I Introduction
Realizing the vision of Internet of Things (IoT) has driven the unprecedented growth of small mobile devices in recent years. This stimulates the explosive data/computation traffic increase that is constantly generated from a wide range of new applications such as online gaming and video streaming. Such mobiles, however, typically suffer from finite computation capabilities and batteries due to their small form factors and low cost. Tackling these challenges gives rise to an emerging technology, called mobileedge computation offloading (MECO), which allows computation data to be offloaded from mobiles to proximate servers such as base stations (BSs) and access points (APs), for achieving desirable low latency and mobile energy savings [1, 2, 3]. In a typical asynchrobous MECO system as shown in Fig. 1, different mobiles generate different amounts of computation data at random time instants, and moreover, have diverse latency requirements depending on the applications. This complicates the multiuser offloading and resource management in MECO systems, which shall be investigated in this work.
Ia Prior Work
Designing efficient MECO systems has attracted extensive attention in recent years. In the pioneering work considering singleuser MECO systems [4], the mobile CPUcycle frequencies and offloading rates were optimized for maximizing the energy savings of local computing and offloading, leading to the optimal binary offloading decision. This work was extended in [5] by powering MECO with wireless energy. In addition, for applications with partitionable data, the performance of energy savings can be further enhanced by partitioning data for local computing and offloading, called partial offloading. A set of partitioning schemes have been proposed, including live prefetching [6], program partitioning [7], and controlling offloading ratio [8].
The offloading design in multiuser MECO systems is more complicated. Particularly, one of the main issues is how to jointly allocate radioandcomputational resources. Most prior work on this topic assumes synchronous MECO, where all the mobiles have the identical dataarrival time instants and deadlines. Under this assumption, the resource allocation for minimizing the total mobileenergy consumption was studied in [9] for both timedivision multiple access (TDMA) and orthogonal frequencydivision multiple access (OFDMA) MECO systems, where the derived optimal policy is shown to have a simple thresholdbased structure. This framework was extended in [10] to design energyefficient multiuser MECO accounting for the nonnegligible edgecloud computing latency by using flowshop scheduling techniques. Further research in this direction considers more complex systems such as multicell MECO [11, 12], and wirelesslypowered MECO [13, 14]. On the other hand, another line of research considers partiallysynchronous MECO, for which the mobiles only share identical dataarrival time instants but may have different computation deadlines. For such systems, a set of offloading scheduling policies have been proposed to minimize the total mobile latency using techniques such as flowshop queuing theory [15], and joint scheduling and data partitioning [16]. In addition, cooperative computing among mobiles was investigated in the recent work [17, 18, 19, 20] for reducing energy consumption and offloading latency via data partitioning and offloading scheduling techniques. Specifically, the peertopeer offloading given the computation deadline was investigated in [19] by using the “stringpulling” approach.^{1}^{1}1Compared with [19], the current work considers heterogenous computation deadlines for different mobiles, which is more complex and thus cannot be directly solved using the “stringpuling” approach or traditional datatransmission techniques. Note that in the above work, the assumption of synchronous or partiallysynchronous MECO is unsuitable for many practical asynchronous MECO systems that consist of mobiles with heterogeneous dataarrival time instants and deadlines. This motivates the current work that studies fully asynchronous MECO systems.
Last, it is worth mentioning that in traditional communication systems without MECO, asynchronous packet transmission with individual latency constraints has been widely studied for designing offline and online scheduling policies[21, 22, 23]. The above work only focuses on data transmissions following the firstcomefirstserve rule. In contrast, for asynchronous MECO systems, the transmission techniques should be integrated with the joint radioandcomputational resource management, local computing, and interwound computation and transmission, which is the new theme of this work.^{2}^{2}2The current work differs from [21, 22, 23] in the problem formulation and transformation, as well as providing new insights for asynchronous offloading.
