Latency Optimization for Resource Allocation in Mobile-Edge Computation Offloading

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1 1 Lateny Optimization for Resoure Alloation in Mobile-Ege Computation Offloaing Jine Ren, Guaning Yu, Yunlong Cai, an Yinghui He arxiv: v1 [s.it] 1 Apr 2017 College of Information Siene an Eletroni Engineering Zhejiang University, Hangzhou , China {renjine, yuguaning, ylai, 2014hyh}@zju.eu.n Abstrat By offloaing intensive omputation tass to the ege lou loate at the ellular base stations, mobile-ege omputation offloaing (MECO has been regare as a promising means to aomplish the ambitious milliseon-sale en-to-en lateny requirement of the fifth-generation networs. In this paper, we investigate the lateny-minimization problem in a multi-user time-ivision multiple aess MECO system with joint ommuniation an omputation resoure alloation. Three ifferent omputation moels are stuie, i.e., loal ompression, ege lou ompression, an partial ompression offloaing. First, lose-form expressions of optimal resoure alloation an minimum system elay for both loal an ege lou ompression moels are erive. Then, for the partial ompression offloaing moel, we formulate a pieewise optimization problem an prove that the optimal ata segmentation strategy has a pieewise struture. Base on this result, an optimal joint ommuniation an omputation resoure alloation algorithm is evelope. To gain more insights, we also analyze a speifi senario where ommuniation resoure is aequate while omputation resoure is limite. In this speial ase, the lose-form solution of the pieewise optimization problem an be erive. Our propose algorithms are finally verifie by numerial results, whih show that the novel partial ompression offloaing moel an signifiantly reue the en-to-en lateny. Inex Terms Mobile ege omputation offloaing (MECO, loal ompression, ege lou ompression, partial ompression offloaing, resoure alloation, pieewise optimization, ata segmentation strategy. I. INTRODUCTION Over the past few years, the explosive popularity of mobile evies, suh as smart-phones, tablets, an wearable evies, has been aelerating the evelopment of the Internet of Things April 4, 2017 DRAFT

2 2 (IoT [1], [2]. Aoring to the preition by Ciso, nearly 50 billion IoT evies will be onnete to the Internet by 2020, most of whih have limite resoures for ommuniation, omputation, an storage [3]. Due to the exponential growth of mobile ata traffi, merely relying on traitional lou omputing is not aequate to realize this ambitious milliseon-sale lateny for ommuniation an omputation in 5G networs. To eep up with this persistent eman an improve the quality of experiene (QoE for users, the emerging tehnology of mobile ege omputing (MEC has been gaining signifiant attention from both aaemia an inustry. MEC offers appliation evelopers an ontent proviers lou-omputing apabilities at the very ege of the mobile networ by implementing MEC servers at ellular base stations (BSs, whih is also referre to as ege lou [4]. Owing to the lose istane from the mobile evie to the lou server, MEC has the potential to fulfill the ritial en-to-en elay requirement of 5G networs. Moreover, through mobile ege omputation offloaing (MECO, the energy onsumption of mobile evies an be also reue by offloaing intensive omputation worloa to the proximate MEC server for exeution [5]. The minimization of en-to-en elay an energy onsumption in the MECO tehnique requires joint alloation of ommuniation an omputation resoures among mobile evies an MEC servers. Reent years have seen lots of stuies on this topi for both single-user [6] [13] an multi-user [14] [19] MECO systems. In [6] an [7], the authors have erive the optimal resoure alloation solution for a single-user MECO system with multiple elasti tass to minimize the average exeution lateny of all tass uner the transmit power onstraint. On the other han, to reue the total energy onsumption uner a given lateny requirement, the authors in [8] have erive the optimal threshol-base offloaing poliy with joint ommuniation an omputation resoure alloation an the authors in [9] have propose the optimal moe seletion between loal omputing an lou omputing. Furthermore, a elay-optimal problem in a singleuser MECO system with a presribe resoure utilization onstraint has been stuie in [10], an a polynomial-time approximate solution with guarantee performane has been evelope herein. Optimal resoure alloation an offloaing eision poliy has been further investigate to minimize the weighte-sum mobile energy onsumption uner the omputation lateny onstraint [14]. Besies, an online joint ommuniation an omputation resoure management algorithm for a multi-user MECO system has been evelope to minimize the long-term average weightesum energy onsumption of mobile evies an the lou server uner the buffer stability onstraint [15]. A stohasti tas arrival moel base on the Lyapunov optimization algorithm has DRAFT April 4, 2017

3 3 been propose to solve the energy-lateny traeoff problem for a multi-user MECO system [16]. In aition, to minimize the total energy onsumption an offloaing lateny, game-theoreti tehniques have been applie to evelop the istribute algorithm, whih is able to ahieve a Nash equilibrium [17]. Moreover, to ope with the bursty tas arrivals, the MEC server an be integrate with uplin-ownlin transmission sheuling to minimize the average lateny [18]. Most aforementione wors on multi-user MECO systems fous on the binary omputation offloaing strategy, i.e., the omputation tas is exeute either at the mobile evie or at the ege lou. Although the pioneering wor in [14] has stuie the energy-effiient partial omputation offloaing, the lateny-minimization issue is not isusse therein, whih is a more urgent esign target for 5G networs. Inspire by this, we investigate the lateny-minimization problem in a multi-user MECO system with partial omputation offloaing in this paper. We assume that mobile evies have a large volume of raw ata that are require to be ompresse an uploae to the ege lou for analysis an storage. This onsiere senario is orresponing to the surveillane an seurity appliation where massive online monitoring ata shoul be timely transmitte to an analyze by a entral unit. Our esign objetive is to minimize the weighte-sum elay of all evies uner the limite ommuniation an omputation resoure onstraints. Aoring to where the ata is ompresse, we propose three ifferent moels: loal ompression where ata is ompresse only at mobile evies, ege lou ompression where ata is transmitte to the ege lou for ompression, an partial ompression offloaing where partial ata is ompresse loally while the other part is ompresse at the ege lou. The main ontributions of this wor are summarize as follows. For the loal ompression moel, we formulate a onvex optimization problem to minimize the weighte-sum elay of all evies uner the ommuniation resoure onstraint. Both lose-form expressions of optimal resoure alloation an minimum weighte-sum elay are erive, an some inherent insights are also highlighte. For the ege lou ompression moel, we analyze the tas ompletion proess by moeling a joint resoure alloation problem with the onstraints of both ommuniation an omputation resoures. Then the lose-form solution an the minimum weighte-sum elay of all evies an be obtaine by utilizing the Lagrange multiplier metho. For the partial ompression offloaing moel, we first formulate a pieewise optimization problem an then erive the optimal ata segmentation strategy in a pieewise struture. Base on this result, we transform the original problem into a pieewise onvex problem April 4, 2017 DRAFT

