PowerDelay Tradeoff in MultiUser MobileEdge Computing Systems
Abstract
Mobileedge computing (MEC) has recently emerged as a promising paradigm to liberate mobile devices from increasingly intensive computation workloads, as well as to improve the quality of computation experience. In this paper, we investigate the tradeoff between two critical but conflicting objectives in multiuser MEC systems, namely, the power consumption of mobile devices and the execution delay of computation tasks. A power consumption minimization problem with task buffer stability constraints is formulated to investigate the tradeoff, and an online algorithm that decides the local execution and computation offloading policy is developed based on Lyapunov optimization. Specifically, at each time slot, the optimal frequencies of the local CPUs are obtained in closed forms, while the optimal transmit power and bandwidth allocation for computation offloading are determined with the GaussSeidel method. Performance analysis is conducted for the proposed algorithm, which indicates that the power consumption and execution delay obeys an tradeoff with as a control parameter. Simulation results are provided to validate the theoretical analysis and demonstrate the impacts of various parameters to the system performance.
Mobileedge computing, dynamic voltage and frequency scaling, power control, bandwidth allocation, Lyapunov optimization, quality of computation experience.
I Introduction
The increasing popularity of smart mobile devices is driving the development of mobile applications, which can be computationintensive, e.g., interactive online gaming, face recognition and 3D modeling. This poses more stringent requirements on the quality of computation experience, which cannot be easily satisfied by the limited processing capability of mobile devices. As a result, new solutions to handle the explosive computation demands and the everincreasing computation quality requirements are emerging [1]. Mobileedge computing (MEC) is such a promising technique to release the tension between the computationintensive applications and the resourcelimited mobile devices [2]. Different from conventional cloud computing systems, where remote public clouds are utilized, MEC offers computation capability within the radio access network. Therefore, by offloading the computation tasks from the mobile devices to the MEC servers, the quality of computation experience, including the device energy consumption and execution latency, can be greatly improved [3].
Nevertheless, the efficiency of computation offloading highly depends on the wireless channel conditions, as offloading tasks requires effective data transmission. Therefore, computation offloading policies for MEC systems have attracted significant attention in recent years [4][11]. For applications with strict deadline requirements, the local execution energy consumption was minimized by adopting dynamic voltage and frequency scaling (DVFS) techniques, and the energy consumption for computation offloading was optimized using data transmission scheduling in [4]. In [5], joint allocation of communication and computational resources for femtocloud computing systems was proposed, where each computation task should be completed before its deadline. In [6], a dynamic computation offloading policy was developed for MEC systems with energy harvesting devices under a strict execution delay requirement. Besides, a decentralized computation offloading algorithm was proposed to minimize the computation overhead for multiuser MEC systems in [7].
Imposing strict execution delay constraints makes the computation offloading design more tractable, as only shortterm performance, e.g., the performance for executing a single task, needs to be considered. However, it may be impractical for applications that can tolerate a certain period of execution latency, such as multimedia streaming. For such type of applications, the longterm system performance is more relevant, where the coupling among the randomly arrived tasks cannot be ignored. In order to minimize the longterm average energy consumption, a stochastic control algorithm was proposed in [8], which determines the offloaded software components. In [9], a delayoptimal stochastic task scheduling algorithm was developed for singleuser MEC systems. Moreover, an online task scheduling algorithm was proposed to investigate the energydelay tradeoff for MEC systems with a multicore mobile device in [10], and this study was later extended to scenarios with heterogeneous types of mobile applications in [11]. Unfortunately, existing works only focused on singleuser MEC systems, and the design methodologies for multiuser MEC systems remain unknown.
In this paper, we consider a general MEC system with multiple mobile devices, where computation tasks arrive at the mobile devices in a stochastic manner. Joint design of local execution and computation offloading strategies will be investigated. With multiple devices, the design becomes much more challenging as intelligent management of the radio resources for computation offloading, e.g., the transmit power and available spectrum, is needed. We formulate a power consumption minimization problem with task buffer stability constraints. An online algorithm is proposed based on Lyapunov optimization, which decides the CPUcycle frequencies for local execution, and the transmit power and bandwidth allocation for computation offloading. In particular, the optimal CPUcycle frequencies are obtained in closed forms, while the optimal transmit power and bandwidth allocation are determined by the GaussSeidel method. Performance analysis is conducted for the proposed algorithm, which explicitly characterizes the tradeoff between the power consumption of the mobile devices and the execution delay. Simulation results verify the theoretical analysis and demonstrate that the proposed algorithm is capable of controlling the power consumption and execution delay performance in multiuser MEC systems.
