Multi-Cell Mobile Edge Computing: Joint Service Migration and Resource Allocation
aa r X i v : . [ c s . I T ] F e b Multi-Cell Mobile Edge Computing: JointService Migration and Resource Allocation
Zezu Liang, Yuan Liu, Tat-Ming Lok, and Kaibin Huang
Abstract
Mobile-edge computing (MEC) enhances the capacities and features of mobile devices by of-floading computation-intensive tasks over wireless networks to edge servers. One challenge faced bythe deployment of MEC in cellular networks is to support user mobility. As a result, offloaded taskscan be seamlessly migrated between base stations (BSs) without compromising the resource-utilizationefficiency and link reliability. In this paper, we tackle the challenge by optimizing the policy formigration/handover between BSs by jointly managing computation-and-radio resources. The objectivesare twofold: maximizing the sum offloading rate, quantifying MEC throughput, and minimizing themigration cost. The policy design is formulated as a decision-optimization problem that accounts forvirtualization, I/O interference between virtual machines (VMs), and wireless multi-access. To solvethe complex combinatorial problem, we develop an efficient relaxation-and-rounding based solutionapproach. The approach relies on an optimal iterative algorithm for solving the integer-relaxed prob-lem and a novel integer-recovery design. The latter outperforms the traditional rounding method byexploiting the derived problem properties and applying matching theory. In addition, we also considerthe design for a special case of “hotspot mitigation”, referring to alleviating an overloaded server/BSby migrating its load to the nearby idle servers/BSs. From simulation results, we observed close-to-optimal performance of the proposed migration policies under various settings. This demonstrates theirefficiency in computation-and-radio resource management for joint service migration and BS handoverin multi-cell MEC networks.
Index Terms
Mobile-edge computing (MEC), service migration, handover, resource management.
Z. Liang and T. M. Lok are with Department of Information Engineering, The Chinese University of Hong Kong, HongKong (e-mail: [email protected]; [email protected]). Y. Liu is with school of Electronic and Information Engineering,South China University of Technology, Guangzhou 510641, China (e-mail: [email protected]). K. Huang is with Departmentof Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong (e-mail: [email protected]).
I. I
NTRODUCTION
Mobile (or multi-access) edge computing (MEC), which provides users computing servicesat the network edge, is envisioned as a key technology in the fifth generation (5G) systemsfor supporting computation-intensive and latency-critical mobile applications [1], [2]. In MECsystems, the computation intensive tasks of mobile users are offloaded to edge servers co-locatedwith base stations (BSs) or access points. This avoids data transportation to the remote cloudcenters, thereby dramatically reduce latency and avoid traffic congestion in the backhaul network.In this work, we address the issue of supporting mobility in an MEC network, referring to acellular network providing MEC services. In a traditional radio access network, a key solutionfor mobility is to handover a travelling user’s wireless link from one BS to another to ensureits reliability [3]. The handover in an MEC network is more complex as it may also involve themigration of computing tasks between servers, called service migration [4]. Making a migrationdecision should account for factors including computation resources and load at two servers/BSsand the migration cost incurred by data transportation across the backhaul network. To tacklethe challenges, we propose in this paper a framework of joint migration-and-handover (JMH) for multi-user multi-cell MEC systems.
A. Resource Management in MEC Networks
Among others, one vein of MEC research that is aligned with the current work is resourcemanagement. It features the joint management of computation and radio resources to achievea high efficiency for computation offloading. A binary offloading policy is proposed in [5] foradapting the offloading decision to a stochastic wireless channel under the criterion of minimizingmobile energy consumption. Peer-to-peer cooperative MEC is proposed in [6] where one mobiledevice serves as a helper for another by offloading the latter’s computation or relaying it toa server. For multi-user MEC systems, the resource management is more complicated since itinvolves resource sharing by multiple offloading users. In [7], a centralized resource allocationscheme is proposed for minimizing sum mobile energy consumption. The design is extended in[8] to the case of asynchronous offloading. On the other hand, distributed resource allocationstrategies can be designed by applying game theory, which is pursued in [9], [10]. In a multi-user and multi-server system, there is an additional issue of load distribution among servers. It isaddressed in [11] by an efficient distributed offloading design based on matching theory and in [12] using the reinforcement-learning approach. It has also been studied in various MEC systemconfigurations like vehicular networks [13] and unmanned aerial vehicle (UAV) systems [14].Another important type of resources in computing is I/O resources such as the bandwidthof a bus connecting a GPU and its associated system. For edge or cloud computing based onvirtualization, tasks are executed simultaneously in the same server in the forms of virtual ma-chines (VMs). The sharing of finite I/O resources by VMs causes mutual computing interference,called I/O interference, which slows down their computing speeds [15]–[17]. Being a potentialperformance bottleneck, I/O interference is extensively studied in the area of cloud computingto understand its effects and find solutions (see e.g., [17]). In contrast, I/O interference is notyet extensively studied in the area of MEC despite some recent work on factoring it into thedesign of offloading policies [18]. In this work, we also consider I/O interference in JMH.In view of prior work, existing results on resource management for MEC focus on theoptimization of offloading policies. In this work, we explore an uncharted direction of resourcemanagement for migration and handover to support mobility in MEC networks. Most existingwork focuses on sharing the resources of a single server (or server cluster) by multiple offloadingusers. In contrast, we focus on the balancing of the resources among servers/BSs by controllingboth migration and handover.
