Energy-Efficient Mobile-Edge Computation Offloading over Multiple Fading Blocks
Rongfei Fan, Fudong Li, Song Jin, Gongpu Wang, Hai Jiang, Shaohua Wu
11 Energy-Efficient Mobile-Edge ComputationOffloading over Multiple Fading Blocks
Rongfei Fan, Fudong Li, Song Jin, Gongpu Wang, Hai Jiang, and Shaohua Wu
Abstract —By allowing a mobile device to offload computation-intensive tasks to a base station, mobile edge computing (MEC)is a promising solution for saving the mobile device’s energy.In real applications, the offloading may span multiple fadingblocks. In this paper, we investigate energy-efficient offloadingover multiple fading blocks with random channel gains. Anoptimization problem is formulated, which optimizes the amountof data for offloading to minimize the total expected energyconsumption of the mobile device. Although the formulatedoptimization problem is non-convex, we prove that the objectivefunction of the problem is piecewise convex, and accordinglydevelop an optimal solution for the problem. Numerical resultsverify the correctness of our findings and the effectiveness ofour proposed method.
Index Terms —Mobile edge computing, fading, mobile com-putation offloading.
I. I
NTRODUCTION
Due to the limited computation resources at a mobiledevice, it is difficult for the mobile device to run computation-intensive applications that require low latency, such as naturallanguage processing, virtual reality (VR), and augmentedreality (AR). Mobile edge computing (MEC) provides asolution, in which a mobile device is allowed to offload partof or all its data to a base station for computing, as thebase station often has high computation capability. After thebase station completes the computing tasks, it feeds back thecomputed results to the mobile device [1].Since the offloading is via wireless links, the wirelesstransmission for offloading involves power consumption andleads to latency. How to minimize power consumption ofthe mobile device while satisfying latency requirements ofcomputing has been investigated in the literature [2]–[10].For a single mobile device with a single antenna, the work in[2] considers that the computation task can be partitioned totwo parts, which are computed locally at the mobile deviceand are offloaded to the base station, respectively. The optimaloffloading ratio (i.e., the percentage of the computation taskthat is offloaded) and the mobile device’s transmit power levelare derived. Also for a single mobile device, the work in [3]
R. Fan and S. Jin are with the School of Information and Electronics, Bei-jing Institute of Technology, Beijing 100081, P. R. China (email: { fanrongfei,15120075709 } @bit.edu.cn). F. Li and H. Jiang are with the Departmentof Electrical and Computer Engineering, University of Alberta, Edmonton,AB T6G 1H9, Canada (email: { fudong1,hai1 } @ualberta.ca). G. Wang iswith the School of Computer and Information Technology, Beijing JiaotongUniversity, Beijing 100044, P. R. China (email: [email protected]). S.Wu is with the Shenzhen Graduate School, Harbin Institute of Technology,Shenzhen, 518055, P. R. China (email: [email protected]). investigates a similar research problem when both the mobiledevice and the base station have multiple antennas. Thework in [4] supposes that the base station’s CPU alternatesbetween busy state and idle state and assumes that the basestation’s idle intervals are known in advance. The offloadeddata in each epoch (during which the base station’s CPUkeeps idle or keeps busy) is optimized so as to minimize themobile device’s total energy consumption. The work in [5]considers multiple single-antenna mobile devices working intime division multiple access (TDMA) mode. Each mobiledevice’s offloading ratio, transmit power allocation, and timeratio for channel access are optimized. The work in [6]considers a multiple-input and multiple-output (MIMO) MECsystem with one and multiple mobile devices. All the datafor computing are offloaded to the base station. Pre-codingmatrix is designed for the single-device case, while pre-coding matrix design and the base station’s computationresource allocation are investigated for the multiple-devicecase. In [7], a mobile device can be served by one of multiplebase stations. Base station selection is investigated so as tominimize the mobile device’s energy consumption. The