackle the computatio offloadig problem i a mixed fog ad cloud computig system, which is composed of a LTE-A smallcell based fog ode, a powerful cloud ceter, ad a group of UEs. The optimizatio problem is formulated ito a mixediteger o-liear programmig MINLP) problem, ad through a joit optimizatio of offloadig decisio makig, computatio resource allocatio, resource block RB) assigmet, ad power distributio, the maximum delay amog all the UEs is miimized. Due to its mixed combiatory, we propose a low-complexity iterative suboptimal algorithm called to solve it. I, first, offloadig decisios are obtaied via biary tailored fireworks algorithm FA); the computatio resources are allocated by bisectio algorithm. Limited by the uplik LTE-A costraits, we allocate feasible RB patters istead of RBs, ad the distribute power amog the RBs of each patter, where Lagragia dual decompositio is adopted. Sice oe UE may be allocated with multiple feasible patters, we propose a ovel heuristic algorithm for each UE to extract the optimal patter from its allocated patters. Simulatio results verify the covergece of the proposed iterative algorithms, ad exhibit sigificat performace gais could be obtaied compared with other algorithms. Idex Terms Computatio offloadig, fireworks algorithm, fog computig, LTE-A, resource allocatio. I. INTRODUCTION With the proliferatio of smart user equipmets UEs) ad the popularity of low-latecy applicatios [1], the curret mobile etworks have bee pushed to their limits. Mobile cloud computig MCC) [] has appeared as a potetial way to cope with the above challeges by offloadig computatios to powerful cloud servers. More recetly, fog computig [3] or mobile edge computig MEC) []) has bee put forwarded as a effective complemet to MCC ad has bee deemed as a importat paradigm ad sceario i 5G [5]. *This work was supported i part by Natioal Natural Sciece Foudatio of Chia ), Natioal Natural Sciece Foudatio of Shaaxi Provice 18JM5), Itergovermetal Iteratioal Cooperatio o Sciece ad Techology Iovatio 1YFE13), the Fudametal Research Fuds for the Cetral Uiversities, ad the 111 Project B838). J. Du, L. Zhao, J. Feg are with State Key Laboratory of ISN, Xidia Uiversity, No. Taibaia-lu, Xi a, 771, Shaaxi, Chia. dujiaboo@13.com; lqzhao@mail.xidia.edu.c; jiefegcl@13.com). X. Chu is with Departmet of Electroic ad Electrical Egieerig, The Uiversity of Sheffield, Mappi Street, Sheffield, S1 3JD, UK. x.chu@sheffield.ac.uk). F. R. Yu is with the Dept. of Systems ad Computer Eg., Carleto Uiversity, Ottawa, ON, Caada Richard.Yu@carleto.ca). C.-L. I is with the Gree Commuicatio Research Ceter, Chia Mobile Research Istitute, Beijig 53, Chia. icl@chiamobile.com). By settig up a virtualized platform betwee UEs ad cloud ceters, fog computig ca provide computatio, storage, ad etworkig services [], [7] to earby UEs, ad thus to further ehace etwork performace i eergy coservatio or delay reductio [5]. Fig.1 shows the typical architecture of a mixed fog/cloud computig system. Utilizig the computatio resources of the fog odes, such as WiFi access poits APs), base statios BSs), or remote radio heads RRHs), fog odes ca offer computatio services at the edge of the etwork [3], []. Fog odes ca commuicate directly with each other, ad all the fog odes are coected to the powerful cloud server through high-speed wired liks [3], []. The cooperatio betwee the cloud server ad the fog odes ca provide users with more efficiet ad appropriate computatio offloadig services. However, this ew architecture brigs may ew problems, e.g., how does the fog cooperate with the cloud, i.e., where should computatio be offloaded to, ad how the resource be allocated, etc., so as to brig the advatages of the ew architecture ito full play. I this paper, i order to eable low-latecy computeitesive user applicatios with fairess amog UEs guarateed, we propose to miimize the maximum delay cosumptio amog all UEs i a LTE-A based mixed fog/cloud computig system by joitly optimizig computatio offloadig, computatio resource allocatio, uplik RB assigmet ad trasmit power allocatio. Sice i the LTE-A uplik, if a UE is assiged with multiple RBs, they must be adjacet RBs [8], [9], so we allocate feasible RB patters to UEs. Each UE the picks out the optimal patter from all the assiged feasible RB patters ad the perform power allocatio o the RBs of the selected patter. As the joit optimizatio problem is a mixed iteger o-liear programmig MINLP) problem, we devote to develop low-complexity suboptimal algorithms to decouple it ito several subproblems to solve. The mai cotributios of this paper are listed as follows. We propose a ovel geeral iterative algorithm framework called biary tailored fireworks algorithm based joit computatio offloadig ad resource allocatio algorithm ) to solve the joit optimizatio problem, where offloadig decisios are first decoupled from the rest of the problem ad obtaied through biary tailored fireworks algorithm. We develop a bisectio algorithm for computatio resource allocatio, which is ested i. We solve uplik RB patter ad power allocatio problem, which is still NP-hard, by relaxig, Lagragia dual

3 decompositio, ad sub-gradiet projectio methods, to obtai the optimal UE ad power allocatio for each feasible RB patter, where each UE may be allocated with multiple feasible patters. We the develop a ovel heuristic algorithm to extract the optimal patter for each UE from all its feasible patters, takig the exclusiveess required by RB allocatio ad higher RB utilizatio ito cosideratio, ad thus to obtai more performace gais. The remaider of this paper is orgaized as follows. Related works are preseted i Sectio II. Sectio III itroduces the system model ad problem formulatio. I Sectio IV, we illustrate the procedure ad geeral structure of. The computatio resource allocatio algorithm is detailed i Sectio V. I Sectio VI, we first preset the RB patter ad power allocatio algorithm, ad the detail the heuristic patter extractig algorithm. Complexity aalysis is preseted i Sectio VII. Simulatio results are provided i Sectio VIII. Fially, the paper is cocluded i Sectio IX. Femto