2606 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 35, NO. 11, NOVEMBER 2017

Optimal Schedule of Mobile Edge Computing forInternet 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 interestto Internet of Things (IoT), where inexpensive simple devicescan get complex tasks offloaded to and processed at powerfulinfrastructure. Scheduling is challenging due to stochastic taskarrivals and wireless channels, congested air interface, and moreprominently, prohibitive feedbacks from thousands of devices.In this paper, we generate asymptotically optimal schedulestolerant to out-of-date network knowledge, thereby relievingstringent requirements on feedbacks. A perturbed Lyapunovfunction is designed to stochastically maximize a network utilitybalancing throughput and fairness. A knapsack problem is solvedper slot for the optimal schedule, provided up-to-date knowledgeon the data and energy backlogs of all devices. The knapsackproblem is relaxed to accommodate out-of-date network states.Encapsulating the optimal schedule under up-to-date networkknowledge, the solution under partial out-of-date knowledge pre-serves asymptotic optimality, and allows devices to self-nominatefor feedback. Corroborated by simulations, our approach is ableto dramatically reduce feedbacks at no cost of optimality. Thenumber of devices that need to feed back is reduced to lessthan 60 out of a total of 5000 IoT devices.

Index Terms— Mobile edge computing, Internet of Things,partial information, Lyapunov optimization.

I. INTRODUCTION

OFFLOADING and executing tasks of wireless termi-nal at the edge of wireless networks, mobile edgecomputing (MEC) is able to bridge the gap between the

Manuscript received April 1, 2017; revised September 12, 2017; acceptedSeptember 25, 2017. Date of publication October 9, 2017; date of currentversion December 1, 2017. This work was supported by the National NaturalScience Foundation of China under Grant 61471060 and Grant 61421061.(Corresponding author: Hui Tian.)

X. Lyu is with the State Key Laboratory of Networking and SwitchingTechnology, Beijing University of Posts and Telecommunications, Beijing100876, China, and also with the Global Big Data Technologies Center,University of Technology Sydney, Sydney, NSW 2007, Australia.

W. Ni is with the Digital Productivity and Service Flagship, CSIRO,Canberra, NSW 2122, Australia.

H. Tian is with the State Key Laboratory of Networking and SwitchingTechnology, Beijing University of Posts and Telecommunications, Beijing100876, China (e-mail: tianhui@bupt.edu.cn).

R. P. Liu is with the Global Big Data Technologies Center, University ofTechnology Sydney, Sydney, NSW 2007, Australia.

X. Wang is with the Key Laboratory for Information Science of Electromag-netic Waves (MoE), Department of Communication Science and Engineering,Fudan University, Shanghai 200433, China.

G. B. Giannakis is with the Department of Electrical and ComputerEngineering and the Digital Technology Center, University of Minnesota,Minneapolis, MN 55455 USA.

A. Paulraj is with the Department of Electrical Engineering, StanfordUniversity, Stanford, CA 94305 USA.

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSAC.2017.2760186

limited capability of individual devices and computationallydemanding mobile applications [1], [2]. It provides an effec-tive means to reduce the cost of wireless terminals withoutcompromising functionalities and services. This is of particularinterest to the future Internet of Things (IoT), where thousandsof inexpensive, computationally incapable devices can bedeployed within the coverage of a base station [3].

Different from cloud computing with dedicated high-speedwired connections, MEC has all devices operate in sharedwireless media which feature unpredictable channel changes,and become increasingly congested with the increase ofdevices [4]. Optimal offloading schedules are obviously impor-tant to MEC, but yet to be established due to sophisticatedstochasticity and changing availability of resources at the MECserver, random data arrival per device, and fluctuating wirelesschannels [5]. Myopically optimized schedules which fail tocapture the stochasticity would be inefficient over longer timehorizons [6]. Optimal schedules have also been hindered by atypical need for explicit knowledge on network states. In thecase of IoT, thousands of devices could be required to feedback their states and channels, whenever the states or channelschange [7]. This would incur prohibitive overhead, congest thechannel, and jeopardize practicality.

Most existing works on MEC has overlooked the stochas-ticity or assumed predictability, and optimized the offloadingschedules deterministically in the presence of full knowledgeon network states. In [8]–[10], non-convex optimization prob-lems were myopically formulated and solved in a centralizedmanner. In [11]–[13], game theoretic approaches or submod-ular optimization methods were taken to produce myopicschedules per slot in a decentralized fashion. In [14] and [15],Markov decision processes (MDP) were formulated to opti-mize the schedules under the assumption of a priori knowledgeon the statistics of stochastic processes, and solved by dynamicprogramming. However, these statistics are typically unknownin prior, and dynamic programming also suffers from the“curse-of-dimensionality” and offers little scalability.

A few recent works have started to take the stochasticity andunpredictability into the consideration and optimize offload-ing schedules foresightedly by using stochastic optimizationtechniques, i.e., Lyapunov optimization [16]–[18]. However,these works encompassed on different network architecturesfrom MEC, e.g., point-to-point link for single device [16] andmulti-hop link in wireless sensor networks [17], [18], and theirresults are inapplicable to MEC with thousands of IoT devicesto be offloaded.

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LYU et al.: OPTIMAL SCHEDULE OF MEC FOR IoT USING PARTIAL INFORMATION 2607

This paper generates new asymptotically optimal offloadingschedules, which are tolerant to partial out-of-date networkknowledge and stochastically maximize a time-average net-work utility balancing system throughput and fairness. Thecontribution of this paper is multi-folded: (a) A perturbedLyapunov technique is proposed to decouple time couplingsof offloading schedule, energy intake, and data admission,and convert the stochastic optimization of MEC offloadingschedules to deterministic optimizations over time slots with-out loss of optimality. An [O(V ), O(1/V )]-tradeoff can beachieved between stability and utility, so that the optimalityloss of the proposed approach (as compared to the offlineoptimum) asymptotically diminishes at the cost of increasingqueue length; (b) Particularly, the deterministic optimizationper slot is meticulously formulated to a linear relaxationof a knapsack problem. The breaking item of the problem,achieving the asymptotical optimality, is identified. Its lowerbound in the presence of partial out-of-date information isderived as the threshold. As a result, only the devices meet-ing the threshold are designed to feed back per slot, hencepreserving the asymptotic optimality with reduced feedbacks;(c) Extensive simulations show that our approach is able tosubstantially improve system utility by 26% and dramaticallyreduce feedbacks per slot by over 98%. Only less than60 out of 5000 devices need to feed back per slot. Thefeedbacks can decline with an increasingly large number ofdevices, due to the increasingly prominent urgency of somedevices.