4 4 an evelop an optimal resoure alloation solution base on the sub-graient algorithm. To yiel more insights into the partial ompression offloaing moel, we investigate a ommon senario where ommuniation resoure is aequate while omputation resoure is limite. In this speifi ase, the elay expression of eah evie an be simplifie an the lose-form solution of the pieewise optimization problem an be erive. It is also verifie by numerial simulation that the propose solution for this speifi senario an ahieve a near-optimal performane in general senarios. The rest of this paper is organize as follows. In Setion II, we introue the multi-user MECO system an the three ifferent ompression moels. In Setion III, we investigate the loal ompression moel an present a lose-form solution to the lateny-minimization problem. The resoure alloation problem for the ege lou ompression moel is analyze in Setion IV. Setion V investigates the partial ompression offloaing moel an the lose-form solution for a speifi senario is also analyze in this setion. Simulation results are presente in Setion VI an the whole paper is onlue in Setion VII. II. SYSTEM MODEL In this setion, we first introue the multi-user MECO system. After that, we analyze the ento-en elay of eah evie in the three moels of loal ompression, ege lou ompression, an partial ompression offloaing, respetively. A. Multi-user MECO System As epite in Fig. 1, we onsier a multi-user MECO system onsisting of one ege lou platform an K single-antenna mobile evies, enote by a set K = {1,2,,K}. The ege lou an be regare as a ata enter that is onnete by mobile evies through wireless hannels. Eah evie has a volume of ata, suh as a raw vieo, that nees to be ompresse an store in the ege lou. 1 The mobile evie in the system an be haraterize by two parameters, i.e., the size of raw vieo L > 0 (in bits an the CPU ompression apaity V (in bits/s. Denote the total ompression apaity of the ege lou as V (in bits/s, whih an be alloate to all evies. That is, evie will be alloate V omputational resoure with 1 In this paper, we tae the vieo ompression as the example for the following analysis whereas our propose framewor an by extene into any ata analytial system. DRAFT April 4, 2017

5 5 onstraint K =1 V V. Aitionally, some other reasonable assumptions use in this paper are esribe as follows. The ege lou has perfet nowlege of the hannel gains an the size of vieos of all evies, whih is require by the entralize sheuler. To guarantee that all vieos an be ompresse at the ege lou simultaneously, we require that all evies an the ege lou utilize the same vieo ompression tehnology, suh as MPEG4. The ompression ratio is enote as β (0,1, i.e., one-bit raw vieo ata will be ompresse into β bit. The elay for vieo segmenting, stithing, an storing an be reasonably neglete sine they are muh shorter than both ommuniation an omputational elays. Vieo 1 Devie 1 Ege Clou Vieo 2 MEC Server Vieo Storage Devie 2 Vieo K Devie K Fig. 1. Multi-user MECO system moel. B. Multiple-Aess Moel We apply a time ivision multiple aess (TDMA metho for the hannel aess. In this metho, one time frame is ivie into K time slots, whih will be alloate to K evies. April 4, 2017 DRAFT

6 6 For onveniene, we normalize the uration of time-slot alloate to evie as t (t [0,1]. Note that the length of eah time frame is short enough (e.g., 10 ms in LTE stanars, whih an be reasonably ignore when alulating the en-to-en elay of eah evie. Let h (i enote the hannel gain of evie in time-slot i, whih is a ranom variable an inepenently an ientially istribute (i.i.. aross the time-slot. Definep as the transmission power of evie. Then the ahievable ata rate (in bits/s in time-slot i an be expresse as ( r (i = Blog 2 1+ p h (i 2, (1 N 0 where B an N 0 are the banwith an the variane of aitive white Gaussian noise (AWGN, respetively. C. Loal Compression Moel In the loal ompression moel, eah raw vieo is ompresse loally an then transmitte to the ege lou for storage. There are two ins of elay in this moel: The elay for ompressing the raw vieo L bits at evie, Domp, = L. The elay for transmitting the ompresse vieo βl bits to the ege lou, D tran,. Assume that it taes at least N of βl bits to the ege lou, where N satisfies { N N = argmin N : V time slots for evie to transmit the ompresse vieo i=1 r (i βl T }, (2 where T is the uration of time-slot alloate to evie. Note that the transmission rate r (i (i = 1,2,,N, as a funtion of the ranom hannel gain h (i, is also a ranom variable an i.i.. aross time slots, therefore N is also a ranom variable. However, aoring to [20] an the martingale theory [21], we an evaluate the average transmission elay as D tran, = E h{t N } = βl E h {t r } = where E h { } is the expetation over the hannel gain h (i. βl t E h {r }, (3 For ease of notation, we efine R = E h {r }, whih an be regare as the average ata rate of evie aross time slots. Moreover, we assume that the transmission an happen only after DRAFT April 4, 2017