enb Fog ode Remote Radio Head RRH) Iter Fog ode lik Backboe Lik Femto Gateway Cloud F-RAN Cloud Ceter Core Network Macro enb Fig. 1: System architecture of a mixed fog/cloud computig system. II. RELATED WORKS I sigle-ue case [], [11], task partitioig ad offloadig decisio is usually optimized i order to maximize eergy savigs [] or to miimize eergy cosumptio [11]. I the most geeral multi-user scearios, computatio resources ad commuicatio resources e.g., badwidth, resource blocks, ad subcarriers) are shared amog UEs. Therefore, except for offloadig decisios, resource allocatio is aother importat issue eeds to be ivestigated. I [1], game theory was utilized i a MCC eviromet. Accordig to other UEs decisios, each UE optimized its offloadig decisio ad thus to miimize its weighted cost. I [13], offloadig decisios were optimized for all UEs to miimize the etwork eergy cosumptio i a MCC system. The authors i [1] ivestigated trasmit power optimizatio uder give offloadig decisios, i order to miimize the system eergy cosumptio. The formulatio i [15] combied task level offloadig decisio optimizatio ad trasmit power allocatio i multiuser MCC ad MEC scearios, respectively, to miimize a weighted system cost of delay ad eergy cosumptio. I [1], except for optimizig offloadig decisios ad trasmit power allocatio, the authors exteded computatio resource allocatio ito the optimizatio framework to further reduce latecy ad eergy cosumptio of all the UEs i a MEC etwork. The followig refereces optimized the allocatio of other forms of radio resources istead of trasmit power. The authors i [17] formulated a joit optimizatio of the offloadig decisio makig, resource block RB) allocatio, ad computatio resource allocatio i the MEC server, with trasmit power give as a costat, to miimize the total weighted cost of delay ad eergy cosumptio of all UEs. I [18], i order to miimize system eergy cosumptio, the authors performed a joit optimizatio of computatio offloadig, subcarrier assigmet, ad computatio resource allocatio i a fog computig system. The authors i [19] formulated a system eergy cosumptio miimizatio problem with the required delay tolerace satisfied i a MCC system, by a joit optimizatio of beamformer desigig, computatio resources allocatio ad offloadig decisio makig. The authors i [] first proposed to joitly optimize the offloadig decisio makig, computatio resource allocatio, ad radio trasmit rate allocatio pioeerigly i a MCC system, i order to coserve eergy while satisfyig user delay costraits, while radio resources were allocated i a coarse-graied uit of bit/s. To summarize, [1], [13] oly optimized offloadig decisios, [1] oly optimized trasmit power allocatio, [15] combied the two aspects for further optimizatio, ad [1] [19] itegrated computatio resource allocatio ito the optimizatio framework besides offloadig decisio optimizatio ad radio resource allocatio. However, i [1] [], radio resource allocatio oly covered a certai dimesio of radio resources, such as trasmit power, RB, or subcarrier allocatio, without a joit optimizatio of multidimesioal wireless resources for further performace gais. What s more, applicatios were offloaded either to the cloud server [1] [15], [19], [] or to the fog ode [1] [18], without a cooperatio betwee the both for providig much stroger offloadig services. Moreover, the above related works [13] [] cocered the system-level performace, without cosiderig that of idividual UEs. Cosequetly, UEs with higher trasmit rate will beefit from computatio offloadig, but at the expese of a performace declie of the UEs with lower trasmit rate, givig rise to ufairess amog UEs. III. SYSTEM MODEL AND PROBLEM FORMULATION I this sectio, we first describe the cocered sceario, the discuss the delay cosumptio i local, fog, ad cloud processig modes, respectively, ad fially we formulate our optimizatio problem uder the cocered sceario. A. Descriptio of the Cocered Sceario We cosider a system comprisig N UEs, a LTE-A small cell based fog ode, ad a distat cloud server. The fog ode ad the cloud are joited by a fiber lik, while all the N UEs are coected to the fog ode through wireless liks sharig K RBs i each trasmissio time iterval TTI, occupyig 1 ms [8], [9]). Each RB is allocated to at most oe UE i

4 3 the small cell [8], [9], [1]. We cosider a quasi-static scee where all UEs ad the wireless chaels keep still withi a offloadig period usually several secods [1], cotaiig several thousads of TTIs). The assumptio holds for may actual applicatios such as face recogitio, atural laguage processig, ad so o, where the iput is ot so large that computatio offloadig could be accomplished withi a short time less tha the time duratio of UEs mobility ad wireless chaels variatio. Thus, i the followig, we cosider the offloadig period as the time uit where the optimizatio is performed [11], [1], [1], [1], [], [3], ad all the TTIs i the same offloadig period adopt the same optimizatio results. Each UE has oly oe iseparable applicatio may be executed locally or remotely i applicatio-level through the followig process. Firstly, it seds a offloadig request icludig the iformatio about the applicatio ad the UE itself) to the maager i the fog ode [3]. The maager collects the iformatio about wireless chael states ad the available resources i the fog ode, together with the offloadig requests, it determies the offloadig decisio where the applicatio be processed, i.e., i the UE locally, i the fog, or i the cloud) ad the associated resource allocatio for each UE. The offloadig decisios are the set back to all the UEs, ad the correspodig resources will be allocated to them i offloadig. As a offloadig request is usually very tiy, we suppose that o buffer is eeded for queueig the computatio requests []. Also, the delay i decisio makig is ot cosidered to eable tractable aalysis [3]. Deote the UE set as N, ad the offloadig decisio of UE as x,y,z. Let x = 1,y = 1,z = 1 represet that the applicatio is processed by UE itself, by the fog ode, or by the cloud, respectively; otherwise, x =,y =,z =. Cosequetly, we have x +y +z = 1, N. 1) The offloadig decisios