The rest of this paper is organized as follows. In Section II,the system model is presented. In Section III, we elaboratethe proposed stochastic online optimization of offloadingschedules, provided up-to-date knowledge on network statesat the MEC server. In Section IV, we extend the approachunder out-of-date knowledge and derive the optimality pre-serving threshold for the self-nomination of devices. Simula-tion results are shown in Section V, followed by conclusionsin Section VI.

II. SYSTEM MODEL

The MEC system that we consider consists of a basestation (BS) and N IoT devices, e.g., smart meters for supplymonitoring, actuators for industrial automation, and sensors forsmart home and healthcare.1 Let N = {1, 2, . . . , N} collect theindexes for N IoT devices. The system operates in a slottedstructure, t ∈ T = {0, 1, . . .} with slot length T . Equippedwith energy harvesting capabilities, the IoT devices can bepowered by renewable energy. An MEC server is deployed atthe BS, and sensory data from the IoT devices can be offloadedthrough wireless links and processed at the MEC server. In thepresence of other uplink traffic to the BS, the number ofavailable subchannels for the IoT devices, denoted by K (t),can vary between time slots. The notations used in this paperare summarized in Table I.

1The consideration of a single BS is elemental, and can be readily applied tothe case of multiple BSs where each BS schedules the devices in its coverageindependently as proposed in this paper. This is due to the typical layeredarchitecture of cellular networks where inter-cell interference is mitigatedthrough (fractional) frequency reuse among BSs.

TABLE I

DEFINITIONS OF NOTATIONS

Independent and identical distributed (i.i.d.) flat block fad-ing channels are assumed, i.e., the channels remain unchangedduring a time slot and vary between slots [19], [20]. Let ci (t)denote the channel capacity in the uplink of device i at slot t .cmini ≤ ci (t) ≤ cmaxi , where cmini and cmaxi are the minimumand maximum capacities of the link, respectively, given finitetransmit powers of IoT devices.

Within a subchannel, the transmissions of multiple IoTdevices can be accommodated via time-division multiple-access. Consider that an IoT device is inexpensive, simpleand narrow-band [3], [7]. It can only access a subchannel atthe same time. Let τi (t) denote the transmission duration ofdevice i within slot t , i.e.,

0 ≤ τi (t) ≤ T, ∀i ∈ N, (1)and the maximum amount of data offloaded from devicei to the MEC server can be given by ̂di (t) = ci (t)τi (t).In some cases, a device can be assigned with differentsubchannels at different times during a slot. The narrow-band device can switch between subchannels to offloadtasks [3].

We define an offloading schedule τ (t) = {τ1(t), . . . , τN (t)}by collecting the schedules of all IoT devices during slot t .The total transmission time of the devices must not exceed thenumber of available subchannel times slot length, given as

∑

i∈N τi (t) ≤ K (t)T . (2)We define D(t) = {τ (t)|0 ≤ τi (t) ≤ T,∑i∈N τi (t) ≤ K (t)T }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 iat slot t . Ai (t) is an i.i.d. random process with the maxi-mum Amaxi . Device i may only be able to admit part of thearrived data, denoted by ai (t), i.e., 0 ≤ ai (t) ≤ Ai (t), basedon the availability of its buffer.

Let Qi (t) be the backlog of the data queue at device i , andit is updated along the time, as given by

Qi (t + 1) = [Qi (t) − di (t)] + ai(t), (3)where

di (t) = min{Qi (t), ci (t)τi (t)}, (4)since the device cannot offload more than what it has. Thecorresponding energy consumption of the device for offloadingthe data, ei (t), can be written as

ei (t) = piτi (t), (5)where pi is the transmit power of device i .

The energy harvesting at an IoT device can also be modeledas a stochastic arrival process [16]–[18], [21]. Let EiH(t)denote the amount of renewable energy harvested by device iduring time slot t . Assume that EiH(t) is i.i.d. with themaximum Ei,maxH . Consider a finite battery at the device.Device i can only store part of the newly harvested energy ateach time slot t , denoted by eiH(t), where 0 ≤ eiH(t) ≤ EiH(t).

Let Ei (t) denote the battery backlog of device i at slot t .It can be updated by

Ei (t + 1) = [Ei(t) − ei (t)] + eiH(t), (6)where ei (t) ≤ Ei (t), since the energy used for offloading datamust not exceed the energy available in the battery.

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 withfi (t) = ξi di (t) CPU cycles or put the task into the queuefor later processing, where ξi is the number of CPU cyclesrequired per task bit of device i for the task.

Let C(t) be the required CPU cycles to process the taskqueued at the MEC server. C(t) can be updated by

C(t + 1) = max [C(t) − F(t), 0] +∑

i∈N fi (t), (7)

where F(t) specifies the total available CPU cycles at slot t . Inthe presence of other concurrent services, F(t) is stochasticwith the maximum Fmax.

∑

i∈N fi (t) gives the total of thenewly offloaded tasks during time slot t .

III. ONLINE OPTIMIZATION OF OFFLOADING IN MEC

Leveraging between throughput and fairness in MEC,the system utility can be defined as

φ (a) =∑

i∈N log(1 + ai ), (8)where X = limT →∞ 1T

∑T −1t=0 E[X (t)] defines the time-

average of any stochastic process X (t), and a ={a1, a2, · · · , aN } collects the time-average total of admitteddata at all N devices. Particularly, φ (a) is a concave loga-rithm function that is able to balance the network throughputand fairness by discouraging individual devices to consumeexcessive energy for little utility gains [22].