workin [8] considers a relay between a mobile device and its basestation, and optimizes the offloading ratio and the transmitpower levels of the mobile device and the relay.Note that in all these works [2]–[8], the wireless channelgain is assumed to keep unchanged in the whole processof data offloading. However, the maximal allowable delayof some applications can be longer than one fading block.For example, the application of VR requires the processingdelay to be within 20-60 ms [11], while the length of onefading block, within which wireless channel gain can beconsidered unchanged, can be at the order of 1 ms [12]. Thus,the data offloading process may span multiple fading blocks,which means that the wireless channel gain varies during thedata offloading process, and therefore, the methods in [2]–[8]cannot be used.To the best of our knowledge, only two works in the lit-erature consider data offloading over multiple fading blocks,and both works suppose there is a single mobile device in thesystem. The work in [9] investigates the problem of optimaloffloading over multiple fading blocks for one mobile devicethat is wireless-powered by the base station. It is assumed thatthe channel state information (CSI) of multiple future fadingblocks is non-causal, i.e., the CSI of multiple future fadingblocks is known in advance, which may not be practical. a r X i v : . [ c s . I T ] A p r The work in [10] also deals with offloading over multiplefading blocks. Data for computation is not partitioned. Thus,the whole computation task is offloaded to the base stationor computed locally, depending on which side leads to lessenergy consumption. The wireless channel gain is assumedto take a “good” state or a “bad” state. The computationtime at the base station is not taken into account (i.e., thecomputation time is assumed to be zero). By approximatingthe energy consumption for offloading one bit of data asa monomial function, a dynamic optimization problem isformulated to minimize the total energy consumption foroffloading, if the mobile device decides to offload rather thanlocally computing its data.In this paper, we also investigate the MEC offloadingover multiple fading blocks. Different from [9], we assumethat the CSI of multiple future fading blocks is causal, i.e.,the CSI of multiple future fading blocks is not known inadvance. In addition, the research problem in [9] is a convexproblem, which is solved by using the KKT condition. Butour considered research problem is a non-convex problem.Differences of our work from [10] are as follows. 1) Weconsider that the data for computation can be partitioned andexecuted in parallel at the mobile device and a base station.This model follows most of existing works [2], [5] and is amore general model compared to the non-partitioning modelin [10]. 2) We take into account computation time at the basestation.The contributions of our work are summarized as follows.Considering computation offloading of a mobile device overmultiple fading blocks, we formulate an optimization problemto minimize the total expected energy consumption of themobile device by deciding the amount of data to be offloaded.The formulated problem is a non-convex optimization prob-lem, and thus, is hard to solve. To solve the non-convex prob-lem, we discover and theoretically prove that the objectivefunction of the problem is piecewise convex, and thus, theformulated problem can be optimally solved by first findingsolutions for individual intervals within which convexity iskept and then picking up the best solution.II. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
Consider one mobile device with a computation-intensivecomputation task. The mobile device is connected to a basestation, and can offload part of its data to the base station forcomputing. After receiving all the offloaded data, the basestation starts computation and then feeds back the computedresults to the mobile device.The wireless channel between the mobile device and thebase station is assumed to be block-faded. Specifically, thechannel gain remains constant in a fading block, but variesrandomly and independently from fading block to fadingblock. Denote the normalized channel gain as h , which isthe ratio of the wireless channel gain to the background noise variance. h is independently and identically distributed