of all UEs is collected i the offloadig matrix Π, which is give by Π = x 1,..., x N y 1,..., y N, where the th colum is the offloadig decisio of UE. z 1,..., z N 3 N The fog ode has the ability to process applicatios, subjectig to its computatio capability. Whe multiple UEs choose fog-processig, the fog ode will allocate computatio resources i CPU cycles/s) to them. Whe cloud-executig is selected by multiple UEs, their applicatios will firstly be trasmitted from the UEs to the fog ode through a shared wireless access lik, ad the be forwarded by the fog ode to the cloud server through a wired fiber lik. Sice the computatio resources i the cloud server is sufficiet, ad the capacity of the wired lik is large eough, the allocatio of those resources the decisios are made at the cloud server) will ot be discussed ad these resources allocated to each UE is give as kow quatities []. However, the limited radio resource eeds to be assiged amog all the remote-executig UEs icludig all the UEs with y = 1 or z = 1). Sice the output of remote processig is usually very tiy, oly the uplik is discussed [11], [1], [1]. The applicatio of UE ca be deoted as Λ = {D,λ }, N, where D represets the iput data size i bits), ad λ deotes processig desity or computatio complexity i CPU cycles/bit) of the applicatio [1]. The umber of CPU cycles C aka.computatio load) required to complete executig the applicatio is modeled asc = D λ, which icreases with both the iput data D ad the processig desity λ. Usig program profilers [], [1], the maager ca obtai D, C ad λ beforehad easily. For each UE there is a cloe i the fog ode, ad the program of Λ is backed up i the cloe [], [7], [11], [3], ad ca be dowloaded easily by the cloud server through the wired lik [], [1], otig that the overhead for settig up ad sychroize the cloe is eglected similar to may existig works [], [7], [11], [3]. Hece, oly the iput data with size D bits will be trasmitted from UE whe offloadig. Feasible RB allocatio patter matrix W = {wk,j } of size K J ca be costructed for each UE, where wk,j = 1 idicates that RB k is allocated to UE i the jth patter, otherwise wk,j =. For example, assumig there are K = RBs, the feasible patter matrix for UE is give by W = K J, N, ) where J is the total umber of feasible patters give by J = 1 K +K)+1 [8]. The uplik RB patter allocatio matrix is formed ass = {s,j } N J, where the biary variables,j = 1 idicates that UE selects patter j, ad otherwise s,j =. If patter j is assiged to UE, usig Shao s formula, the maximum achievable trasmit rate for UE o RB k i patter j ca be expressed as r,kj) = W log 1+ p ),kj)h,kj) σ = W log 1+p,kj) g,kj) ), 3) where W is the badwidth of a RB i Hz); p,kj) is the trasmissio power of UE o RB k i patter j; h,kj) is chael gai, which is assumed to be kow at UE ad remai costat but may chage at the boudary of each offloadig period [1]; g,kj) = h,k σ, ad σ is the power of additive white Gaussia oise. Thus, the maximum achievable trasmit rate ad power of UE o patter j are give by r,j = p,j = r,kj), ) k=1 p,kj), 5) k=1 ad the total achievable trasmit rate ad power of UE are give by r = s,j r,j = j=1 j=1 k=1 s,j r,kj), )

5 p = s,j p,j = j=1 j=1 k=1 s,j p,kj), 7) where p should be less tha the maximum trasmit power p max of UE. It should be oted that i equatios 3), ), ad ), we assume the wireless chael gai of each RB to be a radom value, without specifically cosiderig the impact of the umber of ateas. I the followig we will discuss the delay cosumptio of local, fog ad cloud processig, respectively. B. Delay Uder Differet Offloadig Scearios Deote the local processig capability i CPU cycles/s) of UE as f loc, the the delay of local processig is T loc = C loc. 8) If fog-processig is selected for UE, the, it eeds to trasmit the iput data of size D to the fog ode. Assume fog processig oly starts whe all the iput data has bee received, ad deote the computatio resource allocated to UE by fog i CPU cycles/s), the the delay of UE uder fog processig is give by T fog = D r + C fog. 9) If the applicatio of UE is offloaded to the cloud server, the iput data is first trasmitted to the fog ode, ad the set to the cloud server by the fog ode. Give the rate of the highspeed wired fiber lik ad the cloud processig capability for UE as R fc i bit/s) ad c i CPU cycles/s), respectively, the the wired trasmit delay ad processig delay i the cloud are give by T fc = D /R fc ad T c = C /, c respectively. Thus the total delay of UE for cloud processig is give by T cloud = D r +T fc +T c. ) A summary of the maily used otatios are preseted i TABLE I. C. Problem Formulatio We the formulate the joit optimizatio of computatio offloadig ad resource allocatio ad show its NP-hardess. Based o 8)-), the delay of UE is give by T = T loc x +T fog y +T cloud z. 11) We ited to miimize the maximum delay cosumptio amog all UEs, by joitly optimizig the offloadig decisio Π, the RB patter allocatio S = {s,j } N J, the trasmit power allocatio P = {p,kj) } N K J, ad the computatio resource assigmet f fog = {f fog 1,...,f fog N }. The computatio resources i the fog ode will be allocated oly to the fog-executig UEs, the set of which is deoted as N 1. RBs will be assiged amog all the remote-processig icludig fog-processig ad cloud-processig) UEs, the set of which is deoted as N. The joit optimizatio problem is formulated as P 1 ) : mi max T 1) Π,f fog,s,p N s.t. C1) : x,y,z {,1}, N, C) : x +y +z = 1, N, C3) : F fog, N 1 f fog C) : fog, N 1, C5) : s,j {,1}, j J, N, C) : s,j = 1, j J {j = 1}, N C7) : s,j = 1, N, C8) : j J N j J s,j W 1, k K, C9) : p,kj), N, k K, j J, C) : s,j p,j p max, N. j J TABLE I: Notatio Defiitios Symbol Defiitio T loc, T fog, Delay of UE i local/fog/cloud processig T cloud loc, fog, Processig ability of UE c i local/fog/cloud processig D,C,λ Data size/computatio load/processig desity of the applicatio of UE F fog Total computatio capability of the fog ode p max The maximum trasmit power of UE g,kj) Power gai of UE o RB k i patter j s,j Idicator whether patter j is allocated to UE r,p Wireless trasmit rate/power of UE R fc Wired lik rate