We can formulate P to maximize the system utility as

P: maxτ (t),eH(t),a(t)

φ (a)

s.t. C1: τ (t) ∈ D(t), ∀t ∈ TC2: 0 ≤ eiH(t) ≤ EiH(t), ∀i ∈ N, t ∈ TC3: 0 ≤ ai (t) ≤ Ai (t), ∀i ∈ N, t ∈ TC4: ei (t) ≤ Ei (t), ∀i ∈ N, t ∈ TC5: C, Ei and Qi < ∞, ∀i ∈ N, (9)

where τ (t), eH(t) and a(t) are the offloading schedule,energy intake, and data admission to be optimized, respec-tively. Constraints C1 to C4 are self-explanatory, as discussedin Section II. C5 ensures the stability of all the queues.Particularly, if the queue backlogs are upper bounded, the totaldata admission

∑

i∈N ai is equal to the time-average systemthroughput in the long term [23].

Note that ci (t), Ai (t), EiH(t) and K (t) and computingresources F(t) at the MEC server, are all time-varying withlittle predictability. An offline optimization of P would violatecausality; whereas a myopic optimization per time slot wouldcompromise the optimality in the long term, due to timecoupling of variables in C1 to C4.

A. Perturbed Lyapunov Optimization

Lemma 1: P can be equivalently reformulated as

P1: maxτ (t),eH(t),a(t),γ (t)

φ (γ )

s.t. C1-C5 and C6: γi ≤ aiC7: 0 ≤ γi (t) ≤ Amaxi , ∀i ∈ N, t ∈ T, (10)

where γ (t) is the defined auxiliary variables.Proof: See Appendix A. �

By defining a device-specific virtual queue, C6 can bereformed with improved tractability, where the backlog ofthe virtue queue, denoted by Gi (t), can be updated by thedifference of γi (t) and ai (t), as given by

Gi (t + 1) = max{Gi (t) + γi (t) − ai (t), 0}. (11)Clearly, Gi (t) is stable if and only if C6 is satisfied [23].We can replace C6 with the stability of Gi (t), and P1 can berewritten as

P2: maxτ (t),eH(t),a(t),γ (t)

φ (γ )

s.t. C1-C5 and C7

C8: Gi < ∞, ∀i ∈ N. (12)Stochastic optimization, more specifically, Lyapunov opti-

mization, is able to eliminate time coupling of variables, andenable online decision-makings, while preserving asymptoticoptimality [23]. A perturbed Lyapunov function of P2 can bedefined as

L(t) = 12

{

C(t)2 +∑

i∈N[Qi (t)2 + (Ei (t) − θi )2 + Gi (t)2]

}

,

(13)

LYU et al.: OPTIMAL SCHEDULE OF MEC FOR IoT USING PARTIAL INFORMATION 2609

where θ = {θi ,∀i ∈ N} collects the new perturbingweights (biases) to ensure sufficient energy available in the bat-teries for optimal schedules durations at any slot. Let � (t) ={C(t), Qi (t), Ei (t), Gi (t),∀i ∈ N} collect all the backlogs atslot t . A drift-plus-penalty function can be defined by

�V (t) = � (t) − V E[∑

i∈N log(1 + γi (t))|�(t)]

, (14)

where � (t)�= E [L(t + 1) − L(t)|� (t)] is the conditional

Lyapunov drift, i.e., the conditional expectation of the dif-ference of the perturbed Lyapunov function between twoconsecutive time slots, and V ≥ 0 is a predefined coefficient totune the tradeoff between the system utility and queue stability.

Minimizing an upper bound of �V (t) per slot t can preventunbounded growths of backlogs, while maximizing the time-average conditional expectation of the system utility [23]. Theupper bound can be given in the following lemma.

Lemma 2: For any queue backlogs and actions, �V (t) isupper bounded by

�V (t) ≤ D − V E[∑

i∈N log (1 + γi (t))|� (t)]

+ C(t)E[∑

i∈N fi (t) − F(t)|� (t)]+

∑

i∈N Qi (t)E[ai (t) − di (t)|� (t)]+

∑

i∈N [Ei (t) − θi ]E[eiH(t) − ei (t)|� (t)]

+∑

i∈N Gi (t)E[γi (t) − ai (t)|�(t)] (15)where

D = 12

∑

i∈N {(pi T )2 + (Ei,maxH )

2 + (cmaxi T )2 + 3(Amaxi )2}

+ 12{∑

i∈N (ξi cmaxi T )

2 + (Fmax)2} (16)Proof: See Appendix B. �

Exploiting Lemma 2, we reformulate P2 to minimize (15),subject to the instantaneous constraints in P2, as follows:

P3: maxτ(t),eH(t),a(t),γ (t)

f (γ (t)) + g(τ (t)) − η(eH(t)) − μ(a(t))s.t. C1-C4andC7, (17)

where F(t) is suppressed, since it is independent of thevariables;

f (γ (t)) =∑

i∈N [V log (1 + γi (t)) − Gi (t)γi (t)]; (18)g(τ(t)) =

∑

i∈N{[Qi (t) − C(t)ξi ]di (t) + [Ei (t) − θi ]ei (t)};

(19)

η(eH(t)) =∑

i∈N[Ei (t) − θi ]eiH(t); (20)

μ (a(t)) =∑

i∈N[Qi (t) − Gi (t)]ai (t). (21)Let a∗ denote the optimal solution for P3, and φopt be the

maximum system utility which can only be achieved offlineunder the assumption of full knowledge across all time slots,i.e., the optimum for P. As shown in the following theorem,the optimum for P3, φ (a∗), converges to the optimum for P,φopt, as V increases.

Theorem 1: The gap between the optimum for P3, φ (a∗),and the optimum for P, φopt, is less than D/V , i.e.,

φopt − φ(a∗) ≤ D/V . (22)Proof: See Appendix C. �

Theorem 2: With a∗, the backlogs of all the queues in thesystem is strictly bounded at each time slot t, as given by

Qi (t) ≤ V/ ln 2 + 2Amaxi ; (23)Ei (t) ≤ θi + Ei,maxH ; (24)C(t) ≤ β +

∑

i∈N ξi cmaxi T ; (25)

where

β = maxi∈N{ V/ ln 2 + 2Amaxi

ξi+ E

i,maxH picmini ξi

}. (26)

Proof: See Appendix D. �Theorem 3: The weight perturbation parameter θi can be

adjusted by

θi = [V/ ln 2 + 2Amaxi ]cmaxi /pi + pi T, (27)such that, when a device is scheduled to offload at time slott, there is enough energy in its battery backlog for optimalschedules, i.e., Ei (t) ≥ pi T .

