2606 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 35, NO. 11, NOVEMBER 2017 Optimal Schedule of Mobile Edge Computing for Internet of Things Using Partial Information Xinchen Lyu, Wei Ni, Hui Tian, Ren Ping Liu, Xin Wang, Senior Member, IEEE, Georgios B. Giannakis, Fellow, IEEE, and Arogyaswami Paulraj, Fellow, IEEE

Abstract— Mobile edge computing is of particular interest limited capability of individual devices and computationally to Internet of Things (IoT), where inexpensive simple devices demanding mobile applications [1], [2]. It provides an effec- can get complex tasks offloaded to and processed at powerful tive means to reduce the cost of wireless terminals without infrastructure. Scheduling is challenging due to stochastic task arrivals and wireless channels, congested air interface, and more compromising functionalities and services. This is of particular prominently, prohibitive feedbacks from thousands of devices. interest to the future Internet of Things (IoT), where thousands In this paper, we generate asymptotically optimal schedules of inexpensive, computationally incapable devices can be tolerant to out-of-date network knowledge, thereby relieving deployed within the coverage of a base station [3]. stringent requirements on feedbacks. A perturbed Lyapunov Different from cloud computing with dedicated high-speed function is designed to stochastically maximize a network utility balancing throughput and fairness. A knapsack problem is solved wired connections, MEC has all devices operate in shared per slot for the optimal schedule, provided up-to-date knowledge wireless media which feature unpredictable channel changes, on the data and energy backlogs of all devices. The knapsack and become increasingly congested with the increase of problem is relaxed to accommodate out-of-date network states. devices [4]. Optimal offloading schedules are obviously impor- Encapsulating the optimal schedule under up-to-date network tant to MEC, but yet to be established due to sophisticated knowledge, the solution under partial out-of-date knowledge pre- serves asymptotic optimality, and allows devices to self-nominate stochasticity and changing availability of resources at the MEC for feedback. Corroborated by simulations, our approach is able server, random data arrival per device, and fluctuating wireless to dramatically reduce feedbacks at no cost of optimality. The channels [5]. Myopically optimized schedules which fail to number of devices that need to feed back is reduced to less capture the stochasticity would be inefficient over longer time than 60 out of a total of 5000 IoT devices. horizons [6]. Optimal schedules have also been hindered by a Index Terms— Mobile edge computing, Internet of Things, typical need for explicit knowledge on network states. In the partial information, Lyapunov optimization. case of IoT, thousands of devices could be required to feed back their states and channels, whenever the states or channels I. INTRODUCTION change [7]. This would incur prohibitive overhead, congest the FFLOADING and executing tasks of wireless termi- channel, and jeopardize practicality. Onal at the edge of wireless networks, mobile edge Most existing works on MEC has overlooked the stochas- computing (MEC) is able to bridge the gap between the ticity or assumed predictability, and optimized the offloading schedules deterministically in the presence of full knowledge Manuscript received April 1, 2017; revised September 12, 2017; accepted September 25, 2017. Date of publication October 9, 2017; date of current on network states. In [8]Ð[10], non-convex optimization prob- version December 1, 2017. This work was supported by the National Natural lems were myopically formulated and solved in a centralized Science Foundation of China under Grant 61471060 and Grant 61421061. manner. In [11]Ð[13], game theoretic approaches or submod- (Corresponding author: Hui Tian.) X. Lyu is with the State Key Laboratory of Networking and Switching ular optimization methods were taken to produce myopic Technology, Beijing University of Posts and Telecommunications, Beijing schedules per slot in a decentralized fashion. In [14] and [15], 100876, China, and also with the Global Big Data Technologies Center, Markov decision processes (MDP) were formulated to opti- University of Technology Sydney, Sydney, NSW 2007, Australia. W. Ni is with the Digital Productivity and Service Flagship, CSIRO, mize the schedules under the assumption of a priori knowledge Canberra, NSW 2122, Australia. on the statistics of stochastic processes, and solved by dynamic H. Tian is with the State Key Laboratory of Networking and Switching programming. However, these statistics are typically unknown Technology, Beijing University of Posts and Telecommunications, Beijing 100876, China (e-mail: [email protected]). in prior, and dynamic programming also suffers from the R. P. Liu is with the Global Big Data Technologies Center, University of “curse-of-dimensionality” and offers little scalability. Technology Sydney, Sydney, NSW 2007, Australia. A few recent works have started to take the stochasticity and X. Wang is with the Key Laboratory for Information Science of Electromag- netic Waves (MoE), Department of Communication Science and Engineering, unpredictability into the consideration and optimize offload- Fudan University, Shanghai 200433, China. ing schedules foresightedly by using stochastic optimization G. B. Giannakis is with the Department of Electrical and Computer techniques, i.e., Lyapunov optimization [16]Ð[18]. However, Engineering and the Digital Technology Center, University of Minnesota, Minneapolis, MN 55455 USA. these works encompassed on different network architectures A. Paulraj is with the Department of Electrical Engineering, Stanford from MEC, e.g., point-to-point link for single device [16] and University, Stanford, CA 94305 USA. multi-hop link in wireless sensor networks [17], [18], and their Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. results are inapplicable to MEC with thousands of IoT devices Digital Object Identifier 10.1109/JSAC.2017.2760186 to be offloaded. 0733-8716 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. LYU et al.: OPTIMAL SCHEDULE OF MEC FOR IoT USING PARTIAL INFORMATION 2607