The organization of this paper is as follows. We introduce the system model in Section II. The power consumption minimization problem is formulated in Section III, and an online local execution and computation offloading policy is developed in Section IV. Simulation results will be shown in Section V, and we will conclude this paper in Section VI.
Ii System Model
We consider a mobileedge computing (MEC) system as shown in Fig. 1, where mobile devices running computationintensive applications are assisted by an MEC server. The MEC server could be a small data center installed at a wireless access point deployed by the telecom operator. Therefore, it can be accessed by the mobile devices through wireless channels, and will execute the computation tasks on behalf of the mobile devices [3, 4]. By offloading part of the computation tasks to the MEC server, the mobile devices could enjoy a higher level of quality of computation experience [3].
The available system bandwidth is Hz, which is shared by the mobile devices, and the noise power spectral density at the receiver of the MEC server is denoted as . Time is slotted and the time slot length is . For convenience, we denote the index sets of the mobile devices and the time slots as and , respectively.
Iia Computation Task and Task Queueing Models
We assume the mobile devices are running finegrained tasks [11]: At the beginning of the th time slot, (bits) of computation tasks arrive at the th mobile device, which can be processed starting from the th time slot. Without loss of generality, we assume the ’s in different time slots are independent and identically distributed (i.i.d.) within with .
In each time slot, part of the computation tasks of the th mobile device, denoted as , will be executed at the local CPU, while bits of the computation tasks will be offloaded to and executed by the MEC server. The arrived but not yet executed tasks will be queued in the task buffer at each mobile device, and the queue lengths of the task buffers at the beginning of the th time slot are denoted as with , where evolves according to the following equation:
(1) 
In (1), is the amount of tasks departing from the task buffer at the th device in time slot .
IiB Local Execution Model
In order to process one bit of task input at the th mobile device, CPU cycles will be needed, which depends on the types of applications and can be obtained by offline measurements [12]. Denote the scheduled CPUcycle frequency for the th mobile device in the th time slot as , which cannot exceed . Thus, can be expressed as
(2) 
Accordingly, the power consumption for local execution at the th mobile device is given by
(3) 
where is the effective switched capacitance related to the chip architecture [13].
IiC MEC Server Execution Model
To offload the computation tasks for MEC server execution, the input bits of the tasks need to be delivered to the MEC server. For simplicity, we assume the MEC server is equipped with an core highspeed CPU so that it can execute different applications in parallel, and the processing latency at the MEC server is negligible. We leave the investigation of more general MEC servers to our future work.
The wireless channels between the mobile devices and the MEC server are i.i.d. frequencyflat block fading. Denote the smallscale fading channel power gain from the th mobile device to the MEC server at the th time slot as , which is assumed to have a finite mean value, i.e., . Thus, the channel power gain from the th mobile device to the MEC server can be represented by , where is the pathloss constant, is the reference distance, is the pathloss exponent, and is the distance from mobile device to the MEC server. Hence, the amount of offloaded tasks at the th mobile device in time slot is given by
(4) 
where is the transmit power with the maximum value of , and is the portion of bandwidth allocated to the th mobile device. Denote as the bandwidth allocation vector, which should be chosen from the feasible set [14], i.e.,
(5) 
Iii Problem Formulation
In this section, we will first introduce the performance metrics, namely, the power consumption of the mobile devices and the average queue lengths of the task buffers. A power consumption minimization problem will then be formulated to facilitate the investigation of the powerdelay tradeoff.
The average power consumption of the mobile devices, including the power consumed by the local CPUs and the transmit power for computation offloading, can be expressed as
(6) 
where .
According to the Little’s Law [15], the execution delay is proportional to the average queue length of the task buffer. Hence, we adopt the average queue length of the task buffer as a measurement of the execution delay, which can be written as
(7) 
Denote and . Thus, the power consumption minimization problem is formulated as
(8)  
(9)  
(10)  
(11) 
where (9) and (10) are the CPUcycle frequency constraint and the transmit power constraint, respectively. (11) requires the task buffers to be mean rate stable [16], which ensures that all the arrived computation tasks can be executed with finite delay. In general, is a stochastic optimization problem, for which, the CPUcycle frequency, the transmit power as well as the bandwidth allocation need to be determined for each device at each time slot. This problem is difficult to solve as the optimal decisions are temporally correlated. Also, a joint consideration on the local execution and computation offloading strategies is needed, as both of them affect the system performance. Besides, the spatial coupling of the bandwidth allocation among different mobile devices poses another challenge.