7 7 all ata of the raw vieo is ompletely ompresse. Then the en-to-en elay for evie to omplete its tas an be expresse as D = Domp, +D tran, = L + βl. (4 V t R D. Ege Clou Compression Moel In the ege lou ompression moel, eah evie iretly uploas its raw vieo to the ege lou without any ompression. Thereafter, the ege lou ompresses all raw vieos in parallel by optimally alloating its omputation resoure. Similar to the loal ompression moel, there also exist two ins of elay in this moel: The elay for transmitting the raw vieo L bits to the ege lou, D tran, = L t R. The elay for ompressing the raw vieo L bits at the ege lou, Domp, = L. Corresponingly, we require that the ege lou an start ompressing a raw vieo only after ompletely reeiving its whole ata. Then the en-to-en elay for evie to omplete its tas an be written as D = Dtran, +D omp, = L + L t R V. (5 V E. Partial Compression Offloaing Moel In the loal ompression moel, the loal ompression elay, Domp,, woul be ominant if the spee of evie CPU is limite, i.e., orresponing to the wireless vieo monitoring amera. On the other han, in the ege lou ompression moel, the transmission elay, D tran,, woul be ominant if the hannel banwith is limite. Obviously, both two moels are not optimal in terms of en-to-en elay minimization if the vieo an be partially ompresse at the mobile evie an partially ompresse at the ege lou. Motivate by this, in this subsetion, we propose a partial ompression offloaing moel, in whih eah raw vieo an be partitione into two parts with one ompresse loally while the other offloae for ege ompression. Let us enote the proportion of vieo that is ompresse at the mobile evie as λ [0,1]. Then, we an introue the etaile proeure of partial ompression offloaing in three steps, as epite in Fig. 2. Mobile evie ompresses λ L bits of the raw vieo loally an then transmits the ompresse ata of βλ L bits to the ege lou. April 4, 2017 DRAFT

8 8 Mobile evie transmits the remaining (1 λ L bits to the ege lou. Then the ege lou ompresses this part of raw vieo itself. Finally, the ege lou ombines two parts of ompresse vieo an stories into the atastorage enter. L L V L t R L t R V D D D D D D D D Fig. 2. The whole proess of partial ompression offloaing. In the above three steps, there exist four ins of elay as follows. The elay for ompressing λ L bits at mobile evie : Domp, = λ L. V The elay for transmitting the loal ompresse part: Dtran, = βλ L. t R The elay for transmitting the unompresse part: D tran, = (1 λ L t R. The elay for ompressing (1 λ L bits at the ege lou: Domp, = (1 λ L. Sine eah evie has only one hannel for ata transmission, either loal ompression part or ege lou ompression part an be transmitte at any moment while not simultaneously. Therefore, as epite in Fig. 2, two ases will happen. The first one orrespons to D omp, Dtran,, where the transmission for loal ompresse vieo an start immeiately at the en of V DRAFT April 4, 2017

9 9 the loal ompression. The seon one orrespons to Domp, < D tran,, where the transmission for loal ompresse vieo must wait until the transmission for the ege lou ompression part ens. Therefore, the en-to-en elay of evie in this moel an be written as max { Domp, D = +D tran,,d tran, omp,} +D, if D omp, Dtran,, Dtran, +max { Dtran,,Domp,}, if D omp, < Dtran,. In the next three setions, we will evelop optimal joint ommuniation an omputation resoure alloation algorithms to minimize the weighte-sum elay of all evies for the three ifferent moels, respetively. (6 III. OPTIMAL SOLUTION TO THE LOCAL COMPRESSION MODEL In this setion, we first formulate the lateny-minimization problem for the loal ompression moel an then erive the lose-form expressions for both optimal solution an minimum weighte-sum elay of all evies. A. Problem Formulation We aim at minimizing the weighte-sum elay of all evies, K =1 α D, where the positive weight fators {α } aount for the fairness among evies an satisfy K =1 α = 1. Base on the en-to-en elay expression in (4, we have the following optimization problem for the loal ompression moel. Problem 1: (Loal Compression min {t } s.t. ( L α + βl, (7a V t R t 1, t 0, (7b where (7b is the overall ommuniation resoure onstraint of all evies. =1 =1 B. Optimal Solution It an be easily verifie that Problem 1 is onvex an the Slater s onition an be satisfie, implying that strong uality hols. Thus, Problem 1 an be solve by the Karush-Kuhn-Tuer April 4, 2017 DRAFT

10 10 (KKT onitions. The Lagrange funtion an be expresse as ( L L L = α + βl ( K +ν t V 1, (8 =1 t R =1 { } where ν 0 is the Lagrange multiplier assoiate with the onstraint (7b. Let enote the optimal solution for Problem 1. Then applying KKT onitions leas to the following neessary an suffiient onitions L L t (1 ν ( K = α βl ( 2 +ν R t (1 =1 t (1 1 = 0, =1 > 0, t (1 = 0, = 0, t (1 > 0, Base on these onitions, we an erive the following optimal solution. by t (1 (9 t (1 1, ν 0. (10 Theorem 1: The optimal solution solving Problem 1 of the loal ompression moel is given t (1 = α L R K =1 α L R, K. (11 Remar 1: Theorem 1 reveals that the optimal time-slot alloate to evie is etermine by the orresponing weight fator, size of raw vieo, an hannel apaity. The weight fator, α, an be interprete as the level of importane for evie. The larger the value of α is, the more the time-slot shoul be alloate to evie to minimize the whole system elay. Furthermore, more time slot shoul be alloate to evie if the vieo size, L, beomes larger or the hannel apaity, R, gets smaller. Base on the above solution, we an erive the minimum system elay (i.e., weighte-sum elay of all evies in a lose-form way, as K ( Dsys L = α L βl i + α V i R i = =1 =1 α L V +β i=1 i=1 j=1 j=1 α i α j L i L j R i R j. α j βl j R j (12 DRAFT April 4, 2017