of UE betwee fog ad cloud r,j,p,j Trasmit rate/power of UE o patter j r,kj) Trasmit rate/power of UE p,kj) o RB k i patter j S,P RB Patter/power allocatio matrix x, y, z Offloadig decisios of UE Π Matrix of all UEs offloadig decisios N,N The set/umber of all UEs N 1,N 1 The set/umber of fog-executig UEs N,N The set/umber of remote-executig UEs K, K The set/umber of RBs J,J The set/umber of all feasible patters W Feasible RB patter matrix for UE I,M,γ Number of fireworks/total explosio sparks/ mutatio sparks L Number of iteratios of FA I problemp 1 ), C1) ad C) give each UE the restraits o its offloadig decisios; C3) shows that the total allocated computatio resource should be less tha the total computatio capability F fog i the fog ode; C) idicates that the computatio resource allocated to each fog-processig UE should be oegative; C5) is the biary costrait of RB patter allocatio; C) idicates that except for patter 1, each patter ca be assiged to oly oe UE, while patter 1 ca

6 5 be assiged to ay UE that is ot allocated with RB; C7) requires that each UE ca oly be assiged with oe patter; C8) guaratees that each RB is allocated to oly oe UE; C9) requires the power o each RB is oegative; C) is the maximum trasmit power costrait of each UE. Propositio 1: Problem P 1 ) is NP-hard. Proof: See Appedix A. IV. BINARY TAILORED FIREWORKS ALGORITHM BASED JOINT COMPUTATION OFFLOADING AND RESOURCE ALLOCATION ALGORITHM Sice problem P 1 ) is hard to solve, i this sectio, we decouple it ito two subproblems, i.e., offloadig decisios makig ad resource allocatio, which are solved by the proposed algorithm. Whe a firework is igited, a burst of sparks will fill the surroudig space aroud the firework. Ispired by the pheomeo, the authors i [] proposed a ew kid of heuristic algorithm called fireworks algorithm FA), ad have verified that it performs well i covergece speed ad global searchig, compared with other heuristic algorithms such as geetic algorithm GA) [5] ad particle swarm optimizatio algorithm PSO) []. As good cadidates for solvig MINLP problem, heuristic algorithms have bee used widely i may fields such as radio resource allocatio [] ad fuzzy cotrol [7], etc. However, to the best uderstadig of us, seldom have they bee used efficietly i computatio offloadig, especially fireworks algorithm. I the followig we will itroduce Biary Tailored Fireworks AlgorithmBTFA) firstly, the we propose the geeral framework of our proposed for the cosidered sceario. A. Some Cocepts 1) Fireworks ad Sparks: Fireworks ad the ewly geerated sparks represet feasible solutios i the solutio space. Specifically, a firework/spark idicates a offloadig decisio matrix Π i the cosidered problem. ) Fitess Fuctio ad Fitess Value: Fitess values are employed to evaluate the performace of feasible solutios. We take the objective fuctio i P 1 ) as the fitess fuctio to obtai the fitess value of each firework. 3) Biary Matrix Distace: Biary matrix distace meas the Mahatta distace of two biary matrixes, i.e., the sum of the distace betwee each elemet of the two matrixes. Suppose two matrixes X ad Y are m -dimesioal, the distace betwee the two matrixes is m dx,y) = X i,j Y i,j. 13) B. Overview of FA i=1 j=1 The typical steps for solvig a problem with FA ca be summarized as follows: First, iitialize a swarm of fireworks ad obtai their fitess values accordig to the specified fitess fuctio. The each firework performs explosio operator to geerate some explosio sparks aroud the firework withi a certai amplitude. The umber of sparks ad the explosio amplitude are obtaied accordig to the fitess value of each firework. The better fitess value of a firework, the more explosio sparks it will produce, ad the smaller amplitude will be, ad vice versa. After that several Gaussia mutatio sparks are geerated i order to keep populatio diversity. Afterwards, from the populatio of fireworks ad sparks, several idividuals are picked out as the fireworks of the ext iteratio. The procedure of explosio, mutatio ad selectio is repeated util the algorithm reaches covergece, or reaches the maximum iteratio idex. Fially, from the idividuals obtaied i the last iteratio, the idividual with the best fitess value is selected as the solutio to the costructed problem. C. Operatios of BTFA Give total I fireworks, the major operatios of the biary tailored FA are listed as follows. 1) Explosio: The umber of the explosio sparks for the ith firework Π i is give by χ i = ceil M f max fπ i )+ϵ I, 1) f max fπ i ))+ϵ i=1 where ceil ) is roud up fuctio, M is a parameter costraiig the total umber of explosio sparks, f max = maxfπ i )),i = 1,...,I is the worst fitess value amog all the I fireworks, ad ϵ is a extremely tiy umber which is used to avoid zero-divisio-error. To avoid that oe firework may geerate too less or too may explosio sparks, bouds are defied for each χ i, which is give by ˆχ i = roudam), if χ i < am roudbm), if χ i > bm,a < b < 1, rouds i ), otherwise 15) where roud ) is the roudig off fuctio, a ad b are give costats, ad ˆχ i is the umber of actual geerated explosio sparks. For firework Π i, each explosio spark is geerated like this: 1) choose β colums from the N colums radomly; ) perform cyclic shift o each selected colum; 3) the rest colums are kept uchaged. ) Mutatio: To improve the spark diversity ad thereby to icrease searchig capability, mutatio is itroduced. From the I fireworks, we choose γ of them radomly, each of which will geerate a mutatio spark like this: 1) choose β colums from the N colums radomly; ) for each of the selected colum, reset it with a radom feasible offloadig decisio; 3) the rest colums are kept uchaged. 3) Selectio: After explosio ad mutatio, there exist three differet kids of idividuals, i.e., fireworks, explosio sparks, ad mutatio sparks. The idividual with the best fitess value is always kept as the first firework of the ext geeratio. To keep diversity of the populatio, other I 1 fireworks are selected form the rest idividuals accordig to their selected probabilities, which are obtaied from their biary matrix distace to all other idividuals as follows.