Instead of solving directly, we consider , which is a modified version of by replacing set in (5) by set , with defined as
(12) 
With such modification, the departure function of MEC server execution, , is continuous and differentiable with respect to . In addition, the optimal value of is larger but can be made arbitrarily close to that of by setting () to be sufficiently small. Furthermore, any feasible solution for is also feasible for . Thus, we will focus on in the remainder of this paper.
Iv Online Local Execution and Computation Offloading Policy
In this section, we will propose an online local execution and computation offloading policy to solve based on Lyapunov optimization [16], where a deterministic problem needs to be solved at each time slot. We will then analyze the performance of the proposed algorithm and reveal the powerdelay tradeoff in multiuser MEC systems.
Iva Lyapunov OptimizationBased Online Algorithm
To present the algorithm, we first define the Lyapunov function as
(13) 
Thus, the Lyapunov drift function can be written as
(14) 
Accordingly, the Lyapunov driftpluspenalty function can be expressed as
(15) 
where () is a control parameter in the proposed algorithm. We find an upper bound of under any feasible , , and , as specified in Lemma 1.
Lemma 1
For arbitrary such that , , , and , is upper bounded by
(16) 
where is a constant.
Proof 1
Proof is omitted due to space limitation.
The main idea of the proposed online local execution and computation offloading policy is to minimize the upper bound of in the righthand side of (16) greedily at each time slot. By doing so, the amount of tasks waiting in the task buffers can be maintained at a small level. Meanwhile, the power consumption of the mobile devices can be minimized. The proposed algorithm is summarized in Algorithm 1, where a deterministic optimization problem needs to be solved at each time slot. It is worthy to note that the objective function of corresponds to the righthand side of (16), and all the constraints in except the stability constraints in (11) are retained in . The optimal solution for will be developed in the next subsection.
IvB Optimal Solution For
In this subsection, we will derive the optimal CPUcycle frequencies, transmit powers and bandwidth allocation vector for .
Optimal CPUcycle Frequencies: It is straightforward to show that the optimal CPUcycle frequency for the th mobile device in time slot can be obtained by solving
(17) 
and its optimal solution is achieved at either the stationary point of the objective function or one of the boundary points, which is given by
(18) 
Remark 1
Note that increases with as it is desirable to execute more tasks in order to keep the queue length of the task buffer small. Besides, decreases with both and : With a larger value of , the weight of the power consumption becomes larger, and thus the local CPU slows down its frequency to reduce power consumption; with a larger value of , local execution becomes less efficient as more CPU cycles are needed to process per bit of task input, which leads to a smaller CPUcycle frequency.
Optimal Transmit Power and Bandwidth Allocation: After decoupling from , the optimal and can be obtained by solving
(19) 
It is not difficult to identify that is a convex optimization problem. However, generic convex algorithms suffer from relatively high complexity as they are developed for general convex problems and do not make use of the problem structures [17]. Motivated by this, we propose to solve by optimizing the transmit power and bandwidth allocation in an alternating manner, where in each iteration, the optimal transmit powers are obtained in closed forms and the optimal bandwidth allocation is determined by the Lagrangian method. Since is jointly convex with respect to and , and its feasible region is a Cartesian product of those of and , the alternating minimization procedure is guaranteed to converge to the global optimal solution, which is termed as the GaussSeidel method in literature [18].
1) Optimal Transmit Power: For a fixed bandwidth allocation vector , the optimal transmit power for the th mobile device can be obtained by solving
(20) 
whose optimal solution is achieved at either the stationary point of the objective function or one of the boundary points similar to , and it is given in closed form by
(21) 
2) Optimal Bandwidth Allocation: For a fixed transmit power vector , the optimal bandwidth allocation can be obtained by solving the following problem:
(22) 
which is more challenging as the bandwidth allocation decision is coupled among different mobile devices. Fortunately, the Lagrangian method offers an effective solution for . Specifically, the partial Lagrangian can be written as
(23) 
where is the Lagrangian multiplier associated with constraint . Based on the KarushKuhnTucker (KKT) conditions, the optimal bandwidth allocation and the optimal Lagrangian multiplier should satisfy the following equation set:
(24) 
In (24), if , ; otherwise, denotes the root of for , which is positive and unique as decreases with . Thus, it suggests a bisection search over for the optimal , where , and satisfies . Hence, can be obtained by a bisection search over , and the searching process for the optimal will be terminated when , where is the accuracy of the algorithm. Details of the Lagrangian method for are summarized in Algorithm 2.