11 11 IV. OPTIMAL SOLUTION TO THE EDGE CLOUD COMPRESSION MODEL In this setion, we analyze the lateny-minimization problem for the ege lou ompression moel an evise the joint optimal ommuniation an omputation resoure alloation algorithm. A. Problem Formulation To minimize the weighte-sum elay of all evies, we have the following problem for the ege lou ompression moel. Problem 2: (Ege Clou Compression min {t,v } s.t. ( L α + L, (13a t R V t 1, t 0, (13b =1 =1 =1 V V, V 0, (13 where onstraints (13b an (13 imply that the overall ommuniation an omputation resoures alloate to mobile evies annot exee the orresponing limitations. B. Optimal Solution Fortunately, Problem 2 is also onvex sine eah omponent in (13a is onvex on t an V. Therefore, Problem 2 an be optimally solve using the KKT onitions. The Lagrange funtion an be written as L E = =1 ( L α + L ( K ( K t R V +ξ t 1 +χ V V, (14 =1 =1 where ξ 0 an χ 0 are the Lagrange multipliers assoiate with onstraints (13b an (13, { } respetively. Let t (2,V (2 enote the optimal solution for Problem 2. Then the neessary an suffiient onitions base on the KKT onitions an be expresse as L E = α L > 0, t (2 t (2 ( 2 +ξ = 0, R t (2 = 0, t (2 > 0, (15 April 4, 2017 DRAFT

12 12 L E V (2 ξ ( K = α L ( 2 +χ =1 t (2 V (2 1 = 0, =1 > 0, V (2 = 0, = 0, V (2 > 0, (16 t (2 1, ξ 0, (17 χ ( K =1 V (2 V = 0, =1 V (2 V, χ 0. (18 By solving the above equations, we an obtain the optimal solution for Problem 2, as shown in Theorem 2. Theorem 2: The optimal solution for Problem 2 of the ege lou ompression moel is given by α L t (2 = R, K, K α L =1 R (19 V (2 = α L K =1 α L V, K. Remar 2: From Theorem 2, we an see that the optimal time-slot alloate to eah evie in the ege lou ompression moel has the same expression as that in the loal ompression moel, an the optimal lou ompression apaity alloate to eah evie is etermine by the orresponing weight fator an vieo size. Similarly, the weight fator α is positively relate to the alloate resoures sine it reflets the level of importane for evie. Speially, in ase that eah evie has the same weight, i.e., α = 1, K, the bigger the vieo size is, the K more the time-slot an ege lou ompression apaity shoul be alloate to the evie for ahieving the minimum system elay. Again, we an express the minimum system elay in a lose-form way, as K ( Dsys E = L i L j α i α j + R i=1 i R j=1 j ( = αi α j L i L j i=1 j=1 ( αi K αj L j L i i= R i R j V. j=1 V (20 DRAFT April 4, 2017

13 13 V. OPTIMAL SOLUTION TO THE PARTIAL COMPRESSION OFFLOADING MODEL In the above setions, we have analyze the optimal ommuniation an omputation resoure alloations for the loal ompression moel an the ege lou ompression moel, respetively. In this setion, we shall investigate the lateny-minimization problem for the partial ompression offloaing moel. The problem stuie in this setion is more generi in that it fully utilizes the omputation resoure in both mobile evies an the ege lou, whih an further reue the system elay. In the following, we shall first formulate the lateny-minimization problem an then erive the optimal vieo segmentation strategy in a pieewise struture. After that, we will transform the original problem into a pieewise onvex problem an evelop a sub-graient algorithm to fin the optimal solution effiiently. Finally, the lose-form solution in a speifi senario will be also evise. A. Problem Formulation In the partial ompression offloaing moel, eah raw vieo oul be partially ompresse at the mobile evie an partially ompresse at the ege lou. Therefore, both ommuniation an omputation resoures shoul be jointly alloate an the optimization problem an be formulate as Problem 3: (Partial Compression Offloaing min {t,v,λ } s.t. α D, =1 t 1, t 0, =1 =1 V V, V 0, (21a (21b (21 0 λ 1, K. (21 Notie that D is a pieewise funtion given in (6, whih an be rewritten in a more etaile way, as D = { λ L max V (1 λ L t R + βλ L, (1 λ L + (1 λ L t R t R V { βλ L +max, (1 λ L t R V }, if λ }, if λ < V V V +t R, V +t R. (22 April 4, 2017 DRAFT

14 14 B. Optimal Segmentation Strategy an Problem Transformation It an be seen that the D expression in (22 is ompliate with 3K variables suh that Problem 3 is har to be solve iretly. In the following, we will etermine the optimal λ while eeping t an V fixe. First, let us efine a geometri mean βv V, whih is referre to as the average ompression apaity for ompressing vieo. Corresponingly, t R an be regare as the average ommuniation apaity for transmitting vieo. Then the optimal vieo segmentation strategy has the following pieewise struture, as presente in Lemma 1. Lemma 1: Given the sets of {t } an {V }, the optimal vieo segmentation strategy for eah evie is given by λ = V (t R +V V V (1+β+t R (V +V V V, if t R +t R, if t R < βv V, βv V. (23 Proof: Please refer to Appenix A. Remar 3: The optimal vieo segmentation strategy shown in Lemma 1 is etermine by omparing the average ommuniation apaity with the average ompression apaity of eah evie. In the ase that the average ommuniation apaity ominates the average ompression apaity, i.e.,t R βv V, the omputation resoure is the bottlene of elay minimization for evie, an therefore we shoul mae full use of the omputation resoure. In this ase, λ satisfies Domp, D tran, while D omp, +D tran, = D tran, +D omp,. On the ontrary, in the ase thatt R < βv V, the ommuniation resoure is the main bottlene of elay minimization. Therefore, we nee to fully utilize the ommuniation resoure to minimize the en-to-en elay of eah evie. In this ase, λ fulfills that D omp, = D tran, while D omp, < D tran,. By substituting the optimal vieo segmentation strategy into (22, the en-to-en elay of evie an be written as L (t R +V ( t R +βv t R V D = V (1+β+t R (V +V D,1, if t R L t R +βv t R V +t R D,2, if t R < Then Problem 3 an be equivalently onverte to the following problem. βv V, βv V. (24 DRAFT April 4, 2017