7 Accordig to 13), the biary matrix distace betwee a idividual Π i ad other idividuals is give by RΠ i ) = j KdΠ i,π j ), 1) where K is the populatio of all curret idividuals icludig both fireworks ad sparks. Cosequetly, the selected probability of idividual Π i is give by pπ i ) = RΠ i) RΠ i ). 17) j K D. BTFA Based Joit Computatio Offloadig ad Resource Allocatio Algorithm ) Applyig the proposed biary tailored operators to traditioal fireworks algorithm, ad adoptig the same procedure with FA, we obtai BTFA. Usig BTFA we ca solve the origial formulated problem P 1 ), i.e., the fial offloadig decisio Π ad the correspodig resource allocatio ca be obtaied. Detailed algorithm framework is called BTFA based joit Computatio offloadig ad resource allocatio algorithm ) ad is summarized i Algorithm 1 as folllows. Algorithm 1 Iitializatio: 1: Set N, K, F fog, I, M, β, γ, a ad b. : Iitialize D,C,f loc,p max,r fc,f c of each UE. 3: Geerate I radom fireworks Π 1,...,Π I i the searchig space, perform resource allocatio uder each firework, ad obtai the fitess value of each firework. Iteratio: : while 1 or l <= L do 5: for i <= I do : Obtai the umber of explosio sparks χ i accordig to 1) ad 15). 7: for p <= χ i do 8: Perform explosio to geerate explosio spark p. 9: Perform resource allocatio uder p. : Calculate the fitess value of p. 11: ed for 1: ed for 13: for j <= γ do 1: Geerate mutatio spark j. 15: Perform resource allocatio uder j. 1: Obtai the fitess value of j. 17: ed for 18: The best idividual is cosidered as the first firework of the ext iteratio, ad the other I 1 fireworks are chose from the rest idividuals accordig to the selected probability i 17). 19: ed while : Amog the fireworks selected i the last iteratio, the oe with the miimum fitess value is cosidered as Π. 1: Output: Π ad correspodig resource allocatio. V. COMPUTATION RESOURCE ALLOCATION Next we will tackle the resource allocatio subproblem embedded i Steps 3, 9, ad 15 i Algorithm 1, where both radio ad computatio resource allocatio eed to be determied. I this sectio, we describe how to obtai the computatio resource allocatio, while radio resource allocatio will be preseted i the ext sectio. After offloadig decisio Π has bee obtaied, problem P 1 ) degrades to the joit optimizatio of allocatig computatio resources, RB patters, ad trasmit power as follows P ) : mi max T 18) f fog,s,p N s.t. C3) C). To reduce the computatio complexity, we divide P ) ito two subproblems: computatio resource allocatio ad radio resource assigmet. For computatio resource allocatio a- mog UEs i N 1, assumig the radio resource assigmet S ad P are give, we have P 3 ) : mi max f fog N 1 s.t. C3),C), C y f fog +B ) 19) whereb = Cx +T fc loc +T)z c + Dy+z) r = D r, N 1 is a costat. Lettig Cy +B fog = C +B fog τ, N 1, the o-smooth problem P 3 ) is coverted to Sice P ) : mi τ ) f fog,τ s.t. C3), C), C C1) : fog +B τ, N 1. C f fog, the τ B, so we have C fog, N 1. 1) τ B Cosequetly, we have C fog F fog. ) τ B N 1 N 1 I order to miimize the maximum delay amog all the fog-executig UEs, the UE with the maximum delay which is deoted by UE ) eeds to be allocated with more computatio resources, so less computatio resources will be left for all the other UEs i.e., UEs i the setn 1 UE ), leadig to a icrease i the delay of those UEs. Performig the above process iteratively, all the computatio resources will be distributed evely amog all the fog-executig UEs i the ed. Thus we have C = fog = F fog. 3) τ B N 1 N 1 Thus problem P ) ca be coverted to P 5 ) : mi τ ) τ C s.t. C17) : = F fog. τ B N 1

8 7 Sice the left-had side of C17) is a mootoic decreasig fuctio about τ, bisectio method ca be used to solve problem P 5 ), which is detailed i Algorithm. Algorithm Bisectio Method for Computatio Resource Allocatio Iitializatio: 1: Set τ mi = max{b }, τ max = ) +B F fog. : Set i = 1 ad the precisio ε >. Iteratio: 3: while 1 do : τ i = τ mi +τ max )/. 5: if τ max τ mi ε the : τ = τ i. 7: else 8: if N 1 C τ i B > F fog the N 1 C N 1 9: τ mi = τ i. : else 11: τ max = τ i. 1: ed if 13: ed if 1: i = i+1. 15: ed while 1: Letτ = τ i ad substitutig it ito 3),f fog is obtaied. 17: Output: f fog. VI. COMMUNICATION RESOURCE ASSIGNMENT After computatioal resource allocatio is obtaied, problem P ) degrades to the joit optimizatio of RB patter assigmet ad power allocatio amog all remoteexecutig UEs i N. Deotig the RB patter assigmet ad trasmit power allocatio as S = {s,j } N J ad P = {p,kj) } N K J, the the radio resource allocatio subproblem is give by: ) P ) : mi max D y +z ) +V S,P N r s.t. C5) C), 5) where V = C y + C x + fog loc T fc + T)z c is a costat. However, P ) is still a + T fc + T c )z = C y f fog NP-hard problem as prove i Appedix A. To make the problem tractable, we first relax each s,j to a cotiuous iterval, i.e., s,j 1; the we defie a ew matrix Φ = {ϕ,kj) } N K J = {s,j p,kj) } N K J to replace P = {p,kj) } N K J. Notig D y +z ) r ) ad lettig max D N r +V = D r, N, = τ 1, we have D r + V τ 1, N, the P ) ca be rearraged as P 7 ) : mi S,Φ,τ 1 τ 1 ) s.t. C5) : s,j 1, N, j J, C) C8), C9) : ϕ,kj), N, k K, C) : ϕ,kj) p max, N, j=1 k=1 C18) : r D τ 1 V, N. Propositio : Problem P 7 ) is joitly covex i S ad Φ for give τ 1. Proof: See Appedix B. Sice P 7 ) is covex, Slater s coditio [8] is met ad duality gap ca be assured, so we ca solve it employig Lagragia dual decompositio ad sub-gradiet projectio method [9]. Oce the optimal solutio {S,Φ } to P 7 ) is obtaied, the optimal solutio {S,P } to P ) is obtaied. A. Lagrage Dual Decompositio Based RB Patter ad Power Allocatio To reduce the umber of dual variables ad thus to improve covergece speed, the partial Lagrage fuctio of P 7 ) is give by 7), where µ = {µ }, N ad ω = {ω }, N are Lagrage dual variables correspodig to C) ad C18) i P 7 ), respectively. The Lagrage dual fuctio is give by Dµ,ω) = mi LS,Φ,τ 1,µ,ω), 8) S,Φ,τ 1 {C5) C9)} which ca be decomposed ito J 1 idepedet subproblems except for patter j = 1). The jth subproblem uder give dual variables µ,ω) is give as P 8 ) : mi s j,φ j,τ 1 L j s j,φ j,τ 1 ) 9) s.t. C5) C9), where s j = {s,j } T N 1, Φ j = {ϕ,kj) } N K is the submatrix of S ad Φ for RB patter j, ad L j s j,φ j,τ 1 ) = τ 1 ϕ,kj) 3) + ω N k=1 µ N k=1 s,j W log 1+ ϕ,kj)g,kj) s,j ). From P 8 ) we kow that s j cotais oly oe ozero biary etry, because every patter j ca oly be allocated to oe UE as required i costrait C). For patter j, assumig s,j, N, is kow, we optimize power allocatio for each RB i patter j. Let K ξ,j = ω s,j W log 1+ ϕ ),kj)g,kj) s,j k=1 µ ϕ,kj) +τ 1, 31) k=1