overfading blocks with probability density function (PDF) f ( h ) .Since the maximal allowable latency of the computation taskcan be much longer than the duration of one fading block,the computation offloading process may span multiple fadingblocks.At the beginning of a fading block, the mobile device needsto estimate the normalized channel gain in the fading block(for example, the mobile device sends pilot signals to the basestation; then the base station estimates the normalized channelgain and feeds back to the mobile device). The time overheadto estimate the normalized channel gain is normally veryshort, and thus, is not considered in our work. Nevertheless,it is straightforward to take this time overhead into accountin our system model.Detailed computation and communication models are givenas follows. A. Computation Model
The computation task, denoted as T , can be describedby a profile of three parameters ( T, D, c ) , where T is themaximal allowable latency, D is the amount of computationinput data (in unit of nat for presentation simplicity), and c isthe number of CPU cycles required for computing one inputnat. With such a description, the total number of CPU cyclesrequired for completing task T is c D . The computationtask T is separable. Denote the amount of local task (i.e.,the data to be computed at the mobile device) and externaltask (i.e. the data to be computed at the base station) as D l and D e , respectively. Then we have D = D l + D e . (1)For the mobile device, let f l denote its computation ca-pability, which is in unit of CPU cycles per second and isalso called CPU frequency . Similar to [2], f l is adjustableand is not larger than an upper limit denoted as f Ul . Tocomplete the local task within duration T , the mobile device’scomputation capability should be set as f l = c D l T (recallingthat the number of CPU cycles for computing the local taskis c D l ). Due to the constraint f l ≤ f Ul , we have D l ≤ f Ul Tc . (2) As shown in [7], [9], [10], for a given data size D , the number ofCPU cycles required for computing the data can be written as C D , where C is a random variable whose distribution, denoted as g ( C ) , depends onthe nature of the application. To facilitate the analysis, a fixed value c isselected in [7], [9], [10] to guarantee that Pr ( C D ≤ c D ) ≥ (1 − ε ) where Pr( · ) means probability and ε is a predefined small value. Thus, c can be calculated by using knowledge of g ( C ) . Accordingly, the number ofCPU cycles required for computing the data with size D can be expressedas c D . According to [2], the consumed energy for the mobile deviceto finish the local task with CPU frequency f l within duration T can be written as E l ( D l ) = kf l T = kc D l T (3)where k is a fixed coefficient depending on the architectureof the mobile device’s CPU.At the base station, denote the computation capability forserving the mobile device as f e . Thus, to complete the exter-nal task with c D e CPU cycles, the associated computationaltime can be expressed as T e ( D e ) = c D e f e . (4)Thus, the offloading should be finished within duration T − T e ( D e ) . B. Communication Model
Data offloading is performed over multiple fading blocks.Denote the time duration of one fading block as T f . Similarto [5], the time for feeding back the computed results fromthe base station to the mobile device is negligible and thus, isnot considered. This is reasonable since the computed resultsare usually of small data size and the base station can transmitwith high power. The total number of fading blocks foroffloading D e data nats is a function of D e and can be writtenas N ( D e ) = (cid:24) T − T e ( D e ) T f (cid:25) = (cid:24) T f e − c D e T f f e (cid:25) (5)where (cid:100)·(cid:101) is the ceiling function. For the ease of discussion,these N ( D e ) fading blocks are indexed inversely, i.e., themobile device starts offloading data in fading block N ( D e ) and ends offloading data in fading block 1. Denote the set ofthe N ( D e ) fading blocks as N ( D e ) (cid:44) { , , ..., N ( D e ) } .Note that the mobile device transmits in partial duration infading block 1. Denote transmission time duration in fadingblock n as t n ( D e ) . Then we have t n ( D e ) = T f , for n ∈{ , , ..., N ( D e ) } , and t ( D e ) = T − T e ( D e ) − ( N ( D e ) − T f = (cid:16) T − c D e f e (cid:17) − (cid:108)(cid:16) T − c D e f e (cid:17) /T f (cid:109) T f + T f . (6)In one fading block, recall that h denotes the normalizedchannel gain. Consider transmission of d data nats withtransmission duration t over the spectrum with bandwidth W . By using the Shannon channel capacity formula [12], thetransmit power at the mobile device should be p ( d, h, t, W ) = ( e dtW − h (7)and the consumed energy can be written as e ( d, h, t, W ) = p ( d, h, t, W ) t = ( e dtW − th . (8) The mobile device should offload D e data nats to the basestation within N ( D e ) fading blocks. We should minimizethe energy consumption in the offloading. We have twodefinitions: • J n ( d ) ( n ∈ N ( D e ) ) is defined as the minimum expectedenergy consumption for transmitting totally d data natsfrom fading block n to fading block 1; • J n ( d, h n ) is defined as the minimum expected energyconsumption for transmitting totally d data nats fromfading block n to fading block 1 given that the normal-ized channel gain in fading block n is a specific value h n .Thus, J n ( d ) and J n ( d, h n ) satisfy the following equation J n ( d ) = (cid:82) ∞ J n ( d, h n ) f ( h n ) dh n .According to Bellman’s equation in dynamic programming[13], we have the following iterative calculation formulas: J n ( d ) = (cid:90) ∞ J n ( d, h n ) f ( h n ) dh n , n ∈ N ( D e ) , (9)in which J n ( d, h n ) = min ≤ d n ≤ d (cid:16) e ( d n , h n , T f , W )+ (cid:82) ∞ J n − ( d − d n , h n − ) f ( h n − ) dh n − (cid:17) = min ≤ d n ≤ d ( e ( d n , h n , t n , W ) + J n − ,m ( d − d n )) , ∀ n > . (10)and J ( d, h ) = e ( d, h , t , W ) . In (10), d n means the offloaded data nats within fadingblock n . Then by following the iterative calculation formulasin (9) and (10), J N ( D e ) ( D e ) can be calculated, which isthe minimum expected energy consumption for transmittingtotally D e data nats from fading block N ( D e ) to fading block1. To achieve J N ( D e ) ( D e ) , in fading block n ( n ∈ N ( D e ) ),with h n measured and d remaining data nats to be offloaded,the mobile device only needs to determine d n by solving theoptimization problem defined in (10). C. Problem Formulation
Given the system model, the total energy consump-tion of the mobile device to complete the task T is (cid:0) J N ( D e ) ( D e ) + E l ( D l ) (cid:1) . Our target is to minimize the totalenergy consumption by setting D l and D e optimally beforethe mobile device starts the data offloading, i.e., to solve thefollowing optimization problem Problem 1: min D e ,D l J N ( D e ) ( D e ) + E l ( D l ) s.t. D e ≥ , (11) D l ≥ , (12)Constraints (1) , (2) . III. O
PTIMAL S OLUTION
Problem 1 is not a convex optimization problem since N ( D e ) in J N ( D e ) ( D e ) involves the ceiling function of D e . Inthis section, we will show how to solve Problem 1 optimally.The following lemmas can be expected. Lemma 1:
With d ∈ [0 , D e ] , J n ( d ) and J n ( d, h n ) areconvex with d for n ∈ { , , ..., N ( D e ) − } . Proof:
We use the induction method. For n = 1 , it canbe easily checked that J ( d, h ) = e ( d, h , t , W )= t ( D e ) h (cid:16) e dt De ) W − (cid:17) (13)and J ( d )= (cid:82) ∞ J ( d, h ) f ( h ) dh = t ( D e ) σ W (cid:16) e dt De ) W − (cid:17) (cid:82) ∞ h f ( h ) dh , (14)both of which are convex functions with d .Now, suppose both J n ( d ) and J n ( d, h n ) are convex with d for n ∈ { , , ..., N ( D e ) − } . Then it can be derivedthat J n +1 ( d, h n ) is still convex with d since the infimalconvolution of convex functions is still convex [14, Theorem5.4].Given that J n ( d, h n ) is convex with d , it is straight-forward to see the convexity of J n ( d ) with d for n ∈{ , , ..., N ( D e ) − } since J n ( d ) can be interpreted asnonnegative weighted sum of J n ( d, h n ) with weighting co-efficient f ( h n ) dh n , which is still a convex function [15].This completes the proof. Lemma 2: J N ( D e ) ( D e ) is piecewise convex with D e . Proof:
From (6), we notice that t ( D e ) is piecewise lin-ear decreasing function with D e . Suppose I (cid:44) { , , ..., I ∗ } where I ∗ is the maximal integer i such that ( T − iT f ) islarger than 0. Define D i (cid:44) (cid:16) ( T − ( i + 1) T f ) f e c , ( T − iT f ) f e c (cid:105) , ∀ i ∈ I \ { I ∗ } , (cid:16) , ( T − iT f ) f e c (cid:105) , i = I ∗ . Then when D e falls within D i , we have t ( D e ) = T − iT f − c D e f e . (15)Without loss of generality, we assume D e falls into D i ,where i ∈ I . Thus N ( D e ) = i + 1 . In this case, de-note the optimal solution of d achieving J i +1 ( D e , h i +1 ) as d ∗ i +1 ( D e , h i +1 ) . Suppose D † e ∈ D i and D ‡ e ∈ D i , thus i + 1 = N ( D † e ) = N ( D ‡ e ) . For any α ∈ [0 , , the convexityof J i +1 ( D e , h i +1 ) can be proved in (16) on the top of thenext page, where (a) is due to the convexity of J i ( d, h i ) with d according to Lemma 1 and the convexity of e ( d, h, t, W ) with d , and (b) holds since (cid:0) αd ∗ i +1 ( D † e , h i +1 ) + (1 − α ) d ∗ i +1 ( D ‡ e , h i +1 ) (cid:1) ∈ [0 , αD † e + (1 − α ) D ‡ e ] and d ∗ i +1 ( αD † e + (1 − α ) D ‡ e , h i +1 ) is the optimal solution of d i +1 for achieving J i +1 ( αD † e + (1 − α ) D ‡ e , h i +1 ) .This proves the convexity of J i +1 ( D e , h i +1 ) with D e for D e ∈ D i . Thus J N ( D e ) ( D e , h N ( D e ) ) is a piecewise convexfunction with D e .Then by following the similar discussion in the proof ofLemma 1, the piecewise convexity of J N ( D e ) ( D e ) with D e can be also proved.This completes the proof.According to Lemma 2 and by checking (3), the objectivefunction of Problem 1 is also a piecewise convex functionwith D e . Thus to solve Problem 1, we only need to solve oneconvex optimization problem (whose global optimal solutionis achievable) in each interval of D e , i.e., for D e ∈ D i , ∀ i ∈I , and select the minimum total energy consumption among |I| intervals. Specifically, define the following problem: Problem 2: E ∗ ( i ) (cid:44) min D e ( J i +1 ( D e ) + E l ( D − D e )) s.t. (cid:18) D − f Ul Tc (cid:19) ≤ D e ≤ D, (17) D e ∈ D i . (18)To find E ∗ ( i ) , bisection search method can be used sinceboth J i +1 ( D e ) and E l ( D − D e ) are convex with D e andProblem 2 is a convex optimization problem.Then we only need to select the minimum E ∗ ( i ) among i ∈ I , which is the minimal expected energy consumptionof the mobile device. In other words, Problem 1 is solvedoptimally. IV. N UMERICAL R ESULTS
In this section, numerical results are presented. Similar to[2], [5], [6], system parameters are as follows: c = 40 , T =20 ms, T f = 2 ms, D = 4 × nats, f Ul = 0 . GHz, f e =1 GHz, k = 10 − , and W = 1 MHz. The normalized channelgain h is exponentially distributed (i.e., Rayleigh fading isassumed) with mean value 100.In Fig. 1, the functions J N ( D e ) ( D e ) , J N ( D e ) ( D e , h ) , J n ( D e ) , and J n ( D e , h ) are plotted for n ∈ { , } and h = 100 . It can be seen that the functions J n ( D e ) and J n ( D e , h ) are convex with D e , the functions J N ( D e ) ( D e ) and J N ( D e ) ( D e , h ) are piecewise convex with D e , which supportthe claims in Lemma 1 and Lemma 2, respectively.In Fig. 2, the total energy consumption achieved by ourproposed method is plotted for D ∈ [5 × , × ] . Fig. 2also shows the total energy consumption of the method in [6],which offloads all the data to the base station, and the methodin [10], which computes all data locally or offloads all data tothe base station, whichever has less energy consumption. Itcan be found that as D (the amount of data to be processed)goes higher, the total energy consumption grows, which isintuitive. αJ i +1 (cid:0) D † e , h i +1 (cid:1) + (1 − α ) J i +1 (cid:0) D ‡ e , h i +1 (cid:1) = α · e (cid:0) d ∗ i +1 ( D † e , h i +1 ) , h i +1 , T f , W (cid:1) + (1 − α ) · e (cid:0) d ∗ i +1 ( D ‡ e , h i +1 ) , h i +1 , T f , W (cid:1) + αJ i (cid:0) D † e − d ∗ i +1 ( D † e , h i +1 ) (cid:1) + (1 − α ) J i (cid:0) D ‡ e − d ∗ i +1 ( D ‡ e , h i +1 ) (cid:1) (a) ≥ e (cid:0) αd ∗ i +1 ( D † e , h i +1 ) + (1 − α ) d ∗ i +1 ( D ‡ e , h i +1 ) , h i +1 , T f , W (cid:1) + J i (cid:0) α (cid:0) D † e − d ∗ i +1 ( D † e , h i +1 ) (cid:1) + (1 − α ) (cid:0) D ‡ e − d ∗ i +1 ( D ‡ e , h i +1 ) (cid:1)(cid:1) (b) ≥ e (cid:0) d ∗ i +1 ( αD † e + (1 − α ) D ‡ e , h i +1 ) , h i +1 , T f , W (cid:1) + J i (cid:0) αD † e + (1 − α ) D ‡ e − d ∗ i +1 ( αD † e + (1 − α ) D ‡ e , h i +1 ) (cid:1) = J i +1 (cid:0) αD † e + (1 − α ) D ‡ e , h i +1 (cid:1) . (16) D e (nats) E ne r g y c on s u m p t i on ( J ) -3 J (D e )J (D e )J N(D e ) (D e )J (D e , h)J (D e , h)J N(D e ) (D e , h) Fig. 1: Verification of Lemma 1 and Lemma 2.