B. Service Migration and BS Handover
As a key mechanism for dynamic resource management, service migration has been widelystudied in the area of cloud computing covering a wide range of topics including network loadbalancing [19], hotspot mitigation [20], and I/O interference aware migration [21]. Migrationin cloud computing targets a wired network (e.g., server grid within a data center) where linksare assumed reliable. In contrast, the implementation of service migration in an MEC networkwill be inevitably coupled with the handover of wireless links over BSs. The links experiencefading and each BS serves a dynamic number of users and hence has time-varying availableradio sources besides a random computation load. The coupling between service migration andBS handover calls for their joint design to improve the offloading performance of the MECnetwork, which forms the theme of this work.In traditional wireless networks, BS handover is incurred by deterioration in wireless linkquality of the serving BS and is employed to re-associate with another for higher radio access.However, such handover mechanisms are not sufficient to support efficient computation offloading in an MEC network. On one hand, as mentioned earlier, handover of MEC services is conductedin JMH for ensuring radio and computing reliability, and thereby the migration cost on both sidesshould be taken into account. On the other hand, apart from channel condition, the computationcapabilities of different BSs need to be considered when associating a user with an appropriateBS. The work [22]–[25] investigates BS handover in MEC under mobility consideration, whichhowever does not consider the variation of computation resource by BS handover. In contrast,our proposed JMH framework considers that the computing speeds of two servers/BSs fluctuatecaused by handover. Such an issue is not studied yet in MEC migration/handover.
C. Our Contributions
In this paper, we study the problem of optimal JMH in a multi-user multi-cell MEC systembased on virtualization. The optimization problem aims at maximizing the weighted sum of-floading rates of all the users while minimizing the incurred migration cost as much as possibleby controlling the migration/handover decisions. The problem accounts for both I/O interferenceand multi-user interference.The main contribution of the work lies in developing a practical algorithm for designingthe optimal JMH policy. The said problem is an integer nonlinear program and nonconvex. Toovercome the difficulty, we propose a two-stage solution method. First, the binary constraints ofthe migration decisions are relaxed, allowing fractional programming to be applied to solve therelaxed problem. Next, a novel rounding method based on the problem’s properties is proposedfor recovering the binary decision solution, which outperforms the naive rounding method.The other contribution of the work is to optimize JMH for the hotspot mitigation scenario,referring to alleviating an overloaded server by migrating its load to the helper servers. We showthat when the load of the hotspot server is below a certain threshold, the optimal JMH schemecan effectively address the overloaded condition of the hotspot server via load balance amongservers. When the load exceeds the threshold, all the servers are overloaded after optimal JMHand in this case adding more helper servers is needed.The rest of this paper is organized as follows. In Section II, we present the system model andproblem formulation. We introduce the algorithm to solve the formulated problem in Section IIIand discuss the special case of the hotspot mitigation in Section IV. Finally, simulation resultsand conclusions are provided in Section V and Section VI, respectively. u u u Shared I/O resource
VM1 VM2 VM3 VM3
VM4 u Initial associationHandoverVM migrationBS1BS2BS3
Fig. 1: A multi-cell MEC system, where u and u ’s services are enabled by joint BS handover and VM migration. II. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
A. System Model
As shown in Fig. 1, we consider an MEC system consisting of N BSs and K users, denotedby the set N = { , · · · , N } and set K = { , · · · , K } , respectively. Each BS is integrated with aserver that can provide computing services to users if it hosts the users’ corresponding VMs. Weassume that each user is served by one dedicated VM and each VM only serves the correspondinguser . A VM is a software clone of user’s service environment, which contains the user’s profilesand applications for running user’s offloaded tasks and can be migrated among BSs to continueserving the user wherever it moves. In the proposed JMH framework, when a user switches itsassociation from one BS to another, the user’s corresponding VM is also