Proof: See Appendix E. �Theorems 1, 2 and 3 show an [O(V ), O(1/V )]-tradeoff

between the queue lengths (or in other words, the queueingdelay according to Little’s theorem [24]) and the optimalityloss. Here, the optimality loss is the gap of system utilitybetween the proposed approach and the causality-violatingoffline optimization of the offloading schedule. That is to say,by fine-tuning V , our approach allows the queueing delay tobe adjustable at the cost of the optimal system utility.

B. Optimal Offloading Schedules

Note that τ (t), eH(t), a(t) and γ (t) are decoupled in boththe objective and constraints in P3, and therefore can beoptimized separately as follows to achieve a∗.

1) Optimal Auxiliary Parameter: The subproblem ofoptimizing auxiliary variables can be written as

maxγ (t)

f (γ (t))

s.t. 0 ≤ γi (t) ≤ Amaxi , ∀i ∈ N, (28)where γi (t) is decoupled from γ j (t) for any j = i . Therefore,γi (t) can be optimized independently by solving

maxγi (t)

V log(1 + γi (t)) − Gi (t)γi (t)s.t. 0 ≤ γi (t) ≤ Amaxi (29)

The first-order partial derivation of f (γ (t)) can be givenby ∂ f∂γi = V(1+γi (t)) ln 2 − Gi (t). If

∂ f∂γi

|γi (t)=0 ≤ 0, f (γ (t))decreases monotonically in the region γi (t) ≥ 0 and theoptimum γi (t)∗ = 0. Otherwise, the optimum is either at the

2610 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 35, NO. 11, NOVEMBER 2017

stationary point ∂ f∂γi = 0, i.e., γi (t)∗ = VGi (t) ln 2 − 1, or on theboundary, i.e., γi (t)∗ = Amaxi . Therefore,

γi (t)∗ =

⎧

⎪

⎨

⎪

⎩

0, ifV

ln 2− Gi (t) ≤ 0;

min{ VGi (t) ln 2

− 1, Amaxi }, otherwise.(30)

2) Optimal Energy Intake: Decoupled from (17), thesubproblem of optimizing energy intake can be written as

mineH(t)

∑

i∈N[Ei (t) − θi ]eiH(t)

s.t. 0 ≤ eiH(t) ≤ EiH(t), ∀i ∈ N, (31)where the optimal solution can be readily given by

eiH(t)∗ =

{

0, if Ei (t) ≥ θi ;EiH(t), otherwise.

(32)

The devices consistently push their battery backlogs towards θ .As shown in Theorem 3, sufficient energy is available in thebatteries for optimal schedules by judiciously designing θ .

3) Optimal Data Admission: The decoupled subproblem ofoptimizing data admission can be written as

mina(t)

∑

i∈N[Qi (t) − Gi (t)]ai (t)s.t. 0 ≤ ai (t) ≤ Ai (t), ∀i ∈ N, (33)

where the optimal admission decision can be given by

ai(t)∗ =

{

0, if Qi (t) ≥ Gi (t);Ai (t), otherwise.

(34)

4) Optimal Offloading Schedule: τ (t) is optimized to max-imize g(τ (t)), while satisfying (1) and (2) at each time slot t .From (3) and (4), it is clear that device i should not transmitmore data than Qi (t) at slot t , i.e., τi (t) ≤ Qi (t)/ci (t).Besides, C4 requires that the transmission energy cannotexceed the available energy in the battery backlog, i.e., τi (t) ≤Ei (t)/pi . Therefore, (1) can be tightened to

0 ≤ τi (t) ≤ Ti , ∀i ∈ N, (35)where Ti = min{Qi (t)/ci (t), Ei (t)/pi , T }. By rearrang-ing (19), τ (t) can be optimized by

P4: maxτ(t)

∑

i∈N αi (t)τi (t)

s.t. (2) and (35), (36)

where

αi (t) = [Qi (t) − C(t)ξi ]ci (t) + [Ei (t) − θi ]pi . (37)By interpreting αi (t) as the unit profit of an “item” i andK (t)T as the capacity of a knapsack, P4 becomes the linearrelaxation of a knapsack problem. The optimal solution forlinear relaxation knapsack can be found by selecting the “item”with higher unit profit to fulfill the knapsack capacity [25].

Without loss of generality, we assume the devices are sortedin the descending order of unit profit, i.e., αi (t) ≥ α j (t) fori > j . The optimal solution is τi = Ti if ∑ij=1 Tj ≤ K (t)T .

Define the breaking item to be the first device that is notallocated its full transmission duration specified in (35) [25].The index of the breaking item b can identified, as given by

b = arg mini

∑i

j=1 Tj > K (t)T . (38)

As a result, the optimal offloading schedule can be given by

τi (t)∗ =

⎧

⎪

⎨

⎪

⎩

Ti , if i < b;K (t)T − ∑b−1i=1 Ti , if i = b;0, otherwise,

(39)

which, involving ordering the devices against their unit profits,can be achieved by using the quicksort algorithm with acomplexity of O(N log N) [26].

Algorithm 1 Online Optimization of Offloading in MECAt each device i :1: Acquire Qi (t), Ei (t), ci (t), Ai (t) and EiH(t)2: Perform the optimal solutions given by (30), (32) and (34)3: Feed back Qi (t), Ei (t) and ci (t) to the MEC server4: Update Gi (t + 1) based on (11)

At the MEC server:5: Acquire K (t), F(t) and C(t)6: Collect feedbacks from all the devices and evaluate αi (t)7: Schedule the offloading by (39)

Note that the optimal solutions in (30), (32) and (34) canbe locally computed at individual IoT devices, since the queuebacklogs, wireless channels, data arrivals and energy harvest-ing can be measured at the devices without the coordinationof the MEC server. However, (39) requires the knowledgeon all the IoT devices, i.e., ck(t), Qk(t), Ek(t), and Gk(t)for k ∈ N. At each time slot t , the MEC server needs eachIoT device to feed back its link capacity ck(t) and backlogsQk(t), Ek(t) and Gk(t) to evaluate τi (t)∗ in (39). We proposeto distribute part of these optimizations (30), (32) and (34)at the IoT devices, and the rest (39) at the MEC server, assummarized in Algorithm 1.