This paper generates new asymptotically optimal offloading TABLE I schedules, which are tolerant to partial out-of-date network DEFINITIONS OF NOTATIONS knowledge and stochastically maximize a time-average net- work utility balancing system throughput and fairness. The contribution of this paper is multi-folded: (a) A perturbed Lyapunov technique is proposed to decouple time couplings of offloading schedule, energy intake, and data admission, and convert the stochastic optimization of MEC offloading schedules to deterministic optimizations over time slots with- out loss of optimality. An [O(V ), O(1/V )]-tradeoff can be achieved between stability and utility, so that the optimality loss of the proposed approach (as compared to the offline optimum) asymptotically diminishes at the cost of increasing queue length; (b) Particularly, the deterministic optimization per slot is meticulously formulated to a linear relaxation of a knapsack problem. The breaking item of the problem, achieving the asymptotical optimality, is identified. Its lower bound in the presence of partial out-of-date information is derived as the threshold. As a result, only the devices meet- ing the threshold are designed to feed back per slot, hence preserving the asymptotic optimality with reduced feedbacks; (c) Extensive simulations show that our approach is able to substantially improve system utility by 26% and dramatically reduce feedbacks per slot by over 98%. Only less than 60 out of 5000 devices need to feed back per slot. The feedbacks can decline with an increasingly large number of devices, due to the increasingly prominent urgency of some devices. The rest of this paper is organized as follows. In Section II, Independent and identical distributed (i.i.d.) flat block fad- the system model is presented. In Section III, we elaborate ing channels are assumed, i.e., the channels remain unchanged the proposed stochastic online optimization of offloading during a time slot and vary between slots [19], [20]. Let ci (t) schedules, provided up-to-date knowledge on network states denote the channel capacity in the uplink of device i at slot t. min ≤ ( ) ≤ max min max at the MEC server. In Section IV, we extend the approach ci ci t ci ,whereci and ci are the minimum under out-of-date knowledge and derive the optimality pre- and maximum capacities of the link, respectively, given finite serving threshold for the self-nomination of devices. Simula- transmit powers of IoT devices. tion results are shown in Section V, followed by conclusions Within a subchannel, the transmissions of multiple IoT in Section VI. devices can be accommodated via time-division multiple- access. Consider that an IoT device is inexpensive, simple II. SYSTEM MODEL and narrow-band [3], [7]. It can only access a subchannel at τ ( ) The MEC system that we consider consists of a base the same time. Let i t denote the transmission duration of station (BS) and N IoT devices, e.g., smart meters for supply device i within slot t, i.e., monitoring, actuators for industrial automation, and sensors for 0 ≤ τi (t) ≤ T, ∀i ∈ N, (1) smart home and healthcare.1 Let N ={1, 2,...,N} collect the indexes for N IoT devices. The system operates in a slotted and the maximum amount of data offloaded from device structure, t ∈ T ={0, 1,...} with slot length T . Equipped i to the MEC server can be given by di (t) = ci (t)τi (t). with energy harvesting capabilities, the IoT devices can be In some cases, a device can be assigned with different powered by renewable energy. An MEC server is deployed at subchannels at different times during a slot. The narrow- the BS, and sensory data from the IoT devices can be offloaded band device can switch between subchannels to offload through wireless links and processed at the MEC server. In the tasks [3]. presence of other uplink traffic to the BS, the number of We define an offloading schedule τ(t) ={τ1(t),...,τN (t)} available subchannels for the IoT devices, denoted by K (t), by collecting the schedules of all IoT devices during slot t. can vary between time slots. The notations used in this paper The total transmission time of the devices must not exceed the are summarized in Table I. number of available subchannel times slot length, given as 1 The consideration of a single BS is elemental, and can be readily applied to τi (t) ≤ K (t)T. (2) i∈N the case of multiple BSs where each BS schedules the devices in its coverage independently as proposed in this paper. This is due to the typical layered D( ) ={τ( )| ≤ τ ( ) ≤ , τ ( ) ≤ ( ) } architecture of cellular networks where inter-cell interference is mitigated We define t t 0 i t T i∈N i t K t T through (fractional) frequency reuse among BSs. by collecting all the feasible offloading schedules at time slot t. 2608 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 35, NO. 11, NOVEMBER 2017