15 15 Problem 4: (Equivalent Problem of Problem 3. min {t,v } α D, =1 (25a s.t. (21b an (21. (25b Theorem 3: Problem 4 is a pieewise onvex optimization problem. Proof: Please refer to Appenix B. C. Optimal Resoure Alloation Algorithm The ey hallenge of Problem 4 is that the D expression in (24 is ontinuous but nonifferential (or non-smooth at t R = βv V. Moreover, the partial erivatives of both D,1 an D,2 on t have quarti forms. Therefore, lassial KKT onitions annot be iretly applie to solve this problem an it is rather iffiult to fin its lose-form solution. In the following, we will evelop an effetive algorithm to optimally solve it, whih is base on the sub-graient metho for ommon non-ifferential onvex problems [22]. For ease of notation, we shall first efine the following auxiliary variables. Define a vetor of inepenent resoure variables, as x = [t 1,t 2,,t K,V 1,V 2,,V K ]. Define the weighte-sum elay of all evies, as F = K =1 α D. [ After that, we enote the sub-graient funtion of D as D = D, D ] t V. Sine it has been prove in Theorem 3 that D an be written as max { D,1, D },2, the sub-graient funtion an be haraterize as D,1, if t R > βv t V, [ ] D D,2, D,1, if t R = βv t t t V, D,2, if t R < βv t V, (26 April 4, 2017 DRAFT

16 16 D V D,1 V, if t R > [ ] D,1, D,2, if t V V R = D,2 V, if t R < βv V, βv V, βv V. Base on the above analysis, we introue the following theorem to solve Problem 4. Theorem 4: Problem 4 an be solve by the following iteration (27 x (n+1 = x (n φ n g (n, (28 where φ n is the step size of the n th iteration an g is the sub-graient funtion of K =1 α D, whih is efine as g = where ( K =1 t an ( K ( K =1 α D, subjet to (21b an (21, =1 t, if K =1 t > 1,, if K =1 V > V, ( K =1 V Proof: Please refer to Appenix C. ( K =1 V are utilize as the obstale funtions. Base on Theorem 4, we an effiiently solve Problem 4 by iteratively upating the ommuniation an omputation resoure alloation, whose etaile proeures are presente in Algorithm 1. Algorithm 1 The sub-graient algorithm for the partial ompression offloaing moel 1: Initialize 2: Initialize the maximum onvergene tolerane ǫ > 0. 3: Set the iteration inex n = 0. 4: Set the initial resoure alloation vetor x (0 that subjets to (21b an (21. 5: Calulate F (0 = K =1 α 6: Do 7: Upate the resoure alloation vetor by x (n+1 = x (n φ n g (n. 8: Upate n = n+1. 9: Calulate F (n = K 10: Until F (n F (n 1 ε. =1 α D (0 an g (0 aoring to (24 an (29. D (n an g (n aoring to (24 an (29. (29 Now we isuss the onvergene an the omputational omplexity of Algorithm 1. As we have prove in Appenix C, the vetor x (n will linearly onverge to the optimal solution x DRAFT April 4, 2017

17 17 when ǫ 0 [22]. On the other han, the omputational omplexity of the sub-graient algorithm mainly lies on the require number of iterations until onvergene, whih is etermine by the maximum tolerane ǫ. ( From [23], we an onlue that our propose algorithm has a polynomial 1 time omplexity of O, whih is esirable for pratial implementation. ǫ 2 D. A Speial Case The optimal solution evelope in the above subsetion is not in lose-form. To yiel more insights into the partial ompression offloaing moel, we further investigate a ommon senario where the ommuniation resoure is aequate while the omputation resoure is limite, suh as the typial sensor networ or the mahine-type ommuniations. The ey harateristi of this speifi senario is that the hannel apaity is muh greater than the evie omputation apaity, i.e., R V. Moreover, most urrent mobile evies utilize the MEPG4 vieo ompression tehnology whose ompression ratio,β, is between 1 50 an 1 [24], resulting in the 200 small size of loal ompresse vieo. Uner these onitions, the elay for transmitting the loal ompresse vieo, Dtran,, an be neglete while omparing with the elay for ompressing the loal ompression part of vieo,domp,. That is, βλ L λ L. Therefore, it is straightforwar t R V that the optimal vieo segmentation strategy in this ase satisfies D omp, = D tran, +D omp,. (30 Then applying the etaile elay expressions in Setion II-E into (30, we have the following optimal vieo segmentation strategy. Lemma 2: In the speifi senario of partial ompression offloaing, the optimal vieo segmentation strategy for eah evie is given by λ = V (t R +V V V +t (, K. (31 R V +V Base on Lemma 2, the en-to-en elay of evie an be written as D = L (t R +V V V +t (. (32 R V +V Substituting D into (25a, the onvex Problem 4 an be solve by the KKT onitions. Therefore, the optimal solution for this speifi senario an be erive, as shown in Theorem 5. April 4, 2017 DRAFT