IB Contributions
To the best of the authors’ knowledge, this work was the first attempt on designing the energyefficient offloading controller for practical asynchronous MECO systems with nonidentical taskarrivals and deadlines among mobiles. Compared with synchronous MECO studied in most prior work, the current design eliminates the overhead required for network synchronization, and reduces offloadingandcomputation latency. Towards developing a framework for designing asynchronous offloading, the main contributions of the work are twofold: 1) characterizing the structure of the optimal policy that helps simplify offloadingcontroller design and deepen the understanding of the technology, and 2) proposing the approach for designing practical offloading algorithms via decomposing a complex problem into lowcomplexity convex subproblems. The specific technical contributions and findings are summarized as follows.

General arrivaldeadline orders: Consider the general case with arbitrary orders of dataarrival time instants and deadlines for different mobiles (see Fig. 1). The design of offloading controller is formulated as an optimization problem under the criterion of minimum total mobileenergy consumption and the constraints of timesharing and deadlines. An iterative solution method is proposed to iteratively optimize data partitioning for individual mobiles and multiuser time divisions. The computation complexity is reduced by analyzing the policy structure for each iteration. The analysis reveals that the optimal datapartitioning policy is characterized by a thresholdbased structure. Specifically, each mobile should attempt to increase offloading or reduce it if the computation capacity of the mobile or cloud server becomes a bottleneck as measured using corresponding derived thresholds.

Identical arrivaldeadline orders: To gain more insights, consider the special case where the dataarrival time instants and deadlines of different mobiles follow the identical orders. The optimization problem is decomposed into two sequential problems, corresponding to optimizing the scheduling order and energyefficient joint data partitioning and time division given the optimal order. Thereby, we show that without loss of optimality, the mobiles should be scheduled for offloading according to their dataarrival order. Leveraging this result, the original problem is simplified as the problem of joint optimization of data partitioning and time division. Then the simplified problem is solved using the proposed masterandslave framework, where the slave problem optimizes data partitioning, the master problem corresponds to the energyefficient time division, and both are convex. Interestingly, it is discovered that the optimal timedivision policy attempts to equalize the differences in mobile computation capacities via offloading time allocation to mobiles.

Reverse arrivaldeadline orders: For the same objective as the preceding task, we further consider another special case with the reverse arrivaldeadline orders, where a mobile with later data arrival must complete the computation earlier. The derived optimal scheduling order suggests two nonoverlapping offloading intervals for each mobile. To obtain the optimal offloading durations given the optimal order, we propose a new and simple transformationandscheduling approach. Specifically, the transformation phase converts the original problem into the counterpart with identical arrivaldeadline orders, allowing the use of the previous solution approach. Then given the scheduling order, individual offloading intervals are computed in the the scheduling phase.
The differences between this paper and its conference version [24] are as follows. First, this paper considers the finite computation capacities at the mobiles and edge cloud, while infinite computation capacities are assumed in [24]. Second, several useful discussions are added in this paper to demonstrate the versatility of proposed algorithms. Last, the paper studies the resource management for the case of reverse arrivaldeadline orders, which is not addressed in [24].
Ii System Model
Consider a multiuser MECO system (see Fig. 1), comprising one singleantenna BS connected to an edge cloud and singleantenna mobiles, denoted by a set . Each mobile has oneshot inputdata arrival at a random time instant and is required to complete the computation before a given deadline. We consider asynchronous computation offloading, where the dataarrival time instants and deadlines vary for different mobiles. The input data is partitioned into two parts for parallel computation: one at the mobile’s local CPU and the other offloaded to the BS.^{3}^{3}3For tractability, we assume that the input data can be arbitrarily partitioned following the literature (see e.g., [5]
). This is in fact the case for certain applications such as Gzip compression and feature extraction.