Let Ai (t) denote the arrival of sensory data at device i We can formulate P to maximize the system utility as at slot t. Ai (t) is an i.i.d. random process with the maxi- max P: max φ(a) mum Ai .Devicei may only be able to admit part of the τ(t),e (t),a(t) ( ) ≤ ( ) ≤ ( ) H arrived data, denoted by ai t , i.e., 0 ai t Ai t , based s.t. C1: τ(t) ∈ D(t), ∀t ∈ T on the availability of its buffer. C2: 0 ≤ ei (t) ≤ Ei (t), ∀i ∈ N, t ∈ T Let Qi (t) be the backlog of the data queue at device i,and H H it is updated along the time, as given by C3: 0 ≤ ai (t) ≤ Ai (t), ∀i ∈ N, t ∈ T C4: ei (t) ≤ Ei (t), ∀i ∈ N, t ∈ T Qi (t + 1) =[Qi (t) − di (t)]+ai (t), (3) C5: C, Ei and Qi < ∞, ∀i ∈ N, (9) where where τ(t), e (t) and a(t) are the offloading schedule, d (t) = min{Q (t), c (t)τ (t)}, (4) H i i i i energy intake, and data admission to be optimized, respec- since the device cannot offload more than what it has. The tively. Constraints C1 to C4 are self-explanatory, as discussed corresponding energy consumption of the device for offloading in Section II. C5 ensures the stability of all the queues. the data, ei (t), can be written as Particularly, if the queue backlogs are upper bounded, the total data admission a is equal to the time-average system e (t) = p τ (t), (5) i∈N i i i i throughput in the long term [23]. ( ) ( ) i ( ) ( ) where pi is the transmit power of device i. Note that ci t , Ai t , EH t and K t and computing The energy harvesting at an IoT device can also be modeled resources F(t) at the MEC server, are all time-varying with i ( ) as a stochastic arrival process [16]Ð[18], [21]. Let EH t little predictability. An offline optimization of P would violate denote the amount of renewable energy harvested by device i causality; whereas a myopic optimization per time slot would i ( ) during time slot t. Assume that EH t is i.i.d. with the compromise the optimality in the long term, due to time i,max coupling of variables in C1 to C4. maximum EH . Consider a finite battery at the device. Device i can only store part of the newly harvested energy at i ( ) ≤ i ( ) ≤ i ( ) each time slot t, denoted by eH t ,where0 eH t EH t . A. Perturbed Lyapunov Optimization Let Ei (t) denote the battery backlog of device i at slot t. It can be updated by Lemma 1: P can be equivalently reformulated as ( + ) =[ ( ) − ( )]+ i ( ), φ(γ ) Ei t 1 Ei t ei t eH t (6) P1: max τ(t),eH(t),a(t),γ (t) where ei (t) ≤ Ei (t), since the energy used for offloading data s.t. C1-C5 and C6: γi ≤ ai must not exceed the energy available in the battery. ≤ γ ( ) ≤ max, ∀ ∈ , ∈ T, C7: 0 i t Ai i N t (10) Upon the receipt of the task from device i at time slot t, i.e., di (t) in (4), the MEC server can process the task with where γ(t) is the defined auxiliary variables. fi (t) = ξi di (t) CPU cycles or put the task into the queue Proof: See Appendix A.  for later processing, where ξi is the number of CPU cycles By defining a device-specific virtual queue, C6 can be required per task bit of device i for the task. reformed with improved tractability, where the backlog of Let C(t) be the required CPU cycles to process the task the virtue queue, denoted by Gi (t), can be updated by the queued at the MEC server. C(t) can be updated by difference of γi (t) and ai (t),asgivenby C(t + 1) = max [C(t) − F(t), 0] + fi (t), (7) G (t + ) = {G (t) + γ (t) − a (t), }. i∈N i 1 max i i i 0 (11) ( ) where F t specifies the total available CPU cycles at slot t.In Clearly, Gi (t) is stable if and only if C6 is satisfied [23]. F(t) the presence of other concurrent services, is stochastic We can replace C6 with the stability of Gi (t),andP1 can be max ( ) with the maximum F . i∈N fi t gives the total of the rewritten as newly offloaded tasks during time slot t. P2: max φ(γ ) τ(t),e (t),a(t),γ (t) III. ONLINE OPTIMIZATION OF OFFLOADING IN MEC H s.t. C1-C5 and C7 Leveraging between throughput and fairness in MEC, C8: G < ∞, ∀i ∈ N. (12) the system utility can be defined as i Stochastic optimization, more specifically, Lyapunov opti- φ(a) = log(1 + ai ), (8) i∈N mization, is able to eliminate time coupling of variables, and = 1 T −1 E[ ( )] enable online decision-makings, while preserving asymptotic where X limT →∞ T t=0 X t defines the time- average of any stochastic process X (t),anda = optimality [23]. A perturbed Lyapunov function of P2 can be {a , a , ··· , a } collects the time-average total of admitted defined as 1 2 N data at all N devices. Particularly, φ(a) is a concave loga- 1 2 2 2 2 rithm function that is able to balance the network throughput L(t) = C(t) + [Qi (t) + (Ei (t) − θi ) + Gi (t) ] , 2 and fairness by discouraging individual devices to consume i∈N excessive energy for little utility gains [22]. (13) LYU et al.: OPTIMAL SCHEDULE OF MEC FOR IoT USING PARTIAL INFORMATION 2609

∗ where θ ={θi , ∀i ∈ N} collects the new perturbing Theorem 1: The gap between the optimum for P3, φ(a ), weights (biases) to ensure sufficient energy available in the bat- and the optimum for P, φopt, is less than D/V, i.e., teries for optimal schedules durations at any slot. Let (t) = φopt − φ( ∗) ≤ / . {C(t), Qi (t), Ei (t), Gi (t), ∀i ∈ N} collect all the backlogs at a D V (22) slot t. A drift-plus-penalty function can be defined by  Proof: See Appendix C. ∗ V (t) = (t) − V E log(1 + γi (t))|(t) , (14) Theorem 2: With a , the backlogs of all the queues in the i∈N system is strictly bounded at each time slot t, as given by  where (t) = E [L(t + 1) − L(t)|(t)] is the conditional ( ) ≤ / + max; Qi t V ln 2 2Ai (23) Lyapunov drift, i.e., the conditional expectation of the dif- , E (t) ≤ θ + Ei max; (24) ference of the perturbed Lyapunov function between two i i H consecutive time slots, and V ≥ 0 is a predefined coefficient to C(t) ≤ β + ξ cmaxT ; ∈ i i (25) tune the tradeoff between the system utility and queue stability. i N Minimizing an upper bound of V (t) per slot t can prevent where unbounded growths of backlogs, while maximizing the time- / + max i,max V ln 2 2Ai EH pi average conditional expectation of the system utility [23]. The β = max ∈ { + }. (26) i N ξ minξ upper bound can be given in the following lemma. i ci i Lemma 2: For any queue backlogs and actions,  (t) is V Proof: See Appendix D.  upper bounded by θ Theorem 3: The weight perturbation parameter i can be adjusted by V (t) ≤ D − V E log (1 + γi (t))|(t) i∈N θ =[ / + max] max/ + , + C(t)E[ fi (t) − F(t)|(t)] i V ln 2 2Ai ci pi pi T (27) i∈N + Qi (t)E[ai (t) − di (t)|(t)] such that, when a device is scheduled to offload at time slot i∈N i t, there is enough energy in its battery backlog for optimal + [Ei (t) − θi ]E[e (t) − ei (t)|(t)] ( ) ≥ i∈N H schedules, i.e., Ei t pi T. Proof: See Appendix E.  + Gi (t)E[γi (t) − ai (t)|(t)] (15) i∈N Theorems1,2and3showan[O(V ), O(1/V )]-tradeoff where between the queue lengths (or in other words, the queueing delay according to Little’s theorem [24]) and the optimality 1 2 i,max 2 max 2 max 2 D = {(pi T ) + (E ) + (c T ) + 3(A ) } loss. Here, the optimality loss is the gap of system utility 2 i∈N H i i between the proposed approach and the causality-violating 1 max 2 max 2 + { (ξi c T ) + (F ) } (16) offline optimization of the offloading schedule. That is to say, 2 i∈N i by fine-tuning V , our approach allows the queueing delay to  Proof: See Appendix B. be adjustable at the cost of the optimal system utility. Exploiting Lemma 2, we reformulate P2 to minimize (15), subject to the instantaneous constraints in P2, as follows: B. Optimal Offloading Schedules P3: max f (γ (t)) + g(τ(t)) − η(eH(t)) − μ(a(t)) τ( ), ( ), ( ),γ( ) t eH t a t t Note that τ(t), eH(t), a(t) and γ (t) are decoupled in both s.t. C1-C4andC7, (17) the objective and constraints in P3, and therefore can be optimized separately as follows to achieve a∗. where F(t) is suppressed, since it is independent of the 1) Optimal Auxiliary Parameter: The subproblem of variables; optimizing auxiliary variables can be written as f (γ (t)) = [V log (1 + γi (t)) − Gi (t)γi (t)]; (18) i∈N max f (γ (t)) γ (t) g(τ(t)) = {[Qi (t) − C(t)ξi ]di (t) +[Ei (t) − θi ]ei (t)}; s.t. 0 ≤ γ (t) ≤ Amax, ∀i ∈ N, (28) i∈N i i (19) where γ (t) is decoupled from γ (t) for any j = i. Therefore, i i j η(eH(t)) = [Ei (t) − θi ]e (t); (20) γ ( ) i∈N H i t can be optimized independently by solving μ(a(t)) = [Qi (t) − Gi (t)]ai (t). (21) i∈N max V log(1 + γi (t)) − Gi (t)γi (t) γi (t) Let a∗ denote the optimal solution for P3,andφopt be the . . ≤ γ ( ) ≤ max s t 0 i t Ai (29) maximum system utility which can only be achieved offline under the assumption of full knowledge across all time slots, The first-order partial derivation of f (γ (t)) can be given ∂ f V ∂ f = − ( ) |γ ( )= ≤ (γ ( )) i.e., the optimum for P. As shown in the following theorem, by ∂γ (1+γ (t)) ln 2 Gi t .If ∂γ i t 0 0, f t ∗ i i i the optimum for P3, φ(a ), converges to the optimum for P, decreases monotonically in the region γi (t) ≥ 0andthe opt ∗ φ ,asV increases. optimum γi (t) = 0. Otherwise, the optimum is either at the 2610 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 35, NO. 11, NOVEMBER 2017

∂ f ∗ V stationary point = 0, i.e., γi (t) = − 1, or on the Define the breaking item to be the first device that is not ∂γi Gi (t) ln 2 ∗ max allocated its full transmission duration specified in (35) [25]. boundary, i.e., γi (t) = A . Therefore, ⎧ i The index of the breaking item b can identified, as given by ⎪ V ⎨0, if − G (t) ≤ 0; i ∗ i γ ( ) = ln 2 b = arg min Tj > K (t)T. (38) i t V j=1 ⎩⎪ { − , max}, i min 1 Ai otherwise. Gi (t) ln 2 As a result, the optimal offloading schedule can be given by (30) ⎧ ⎪ , < ; ⎨Ti if i b 2) Optimal Energy Intake: Decoupled from (17), the ∗ b−1 τi (t) = K (t)T − T , if i = b; (39) subproblem of optimizing energy intake can be written as ⎩⎪ i=1 i 0, otherwise, i min [Ei (t) − θi ]e (t) ( ) ∈ H eH t i N which, involving ordering the devices against their unit profits, . . ≤ i ( ) ≤ i ( ), ∀ ∈ , s t 0 eH t EH t i N (31) can be achieved by using the quicksort algorithm with a complexity of O(N log N) [26]. where the optimal solution can be readily given by ∗ 0, if E (t) ≥ θ ; Algorithm 1 Online Optimization of Offloading in MEC ei (t) = i i (32) H i ( ), EH t otherwise. At each device i: i 1: Acquire Qi (t), Ei (t), ci (t), Ai (t) and E (t) The devices consistently push their battery backlogs towards θ. H 2: Perform the optimal solutions given by (30), (32) and (34) As shown in Theorem 3, sufficient energy is available in the 3: Feed back Qi (t), Ei (t) and ci (t) to the MEC server batteries for optimal schedules by judiciously designing θ. 4: Update Gi (t + 1) based on (11) 3) Optimal Data Admission: The decoupled subproblem of At the MEC server: optimizing data admission can be written as ( ) ( ) ( ) 5: Acquire K t , F t and C t α ( ) [Q (t) − G (t)]a (t) 6: Collect feedbacks from all the devices and evaluate i t min ∈ i i i a(t) i N 7: Schedule the offloading by (39) s.t. 0 ≤ ai (t) ≤ Ai (t), ∀i ∈ N, (33) where the optimal admission decision can be given by Note that the optimal solutions in (30), (32) and (34) can be locally computed at individual IoT devices, since the queue , ( ) ≥ ( ); ∗ 0 if Qi t Gi t backlogs, wireless channels, data arrivals and energy harvest- ai (t) = (34) Ai (t), otherwise. ing can be measured at the devices without the coordination of the MEC server. However, (39) requires the knowledge 4) Optimal Offloading Schedule: τ(t) is optimized to max- on all the IoT devices, i.e., c (t), Q (t), E (t),andG (t) imize g(τ(t)), while satisfying (1) and (2) at each time slot t. k k k k for k ∈ N. At each time slot t, the MEC server needs each From (3) and (4), it is clear that device i should not transmit IoT device to feed back its link capacity c (t) and backlogs more data than Q (t) at slot t, i.e., τ (t) ≤ Q (t)/c (t). k i i i i Q (t), E (t) and G (t) to evaluate τ (t)∗ in (39). We propose Besides, C4 requires that the transmission energy cannot k k k i to distribute part of these optimizations (30), (32) and (34) exceed the available energy in the battery backlog, i.e., τ (t) ≤ i at the IoT devices, and the rest (39) at the MEC server, as E (t)/p . Therefore, (1) can be tightened to i i summarized in Algorithm 1. 