IV. OPTIMIZATION OF OFFLOADING SCHEDULESUNDER OUT-OF-DATE BACKLOGS

As discussed, the link capacity and backlogs of each IoTdevice are needed in the MEC server to evaluate (39) foroptimal offloading schedules. Consider practical IoT, however,the number of IoT devices can be hundreds to thousands,while the number of available subchannels can be far less,i.e., K (t) N . Getting each device to feed back as perslot could be infeasible due to explosive overhead, and alsoinefficient since the majority of the feedbacks would notcontribute to the optimization of the schedule.

From (39), it is clear that the IoT devices with high unitprofits are given priority to offload. Particularly, the optimalityof (39) would not be compromised, if the devices with unitprofits less than the breaking item, i.e., αi (t) < αb(t), do notfeed back. To this end, there is an opportunity arising tosubstantially 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 theMEC 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 knowledgeis established, which captures the case of practical interestwhere only part of IoT devices feed back their backlogs.Using this, we derive a threshold for αi (t), denoted by αb(t),which enables a small number of devices to self-nominate tofeed back, while preserving the asymptotic optimality statedin Theorem 1. Following are the details.

From (37), the MEC server evaluates αi (t) based on C(t),ci (t), Qi (t) and Ei (t), where C(t) is known to the serverwhereas the other three parameters need device i to feed back.In the absence of up-to-date feedbacks from most devices,we define ̂Qi (t) and ̂Ei (t) for the data and energy backlogsof device i , respectively, to capture the out-of-date knowledgeat the MEC server.

The MEC server can update ̂Qi (t) and ̂Ei (t), as follows.In the case that device i does not feed back at slot t ,the MEC server keeps the out-of-date backlogs unchanged, i.e.,̂Qi (t + 1) = ̂Qi (t) and ̂Ei (t + 1) = ̂Ei (t). In the case thatdevice i feeds back at slot t , the MEC server refreshes thebacklogs up-to-date, as given by

̂Qi (t + 1) = Qi (t) − di(t) + ai (t),̂Ei (t + 1) = Ei (t) − ei (t) + eiH(t). (40)

From (3) and (6), we can see that these backlogs do notexceed the actual backlogs, i.e., ̂Qi (t) ≤ Qi (t) and ̂Ei (t) ≤Ei (t) for i ∈ N and t ∈ T. In the absence of feedback fromdevice i , the channel capacity of the device can be replacedby the minimum capacity of the link cmini .

By referring to (37), the MEC server can evaluate theunit profit of device i based on the out-of-date backlogs,as given by

α̂i (t) = [̂Qi (t) − C(t)ξi ]cmini + [̂Ei (t) − θi ]pi . (41)

Clearly, α̂i (t) ≤ αi (t), ∀i ∈ N, t ∈ T. The unit profit ofthe breaking item αb(t) can be achieved by sorting devicesin the descending order of α̂i (t) and identifying the breakingitem based on (38). Algorithm 1 can be slighted modified toaccommodate the reduced feedbacks. Specifically, the MECserver broadcasts αb(t) as the threshold before step 1 basedon its out-of-date information. Device i compares the thresholdwith its unit profit αi (t), and self-nominates to feed back instep 3, if αi (t) ≥ αb(t). Algorithm 1 resumes in the rest steps.

Corollary 1: The out-of-date knowledge on the backlogs ofdevices does not compromise the optimality of P3.

Proof: The optimal eH(t), a(t) and γ (t) for P3 areobtained based on the real-time measurement and calculationat each device. We only need to prove the optimality of τ (t)for P4. Since αb(t) ≤ αb(t) always holds, the optimal set ofdevices to be selected in (39) to offload, i.e., {i |αi (t) ≥ αb(t)},is a subset of the devices feeding back their link capacity andup-to-date backlogs, i.e., {i |αi (t) ≥ αb(t)}. This preserves theoptimality of τ (t). �

Fig. 1. The changes of throughput and the corresponding average numberof device as N increases from 100 to 5000 where V = 40.

V. SIMULATION RESULTS

In this section, we evaluate the proposed optimal offloadingschedules of MEC system in Matlab with T = 1 sec.Focused on scheduling a large number of devices using partialout-of-date knowledge, we assume that the uploaded taskscan be computed correctly, and the randomness of externali-ties, such as sensory data, energy arrival and channel condi-tions, can be parameterized. Particularly, the computationalresource F(t) ∼ U[0, 3] GHz, the arrival of sensory dataAi (t) ∼ U[0, 10] kbps, the number of available subchannelsK (t) ∼ U[0, 30], where U[a, b] denotes a random uniformdistribution within [a, b]. We also assume that the arrivalof renewable energy EiH(t) ∼ U[0, 20] mJ/s, the transmitpower pi ∼ U[10, 23] dBm, and the average link capacityci ∼ U[20, 80] kbps, where ci (t) ∼ U[0.5 ci , 2 ci ]. Thenumber of devices N is set to be 500, unless otherwisespecified. Every result of the simulations is the average of5000 independent runs.

For comparison purpose, we also simulate (a) RoundRobin (RR) method which gives no priority to any devices;and (b) Proportional Fair (PF) based on the popular weightedfair queueing model, where devices are prioritized and sched-uled based on ci (t)/

∑t−1τ=0 di (t). We note that, in the case of

RR, devices are scheduled in a predetermined order and onlya subset of devices to be scheduled at the upcoming slot needto feed back their states on channels, battery levels and queuebacklogs. In the case of PF, however, each device needs tofeed back its priority and states, based on which the MECserver selects the devices to offload. In other words, PF stillrequires explicit up-to-date knowledge on all the devices.