9 8 LS,Φ,τ 1,µ,ω) = τ 1 + N µ p max j=1 k=1 ϕ,kj) + N ω j=1 k=1 s,j W log 1+ ϕ ),kj)g,kj) s,j D y +z ) τ 1 V 7). P 8 ) reduces to the followig problem P 9 ) : Γ,j = mi φ,kj) ξ,j 3) s.t. C8 ) : ϕ,kj), N, k K. ξ Let,j φ,kj) =, the optimal power allocatio for each RB i patter j is obtaied as follows p,kj) = ϕ,kj) s,j = ω W µ l 1 g,kj) ) +, 33) where x + max{,x}. By substitutig p,kj) i place of φ,kj) s,j i 31), we obtai Γ,j as ) Γ,j = ξ,j p,kj) = p,kj). 3) Performig the procedure 31)-3) for each UE i N, we obtai Γ j = {Γ,j }, N. The UE with the miimum Γ,j, N, is selected as the optimal UE for patter j. We allocate patter j to UE, ad set s,j = 1. Thus, the optimal solutio s j = {s,j } to the jth sub-problem i P 8) is give by s,j = { 1, = argmi {Γ,j }, otherwise. 35) Performig the procedure 9)-35) for every patter j J, we obtai S = {s 1,...s J }. B. Heuristic Algorithm to Extract the Optimal Patter HAEOP) Note that i S, oe UE may be allocated with more tha oe patter, while i the LTE-A uplik oe UE ca be allocated with at most oe patter as i C7)), ad the patters allocated to differet UEs should ot cotai the same RBs as i C8)), otherwise coflict will occur. A coflict table of each RB i -RB case is listed below i Table II. TABLE II: Coflict table i -RB case Idex of RB Correspodig coflictig patters 1,,9,11 3,,7,9,,11 3,7,8,9,,11 5,8,,11 We propose a heuristic algorithm called heuristic algorithm to extract the optimal patter HAEOP) for each UE to pick out the optimal patter from their feasible patters subjectig to costraits C7) ad C8). It is give i Algorithm 3 ad explaied below. I the sortig process Step ), we will give a higher priority to the UE with less feasible patters, sice UEs with Algorithm 3 HAEOP 1: Iput: The obtaied S. : List the coflict table for each RB accordig to W. 3: List the iitial feasible patter set for each remoteprocessig UE accordig to S. : Sort the N remote-processig UEs accordig to the umber of their iitial feasible patters. The less it is, the top the UE. If multiple UEs have the same umber of feasible patters, compare the miimum feasible patter idex. The smaller it is, the top the UE. 5: Start the first roud of patter selectio amog all the valid remote UEs. If a UE has o feasible patter, it is deemed as a ivalid UE, ad allocate patter j = 1 to it. If there s oly oe remote UE, choose patter j = J as its fial scheme, the break. Else, the first UE chooses the miimum patter idex from all its feasible patters. : For the middle N 1) UEs, perform patter selectio accordig to the followig rules. i) First each UE obtais all patters coflictig with ay its previous UEs; ii) take out all the coflict patters from its iitial feasible patter set, ad the rest costitutes its ew feasible patter set; iii) chooses the patter with the miimum idex from its ew feasible patter set; if the set is empty, allocate patter j = 1 to the UE. 7: The last UE first performs the same procedure as the previous N 1) UEs did to obtai its feasible patter set. If the set is oempty, choose the maximum idex from this set; else choose patter j = 1. 8: Calculate the total umber of occupied RBs accordig to the chose patter of each UE. If it is less tha K, start the ext roud of patter selectio: the first UE chooses its ext feasible patter; the the rest valid UEs repeat the same procedure as Steps ad 7 above, util either of the followig two termial coditios are satisfied: i) the total umber of occupied RBs equals to K, the the patters selected i this roud are cosidered as the optimal patter allocatio scheme; or ii) if ay valid UE has o feasible patter, the the patters selected i the previous roud are cosidered as the optimal scheme. 9: Output: The optimal patter S = {s,j }. more feasible patters have a higher probability of fidig a feasible patter after all other UEs have performed their patter selectio. Accordig to W, a patter with a smaller idex cotais less RBs. So i Steps 5-, the first 1 N 1) UEs will choose the patter with the smallest idex amog their feasible patters, so that the remaiig UEs may have more chace of

10 9 fidig a feasible patter. I Step 7, the last UE will choose the feasible patter with the maximum idex i order to maximize RB utilizatio. If ay UE is left with o feasible patter i the first roud of selectio, i.e., all its iitial feasible patters coflict with the chose patters of its previous UEs, the the UE is allocated with patter j = 1 o RBs) ad is called a ivalid UE i.e., it fails i task offloadig), otherwise it is called a valid UE. Step 8 termiates uder oe of the two coditios: 1) someoe is left with o feasible patter, the the patters of all remote-processig UEs selected i the previous roud are cosidered as the optimal patter allocatio scheme; ) all the K RBs have bee allocated, the the patters selected i this roud are cosidered as the optimal scheme. We deote the optimal patter allocatio matrix as S = {s,j }. After the optimal RB patter allocatio S is obtaied, we perform the optimal power allocatio for each UE o the RBs i its selected patter j as follows p,kj ) = { p,kj), = ad j = j, otherwise C. Lagrage Multipliers Update. 3) After solvig all subproblems i P 8 ), S ad P ca be obtaied for give µ ad ω. The dual variables µ ad ω ca be updated by resolvig the dual problem of P 7 ), which is give by Algorithm Joit Uplik RB Patter assigmet ad Power Allocatio Iitializatio: 1: Set µ),ω) ad the precisio δ, set t =. Iteratio: : while 1 do 3: for each patter j = 1 to K + K +1 do : for each do 5: if k j the : Calculate p,kj) via 33) ad obtai Γ j by 3). 7: Obtai s,j accordig to 35), ad sj = {s,j }. 8: Extract S = {s,j } from S = {s 1,...,s J } usig Algorithm 3. 9: For = ad j = j, set p,kj ) = p,kj) ad s,j = 1. : ed if 11: ed for 1: ed for 13: Update dual variables µ, ω from ) ad 1), respectively. 1: t = t+1. 15: if µt + 1) µt) < δ, ωt + 1) ωt) < δ the 1: break. 