0 ≤ τi (t) ≤ Ti , ∀i ∈ N, (35) IV. OPTIMIZATION OF OFFLOADING SCHEDULES where Ti = min{Qi (t)/ci (t), Ei (t)/pi , T }. By rearrang- ing (19), τ(t) can be optimized by UNDER OUT-OF-DATE BACKLOGS As discussed, the link capacity and backlogs of each IoT P4: max αi (t)τi (t) τ(t) i∈N device are needed in the MEC server to evaluate (39) for s.t. (2) and (35), (36) optimal offloading schedules. Consider practical IoT, however, the number of IoT devices can be hundreds to thousands, where while the number of available subchannels can be far less, i.e., K (t) N. Getting each device to feed back as per αi (t) =[Qi (t) − C(t)ξi ]ci (t) +[Ei (t) − θi ]pi . (37) slot could be infeasible due to explosive overhead, and also By interpreting αi (t) as the unit profit of an “item” i and inefficient since the majority of the feedbacks would not K (t)T as the capacity of a knapsack, P4 becomes the linear contribute to the optimization of the schedule. relaxation of a knapsack problem. The optimal solution for From (39), it is clear that the IoT devices with high unit linear relaxation knapsack can be found by selecting the “item” profits are given priority to offload. Particularly, the optimality with higher unit profit to fulfill the knapsack capacity [25]. of (39) would not be compromised, if the devices with unit Without loss of generality, we assume the devices are sorted profits less than the breaking item, i.e., αi (t)<αb(t), do not α ( ) ≥ α ( ) in the descending order of unit profit, i.e., i t j t for feed back. To this end, there is an opportunity arising to > τ = i ≤ ( ) i j. The optimal solution is i Ti if j=1 Tj K t T . substantially reduce feedbacks at every slot. LYU et al.: OPTIMAL SCHEDULE OF MEC FOR IoT USING PARTIAL INFORMATION 2611

In this section, we generalize the proposed online optimiza- tion with up-to-date knowledge on devices’ backlogs at the MEC server, to the scenario where the knowledge is out- of-date at the MEC server from slot to slot. Lyapunov opti- mization under partial information and out-of-date knowledge is established, which captures the case of practical interest where only part of IoT devices feed back their backlogs. α ( ) α ( ) Using this, we derive a threshold for i t , denoted by b t , which enables a small number of devices to self-nominate to feed back, while preserving the asymptotic optimality stated Fig. 1. The changes of throughput and the corresponding average number in Theorem 1. Following are the details. of device as N increases from 100 to 5000 where V = 40. From (37), the MEC server evaluates αi (t) based on C(t), ci (t), Qi (t) and Ei (t),whereC(t) is known to the server V. S IMULATION RESULTS whereas the other three parameters need device i to feed back. In the absence of up-to-date feedbacks from most devices, In this section, we evaluate the proposed optimal offloading we define Qi (t) and Ei (t) for the data and energy backlogs schedules of MEC system in Matlab with T = 1 sec. of device i, respectively, to capture the out-of-date knowledge Focused on scheduling a large number of devices using partial at the MEC server. out-of-date knowledge, we assume that the uploaded tasks The MEC server can update Qi (t) and Ei (t), as follows. can be computed correctly, and the randomness of externali- In the case that device i does not feed back at slot t, ties, such as sensory data, energy arrival and channel condi- the MEC server keeps the out-of-date backlogs unchanged, i.e., tions, can be parameterized. Particularly, the computational Qi (t + 1) = Qi (t) and Ei (t + 1) = Ei (t). In the case that resource F(t) ∼ U[0, 3] GHz, the arrival of sensory data device i feeds back at slot t, the MEC server refreshes the Ai (t) ∼ U[0, 10] kbps, the number of available subchannels backlogs up-to-date, as given by K (t) ∼ U[0, 30],whereU[a, b] denotes a random uniform distribution within [a, b]. We also assume that the arrival ( + ) = ( ) − ( ) + ( ), i ( ) ∼ [ , ] Qi t 1 Qi t di t ai t of renewable energy EH t U 0 20 mJ/s, the transmit ( + ) = ( ) − ( ) + i ( ). power pi ∼ U[10, 23] dBm, and the average link capacity Ei t 1 Ei t ei t eH t (40) ci ∼ U[20, 80] kbps, where ci (t) ∼ U[0.5 ci , 2 ci ].The From (3) and (6), we can see that these backlogs do not number of devices N is set to be 500, unless otherwise exceed the actual backlogs, i.e., Qi (t) ≤ Qi (t) and Ei (t) ≤ specified. Every result of the simulations is the average of Ei (t) for i ∈ N and t ∈ T. In the absence of feedback from 5000 independent runs. device i, the channel capacity of the device can be replaced For comparison purpose, we also simulate (a) Round min Robin (RR) method which gives no priority to any devices; by the minimum capacity of the link ci . By referring to (37), the MEC server can evaluate the and (b) Proportional Fair (PF) based on the popular weighted unit profit of device i based on the out-of-date backlogs, fair queueing model, where devices are prioritized and sched- ( )/ t−1 ( ) as given by uled based on ci t τ=0 di t . We note that, in the case of RR, devices are scheduled in a predetermined order and only α ( ) =[ ( ) − ( )ξ ] min +[ ( ) − θ ] . i t Qi t C t i ci Ei t i pi (41) a subset of devices to be scheduled at the upcoming slot need to feed back their states on channels, battery levels and queue Clearly, αi (t) ≤ αi (t), ∀i ∈ N, t ∈ T. The unit profit of backlogs. In the case of PF, however, each device needs to α ( ) the breaking item b t can be achieved by sorting devices feed back its priority and states, based on which the MEC in the descending order of αi (t) and identifying the breaking server selects the devices to offload. In other words, PF still item based on (38). Algorithm 1 can be slighted modified to requires explicit up-to-date