D(nats) -4 -3 -2 -1 T o t a l ene r g y c on s u m p t i on ( J ) ProposedThe method in [7]The method in [11]
Fig. 2: Total energy consumption versus D .In Fig. 3, the total energy consumption achieved by ourproposed method and the methods in [6], [10] are plottedversus the deadline T . It can be seen that as T increases, thetotal energy consumption tends to decrease. This is due to thefact that as the deadline T for processing data becomes lessurgent, the mobile device can have more time to complete itslocal computing and its data offloading, both of which can T(seconds) -4 -2 T o t a l ene r g y c on s u m p t i on ( J ) ProposedThe method in [7]The method in [11]
Fig. 3: Total energy consumption versus T . T f (seconds) T o t a l ene r g y c on s u m p t i on ( J ) -3 ProposedThe method in [7]The method in [11]
Fig. 4: Total energy consumption versus T f .help to save energy consumption.In Fig. 4, T is set as 40ms. The total energy consumptionachieved by our proposed method and the methods in [6],[10] are plotted versus T f . It shows that as the duration ofone fading block T f decreases, the total energy consumptiondecreases. This can be explained as follows. When thereare more fading blocks within duration T , the mobile userhas more flexibility in adjusting the transmit power over f e (Hz) O p t i m a l D e ( na t s ) Mean of h:10Mean of h:100Mean of h:1000
Fig. 5: Optimal offloading data D e versus f e .time, i.e., transmit with high power in case of good channelcondition and transmit with low power in case of bad channelcondition. Thus, more efficient utilization of the transmitpower is achieved, which leads to energy saving.From Fig. 2, Fig. 3, and Fig. 4, it can be seen that ourproposed method always has less energy consumption thanthe methods in [6], [10]. The reason is as follows. Themethods in [6], [10] set D e = D or D l = D , which meansthat they only provide a feasible solution to Problem 1. Ourmethod provides the optimal solution to Problem 1, and thus,has superior performance than the methods in [6], [10].Fig. 5 plots the optimal offloading data D e under variouscomputation capability f e and various mean values of nor-malized channel gain h . From Fig. 5, it can be seen that as thecomputation capability f e goes up, the mobile device tendsto offload more data to the base station until all the data isoffloaded to the base station. It can be also seen that as themean of h grows, the mobile device would like to offloadmore data to the base station. Indeed, with better wirelesschannel, the mobile device can send more data to the basestation without increasing its power consumption.V. C ONCLUSION
In this paper, we study the energy-efficient offloading inMEC over multiple fading blocks. An offloading strategy isproposed, which targets minimizing the total expected energyconsumption of the mobile device, by selecting the amountof data nats for offloading. An optimization problem isformulated, which is non-convex. We show that the objectivefunction of the optimization problem is piecewise convex anddevelop an optimal solution for the problem. Since privacyis an important issue for the application of MEC [16], [17],in the future, by combining some interesting discussions onsecurity and privacy over data offloading as in [18]–[20], wewish to extend our work in MEC. R