18 18 Theorem 5: The optimal solution for the speifi senario of partial ompression offloaing is given by ( V α L R t = θ t V R = R (V +V + V ( α L V ω t R +V, K, +, K, where (y + = max{y,0},θ an ω are the optimal value of Lagrange multipliers that satisfy the ative ommuniation an omputation resoure onstraints K =1 t = 1 an K =1 V = V, respetively. Proof: Please refer to Appenix D. Remar 4: Theorem 5 reveals that the optimal time-slot an lou ompression apaity alloate to evie is etermine by the orresponing weight fator, size of raw vieo, hannel apaity, an loal ompression apaity. As the vieo size L inreases, more ommuniation an omputation resoures will be alloate to this evie. However, less ommuniation resoure will be alloate if the ommuniation apaity beomes larger. This result is onsistent with the intuition that, to reue the weighte-sum elay of all evies, the BS shoul alloate more ommuniation resoure to those evies with ba hannels. Moreover, in ase that the loal ompression apaity V (33 gets smaller, more ommuniation an omputation resoures shoul be alloate to this evie uner the riterion of minimizing the system elay. VI. NUMERICAL RESULTS In this setion, we will present numerial results to verify our analysis an valiate the performane of the propose algorithms. The simulation settings are as follows unless otherwise state. The BS has a raius of 250 m. Eah mobile evie is ranomly loate in the system an an assoiate with the BS through one wireless hannel. The weights for all evies are the same, i.e., α = 1 for all suh that the system elay represents the average en-to-en elay K of all evies. The hannel gains between mobile evies an the ege lou are generate aoring to i.i.. Rayleigh ranom variables with unit varianes. The transmission power is set equal for eah evie, i.e., p = 24 Bm, K. The total banwith B =10 MHz. For eah ompression tas, the vieo size an the evie ompression apaity follow the uniform istribution withl [10, 100] Mbits anv [0.5, 2] Mbps, respetively. All ranom variables DRAFT April 4, 2017

19 19 are inepenent for ifferent evies, moeling heterogeneous mobile ompression apaity. The total ompression apaity of the ege lou V is selete as 40 Mbps an the ompression ratio β is set as Other major simulation parameters are liste in Table I. TABLE I SIMULATION PARAMETERS Parameter Value Cell raius 250 m Banwith, B 10 MHz Noise power ensity, N Bm/Hz Path loss exp. 4 Transmission power, p 24 Bm Raw vieo size, L [10, 100] Mbits Devie ompression apaity, V [0.5, 2] Mbps Ege lou ompression apaity, V 40 Mbps A. Performane Comparison among Three Moels We first ompare the minimum system elays of loal ompression, ege lou ompression, an partial ompression offloaing. Fig. 3(a epits the minimum system elay versus the number of mobile evies in the three ifferent moels. First, the system elays of the ege lou ompression an partial ompression offloaing moels inrease with the number of mobile evies ue to the limite omputation resoure, while the system elay of the loal ompression moel is approximately invariant sine the ommuniation resoure is relatively aequate in our simulation. Seonly, by omparing the urves of loal ompression an ege lou ompression, we an observe that the ege lou ompression performs better than the loal ompression only when the number of evies is small. The reason an be explaine as follows. In ase that the number of evies is small, the lou ompression apaity alloate to eah evie woul be larger than the loal ompression apaity. In this ase, it is better to offloa omputation worloa to the ege lou for ompression than loal ompression from the perspetive of elay minimization. On the other han, as the number of evies grows, the lou ompression apaity alloate to eah evie woul be smaller than the loal ompression apaity, leaing to the better performane of loal ompression moel. Thirly, the partial ompression offloaing moel has the best performane among the three moels sine it jointly utilizes the ommuniation an omputation resoures. April 4, 2017 DRAFT

20 Loal Compression Ege Clou Compression Partial Compression Offloaing Solution in Theorem 5 System elay(s Number of evies (a System elay with the number of evies Loal Compression Ege Clou Compression Partial Compression Offloaing Solution in Theorem 5 System elay(s Average evie ompression apaity (Mbps (b System elay with the evie ompression apaity. Fig. 3. System elay of three moels. The performane gap between the ege lou ompression an partial ompression offloaing moels beomes more evient with the growing number of evies, whih iniates that when the number of users beomes large, using the partial ompression offloaing moel an greatly reue the system elay an improve the QoE for users. Finally, the lose-form solution in Theorem 5 an ahieve a near-optimal performane while outperforms both the loal ompression an ege lou ompression moels. It is beause uner our simulation settings, the ompression apaity of mobile evies is muh smaller than the orresponing ommuniation apaity. This result DRAFT April 4, 2017

21 21 emonstrates the effetiveness of our erivation in Theorem 5. Fig. 3(b shows the minimum system elay versus the average evie ompression apaity in the three ifferent moels. In this simulation, we assume 20 mobile evies in the system while varying the average loal ompression apaity of all evies from 0.75 Mbps to 2 Mbps. From the figure, we an observe that the system elays of the loal ompression an partial ompression offloaing moels erease with the average evie ompression apaity sine both use the loal omputation resoure for vieo ompression. Furthermore, the solution in Theorem 5 has a very lose-to-optimal performane espeially when the evie ompression apaity is small, emonstrating its auray an appliability in our system. B. Optimal Resoure Alloation in Partial Compression Offloaing Moel Next, we analyze the impat of vieo size an evie ompression apaity on the optimal resoure alloation in the partial ompression offloaing moel. In this simulation, we assume five evies in the system an eep the vieo size an loal ompression apaity of evies 2-5 fixe while varying those parameters of evie 1. The etaile simulation parameters for all evies are summarize in Table II. TABLE II SIMULATION PARAMETERS OF FIVE DEVICES Devie Vieo Size Loal ompression apaity Mbits Mbps 2 90 Mbits 1.2 Mbps 3 80 Mbits 1.3 Mbps 4 70 Mbits 1.4 Mbps 5 60 Mbits 1.5 Mbps Fig. 4 illustrates the optimal time-slot an lou omputation resoure alloations with ifferent vieo sizes of evie 1, where the loal ompression apaity of evie 1 is fixe to 1.1 Mbps. It an be observe that the optimal resoures t 1 an V 1 alloate to evie 1 inrease with its vieo size. On the other aspet, the resoures assigne to other evies will onsequently erease. This is rather intuitive ue to the fat that more resoures shoul be alloate to evie 1 to minimize the system elay as its vieo size inreases. Furthermore, it is shown that the optimal ommuniation an omputation resoure alloations have almost the same tren, as isplaye in Fig. 4(a an Fig. 4(b. The reason for this outome is lear sine the ommuniation an omputation resoures have the same effet on omputing the en-to-en elay of eah evie. April 4, 2017 DRAFT