In the messagepassing phase prior to computation offloading, each mobile feeds back to the BS its state parameters, including the estimated channel gain, dataarrival time instant and deadline (acquired by CPU profiling or CPUutilization prediction techniques
[25, 26]). Using the information, the BS determines the energyefficient resourcemanagement policy for controlling the mobiles’ offloaded bits and durations, and then broadcasts the control policy to mobiles.Iia Model of InputData Arrivals
The asynchronous data arrivals for the mobiles are modeled as follows. As shown in Fig. 1, each mobile, say mobile , needs to complete a computation task with bit input data within the time interval , where is the dataarrival time instant and is the computation deadline. The required computation latency for mobile , denoted by , is thus given by , in second (s). Without loss of generality, assume that and .^{4}^{4}4We assume that for , such that the computing intervals of each mobile always overlaps with that of others (see Fig. 1). Otherwise, the total duration can be decoupled into several nonoverlapping durations. To facilitate the exposition in the sequel, we define two useful sets as below.
Definition 1 (EpochSet, UserSet).
Let with denote a sequence of ordered time instants and the permutation matrix given by
such that , and . The time interval between two consecutive time instants is called an epoch with length for . For each mobile, say mobile , let denote its epochset
which specifies the indexes of epochs that constitute the computing interval of mobile
. For each epoch, say epoch , define the userset as the indexes of mobiles whose computing intervals cover epoch .For an example shown in Fig. 1, the epoch set for mobile is , and the userset for epoch is . If given , and , can be constructed as where the vector is the
th column of the identity matrix
.IiB Models of Local Computing and Computation Offloading
Let denote the offloaded bits of mobile during epoch . To finish the computation before the deadline, the remaining bit data is computed by the mobile’s CPU. The models of local computing and computation offloading are described as follows.
IiB1 Local Computing
Based on the model in [27], let denote the number of CPU cycles required for computing bit data for mobile , which may be different for different mobiles depending on their specific computingtask complexities. During the computing duration , since operating at a constant CPUcycle frequency is most energyefficient for local computing [28], the CPUcycle frequency for mobile is chosen as . Following the model in [29], under the assumption of low CPU voltage, the energy consumption for each CPU cycle can be modeled by , where is a constant determined by the circuits. Then the localcomputing energy consumption for mobile , denoted by , is obtained as:
Let denote the maximum CPU frequency of mobile . Then we have . As a result, the offloaded data size of mobile is lowerbounded as , where .
IiB2 Computation Offloading
For each mobile, computation offloading comprises three sequential phases: 1) offloading data from the mobile to the edge cloud, 2) computation by the edge cloud, and 3) downloading of computation results from the edge cloud to the mobile. Assume that the edge cloud assigns an individual virtual machine (VM) for each mobile using VM multiplexing and consolidation techniques that allow for multitask parallel computation [30]. Based on the model in [9], the finite VM computation capacity for each mobile can be reflected by upperbounding the number of offloaded CPU cycles, denoted by , for which the required computation time remains negligible compared with the total computation latency . Mathematically, it enforces that
. Moreover, assuming relatively small sizes of computation results for applications (such as face recognition, object detection in video, and online chess game) and high transmission power at the BS, downloading is much faster than offloading and consumes negligible mobile energy.