EFERENCES[1] X. Ma, Y. Zhao, L. Zhang, H. Wang, and L. Peng, “When mobileterminals meets the clouds: Computation offloading as the bridge,”
IEEE Netw. , vol. 27, no.5, pp. 28-33, Sep./Oct. 2013.[2] Y. Wang, M. Sheng, X. Wang, L. Wang, and J. Li, “Mobile-edge com-puting: Partial computation offloading using dynamic voltage scaling,”
IEEE Trans. Commun. , vol. 64, no. 10, pp. 4268-4282, Oct. 2016.[3] O. Mu ˜ noz, A. Pascual-Iserte, and J. Vidal, “Optimization of radio andcomputational resources for energy efficiency in latency-constrainedapplication offloading,” IEEE Trans. Veh. Technol. , vol. 64, no. 10,pp.4738-4755, Oct. 2015.[4] C. You and K. Huang, “Exploiting non-causal CPU-state informationfor energy-efficient mobile cooperative computing,”
IEEE Trans. Wire-less Commn. , vol. 17, no. 6, pp. 4104-4117, Jun. 2018.[5] C. You, K. Huang, H. Chae, and B. H. Kim, “Energy-efficient resourceallocation for mobile-edge computation offloading,”
IEEE Trans. Wire-less Commn. , vol. 16, no. 3, pp. 1397-1411, Mar. 2017.[6] S. Sardellitti, G. Scutari, and S. Barbarossa, “Joint optimization of radioand computational resources for multicell mobile-edge computing,”
IEEE Trans. Signal Inf. Process. Netw. , vol. 1, no. 2, pp. 89-103,June 2015.[7] Z. Sheng, C. Mahapatra, V. C. M. Leung, M. Chen, and P. K. Sahu,“Energy efficient cooperative computing in mobile wireless sensornetworks,”
IEEE Trans. Cloud Comput. , vol. 6, no. 1, pp. 114-126,Jan.-Mar., 2018.[8] X. Cao, F. Wang, J. Xu, R. Zhang, and S. Cui, “Joint computation andcommunication cooperation for energy-efficient mobile edge comput-ing,”
IEEE Internet of Things Journal , accepted.[9] C. You, K. Huang, and H. Chae, “Energy efficient mobile cloudcomputing powered by wireless energy transfer,”
IEEE J. Sel. AreasCommun. , vol. 34, no. 5, pp. 1757-1771, May 2016.[10] W. Zhang, Y. Wen, K. Guan, D. Kilper, H. Luo, and D. O. Wu, “Energy-optimal mobile cloud computing under stochastic wireless channel,”
IEEE Trans. Wireless Commn. , vol. 12, no. 9, pp. 4569-4581, Sept.2013.[11] J. Zhao, R. S. Allison, M. Vinnikov, and S. Jennings, “Estimating themotion-to-photon latency in head mounted displays,” in
Proc. 5th IEEEVirtual Reality , Los Angeles, CA, U.S., Mar. 2017.[12] D. Tse, and P. Viswanath,
Fundamentals of Wireless Communication .New York: Cambridge University Press, 2005.[13] D. P. Bertsekas,
Dynamic Programming and Optimal Control .Nashua: Athena Scientific, 2017.[14] R. T. Rockafellar,
Convex Analysis . Princeton: Princeton Univ. Press,1970.[15] S. P. Boyd and L. Vandenberghe,
Convex Optimization . Cam-bridge: Cambridge University Press, 2004.[16] W. Tu and L. Lai, “Keyless authentication and authenticated capacity,
IEEE Trans. Inf. Theory , vol. 64, no. 5, pp. 3696-3714, May 2016.[17] W. Tu, M. Goldenbaum, L. Lai, and H. V. Poor, “On simultaneouslygenerating multiple keys in a joint source-channel model,
IEEE Trans.Inf. Forensics Security , vol. 12, no. 2, pp. 298-308, Feb. 2017.[18] W. Tu and L. Lai, “On the simulatability condition in key generationover a non-authenticated public channel, in
Proc. IEEE InternationalSymposium on Information Theory (ISIT) , pp. 720-724, Hongkong,China, 2015.[19] W. Tu and L. Lai, “Function computation with privacy constraints, in
Proc. 51st Asilomar Conference on Signals, Systems, and Computers ,pp. 1672-1676, Pacific Grove, CA, 2017.[20] W. Tu and L. Lai, “Keyless authentication over noisy channel, in