migrated between thetwo BSs (e.g., see users u and u in Fig. 1). We assume a time-slotted model that the user-BSassociations and channel gains remain unchanged in each slot but can be varied from one slot toanother. The channel gains are considered to include path loss and shadowing while neglect thesmall-scale (fast) fading, in view of the fact that small-scale fading has little effect on referencesignal receiving quality (RSRQ) that is the measurement for handover and dominated by pathloss in practice. Moreover, the effect of small-scale fading can be averaged out by employinga sufficiently long channel code in practice [26], [27]. Thus, the channel gains can be regardedas static within each slot but may vary from one slot to another. Let x k,n denote the JMH An VM dedicated to serving a user is commonly used for the user’s service with the requirements in user-customizedcompute environment, data security and confidentiality, and/or performance stability. For example, each Google Docs servicegets an VM/container to serve a specific user. decision for the service migration, with x k,n = 1 indicating the service of user k is placed atBS n and x k,n = 0 otherwise. We assume that each user can associate with only one BS, thus, P n ∈N x k,n = 1 , ∀ k ∈ K . The JMH process can incur system overheads, such as consumingbackhaul bandwidth to transfer VM data and the handover signaling. To account for this, weconsider a fixed cost c k,j,n occurs when user’s k service is migrated from BS j to BS n , with n = j . We assume c k,j,n = 0 if n = j (i.e., no cost occurs if not migrated). Then, given thecurrent service locations { x k,n } , the JMH cost of each user in next time slot is given by C k = X n ∈N X j ∈N x k,j x k,n c k,j,n , (1)where x k,j x k,n indicates whether user k ’s service is initially placed on BS j and to be migratedto BS n (i.e., x k,j x k,n = 1 ) or not (i.e., x k,j x k,n = 0 ). For simplicity, we assume that themigration/handover time is negligible compared with the slot length.
1) Communication Model:
Denote the uplink channel gain from user k to BS n as g k,n , thetransmit power of user k as p k , and the noise power of BS n as σ n . For the ease of problemanalysis, we consider the simple case that users offload data at the same band, i.e., frequencyreuse factor of . The extension to radio resource allocation among users will be elaborated inSection III-E. Then, the achievable uplink transmission rate for user k offloading tasks to BS n ,denoted as r k,n , is r k,n = B log p k g k,n σ n + P j ∈K\{ k } p j g j,n ! . (2)where B denotes the system bandwidth. As the transmit powers are assumed to be fixed, eachuser’s transmission rate r k,n is deterministic through (2). We ignore the result downloading phasebecause of the relative much smaller sizes of computation results [23].
2) Computation Model:
The MEC server at each BS accommodates users’ offloaded tasksinto their own VMs and executes them in parallel, namely parallel computing . We consider theI/O interference in parallel computing [15], [16] and adopt a model developed in the literature[28] to characterize the computation rate. Specifically, let f k,n denote the expected computationrate (offloaded bits per second) of user k ’s VM when running in isolation at BS n . Following [28], we define d n > as a performance degradation factor at BS n to specify the computationrate reduction of a VM when multiplexed with another VM. With one-to-one correspondencebetween each VM and each user as mentioned earlier, the number of VMs hosted at a BS isequal to the number of the associated users P k ∈K x k,n . Therefore, given P k ∈K x k,n associatedusers at BS n , the actual computation rate for execution of user k ’s task is F k,n = f k,n (1 + d n ) − P k ∈K x k,n . (3)Here (3) indicates that the computation rate of each user decreases as the number of co-locatedusers at a BS increases. This implies a tradeoff that accommodating more users (or consolidatingmore VMs) at a BS can increase multiplexing gain in parallel computing but degrades thecomputation rates for individual users due to the I/O interference.On the other hand, consideringthe finite computation capacity of a BS, we assume that the number of multiplexed VMs (orequivalently, the number of users) at a BS is bounded by a number M n , i.e., P k ∈K x k,n ≤ M n .After characterizing user’s communication rate in (2) and computation rate in (3), we use theoffloading rate as a metric to measure the computation offloading performance. Here, the of-floading rate is defined as the number of user’s offloadable bits per unit time, which is given by R k,n = 1 (cid:30)(cid:18) r k,n + 1 F k,n (cid:19) , (4)where the first and second terms in the denominator are the inverse of the transmission rate andcomputation rate, respectively, denoting the corresponding required time for transmitting andcomputing -bit. B. Problem Formulation