^{5}^{5}5For dataintensive applications such as virtual/augmented reality, the energy consumption and latency for result downloading is nonnegligible. In these cases, we expect that the current framework for offloading can be modified and applied to designing asynchronous downloading control as well. Under these conditions, the second and third phases are assumed to have negligible durations compared with the first phase. Assume that the mobiles access the cloud based on TDMA. Specifically, for each epoch, say epoch , the mobiles belonging to the userset timeshare the epoch duration . For these mobiles, let denote the allocated offloading duration for mobile , where corresponds to no offloading. For the case of offloading (), let denote the channel power gain between mobile and the BS, which is assumed to be constant during the computation offloading for each mobile. Based on a widelyused empirical model in [31, 32, 4, 6], the transmission power, denoted by , can be modeled by a monomial function with respect to the achievable transmission rate (in bits/s) :(1) 
where denotes the energy coefficient incorporating the effects of bandwidth and noise power, and
is the monomial order determined by the adopted coding scheme. Though this assumption may restrict the generality of the problem, it leads to simple solutions in (semi) closed forms as shown in the sequel and provides useful insights for practical implementation. Moreover, it provides a good approximation for the transmission power of practical transmission schemes. For example, considering the coding scheme for the targeted bit error probability less than
[33], Fig. 2 gives the normalized signal power per symbol versus the rate, where the monomial order of can fairly approximate the transmission power.^{6}^{6}6In practice, the value of can be determined by curvefitting using experimental data. Note that it is possible to achieve better curvefitting performance by using the polynomial function, for which the proposed iterative design in the sequel can be extended to solve the corresponding convex optimization problem with key procedures remaining largely unchanged. Thus, the offloading energy consumption can be modeled by the following monomial function with respect to and :(2) 
Note that if , we have and thus . The total energy consumption of mobile for transmitting the offloaded input data, denoted by , is given by: .
Iii Problem Formulation
In this section, the energyefficient asynchronous MECO resource management is formulated as an optimization problem that jointly optimizes the data partitioning and time divisions for the mobiles. The objective is to minimize the total mobileenergy consumption: . For each epoch, the multiuser offloading should satisfy the timesharing constraint:
(3) 
For each user, the total offloaded data size and computation are constrained by:
(Data constraint)  (4)  
(Local computation capacity constraint)  (5)  
(VM computation capacity constraint)  (6) 
Note that the deadline constraint for each mobile is enforced by setting the localcomputing data size as bits. Under these constraints, the optimization problem is readily formulated as:
(P1)  
s.t.  
where . One can observe that Problem P1 is feasible if and only if , which is equivalent to . Next, note that the variables and are coupled in the objective function. To overcome this difficulty, one important property of Problem P1 is provided in the following lemma, which can be proved in Appendix A.
Lemma 1.
Problem P1 is a convex optimization problem.
Thus, Problem P1 can be directly solved by the Lagrange method that involves the primal and dual problem optimizations [34]. This method, however, cannot provide useful insights on the structure of the optimal policy, since it requires the joint optimization for the data partitioning and time division that have no closed form. To address this issue, in the following sections, we first study the optimal resourcemanagement policy for the general case where deadlines of mobiles are arbitrary (e.g., ) by using the block coordinate decent (BCD) optimization method [35]. Subsequently, we derive more insightful structures of the optimal policy for two special cases, namely asynchronous MECO with the identical and reverse arrivaldeadline orders. Recall that for the dataarrival order, we have without loss of generality. The socalled identical and reverse arrivaldeadline orders refer to the cases where it satisfies and , respectively, as illustrated in Fig. 3.
Iv Optimal Resource Management with General ArrivalDeadline Orders
This section considers the asynchronous MECO with general arrivaldeadline orders and designs the energyefficient resourcemanagement policy. To characterize the structures of the optimal policy, we propose an iterative algorithm for solving Problem P1 by applying the BCD method. Specifically, given any offloading durations for all the mobiles , we optimize the offloaded data sizes for each mobile, corresponding to energyefficient data partitioning. On the other hand, the offloading durations of the mobiles, , are optimized given any offloaded data sizes , referred to as energyefficient time division.
Iva EnergyEfficient Data Partitioning
This subsection aims at finding the optimal offloaded data sizes for the mobiles, given any feasible offloading time divisions . For each mobile , let denote its offloading epoch set comprising the epoch indexes for which . Mathematically, . Then it can be easily observed that Problem P1 reduces to parallel subproblems, each corresponding to one mobile as:
(P2) 
Problem P2 can be easily proved to be a convex optimization problem. Applying the Lagrange method leads to the optimal datapartitioning policy as follows, which is proved in Appendix B.