22 Time-slot fration Devie 1 Devie 2 Devie 3 Devie 4 Devie Vieo size L 1 (Mbits (a Communiation resoure alloation. 12 Clou ompression apaity (Mbps Devie 1 Devie 2 Devie 3 Devie 4 Devie Vieo size L 1 (Mbits Fig. 4. Optimal resoure alloation with ifferent vieo sizes. (b Computation resoure alloation. Fig. 5 presents the optimal time-slot an ege lou ompression apaity with ifferent loal ompression apaities of evie 1, where the vieo size of evie 1 is fixe to 100 Mbits. It an be observe that the optimal resoures t 1 an V 1 alloate to evie 1 erease with its loal ompression apaity. On the other aspet, the resoures alloate to other evies will onsequently inrease. The reason is that, more resoures shoul be alloate to those evies with lower ompression apaity to reue the weighte-sum elay of all evies. In aition, both optimal ommuniation an omputation resoure alloations have an approximately linear DRAFT April 4, 2017

23 Time-slot fration Devie ompression apaity V 1 (Mbps (a Communiation resoure alloation. Devie 1 Devie 2 Devie 3 Devie 4 Devie 5 12 Clou ompression apaity (Mbps Devie ompression apaity V 1 (Mbps Devie 1 Devie 2 Devie 3 Devie 4 Devie 5 (b Computation resoure alloation. Fig. 5. Optimal resoure alloation with ifferent loal ompression apaities. tren, as shown in Fig. 5(a an Fig. 5(b. This is ue to the fat that the ommuniation apaity t R, the evie ompression apaity V, an the lou ompression apaity V have the same effet on alulating the en-to-en elay of eah evie. VII. CONCLUSION This paper investigates joint ommuniation an omputation resoure alloation for a TDMAbase multi-user MECO system. Our optimization aims to improve the QoE for users by April 4, 2017 DRAFT

24 24 minimizing the weighte-sum elay of all evies. Three moels, namely loal ompression, ege lou ompression, an partial ompression offloaing, are stuie an ompare. The optimal solutions for both loal ompression an ege lou ompression are firstly ahieve in loseform, an some inherent insights are also highlighte. In the partial ompression offloaing moel, we erive the optimal vieo segmentation strategy for eah evie in a lose-form expression. Then we formulate a pieewise onvex optimization problem, whih an be effiiently solve by a evelope sub-graient metho. Moreover, to gain some insights, we onsier a speifi senario in whih ommuniation apaity is muh greater than evie ompression apaity. In this speifi senario, the lose-form solution an be erive. Finally, numerial results emonstrate that the partial ompression offloaing an effiiently reue the en-to-en lateny as ompare against the other two moels. Future wors may onsier lateny-minimization ommuniation an omputation resoure alloation problem with non-orthogonal hannel aess where o-hannel interferene exists. In suh a ase, our analytial result for the elay performane remains unhange but the hannel apaity expression for eah evie will be more ompliate. Non-onvex optimization tools shoul be utilize to eal with this senario. Another interesting iretion of our future wor is to investigate energy-effiieny optimization problem for the multiuser MECO system, i.e., minimizing the overall energy onsumption by jointly alloating ommuniation an omputation resoures. APPENDIX A PROOF OF LEMMA 1 To prove Lemma 1, we first alulate a ritial ase for evie that satisfies Domp, = Dtran,, (34 Dtran, = D omp,, whih means that the elay for ompressing the loal part ata equals to that for transmitting the ege lou part ata, an the elay for ompressing the ege lou part ata also equals to that for transmitting the loally ompresse ata. Then applying the etaile elay expressions into (34, we an obtain the onition to reah this ritial ase, as t R = βv V. (35 DRAFT April 4, 2017

25 25 In aition, let us efine λ (1 = V t R +V whih satisfies D omp, = D tran, an λ(2 = t R t R +βv whih satisfies Dtran, = D omp,. Then we an prove Lemma 1 by the following analysis. A. Case A: t R βv V In this ase, λ (1 λ (2. When λ [ 0,λ (1, Domp, < D tran, an D tran, < D omp,. Therefore, we have D = Dtran, +D omp, = (1 λ L t R + (1 λ L, whih ereases with λ V. When [ ] λ λ (1,λ(2, Domp, D tran, an D tran, D omp,. Thus we have D = max(domp, + D tran,,d tran, +D omp,. Sine D omp, +D tran, = λ L V D omp, = (1 λ L t R + (1 λ L V + βλ L t R inreases with λ while D tran + ereases with λ, the elay of evie ahieves to the minimum V(t R +V V V (1+β+t R (V +V when Domp, + D tran, = D tran, + D omp,, whih results in λ = [ ] λ (1,λ(2. Finally when λ we have D = D omp, +D tran, = λ L V ( λ (2,1 ], D omp, > D tran, an D tran, > D omp,. Therefore, + βλ L t R, whih inreases with λ. Base on the above analysis, the optimal vieo segmentation strategy in this ase is λ = V (t R +V V V (1+β+t R (V +V. B. Case B: t R < βv V In this ase, λ (1 > λ (2. When λ [ 0,λ (2, we have D omp, < D tran, an D tran, < Domp,. Therefore, we have D = Dtran, + D omp, = (1 λ L t R + (1 λ L, whih ereases [ ] V with λ. When λ λ (2,λ(1, Domp, D tran, an D tran, D omp,. Thus we have D = Dtran, + D tran, = L t R (1+(β 1λ, whih also ereases with λ beause 0 < β < 1. ( Finally when λ λ (1 ],,1 Domp, > D tran, an D tran, > D omp,. Therefore, we have D = D omp, +D tran, = λ L V + βλ L t R, whih inreases with λ. Base on the above isussion, the optimal vieo segmentation strategy in this ase is λ = λ(1 = V V +t R. APPENDIX B PROOF OF THEOREM 3 Note that all onstraints in Problem 4 are affine. Therefore, Problem 4 is onvex if the objetive funtion is onvex. In the following, we first prove that D is a ontinuously pieewise onvex funtion. The Hessian of D,1 is H = H 11 H 12 H 21 H 22 = 2 D,1 t 2 2 D,1 V t 2 D,1 t V 2 D,1 (V 2. (36 April 4, 2017 DRAFT