Fig. 1 plots the throughput of the proposed approach andthe corresponding average number of devices that feed backper slot, where N increases from 100 to 5000. We can seein Fig. 1(a) that the throughput of the proposed approachconverges as the number of devices gets large, e.g., N > 1000.This is due to the limited wireless link capacity and computa-tional resources of the network. Nevertheless, our approachprovides higher throughput than RR and PF, by prudentlyleveraging the data queues and energy battery.

More importantly, we can see in Fig. 1(b) that the numberof devices self-nominated to feed back their backlogs andchannels is substantially reduced in the proposed approachbased on partial and out-of-date information. Particularly,the number of self-nominating devices is about twice thenumber 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 proposedapproach, RR and PF, where V ranges from 0 to 40 and N = 500.

Fig. 3. The comparison of the simulation results on the maximum backlogsof the data and energy queues where V increases from 0 to 40, whereN = 500.

ing to see in Fig. 1(b) that there is a surge of throughputaround N = 200, after which the number of self-nominatingdevices declines with the increasing number of devices. Thereason is that the total data arrival of all devices is low andlikely to be instantly offloaded in the case of N < 200.In contrast, the total data arrival is so high in the case ofN > 200, and the data queues build up at the devices.At every slot, the urgency of a small number of devices foroffloading tasks becomes increasingly prominent, leading to agrowth of αb(t) and subsequent decrease of self-nominatingdevices.

Fig. 2 plots the system throughput and fairness of theproposed approach, as V increases [27]. Here, V ∈ [0, 40]is displayed to provide a reasonable dynamic range to showthe impact of V on both the queue length and throughput.We can see that both the throughput and Jain’s index increaserapidly with V when V ≤ 5, and then slow down increasingand start to stabilize when V ≥ 30. As revealed in Theorem 1,the reason for this is that increasing V leads the utility togrow, which leverages both the throughput and fairness. Forcomparison, we also plot the throughput and Jain’s indexof RR and PF, both of which are independent of V andtherefore remain unchanged. We can see that the proposedscheme approach is able to outperform RR and PF by 18% interms of throughput and up to 4.5% in terms of Jain’s index;see V = 40.

Fig. 3 compares the simulation results on the maximumbacklogs of the data and energy queues where V increasesfrom 0 to 40. We can see that the maximum backlog of batterytightly fits its upper bound given in (15), while the maximumbacklog of data is consistently lower than its upper bound

Fig. 4. The time variation of the average data backlog of the proposedapproach, RR and PF along the time, 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 andV = 40.

by about 5 kbits. This corroborates the upper bounds andthe theorem – Theorem 2. The consistent differences betweenthe backlogs and upper bounds also provide opportunities topredict the value for V and the expected maximum utility byusing the upper bounds (15), given the sizes of data buffer andbattery at the devices.

Fig. 4 plots the time variation of the data backlog of theproposed approach in comparison with those of RR and RFalong the time. We can see that the average data backlogsare able to quickly stabilize, as the time elapses. Moreover,a small value of V is able to further speed up the stabilizationand also reduce the backlog, as compared to a large V value.This finding is consistent with Fig. 3. In Fig. 4, we also see thatthe average backlogs of both RR and PF continue increasingalmost linearly to the time, and have yet to stable by the end ofthe simulation time. The superiority of the proposed approachin terms of system stability is confirmed.

Fig. 5 compares the throughput between the proposedapproach, RR and PF, where the data arrival at the devices andthe available computing resources at the MEC server are variedseparately. We show in Fig. 5(a) that the throughput of theproposed approach increases with the growth of data arrival.Particularly, the throughput grows quickly and then starts tostabilize when Amaxi ≥ 5 kbps. This is because the data arrivalof Amaxi = 5 kbps starts to congest the network, and to saturatethe offloading throughput. RR and PF also experience thecongestion of the network and the saturation of the offloadingthroughput. However, the saturated throughput of RR and PFis 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 saturationin Fig. 5(b), where the computing resources are increasinglyavailable at the MEC server and the offloading throughput isrestrained by the wireless link capacity of the network.

VI. CONCLUSION

In this paper, Lyapunov optimization techniques weretaken to generate asymptotically optimal offloading sched-ules for MEC under partial network knowledge. Particularly,we decomposed the Lyapunov optimization problem intoa knapsack problem which can be solved for asymptoti-cally optimal schedules, given up-to-date network knowledge.We proceeded to evaluate the solution for the knapsack prob-lem under out-of-date knowledge, and prove that the solution,encapsulating the asymptotically optimal schedules, preservesthe asymptotic optimality. It can be used for devices to self-nominate for feedback and facilitate scheduling. Simulationsshow that the proposed approach is able to reduce feedbacksby over 98% without compromising the asymptotic optimality,e.g., only require 60 out of 5000 devices to feed back per slot.

APPENDIX A

Given that C1 to C5 are part of P1. We only need toprove that the maximum for P1 is no less than that for P inthe presence of C6 and C7. Let φ (γ ) denote the maximumutility of P1, where a∗ is the corresponding time-averagedata admission. Since φ(x) is a monotonically increasingfunction, C6 guarantees that φ (a∗) ≥ φ (γ ). Using Jensen’sinequality [28], we have φ(γ ) ≥ φ (γ ) and, in turn, φ (a∗) ≥φ (γ ). Let φopt denote the solution for P, where aopt(t) isthe corresponding data admission at slot t . Clearly, underC6 and C7 in P1, performing the strategy γ (t) = aopt(t)for all time slots t , achieves φ (γ ) = φopt. As a result,φ (a∗) ≥ φopt, i.e., P1 is equivalent to P.