17: ed if 18: ed while 19: Output: S = {s,j }, P = {p,kj ) }. P ) : max Dµ,ω) 37) µ,ω s.t. µ,ω. From 7) ad 8), we kow that P ) is covex, because Dµ,ω) is a liear fuctio about the dual variables µ ad ω. By utilizig sub-gradiet projectio method, we solve P ) i a iterative maer to obtai dual optimum µ ad ω. Propositio 3: The sub-gradiets of Dµ, ω) at the tth iteratio are give i equatios 38) ad 39), where p,kj) ad s,j is the optimal solutio to dual fuctio 8) for a give set of dual variables µ ad ω. Proof: See Appedix C. Based o 38) 39), the Lagrage multipliers are updated with the sub-gradiet projectio method [3] as follows µ t+1) = [µ t) ht) µ t)] +,, ) ω t+1) = [ω t) jt) ω t)] +,, 1) Formulate the MINLP problem of joit optimizatio of offloadig decisio, computatio resource allocatio, RB patter assigmet, ad power allocatio i a mixed fog/cloud computig LTE-A based system Iitalize I offloadig decisio matrixes as iitial Iitialize I offloadig fireworks decisio matrixes as iitial fireworks, perform resource allocatio uder each firework, ad calculate the fitess value for each firework Algorithm 1) Yes Yes Explosio Obtai the umber of explosio sparks of firework i accordig to its fitess value Perform explosio for firework i to geerate explosio sparks Resource allocatio Calculate fitess value for each explosio spark Iteratio ed? No i<=i =? No Mutatio Geerate mutatio sparks Resource allocatio Calculate fitess value for each mutatio spark Selectio From the set of fireworks, explosio sparks, ad mutatio sparks, choose I idividuals as the fireworks for the ext iteratio Amog fireworks selected i the last iteratio, the oe with the miimum fitess value as the optimal firework fial offloadig decisio) Bisectio method based computatio resource allocatio Algorithm ) Lagrage dual decompositio based RB patter ad trasmit power allocatio Algorithm ) HAEOP optimal patter extractig for each UE Algorithm 3, embedded i Algorithm ) Resource allocatio procedure uder each firework or spark embedded i Algorithm 1) where t is the iteratio idex; ht) ad jt) are positive step sizes. I this paper we adopt square summable but ot summable step sizes [3], where ht) = 1/ t), ad jt) = 1/ 1 t). The Lagrage multipliers are updated iteratively util the required precisio is satisfied. The procedure for joit RB patter ad power allocatio is summarized i Algorithm. For a geeral uderstadig, the work flow chart of our system is give i Fig., the mai body of which is our proposed. Fial offloadig decisio ad the correspodig resource allocatio scheme a) Mai framework of Fig. : The work flow chart of our system. VII. COMPLEXITY ANALYSIS b) Detailed procedure of resource allocatio The computatioal complexity of i Algorithm 1 maily comes from the resource allocatio procedures i Steps 9 ad 15 i the while loop. I Step 9, the resource allocatio

11 ω t) = j=1 µ t) = p max s,jr,j D y +z ) τ 1 V = j=1 k=1 s,jp,j, 38) j=1 ) s,jw log 1+p,kj) g,kj) D y +z ) τ 1 V, 39) procedure is performed I i=1 iχ i times for the I iχ i explosio sparks, respectively. I Step 15, the resource allocatio procedure is performed γ times for the γ mutatio sparks, respectively. For otatioal simplicity, we defie the total umber of explosio ad mutatio sparks as Ξ = I iχ i +γ. I each resource allocatio procedure, computatio resources are allocated usig Algorithm, ad the radio resources are allocated employig)) Algorithm. I Algorithm, log τ max τ mi ε it requires O iteratios for the bisectio method to coverge. I Algorithm, the complexity maily comes from the extractig of RB patter i Step 8, i.e., Algorithm 3. The sub-gradiet projectio method i the outer while loop that eeds O ) 1 δ iteratios to coverge [8], the K + K + 1 iteratios i the outer for loop, ad the at most N iteratios i the ier for loop. The complexity of Algorithm 3 maily comes form the patter extractig procedure i its Steps 5 8. Sice there are at most N = N remote-processig UEs, the complexity of the first roud of RB patter extractig i Steps 5 7 is ON). Assumig that the first remoteprocessig UE posses C feasible patters, sice all the remoteprocessig UEs are sorted accordig to the ascedig order of their umber of feasible patters, C is far less tha N, ad thus the complexity of RB patter extractig i Steps 5 8 is OCN) = ON). Cosequetly, the complexity of Algorithm 3 is ON). Hece, the complexity of Algorithm ) 1 is O δ K + K +1) N N = O 1 δ K N ). Therefore, the complexity )) of each resource allocatio procedure is τ O log max τ mi ε +O 1 δ K N ) = O 1 δ K N ). i=1 i=1 Based o the above aalysis ad give that the outer while loop i Algorithm 1 rus for L times, the complexity of is O 1 δ ΞLK N ). VIII. RESULTS AND DISCUSSIONS I this sectio, simulatio results are preseted to evaluate the performace of the proposed algorithms. The followig parameters remai uchaged through our simulatios: L =, K = 15 [9], W = 18 KHz [9], p max = W []. The followig parameters are set as default uless otherwise specified: N =, I =, M =, γ = 1, a =., b =.8, F fog = 5 9 cycles/s [18], c = 9 cycles/s [1], loc is uiformly distributed i [5,] M cycles/s, ad R fc = 15 b/s []. For simplicity, the wireless chael gai g,kj) = h,kj) σ is assumed to take values i [5,1] radomly [9]. We adopt face recogitio [1] as the default applicatio, where D =. MB ad λ = 97. cycles/bit [1]. Next, we verify the performace gai obtaied by our proposed algorithms ad the followig schemes are compared. The proposed scheme ): The scheme obtais offloadig decisios ad resource allocatio usig FAJO- RA i Algorithm 1, where i each iteratio computatio resource is allocated usig Algorithm, RB patter ad power are allocated usig Algorithm, ad each UE picks out the optimum RB patter usig Algorithm 3. Radio ad computatio resource allocatio optimizatio RCRA): RB patter ad power are allocated usig Algorithms 3 ad, computatio resource are allocated usig Algorithm, ad offloadig decisios are obtaied radomly. Radio resource allocatio with HAEOP RRA-H): RB patter ad power are allocated employig Algorithms 3 ad, while offloadig decisios ad computatio resource allocatio are obtaied radomly. Radio resource allocatio ad radom patter extractig RRA-R): RB patter ad power are allocated usig Algorithm, ad from the allocated patters each UE selects oe of which radomly. Offloadig decisios ad computatio resource allocatio are obtaied radomly. Local processig Local): All UEs process their applicatios locally without optimizatio. Moreover, i the followig Figs. 8 ad 9, aother algorithm amed SDR based offloadig decisio optimizatio ad optimized resource allocatio algorithm SDR-ODRA) was cosidered as a bechmark to demostrate the performace of our proposed BTFA based offloadig decisio makig algorithm. I SDR-ODRA, the offloadig decisios are obtaied usig the SDR based algorithm i [], where i each iteratio computatio resource is allocated usig Algorithm, RB patter ad power are allocated usig Algorithm, ad each UE picks out the optimum RB patter usig Algorithm 3. Remark: SDR based offloadig decisio makig algorithm was first ovelly proposed by the authors i [], ad ow has bee widely used i may existig works. Similar to our previous work [3], the umber of rus i.e., radomizatio trails) [3] i SDR-ODRA is set as. Five metrics are adopted, icludig: 1) three kids of delay, i.e., the maximum, miimum ad average delay of all UEs, which are deoted as T max i.e., objective value), T mi, ad T av, respectively; ) the umber of beefited UEs, where a beefited UE meas the UE whose delay cosumptio is reduced compared with local processig; 3) the probability of failure i offloadig, where a UE fails i offloadig meas it is allocated with patter j = 1.