26 26 We an prove that the Hessian in (36 is positive-efinite by proving all the leaing prinipal mirrors of H are positive, as ( ( β(1+β V 1 = H 11 = 2L V 2 ( (1+βV V +3t ( R V +V t 3 R ( (1+βV V +t ( R V +V 3 + (t R 2( ( V +V 3βV V ( ( V +V +t R β ( V 2 +(V 2 t 3 R ( (1+βV V +t ( R V +V 3 > 0, (37 2 = H 11 H 22 H 12 H 21 ( ( ( = 4L 2 t R +βv 2 β(1+β V 2V ( (1+βV V +t ( R 2V +3V t 3 R ( (1+βV V +t ( R V +V 5 + (t R 2( t R (β ( V 2 +(V 2 ( (V +βv 2 +4V V +3(V 2 t 3 R ( (1+βV V +t ( R V +V 5 > 0. (38 Therefore, D,1 is stritly onvex on both t an V. Next, we prove that D,2 is also onvex on t an V. The seon-orer partial erivative of D,2 on t fulfills 2 t R +β ( V 3 2 D,2 t 2 = 2L ((t R 3 +3βV (t R 2 +3β ( V t 3 R ( t R +V 3 > 0. (39 Sine D,2 oes not hange over V, D,2 is onvex on t an V. Moreover, it an be easily verifie that D,1 = D,2, D,1 D,2, an D,1 t t V D,2 V at t R = βv V. Therefore D is a ontinuous an pieewise funtion. Then omputing the ifferene between D,1 an D,2, we have D,1 D,2 = L t R ( ( t R +βv (t R 2 βv ( V V (1+β+t ( R V +V V ( t R +V. (40 Therefore, when t R βv V, D,1 D,2, otherwise D,1 < D,2. Aoring to (24, D an be rewritten as max{ D,1, D,2 }, whih is onvex on t an V sine the pointwise maximum preserves onvexity. Furthermore, the objetive funtion K =1 α D is the summation of a set of onvex funtions, whih is also onvex. This ens the proof. DRAFT April 4, 2017

27 27 APPENDIX C PROOF OF THEOREM 4 In the following, we will prove that the iterative metho in Theorem 4 oul onverge to the optimal resoure alloation solution, enote as x. First, the Euliean istane between the (n+1 th iteration solution an the optimal solution an be alulate as x (n+1 x 2 = 2 x (n φ n g (n x 2 2 (41 = x (n x 2 2φ 2 ng ( (nt x (n x +φ 2 n g (n 2 2 (42 x (n x 2 2φ 2 n( F(x (n F +φ 2 n g (n 2 2 (43 x (0 x n 2 2 ( φ 2 i F(x (i F(x + i=0 n i=0 φ 2 i g (i 2 2, (44 where x 2 is the Euliean norm of x an g (nt represents the transpose of g (n. Notie that the inequality operation (43 is base on the onvexity of K =1 α D, as F(x F(x (n + g (nt ( x x (n. Let us enote F (n best = minn i=0f ( x (i. Then we have n ( ( φ i F x (i F (x i=0 ( F (n best F (x ( n φ i. (45 Substituting (45 into (44, we an erive the upper boun ifferene between F (n best an F(x, as F (n best F (x where G = max n i=0 g (i 2. 2 i=0 x (0 x 2 2 x (n+1 x 2 + n 2 i=0 φ2 i g (i n i=0 φ i (46 x (0 x 2 x (n+1 x 2 +G n 2 2 i=0 φ2 i 2 n i=0 φ, (47 i Uner suh irumstanes, if we selet φ n that satisfies n=0 φ n = an n=0 φ2 n <, suh as φ n = 1 n+1, F(n best F(x will graually onverge to zero. Moreover, to aelerate the onvergene spee, we an selet the Polya step size φ n = F ( x (n F (n best +γ n g (n 2, where γ n 2 satisfies n=0 γ n = an n=0 γ2 n < [22]. Then the iteration will linearly onverge to the optimal solution x. This ens the proof. April 4, 2017 DRAFT

28 28 V APPENDIX D PROOF OF THEOREM 5 It an be easily verifie that the en-to-en elay expression (32 is stritly onvex on t an using the erivation metho in Appenix B. Therefore, we an utilize the KKT onitions to erive the lose-form solution. Let {t,v } enote the optimal solution for this speifi senario. Then the Lagrange funtion of K =1 α D an be expresse as ( K ( K L P = α D +θ t 1 +ω V V. (48 =1 =1 Applying the KKT onitions leas to the following neessary an suffiient onitions ( L α L R V 2 P > 0, t t = ( t R V +t R V +V V 2 +θ = 0, (49 = 0, t > 0, L P V = =1 ( 2 α L t R ( t R V +t R V +V V 2 +ω > 0, V = 0, = 0, V > 0, (50 ( K θ t 1 = 0, ω ( K =1 =1 V V = 0, t 1, θ 0, (51 =1 =1 V V, ω 0. (52 Base on the above onitions, we an erive the optimal resoure alloation solution for the speial ase, as This ens the proof. ( V α L R t = θ t V R = R (V +V + V ( α L V ω t R +V, K, +, K. (53 REFERENCES [1] J. Gubbi, R. Buyya, S. Marusi, an M. Palaniswmi, Internet of Things (IoT: A vision, arhitetural elements, an future iretions, ELSEVIER Future Gener. Compt. Syst., vol. 29, no. 7, pp , Sep DRAFT April 4, 2017

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