APPENDIX B

Taking squares on both sides of (3), (7), (6) and (11), andthen exploiting (max[a − b, 0] + c)2 ≤ a2+b2+c2+2a(c−b)for any a, b, c ≥ 0, we have:Qi (t + 1)2−Qi (t)2 ≤ ai (t)2+ di (t)2+ 2Qi (t)[ai (t)− di (t)],

(42)

C(t + 1)2 − C(t)2 ≤ (∑

k∈N fk(t))2 + F(t)2

+ 2 C(t)[∑

k∈N fk(t) − F(t)], (43)Ei (t + 1)2 − Ei (t)2 ≤ eiH(t)2 + ei (t)2

+ 2[Ei(t) − θi ][eiH(t) − ei (t)], (44)Gi (t + 1)2− Gi (t)2 ≤ ai (t)2+ γi (t)2+ 2Gi (t)[γi (t) − ai (t)].

(45)

Plugging (42)-(45) into (14) yields the upper bound inLemma 2 with D satisfying

D ≥ 12

∑

i∈N{E[ei (t)2] + E[eiH(t)2] + E[γi (t)2] + 2E[ai(t)2]

+ E[di(t)2]} + 12{E[(

∑

i∈N fi (t))2] + E[F(t)2]}. (46)

Then, (16) provides an upper bound for (46) by replacing allthe expectations with the maximums in (46).

APPENDIX C

Let π denote any feasible control strategy for P. Theminimum upper bound of �V (t) in (15) can be given by

�V (t) ≤ D − V E[∑

i∈N log (1 + γi (t))|� (t), π]

+ C(t)E[∑

i∈N fi (t) − F(t)|� (t), π]+

∑

i∈N Qi (t)E[ai (t) − di (t)|� (t), π]+

∑

i∈N [Ei (t) − θi ]E[eiH(t) − ei (t)|� (t), π]

+∑

i∈N Gi (t)E[γi (t) − ai (t)|�(t), π]. (47)According to [23], for any σ > 0, there exists at least onerandomized stationary control policy π∗ (i.e., the policy isindependent from the backlogs), such that:

−φ (γ (t)|π∗) ≤ −φopt + σ,E[

∑

i∈N fi (t) − F(t)|π∗] ≤ σ,

E[ai (t) − di (t)|π∗] ≤ σ,E[eiH(t) − ei (t)|π∗] ≤ σ,E[γi (t) − ai (t)|π∗] ≤ σ. (48)

Plugging (48) into (47) and then taking σ → 0 yields�V (t) ≤ D − V φopt. (49)

Taking expectation on both sides of (49), we obtain

E[L(t + 1) − L(t)] − V E[φ(γ (t))] ≤ D − V φopt. (50)Summing up all the telescoping series of (50) over{0, 1, · · · , T − 1}, we can obtainE[L(T )] − E[L(0)] − V

∑T −1t=0 E[φ(γ (t))] ≤ DT − V T φ

opt.

(51)

Consider that all queues are empty initially. Given the fact thatE[L(T )] ≥ 0 and L(0) = ∑i∈N θ2i , we have

1

Tlim

T →∞∑T −1

t=0 E[φ(γ (t))] ≥ φopt − D

V, (52)

i.e., φ(γ (t)) ≥ φopt − DV . In Lemma 1, we have proved thatφ (a∗) ≥ φ(γ (t)). Therefore, we obtain φopt − φ (a∗) ≤ DV .

APPENDIX D

We first prove the upper bound of the auxiliary variablebacklogs, i.e., Gi (t) ≤ Vln 2 + Amaxi through mathematicalinduction. Specifically, the upper bound holds at slot t = 0given the fact that Gi (t) = 0,∀i ∈ N. Suppose that theupper bound holds at time slot t . Then, if Gi (t) ≥ Vln 2 ,γi (t) = 0 according to (30), and Gi (t) cannot increase atslot t , i.e., Gi (t + 1) ≤ Gi (t). Otherwise, if Gi (t) < Vln 2 ,the difference between Gi (t) and Gi (t + 1) is less than A maxiaccording to (11) and C7, i.e., Gi (t + 1) ≤ Vln 2 + Amaxi . As aresult, the upper bound also holds at slot t +1. This concludesthe 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)holds for t = 0, we suppose that it holds at slot t . From (34),if Qi (t) ≥ Gi (t), device i does not admit new data, andtherefore, Qi (t + 1) ≤ Qi (t) ≤ Gi (t) + Amaxi . On the otherhand, if Qi (t) < Gi (t), the increased backlog must not exceedAmaxi , i.e., Qi (t + 1) ≤ Gi (t) + Amaxi . Substituting the upperbound of Gi (t), we have Qi (t + 1) ≤ Vln 2 + 2 Amaxi . In otherwords, (23) holds at slot t + 1. By mathematical induction,(23) is proved.

Next, we proceed to prove (24). Since (24) holds at slott = 0, we suppose that it holds at slot t . If Ei (t) ≥ θi , device imust not harvest energy according to (32), i.e., Ei (t + 1) ≤Ei (t) ≤ θi + Ei,maxH . Otherwise, if Ei (t) < θi , we have Ei (t +1) ≤ θi + Ei,maxH due to eiH ≤ Ei,maxH . In other words, (24)holds at slot t + 1. This concludes the proof of (24).

Finally, we prove (25). Since it holds at slot t = 0,we suppose that it holds at slot t . If C(t) ≥ β, we haveαi (t) ≤ 0 for any device i ∈ N, i.e., the MEC server does notschedule any device according to (39), resulting in C(t +1) ≤C(t). If C(t) < β, we have C(t + 1) ≤ β + ∑i∈N ξi cmaxi Tby substituting the maximum link capacity. As a result, (25)holds at t + 1. This concludes the proof of (25).

APPENDIX E

When θi is adjusted according to (27) and Ei (t) < pi T at

slot t , we have Ei (t) − θi < −[V/ ln 2+2 Amaxi ]cmaxi

pi. Plugging

this into (37), we obtain αi (t) < [Qi (t) − C(t)ξi ]ci (t) −[V/ ln 2+2 Amaxi ]cmaxi < Qi (t)ci (t)−[V/ ln 2+2 Amaxi ]cmaxi ,i.e., αi (t) < 0 from (23). This proves that a device is notscheduled if Ei (t) < pi T , given θi in (27).

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Xinchen Lyu received the B.E. degree from theBeijing University of Posts and Telecommunica-tions (BUPT), Beijing, China, in 2014. He is cur-rently pursuing the Cotutelle Ph.D. degree withBUPT and the University of Technology Sydney. Hisresearch interests include mobile edge computingand radio resource management.