knowledge on all the devices. accommodate the reduced feedbacks. Specifically, the MEC Fig. 1 plots the throughput of the proposed approach and α ( ) server broadcasts b t as the threshold before step 1 based the corresponding average number of devices that feed back on its out-of-date information. Device i compares the threshold per slot, where N increases from 100 to 5000. We can see with its unit profit αi (t), and self-nominates to feed back in in Fig. 1(a) that the throughput of the proposed approach α ( ) ≥ α ( ) > step 3, if i t b t . Algorithm 1 resumes in the rest steps. converges as the number of devices gets large, e.g., N 1000. Corollary 1: The out-of-date knowledge on the backlogs of This is due to the limited wireless link capacity and computa- devices does not compromise the optimality of P3. tional resources of the network. Nevertheless, our approach Proof: The optimal eH(t), a(t) and γ (t) for P3 are provides higher throughput than RR and PF, by prudently obtained based on the real-time measurement and calculation leveraging the data queues and energy battery. at each device. We only need to prove the optimality of τ(t) More importantly, we can see in Fig. 1(b) that the number α ( ) ≤ α ( ) for P4.Since b t b t always holds, the optimal set of of devices self-nominated to feed back their backlogs and devices to be selected in (39) to offload, i.e., {i|αi (t) ≥ αb(t)}, channels is substantially reduced in the proposed approach is a subset of the devices feeding back their link capacity and based on partial and out-of-date information. Particularly, { |α ( ) ≥ α ( )} up-to-date backlogs, i.e., i i t b t . This preserves the the number of self-nominating devices is about twice the optimality of τ(t).  number of devices to be optimally scheduled. It is interest- 2612 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 35, NO. 11, NOVEMBER 2017

Fig. 2. The system throughput and Jain’s fairness index of the proposed approach, RR and PF, where V ranges from 0 to 40 and N = 500. Fig. 4. The time variation of the average data backlog of the proposed approach, RR and PF along the time, where N = 500.

Fig. 3. The comparison of the simulation results on the maximum backlogs of the data and energy queues where V increases from 0 to 40, where N = 500. Fig. 5. The comparison of throughput between the proposed approach, RR and PF, versus data arrival and CPU resources, where N = 500 and V = 40. ing to see in Fig. 1(b) that there is a surge of throughput around N = 200, after which the number of self-nominating devices declines with the increasing number of devices. The by about 5 kbits. This corroborates the upper bounds and reason is that the total data arrival of all devices is low and the theorem Ð Theorem 2. The consistent differences between likely to be instantly offloaded in the case of N < 200. the backlogs and upper bounds also provide opportunities to In contrast, the total data arrival is so high in the case of predict the value for V and the expected maximum utility by N > 200, and the data queues build up at the devices. using the upper bounds (15), given the sizes of data buffer and At every slot, the urgency of a small number of devices for battery at the devices. offloading tasks becomes increasingly prominent, leading to a Fig. 4 plots the time variation of the data backlog of the α ( ) growth of b t and subsequent decrease of self-nominating proposed approach in comparison with those of RR and RF devices. along the time. We can see that the average data backlogs Fig. 2 plots the system throughput and fairness of the are able to quickly stabilize, as the time elapses. Moreover, proposed approach, as V increases [27]. Here, V ∈[0, 40] a small value of V is able to further speed up the stabilization is displayed to provide a reasonable dynamic range to show and also reduce the backlog, as compared to a large V value. the impact of V on both the queue length and throughput. This finding is consistent with Fig. 3. In Fig. 4, we also see that We can see that both the throughput and Jain’s index increase the average backlogs of both RR and PF continue increasing rapidly with V when V ≤ 5, and then slow down increasing almost linearly to the time, and have yet to stable by the end of and start to stabilize when V ≥ 30. As revealed in Theorem 1, the simulation time. The superiority of the proposed approach the reason for this is that increasing V leads the utility to in terms of system stability is confirmed. grow, which leverages both the throughput and fairness. For Fig. 5 compares the throughput between the proposed comparison, we also plot the throughput and Jain’s index approach, RR and PF, where the data arrival at the devices and of RR and PF, both of which are independent of V and the available computing resources at the MEC server are varied therefore remain unchanged. We can see that the proposed separately. We show in Fig. 5(a) that the throughput of the scheme approach is able to outperform RR and PF by 18% in proposed approach increases with the growth of data arrival. terms of throughput and up to 4.5% in terms of Jain’s index; Particularly, the throughput grows quickly and then starts to = max ≥ see V 40. stabilize when Ai 5 kbps. This is because the data arrival max = Fig. 3 compares the simulation results on the maximum of Ai 5 kbps starts to congest the network, and to saturate backlogs of the data and energy queues where V increases the offloading throughput. RR and PF also experience the from 0 to 40. We can see that the maximum backlog of battery congestion of the network and the saturation of the offloading tightly fits its upper bound given in (15), while the maximum throughput. However, the saturated throughput of RR and PF backlog of data is consistently lower than its upper bound is much lower