12 11 A. Covergece of Algorithms 1, ad Objective value s) ζ Number of iteratios Fig. 3: Covergece of Algorithm 1 ) F fog =1* 9 F fog =3* 9 F fog =5* 9 F fog =8* Number of iteratios Fig. : Covergece of Algorithm. fog-processig UEs, ad Fig. shows that τ decreases uder differet F fog after each iteratio util covergece. Fig. 5 shows that the dual variables i Algorithm coverge fast. Accordig to the three figures, we kow that the proposed algorithms are cost-efficiet i solvig the NP-hard problem P 1 ). Objective value T max s) Exhaustive Number of UEs a) Executio time of algorithm s) 5 3 Exhaustive Number of UEs b) Fig. : Effectiveess ad complexity of. Fig. shows the comparisos i effectiveess ad complexity betwee the proposed algorithm ad exhaustive algorithm, where offloadig decisios are obtaied by exhaustive search, ad computatio ad commuicatio resource allocatio employ our proposed Algorithms, 3, ad. From the Fig. a), it ca be kow that is slightly iferior to exhaustive search i performace, i.e., a little icrease i objective value. However, the complexity compariso i Fig. b) idicates that the executio time of exhaustive algorithm icreases expoetially with the umber of UEs, while oly takes a little executio time eve with more UEs, idicatig that is good i scalability. Dual variables µ Dual variables ω 15 x 5 µ 3 µ 5 15 x 7 ω 3 ω 5 15 Number of iteratios Fig. 5: Covergece of Algorithm. Fig. 3 verifies the covergece of the outer loop of i Algorithm 1, from which we ca see coverges fast withi iteratios. Figs. ad 5 evaluate the covergece rate of the mai loop of Algorithms ad, respectively, both of which are embedded i Steps 3, 9 ad 15 i Algorithm 1. As discussed i Sectio V, τ is the maximum delay of all µ 1 µ ω 1 ω B. Effectiveess of Algorithm 3 HAEOP) 15 5 Maximum delay s) Miimum delay s) Average delay s) Number of beefited UEs Probability of failure i offloadig %) Fig. 7: Performace evaluatio of Algorithm 3. I Fig. 7 the performace of HAEOP i Algorithm 3 is evaluated by comparig RRA-H ad RRA-R, where all the parameters adopt their default values. As show i the first sub-figure, the three delays T max, T mi ad T av of RRA-H

13 1 are always much shorter tha that of RRA-R. The secod subfigure idicates RRA-H could beefit more UEs ad reduce the probability of failures i offloadig effectively. The reaso for Fig. 7 is that: i RRA-R, each UE selects a radom patter from its feasible patters, thus its chose patter may cotai the same RB with the patters chose by its previous UEs, ad cosequetly coflict will happe, leadig to higher failure probability ad less beefited UEs. While i RRA-H, sice HAEOP is adopted i patter selectio, each UE picks out the optimum feasible patter, cosiderig exclusiveess of RBs, thus failures could be avoided effectively, ad therefore more UEs will be beefited as is show i the secod sub-figure. O the other had, HAEOP takes RB utilizatio ito accout, thus the maximum RB utilizatio ca be obtaied uder the fial selected patter allocatio scheme. Cosequetly, T max, T mi ad T av ca be reduced greatly as show i the first sub-figure. C. Performace Comparisos versus Differet Applicatio parameters Objective value T max s) SDR ODRA RCRA Local Processig desity λ CPU cycles/bit) Fig. 8: Objective value T max compariso uder differet processig desity λ. Objective value T max s) SDR ODRA RCRA Local Iput data size D MByte) Fig. 9: Objective value T max compariso uder differet iput data size D. Figs. 8 ad 9 shows how applicatio parameters icludig processig desity λ ad iput data size D affect the objective value T max, respectively. The two figures are i accordace with our ituitio that more iput data D or the higher computatio complexity λ, higher delay will be brought i, ad cosequetly a lager value of T max. Moreover, as a joit optimizatio of offloadig decisios ad resource allocatio, performs always the best, followed by SDR-ODRA, RCRA, RRA-H, ad RRA-R successively, ad Local is the worst i performace. However, some differeces exist betwee the two figures. I Fig. 8, D takes the default value. MB, which is relatively large. Whe λ is very small, local processig is usually a good choice, while offloadig will cosume more time i data trasmissio. I Fig. 9, λ = 97. cycles/bit. Whe D is very small, the computatio workloadc is also very small, so all the algorithms will cosumes quite less time, ad therefore T max is very small for all the algorithms. It should be oted that, although SDR-ODRA ca obtai almost the same performace as, the computatioal complexity of SDR-ODRA is much higher tha i the offloadig decisio makig process. I SDR-ODRA, offloadig decisios are obtaied usig CVX ad radomizatio, where iterior poit method is adopted, leadig to higher complexity. I, the outer fireworks algorithm eed several iteratios to coverge, ad i each iteratio each spark i.e., a offloadig decisio) ca be geerated usig fireworks operators with very tiy complexity. D. Performace Comparisos versus Chael State Objective value s) ~ ~ 8~ 1~1 1~18 Wireless chael gai g,kj) RCRA Local Fig. : Objective value T max compariso uder differet differet wireless chael gai g,kj). Figs. ad 11 display how chael state affects the objective value T max, icludig the wireless access chael gai g,kj) betwee UEs ad the fog ode, ad the wired lik rate R fc betwee fog ad cloud, respectively. As chael state has o ifluece o local processig, its objective value always keeps still. However, whe the chael state gets better ad better, less time will be cosumed i data trasmissio for all other algorithms, leadig to a decrease i T max for them. From the two figures we ca also fid that always performs the best, with its objective value T max far less tha other algorithms.