Proposition 1 (EnergyEfficient Data Partitioning).
For each mobile, say mobile , given the offloading time divisions , the optimal datapartitioning policy for different epochs for solving Problem P2, denoted by , is given by
(7) 
for , where
(8) 
, , and is the solution to , with
(9) 
Proposition 1 shows that the optimal offloaded data sizes, , are determined by a single parameter . Specifically, as , where and defined in (9) monotonically decreases with , and thus can be uniquely determined and efficiently computed using the bisectionsearch algorithm [34]. In addition, it can be proved by contradiction that for the case of , the optimal offloaded data size in each epoch, , is monotonicallyincreasing with , and , and monotonicallydecreasing with and . This is consistent with the intuition that it is desirable to offload more bits as the channel condition improves, the localcomputing complexity increases, the allocated offloading time duration increases, or the computation deadline requirement becomes more stringent. Moreover, when the monomial order increases (e.g., when the offloading wireless transmission targets for a lower error probability), it is more energyefficient to reduce the offloaded data size since the required transmission power increases with .
Remark 1 (Identical Offloading Rates).
It can be inferred from Proposition 1 that given the optimal time divisions , for each mobile, the optimal offloading rates in different epochs are identical. This is expected, since for each mobile, the channel power gain, bandwidth and noise power are the same in different epochs.
To further characterize the effects of offloading duration and computation capacities of the mobile and cloud on the datapartitioning policy, we define an auxiliary function for each mobile , denoted as for simplicity, as the root of the following equation with respect to .
(10) 
Two useful properties of can be easily derived: 1) , and 2) is monotonically decreasing with the total offloading duration . Then the optimal datapartitioning policy in Proposition 1 can be restated as follows, which is proved in Appendix C.
Corollary 1.
Corollary 1 shows that each mobile should perform the mobileconstrained minimum or cloudconstrained maximum computation offloading (with the total offloaded data sizes being and , respectively), if the mobile or VM server becomes a bottleneck with insufficient computation capacities less than the given thresholds, respectively. It is worth mentioning that if both the mobile and VM have insufficient capacities, computing the inputdata by the deadline is infeasible. Moreover, it can be observed that as the total offloading duration grows, decreases and increases, meaning that the mobile tends to offload more data provisioned with a longer offloading duration.
IvB EnergyEfficient Time Division
For given offloaded data sizes , this subsection focuses on optimizing the timedivision policy, , in all epochs to minimize the total mobileenergy consumption. For each epoch , let denote the offloading userset comprising the mobile indexes for which . Mathematically, . Since the timesharing constraints can be decoupled for different epochs, Problem P1 reduces to solving the following parallel subproblems:
(P3) 
Problem P3 is a convex optimization problem and its optimal solution can be easily derived by using the Lagrange method, which is given in the following proposition.
Proposition 2 (EnergyEfficient Time Division).
For each epoch, say epoch , given any offloaded data sizes , the optimal timedivision policy for different mobiles for solving Problem P3, denoted by , is given by
(12) 
where .
Proposition 2 shows that the optimal offloading duration for each mobile is proportional to the epoch duration by a proportional ratio , which is determined by the offloaded data size and channel gain. Specifically, to minimize the total mobileenergy consumption in each epoch, the mobile with a larger offloaded data size and poorer channel should be allocated with a longer offloading duration.
Last, based on the results obtained in these two subsections, the optimal solution to Problem P1 can be efficiently computed by the proposed iterative algorithm using the BCD method, which is summarized in Algorithm 1. Since Problem P1 is jointly convex with respect to the data partitioning and time divisions , iteratively solving Problem P2 and P3 can guarantee the convergence to the optimal solution to Problem P1.
Remark 2 (LowComplexity Algorithm).