Remark 2
One main benefit of the proposed online algorithm is that it does not require prior information on the computation task arrival and wireless channel fading processes, which makes it also applicable for unpredictable environments. Besides, the proposed algorithm is of low complexity, as at each time slot, the optimal CPUcycle frequencies are obtained in closed forms, while the computation offloading policy is determined by an efficient alternating minimization algorithm. Furthermore, as will be shown in the next subsection, the achievable performance of the proposed algorithm can be analytically characterized and thus facilitates the analysis on the powerdelay tradeoff in multiuser MEC systems.
IvC Performance Analysis
In this subsection, we will provide the main theoretical result in this paper, which characterizes the upper bounds for the power consumption of the mobile devices and the average sum queue length of the task buffers. Also, the tradeoff between the power consumption and execution delay will be revealed.
Theorem 1
Assume that is feasible, we have:

The average power consumption of the mobile devices under the proposed algorithm satisfies:
(25) where is the optimal value of .

For arbitrary , is mean rate stable.

Suppose there exist and () that satisfy the Slater conditions [16], then the average sum queue lengths of the task buffers satisfies:
(26)
Proof 2
Proof is omitted due to space limitation.
Remark 3
Theorem 1 shows that under the proposed online local execution and computation offloading policy, the worstcase power consumption of the mobile devices decreases inversely proportional to , while the upper bound of the execution delay increases linearly with , i.e., there exists an tradeoff between these two objectives. Thus, we can balance the power consumption and execution delay by adjusting : For delaysensitive types of applications, we can use a small value of ; while for energysensitive networks and delaytolerant applications, a large value of can be adopted.
V Simulation Results
In simulations, we assume mobile devices are located at an equal distance of m from the MEC server. The smallscale fading channel power gains are exponentially distributed with unit mean. Besides, , ms, MHz, dBm/Hz, dB, m, , GHz, mW, is uniformly distributed within , and cycles/bit, [12]. The simulation results are averaged over 5000 time slots.
We first show the relationship between the power consumption of the mobile devices/average queue length of the task buffers and the control parameter in Fig. 2. We see from Fig. 2a) that the power consumption decreases inversely proportional to and converges to when is sufficiently large. Meanwhile, as shown in Fig. 2b), the average queue length of the task buffers increases linearly with and becomes unbounded when goes to infinity. These results verify the tradeoff between the power consumption and execution delay as shown in Theorem 1.
In Fig. 3, we show the relationship between the power consumption and execution delay for scenarios with and without MEC^{1}^{1}1The average execution delay is calculated by (time slots) according to the Little’s Law.. It is observed that by increasing from to , the power consumption of the mobile devices decreases significantly for both cases. However, the behaviors of the execution delay are substantially different: With MEC, the execution delay decreases sharply from 33.2 to 1.05 ms as decreases, while without MEC, the execution delay has minor changes at around ms. This is because without the aid of the MEC server, the devices cannot stabilize their task buffers even with a small , where the local CPUs operate at their maximum frequencies. Therefore, we verify the benefits of MEC for improving the quality of computation experience.
By varying and , we show the relationship between the power consumption and execution delay in Fig. 4. In general, the average execution delay increases as the power consumption decreases, which indicates that a proper should be chosen to balance the two desirable objectives. For instance, with and kbits, if the average execution delay requirement is 20 ms, can be chosen, and the power consumption will be 0.1 W. Besides, with a given execution delay, the power consumption increases with the computation task arrival rate (the number of mobile devices), which agrees with the intuitions as the workload of the MEC system becomes heavier, more power is needed to stabilize the task buffers. In addition, when goes to infinity, doubling the computation task arrival rates results in a higher power consumption than doubling the number of mobile devices, which is due to the increased multiuser diversity gain and the availability of extra local CPUs.