than that of the proposed approach. We can LYU et al.: OPTIMAL SCHEDULE OF MEC FOR IoT USING PARTIAL INFORMATION 2613 also see the network congestion and throughput saturation Then, (16) provides an upper bound for (46) by replacing all in Fig. 5(b), where the computing resources are increasingly the expectations with the maximums in (46). available at the MEC server and the offloading throughput is restrained by the wireless link capacity of the network. APPENDIX C Let π denote any feasible control strategy for P.The VI. CONCLUSION minimum upper bound of V (t) in (15) can be given by In this paper, Lyapunov optimization techniques were taken to generate asymptotically optimal offloading sched- V (t) ≤ D − V E log (1 + γi (t))|(t), π i∈N ules for MEC under partial network knowledge. Particularly, + C(t)E[ fi (t) − F(t)|(t), π] we decomposed the Lyapunov optimization problem into i∈N a knapsack problem which can be solved for asymptoti- + Q (t)E[a (t) − d (t)|(t), π] ∈ i i i cally optimal schedules, given up-to-date network knowledge. i N i + [Ei (t) − θi ]E[e (t) − ei (t)|(t), π] We proceeded to evaluate the solution for the knapsack prob- i∈N H lem under out-of-date knowledge, and prove that the solution, + Gi (t)E[γi (t) − ai (t)|(t), π]. (47) encapsulating the asymptotically optimal schedules, preserves i∈N the asymptotic optimality. It can be used for devices to self- According to [23], for any σ>0, there exists at least one nominate for feedback and facilitate scheduling. Simulations randomized stationary control policy π∗ (i.e., the policy is show that the proposed approach is able to reduce feedbacks independent from the backlogs), such that: by over 98% without compromising the asymptotic optimality, −φ(γ ( )|π∗) ≤−φopt + σ, t e.g., only require 60 out of 5000 devices to feed back per slot. ∗ E[ fi (t) − F(t)|π ]≤σ, i∈N APPENDIX A ∗ E[ai (t) − di (t)|π ]≤σ, Given that C1 to C5 are part of P1. We only need to E[ i ( ) − ( )|π∗]≤σ, eH t ei t prove that the maximum for P1 is no less than that for P in ∗ E[γ (t) − a (t)|π ]≤σ. (48) thepresenceofC6 and C7.Letφ(γ ) denote the maximum i i utility of P1,wherea∗ is the corresponding time-average Plugging (48) into (47) and then taking σ → 0 yields data admission. Since φ(x) is a monotonically increasing  ( ) ≤ − φopt. function, C6 guarantees that φ(a∗) ≥ φ(γ ). Using Jensen’s V t D V (49) ∗ inequality [28], we have φ(γ ) ≥ φ(γ ) and, in turn, φ(a ) ≥ Taking expectation on both sides of (49), we obtain φ(γ ).Letφopt denote the solution for P,whereaopt(t) is E[ ( + ) − ( )]− E[φ(γ( ))]≤ − φopt. the corresponding data admission at slot t. Clearly, under L t 1 L t V t D V (50) γ ( ) = opt( ) C6 and C7 in P1, performing the strategy t a t Summing up all the telescoping series of (50) over φ(γ ) = φopt for all time slots t, achieves . As a result, {0, 1, ··· , T − 1}, we can obtain φ(a∗) ≥ φopt,i.e.,P1 is equivalent to P. T −1 E[L(T )]−E[L(0)]−V E[φ(γ(t))]≤DT − VTφopt. t=0 APPENDIX B (51) Taking squares on both sides of (3), (7), (6) and (11), and then exploiting (max[a − b, 0]+c)2 ≤ a2+b2+c2+2a(c−b) Consider that all queues are empty initially. Given the fact that E[ ( )]≥ ( ) = θ 2 for any a, b, c ≥ 0, we have: L T 0andL 0 i∈N i ,wehave 2 2 2 2 1 T −1 D Qi (t + 1) − Qi (t) ≤ ai (t) + di (t) + 2Qi (t)[ai (t)− di (t)], lim E[φ(γ(t))]≥φopt − , (52) T T →∞ t=0 V (42) opt D 2 2 2 2 i.e., φ(γ(t)) ≥ φ − . In Lemma 1, we have proved that C(t + 1) − C(t) ≤ ( fk (t)) + F(t) V k∈N φ( ∗) ≥ φ(γ( )) φopt − φ( ∗) ≤ D a t . Therefore, we obtain a V . + 2 C(t)[ fk (t) − F(t)], (43) k∈N ( + )2 − ( )2 ≤ i ( )2 + ( )2 APPENDIX D Ei t 1 Ei t eH t ei t i We first prove the upper bound of the auxiliary variable + 2[Ei (t) − θi ][e (t) − ei (t)], (44) H backlogs, i.e., G (t) ≤ V + Amax through mathematical ( + )2 − ( )2 ≤ ( )2 + γ ( )2 + ( )[γ ( ) − ( )]. i ln 2 i Gi t 1 Gi t ai t i t 2Gi t i t ai t induction. Specifically, the upper bound holds at slot t = 0 (45) given the fact that Gi (t) = 0, ∀i ∈ N. Suppose that the t G (t) ≥ V Plugging (42)-(45) into (14) yields the upper bound in upper bound holds at time slot . Then, if i ln 2 , γ ( ) = ( ) Lemma 2 with D satisfying i t 0 according to (30), and Gi t cannot increase at ( + ) ≤ ( ) ( )< V slot t, i.e., Gi t 1 Gi t .Otherwise,ifGi t ln 2 , 1 2 i 2 2 2 max D ≥ {E[ei (t) ]+E[e (t) ]+E[γi (t) ]+2E[ai (t) ] the difference between G (t) and G (t + 1) is less than A 2 H i i i i∈N ( + ) ≤ V + max according to (11) and C7, i.e., Gi t 1 ln 2 Ai .Asa 1 + + E[d (t)2]} + {E[( f (t))2]+E[F(t)2]}. result, the upper bound also holds at slot t 1. This concludes i ∈ i (46) 2 i N the proof of the upper bound of Gi (t). 2614 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 35, NO. 11, NOVEMBER 2017

Given the upper bound of Gi (t), we prove (23). Since (23) [11] X. Chen, “Decentralized computation offloading game for mobile holds for t = 0, we suppose that it holds at slot t. From (34), cloud computing,” IEEE Trans. Parallel Distrib. Syst., vol. 26, no. 4, ( ) ≥ ( ) pp. 974Ð983, Apr. 2015. if Qi t Gi t ,devicei does not admit new data, and [12] X. Chen, L. Jiao, W. Li, and X. Fu, “Efficient multi-user computation ( + ) ≤ ( ) ≤ ( ) + max therefore, Qi t 1 Qi t Gi t Ai . On the other offloading for mobile-edge cloud computing,” IEEE/ACM Trans. Netw., vol. 24, no. 5, pp. 2795Ð2808, Oct. 2016. hand, if Qi (t)