Given offloading timedivisions, the computation complexity for the optimal data partitioning is up to , where characterizes the complexity order for the onedimensional search. Given offloaded data sizes, the optimal timedivision policy has the complexity order of owing to the closedform expression. Thus, the total computation complexity for the proposed BCD algorithm is accounting for the iterative procedures. Simulation results in the sequel show that the proposed method can greatly reduce the computation complexity, especially for larger number of mobiles and epochs compared with the general convex optimization solvers, e.g. CVX, which is based on the standard interiorpoint method that has the complexity order of [36].
IvC Extension: Asynchronous MECO Based on Exponential Offloading EnergyConsumption Model
In this subsection, the solution approach developed in the preceding subsections is extended to the case with the exponential offloading energyconsumption model. Specifically, based on Shannon’s equation, the achievable rate can be expressed as where denotes the bandwidth, and the noise power. Since constantrate transmission is the most energyefficient transmission policy [5], it follows that the energy consumption for offloading bit data with duration is given by
(13) 
where the function is defined as . Based on this model, Problem P1 is modified by replacing the objective function with the following and the resulting new problem is denoted as Problem P4.
(14) 
By following the similar procedure as for deriving Lemma 1, it can be shown that Problem P4 is a convex optimization problem. To characterize its optimal policy structure, we apply the BCD method to derive the energyefficient datapartitioning and timedivision policies as detailed in the following.
IvC1 EnergyEfficient Data Partitioning
For any given offloading division , Problem P4 reduces to parallel subproblems:
(P5) 
where is similarly defined as in Problem P2. Problem P5 is a convex optimization problem. Directly applying Lagrange methods yields the optimal solution as below.
Proposition 3.
Consider asynchronous MECO based on the exponential offloading energyconsumption model. For any given offloading time division , the optimal offloading data size for each mobile is given by
(15) 
for , where
(16) 
, and is the solution to with
This proposition shows that given the offloading time division, if , the optimal offloading policy for the offloading data size has a thresholdbased structure. Specifically, the mobile offloads partial input data or performs full local computing if is above or below the threshold , respectively. This is expected since offloading can reduce energy consumption only under the conditions of a good channel, stringent latency requirement or high localcomputing complexity.
IvC2 EnergyEfficient Time Division
Similar to Section IVB, for any given offloading data sizes , Problem P4 reduces to the following parallel subproblems:
(P6) 
It can be proved that Problem P6 is a convex optimization problem. Define a function as . Following the similar procedure as for deriving Proposition 2, the optimal timedivision policy for this case is characterized as below.
Proposition 4.
Consider asynchronous MECO based on the exponential offloading energyconsumption model. For each epoch, say epoch , given any offloading data sizes , the optimal offloading time division for solving Problem P6, denoted by , is given by
(17) 
where is the inverse function of given by , and satisfies .
Last, combining the results of the optimal data partitioning and time division, the optimal solution to Problem P4 can be obtained by an iterative algorithm using the BCD method, which is similar to Algorithm 1 and omitted for brevity.
IvD Discussions
Extension of the proposed BCD solution approach to other more complicated scenarios are discussed as follows.

Robust design: To cope with imperfect mobile prediction and estimation in practice, the current framework can be modified as follows by applying robust optimization techniques. Based on a model of bounded uncertainty (see e.g., [37]
), the systemstate parameters, including channel gain, dataarrival time and deadline, can be added with unknown bounded random variables representing estimationorprediction errors. Then using the worstcase approach
[37], Problem P1 can be modified by replacing these parameters with their “worse cases” and then solved using the same approach, giving a robust offloading policy. 
Online design: Similar to the online design approach in [22], upon new inputdata arrivals or variations of mobiles’ information, the proposed control policies can be adjusted by updating information (e.g., new channel gains) and applying the current offline framework to determine the updated datapartitioning and timedivision policies. Note that reusing the former results as the initial policy in the iterative recalculation is expected to reduce the computation complexity in temporallycorrelated channels. Moreover, the disruptions of task computing can be avoided by continuing the former policy until obtaining the updated one. Last, assuming instantaneous mobiles’ information available at the BS, the policyupdate approach can also be used for designing the greedy online policy. For frequent arrivals, the computation complexity can be reduced by designing a random policyupdate approach, where the update probability depends on instantaneous mobiles’ information.