Vi Conclusions
In this paper, we investigated the powerdelay tradeoff in a multiuser mobileedge computing system. A power consumption minimization problem with task buffer stability constraints was formulated, and an online algorithm that decides the local execution and computation offloading policy was derived based on Lyapunov optimization. Performance analysis was conducted for the proposed algorithm, which explicitly characterizes the tradeoff between the power consumption and execution delay performance. Simulation results validated the theoretical analysis, and showed that the proposed algorithm is capable of balancing the power consumption of the mobile devices and the quality of computation experience. For future investigation, it would be interesting to extend the findings in this work to scenarios with fairness considerations among multiple devices.
References
 [1] J. Gubbi, R. Buyya, S. Marusic, and M. Palaniswmi, “Internet of Things (IoT): A vision, architectural elements, and future directions,” ELSEVIER Future Gener. Comp. Syst., vol. 29, no. 7, pp. 16451660, Sep. 2013.
 [2] European Telecommunications Standards Institute (ETSI), “Mobileedge computingIntroductory technical white paper,” Sep. 2014.
 [3] M. Satyanarayanan, P. Bahl, R. Caceres, and N. Davies, “The case for vmbased cloudlets in mobile computing,” IEEE Pervasive Comput., vol. 8, no. 4, pp. 1423, Oct. 2009.
 [4] W. Zhang, Y. Wen, K. Guan, D. Kilper, H. Luo, and D. Wu, “Energyoptimal mobile cloud computing under stochastic wireless channel,” IEEE Trans. Wireless Commun., vol. 12, no. 9, pp. 45694581, Sep. 2013.
 [5] O. Munoz, A. Iserte, and J. Vidal, “Optimization of radio and computational resources for energy efficiency in latencyconstrained application offloading,” IEEE Trans. Veh. Technol., vol. 64, no. 10, pp. 47384755, Oct. 2015.
 [6] Y. Mao, J. Zhang, and K. B. Letaief, “Dynamic computation offloading for mobileedge computing with energy harvesting devices,” IEEE J. Sel. Areas Commun., to appear.
 [7] X. Chen, “Decentralized computation offloading game for mobile cloud computing,” IEEE Trans. Parallel Distrib. Syst., vol. 26, no. 4, pp. 974983, Apr. 2015.
 [8] D. Huang, P. Wang, and D. Niyato, “A dynamic offloading algorithm for mobile computing,” IEEE Trans. Wireless Commun., vol. 11, no. 6, pp. 19911995, Jun. 2012.
 [9] J. Liu, Y. Mao, J. Zhang, and K. B. Letaief, “Delayoptimal computation task scheduling for mobileedge computing systems,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), Barcelona, Spain, Jul. 2016.
 [10] Z. Jiang and S. Mao, “Energy delay tradeoff in cloud offloading for multicore mobile devices,” in Proc. IEEE Global Commun. Conf. (GLOBECOM), San Diego, CA, Dec. 2015.
 [11] J. Kwak, Y. Kim, J. Lee, and S. Chong, “DREAM: Dynamic resource and task allocation for energy minimization in mobile cloud systems,” IEEE J. Sel. Areas Commun., vol. 33, no. 12, pp. 25102523, Dec. 2015.
 [12] A. P. Miettinen and J. K. Nurminen, “Energy efficiency of mobile clients in cloud computing,” in Proc. USENIX Conference on Hot Topics in Cloud Computing (HotCloud), Boston, MA, Jun. 2010.
 [13] T. D. Burd and R. W. Brodersen, “Processor design for portable systems,” Kluwer J. VLSI Signal Process. Syst., vol. 13, no. 2/3, pp. 203221, Aug. 1996.
 [14] Z. Wang, V. Aggarwal, and X. Wang, “Joint energybandwidth allocation in multiple broadcast channels with energy harvesting,” IEEE Trans. Commun., vol. 63, no. 10, pp. 38423885, Oct. 2015.
 [15] S. M. Ross, Introduction to probability models. Academic Press, 2014.
 [16] M. J. Neely, Stochastic network optimization with application to communication and queueing systems. Morgan & Calypool, 2010.
 [17] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge University Press, 2004.
 [18] L. Grippo and M. Sciandron, “On the convergence of the block nonlinear GaussSeidel method under convex constraints,” ELSEVIER Oper. Res. Lett., vol. 26, no. 3, pp. 127136, Apr. 2000.