Wei Ni received the B.E. and Ph.D. degrees in elec- Xin Wang received the B.Sc. and M.Sc. tronic engineering from Fudan University, Shanghai, degrees from Fudan University, Shanghai, China, China, in 2000 and 2005, respectively. He is cur- in 1997 and 2000, respectively, and the Ph.D. rently a Senior Scientist, and the Team and Project degree from Auburn University, Auburn, AL, USA, Leader at CSIRO, Australia. He also holds hon- in 2004, all in electrical engineering. orary positions at the University of New South He was a Post-Doctoral Research Associate Wales, Macquarie University, and the University with the Department of Electrical and Computer of Technology Sydney. Prior to this, he was a Engineering, University of Minnesota, Minneapolis, Post-Doctoral Research Fellow with Shanghai Jiao- MN, USA, from 2004 to 2006. In 2006, he joined the tong University (2005Ð2008), a Research Scientist Department of Computer and Electrical Engineering and the Deputy Project Manager at the Bell Labs and Computer Science, Florida Atlantic University, R&I Center, Alcatel/Alcatel-Lucent (2005Ð2008), and a Senior Researcher at Boca Raton, FL, USA, where he is currently an Associate Professor. He is Devices R&D, Nokia (2008Ð2009). His research interests include optimiza- also a Distinguished Professor with the Department of Communication tion, game theory, graph theory, and their applications to network and security. Science and Engineering, Fudan University. His research interests include Dr. Ni has been serving as an Editor for the Hindawi Journal of Engineering stochastic network optimization, energy-efficient communications, cross-layer since 2012, a Secretary for the IEEE NSW VTS Chapter since 2015, design, and signal processing for communications. the Track Chair for VTC-Spring 2016 and 2017, and the Publication Chair for BodyNet 2015. He served as the Student Travel Grant Chair for WPMC 2014, Georgios B. Giannakis (F’97) received the Diploma a Program Committee Member for CHINACOM 2014, and a TPC Member degree in electrical engineering from the National of the IEEE ICC14, ICCC15, EICE14, and WCNC10. Technical University of Athens, Greece, in 1981, and the M.Sc. degree in electrical engineering and mathematics and Ph.D. degree in electrical engi- Hui Tian received the M.S. degree in micro- neering from the University of Southern California, electronics and the Ph.D. degree in circuits and sys- in 1983 and 1986, respectively. He was with the tems from the Beijing University of Posts and University of Virginia from 1987 to 1998. Since Telecommunications (BUPT), China, in 1992 and 1999, he has been a Professor with the University 2003, respectively. She is currently a Professor of Minnesota, where he was an Endowed Chair in with BUPT, the Network Information Processing wireless telecommunications, a University of Min- Research Center Director of the State Key Lab- nesota McKnight Presidential Chair in ECE, and the Director of the Digital oratory of Networking and Switching Technology, Technology Center. and the MAT Director of the Wireless Technology His general interests span the areas of communications, networking and Innovation Institute. Her current research interests statistical signal processing subjects on which he has authored over 400 journal mainly include radio resource management, cross papers, 680 conference papers, 25 book chapters, two edited books and two layer optimization, M2M, cooperative communication, mobile social network, research monographs (h-index 119). Current research focuses on learning from and mobile edge computing. big data, wireless cognitive radios, and network science with applications to social, brain, and power networks with renewables. He is the (co-)inventor Ren Ping Liu (SM’13) is currently a Professor of 28 patents issued. He is a fellow of the EURASIP. He was a (co-)recipient with the School of Electrical and Data Engineering, of eight best paper awards from the IEEE Signal Processing (SP) and University of Technology Sydney, where he leads Communications Societies, including the G. Marconi Prize Paper Award the Network Security Laboratory, Global Big Data in Wireless Communications. He also received the Technical Achievement Technologies Centre. He is also the Research Pro- Award from the SP Society (2000) and EURASIP (2005), the Young Faculty gram Leader of Digital Agrifood Technologies in Teaching Award, the G. W. Taylor Award for Distinguished Research from the Food Agility CRC, a government/research/industry University of Minnesota, and the IEEE Fourier Technical Field Award (2015). initiative to empower Australia’s food industry He has served the IEEE in a number of posts, including that of a Distinguished through digital transformation. Prior to that, he was a Lecturer for the IEEE-SP Society. Principal Scientist with CSIRO, where he led wire- less networking research activities. He specializes Arogyaswami Paulraj (F’15) received the B.E. in protocol design and modeling, and has delivered networking solutions degree from the Naval College of Engineering, to a number of government agencies and industry customers. He has over Lonavala, India, in 1966, and the Ph.D. degree from 100 research publications, and has supervised over 30 Ph.D. students. His IIT Delhi, New Delhi, India, in 1973. He is a Pio- research interests include Markov analysis and QoS scheduling in WLAN, neer of MIMO wireless communications. He joined VANET, IoT, LTE, 5G, SDN, and network security. He received the Australian Stanford University, Stanford, CA, in 1972, after Engineering Innovation Award and the CSIRO Chairman Medal. a 20-year industry/military career in India. He has He received the B.E. (Hons.) and M.E. degrees from the Beijing University authored over 400 research publications and two text of Posts and Telecommunications, China, and the Ph.D. degree from The books, and is a co-inventor in 66 U.S. patents. He is University of Newcastle, Australia. He served as the technical program a member of eight national academies, including committee chair and the organizing committee chair for a number of IEEE those of USA and China. His recognitions include conferences. He is the Founding Chair of the IEEE NSW VTS Chapter. the 2014 Marconi Prize and the 2011 IEEE Alexander Graham Bell medal.