Timevarying channels: Assuming blockfading channels where the channel gain is fixed in each fading block and independent and identically distributed (i.i.d.) over different blocks, the solution approach can be easily modified that essentially involves redefining the epochset as the fadingblock indexes within the computation duration and the corresponding userset in Definition 1. Then Problem P1 can be extended by replacing in the objective function with that denotes the channel gain of mobile in epoch . This problem can be solved using the same solution approach developed in the paper.

Nonnegligible cloudcomputation time: In the case of nonnegligible cloudcomputation time, the current problem in Problem P1 can be modified to include the said time in the deadline constraint as a function of the number of offloaded bits. For example, following the model in [9], the cloudcomputation time is a linear function of the number of offloaded bits scaled by the fixed cloudcomputation duration per bit. Though this entails more complex problems, the general solution approaches developed in this paper for asynchronous MECO should be still largely applicable albeit with possible modifications by leveraging results from existing work that considers cloudcomputation time (see e.g., [9]).

OFDMA MECO: Consider the asynchronous MECO system based on OFDMA. Similar to [9], the corresponding energyefficient resource management can be formulated as a mixinteger optimization problem where the integer constrains arise from subchannel assignments. Though the optimal solution is intractable, following a standard approach, suboptimal algorithms can be developed by relaxing the integer constraints and then rounding the results to give subchannel assignments.

Binary offloading: The current results can be used to design the asynchronous MECO based on binary offloading. Note that the corresponding problem is a mixedinteger optimization problem, which is difficult to solve. To address this issue, a greedy and lowcomplexity algorithm can be designed by using probabilistic offloading. Particularly, with the obtained results for partial offloading in this work, the offloading probability for each mobile can be set as the ratio between offloaded and total data sizes. Then a set of resourcemanagement samples can be generated, each randomly selecting individual mobiles for offloading following the obtained probability. Last, the sample yielding the minimum total mobile energy consumption gives the greedy policy. It is worthy mentioning that the policy can be further improved by using the crossentropy method, which adjusts the offloading probability based on the outcomes of samples, but it will result in higher computation complexity [38].
V Optimal Resource Management with Identical ArrivalDeadline Orders
To gain further insights for the structure of the optimal resourcemanagement policy, this section considers the special case of asynchronous MECO with identical arrivaldeadline orders, i.e., a mobile with earlier data arrival also needs to complete the computation earlier. This case arises when the mobiles have similar computation tasks (e.g., identical online gaming applications) but with random arrivals. For this case, the solution to Problem P1 can be further simplified by firstly determining an optimal scheduling order and then designing energyefficient joint datapartitioning and timedivision policy given the optimal order. Note that this design approach does not require the resource management in each epoch. We consider that the mobiles and VMs have unbounded computation capacities and the monomial order , since it can fairly approximate the transmissionenergy consumption in practice.^{7}^{7}7The results can be extended to derive the suboptimal policy for the case of by using approximating techniques, although the corresponding optimal policy has no closed form which can be computed by iterative algorithms. More importantly, it will lead to useful insights into the structure of the optimal policy as shown in the sequel that the optimal timedivision policy admits a defined effective computingpower balancing structure. Moreover, the optimal policy is simplified for a twouser case.
First, we define the offloading scheduling order as follows.
Definition 2 (Offloading Scheduling Order).
Let denote the offloading scheduling order with for . Under this order, mobile is firstly scheduled for offloading, followed by mobile , mobile until mobile . Note that in general since each mobile can be scheduled more than once.