LYU et al.: OPTIMAL SCHEDULE OF MEC FOR IoT USING PARTIAL INFORMATION 2615

Wei Ni received the B.E. and Ph.D. degrees in elec-tronic engineering from Fudan University, Shanghai,China, in 2000 and 2005, respectively. He is cur-rently a Senior Scientist, and the Team and ProjectLeader at CSIRO, Australia. He also holds hon-orary positions at the University of New SouthWales, Macquarie University, and the Universityof Technology Sydney. Prior to this, he was aPost-Doctoral Research Fellow with Shanghai Jiao-tong University (2005–2008), a Research Scientistand the Deputy Project Manager at the Bell Labs

R&I Center, Alcatel/Alcatel-Lucent (2005–2008), and a Senior Researcher atDevices R&D, Nokia (2008–2009). His research interests include optimiza-tion, game theory, graph theory, and their applications to network and security.

Dr. Ni has been serving as an Editor for the Hindawi Journal of Engineeringsince 2012, a Secretary for the IEEE NSW VTS Chapter since 2015,the Track Chair for VTC-Spring 2016 and 2017, and the Publication Chair forBodyNet 2015. He served as the Student Travel Grant Chair for WPMC 2014,a Program Committee Member for CHINACOM 2014, and a TPC Memberof the IEEE ICC14, ICCC15, EICE14, and WCNC10.

Hui Tian received the M.S. degree in micro-electronics and the Ph.D. degree in circuits and sys-tems from the Beijing University of Posts andTelecommunications (BUPT), China, in 1992 and2003, respectively. She is currently a Professorwith BUPT, the Network Information ProcessingResearch Center Director of the State Key Lab-oratory of Networking and Switching Technology,and the MAT Director of the Wireless TechnologyInnovation Institute. Her current research interestsmainly include radio resource management, cross

layer optimization, M2M, cooperative communication, mobile social network,and mobile edge computing.

Ren Ping Liu (SM’13) is currently a Professorwith the School of Electrical and Data Engineering,University of Technology Sydney, where he leadsthe Network Security Laboratory, Global Big DataTechnologies Centre. He is also the Research Pro-gram Leader of Digital Agrifood Technologies inFood Agility CRC, a government/research/industryinitiative to empower Australia’s food industrythrough digital transformation. Prior to that, he was aPrincipal Scientist with CSIRO, where he led wire-less networking research activities. He specializes

in protocol design and modeling, and has delivered networking solutionsto a number of government agencies and industry customers. He has over100 research publications, and has supervised over 30 Ph.D. students. Hisresearch interests include Markov analysis and QoS scheduling in WLAN,VANET, IoT, LTE, 5G, SDN, and network security. He received the AustralianEngineering Innovation Award and the CSIRO Chairman Medal.

He received the B.E. (Hons.) and M.E. degrees from the Beijing Universityof Posts and Telecommunications, China, and the Ph.D. degree from TheUniversity of Newcastle, Australia. He served as the technical programcommittee chair and the organizing committee chair for a number of IEEEconferences. He is the Founding Chair of the IEEE NSW VTS Chapter.

Xin Wang received the B.Sc. and M.Sc.degrees from Fudan University, Shanghai, China,in 1997 and 2000, respectively, and the Ph.D.degree from Auburn University, Auburn, AL, USA,in 2004, all in electrical engineering.

He was a Post-Doctoral Research Associatewith the Department of Electrical and ComputerEngineering, University of Minnesota, Minneapolis,MN, USA, from 2004 to 2006. In 2006, he joined theDepartment of Computer and Electrical Engineeringand Computer Science, Florida Atlantic University,

Boca Raton, FL, USA, where he is currently an Associate Professor. He isalso a Distinguished Professor with the Department of CommunicationScience and Engineering, Fudan University. His research interests includestochastic network optimization, energy-efficient communications, cross-layerdesign, and signal processing for communications.

Georgios B. Giannakis (F’97) received the Diplomadegree in electrical engineering from the NationalTechnical University of Athens, Greece, in 1981,and the M.Sc. degree in electrical engineering andmathematics and Ph.D. degree in electrical engi-neering from the University of Southern California,in 1983 and 1986, respectively. He was with theUniversity of Virginia from 1987 to 1998. Since1999, he has been a Professor with the Universityof Minnesota, where he was an Endowed Chair inwireless telecommunications, a University of Min-

nesota McKnight Presidential Chair in ECE, and the Director of the DigitalTechnology Center.

His general interests span the areas of communications, networking andstatistical signal processing subjects on which he has authored over 400 journalpapers, 680 conference papers, 25 book chapters, two edited books and tworesearch monographs (h-index 119). Current research focuses on learning frombig data, wireless cognitive radios, and network science with applications tosocial, brain, and power networks with renewables. He is the (co-)inventorof 28 patents issued. He is a fellow of the EURASIP. He was a (co-)recipientof eight best paper awards from the IEEE Signal Processing (SP) andCommunications Societies, including the G. Marconi Prize Paper Awardin Wireless Communications. He also received the Technical AchievementAward from the SP Society (2000) and EURASIP (2005), the Young FacultyTeaching Award, the G. W. Taylor Award for Distinguished Research from theUniversity of Minnesota, and the IEEE Fourier Technical Field Award (2015).He has served the IEEE in a number of posts, including that of a DistinguishedLecturer for the IEEE-SP Society.

Arogyaswami Paulraj (F’15) received the B.E.degree from the Naval College of Engineering,Lonavala, India, in 1966, and the Ph.D. degree fromIIT Delhi, New Delhi, India, in 1973. He is a Pio-neer of MIMO wireless communications. He joinedStanford University, Stanford, CA, in 1972, aftera 20-year industry/military career in India. He hasauthored over 400 research publications and two textbooks, and is a co-inventor in 66 U.S. patents. He isa member of eight national academies, includingthose of USA and China. His recognitions include

the 2014 Marconi Prize and the 2011 IEEE Alexander Graham Bell medal.

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