Note that given a scheduling order (e.g., ), one specific mobile (e.g., mobile ) can be repeatedly scheduled, corresponding to computation offloading in multiple nonoverlapping epochs. Recall that Problem P1 optimizes the offloading time divisions and offloaded data sizes for the mobiles in all epochs. Specifically, for each epoch, the derived timedivision policy only determines the offloading durations allocated for different mobiles, without specifying the scheduling order. In other words, if considering the scheduling order, one timedivision policy resulted from the solution to Problem P1 can correspond to multiple scheduling orders as illustrated in Fig. 4. On the other hand, if given the scheduling order, the timedivision policy for solving Problem P1 can be uniquely determined.
Based on the above definition and discussions, in the following subsections, we first derive one optimal scheduling order and then optimize the joint datapartitioning and timedivision policy given the optimal order.
Va Optimal Scheduling Order
Recall that given the identical arrivaldeadline orders, we have and . This means that mobile has earlier data arrival than mobile and also requires the computation to be completed earlier. Using this key fact, we characterize one optimal offloading scheduling order as follows, which is proved in Appendix D.
Lemma 2 (Optimal Scheduling Order).
For the case of identical arrivaldeadline order, one optimal scheduling order that can lead to the optimal solution to Problem P1 is .
Lemma 2 shows that for the case of identical arrivaldeadline orders, there exists one optimal deterministic and simple scheduling order that entails sequential transmission by mobiles following their dataarrival order. The intuitive reason behind the optimality of such an order is that the mobile with an earlier inputdata arrival has a more pressing deadline. On the other hand, for the case with general arrivaldeadline orders, the optimal scheduling has no clear structure, due to the irregularity in data arrivals and deadlines across mobiles.
VB EnergyEfficient Data Partitioning and Time Division Given the Optimal Scheduling Order
Given the optimal scheduling order in Lemma 2, this subsection aims to jointly optimize the offloaded data sizes and offloading durations for the mobiles for achieving the minimum total mobileenergy consumption.
Note that, instead of partitioning each epoch duration for relevant mobiles, the introduced scheduling order helps provide an alternative design approach that can directly partition the total time interval for the mobiles given the optimal scheduling order. This approach yields new insights for the policy structure as elaborated in the sequel. Specifically, let , and denote the startingtime instant, total offloading duration and offloaded data size for mobile , respectively. The offloading for the mobiles should satisfy the following constraints. First, under the data causality constraint which prohibits input data from being offloaded before it arrives, we have
(18) 
Next, the deadline constraint requires that
(19) 
In addition, the timesharing constraint in (3) reduces to the time nonoverlapping constraint as:
(20) 
where is defined as . Based on Lemma 2 and above constraints, the solution to Problem P1 assuming can be derived by solving the following problem:
(P4)  
s.t. 
Problem P4 can be proved to be a convex optimization problem using the similar method as for deriving Lemma 1. One important property of Problem P4 is given below, which can be proved by contradiction and the proof is omitted for brevity.
Lemma 3.
For the case of identical arrivaldeadline orders, the optimal offloading startingtime instants and durations for solving Problem P4, denoted by , satisfy the following:
(21) 
Lemma 3 indicates that the multiuser offloading should fully utilize the whole time duration, which is expected since offloadingenergy consumption decreases with the offloading duration. Using Lemma 3, Problem P4 can be rewritten as follows.
(P5)  
s.t.  
Note that given the constraint of , the data causality constraint is always satisfied since . Moreover, indicates the deadline constraint. It can be easily proved that Problem P5 is a convex optimization problem. To characterize the structure of the optimal policy, we decompose Problem P5 into two subproblems, namely the slave problem corresponding to the energyefficient data partitioning given offloading durations and the master one for the energyefficient time division.
VB1 Slave Problem for EnergyEfficient Data Partitioning Given Offloading Durations
For any given offloading durations , Problem P5 reduces to the slave problem that optimizes the offloaded data sizes . It is easy to see that this slave problem can be decomposed into parallel subproblems as
Comments
There are no comments yet.