We consider a problem of service migration among BSs under the consideration of jointcomputation-and-radio resource management. Specifically, given the initial offloading-serviceplacement, we aim to find the optimal JMH decisions that maximize the weighted sum offloadingrate while reduce the total incurred JMH cost at the same time. The problem is formulated as ( P1 ) : max X X k ∈K ω k X n ∈N x k,n R k,n − λ X k ∈K X n ∈N X j ∈N x k,j x k,n c k,j,n (5) To be more specific, d is defined as the percentage increase in the computing time (i.e., /f ) of a VM when multiplexedwith another VM on the same server. We assume the server can coordinate the workloads and resource demands of VMs toreach a homogenous factor d [28]. Then, indicating with T the computing time of a VM in isolation, the expected computingtime of a VM when multiplexed with an additional VM can be expressed as T = T · (1 + d ) . Generalizing, we can derive theexpected computing time of a VM when multiplexed with additional i − VMs as T i = T · (1 + d ) i − . s . t . X n ∈N x k,n = 1 , ∀ k ∈ K , (6) X k ∈K x k,n ≤ M n , ∀ n ∈ N , (7) x k,n ∈ { , } , ∀ k ∈ K , ∀ n ∈ N , (8)where X = { x k,n } and ω k ≥ denotes a weight assigned to user k ’s offloading rate. λ ≥ is aweight factor for adjusting the sum offloading rate and JMH cost, which is determined by thesystem operator according to the system preference . The objective (5) is to optimize the tradeoffbetween the weighted sum of users’ offloading rates and the required JMH cost, which can beregraded as the utility by JMH. Constraint (6) states that each user is associated with only oneBS. Constraint (7) ensures that the number of users (or VMs) served by a BS does not exceedthe maximum number. Clearly, P n ∈N M n ≥ K should be satisfied for problem feasibility.Due to the binary variables X , Problem (P1) is an integer nonlinear programming problemthat is difficult to solve exactly. The brute force algorithm has a complexity of O ( N K ) , whichis prohibitive when the size of the cellular network is large. To this end, we design a low-complexity and suboptimal algorithm in the next section, which is shown to have close-to-optimalperformance in the simulations.III. A LGORITHM D EVELOPMENT
In this section, we develop an efficient algorithm to solve Problem (P1), which proceeds in twostages: First, by integer relaxation and some mathematic manipulation, we transform Problem(P1) into a sequence of convex problems that can be optimally solved. In the second stage, asthe obtained results may be fractional, we propose a new method to recover a feasible integersolution. Last, we extend the algorithm framework to the radio-resource allocation case.
A. Continuous Relaxation and Fractional Programming Transform
To make Problem (P1) more tractable, we first relax the binary variable x k,n into [0 , .Moreover, we introduce a new variable y n to replace the term P k ∈K x k,n in (4) and in (7), To set λ , we can first measure the statistical performance of sum offloading rate and the total JMH cost for different valuesof λ , which is obtained by running the proposed Algorithm 2 for given a set of λ ’s and channel realizations. Note that this stepcan be performed a priori. After that, we can choose a proper λ according to the system preference. which implies the computation load (or the number of VM) that BS n accommodates. Then, therelaxed Problem (P1) can be written as the following equivalent form: ( P1 ′ ) : max X , y X k ∈K X n ∈N x k,n ω k r k,n + (1+ d n ) yn − f k,n + X k ∈K X n ∈N x k,n z k,n (9) s . t . X n ∈N x k,n = 1 , ∀ k ∈ K , (10) X k ∈K x k,n ≤ y n , ∀ n ∈ N , (11) ≤ x k,n ≤ , ∀ k ∈ K , ∀ n ∈ N , (12) ≤ y n ≤ M n , ∀ n ∈ N , (13)where y = { y n } , and z k,n , Z − λ P j ∈N x k,j c k,j,n ≥ in (9) is an aggregated term related to thecost, in which Z is a sufficiently large constant for ensuring all the z k,n ’s being non-negative, e.g.,set Z ≥ max k,n { λ P j ∈N x k,j c k,j,n } . Adding a common Z to each term serves for rewriting theobjective function (5) as a form of sum of non-negative functions, so as to meet the requirementof the sum-of-ratios algorithm design. Evidently, since R k,n monotonically decreases with y n ,the auxiliary variable y n always achieves its lower bound P k ∈K x k,n in (11) for optimality, i.e.,the equality holds in (11). Due to the integer relaxation at constraints (12), Problem (P1 ′ ) yieldsthe upper bound of the original Problem (P1).Based on the above transformation, Problem (P1 ′ ) becomes a continuous optimization problemwith the sum-of-ratios objective. According to [29], we can transform Problem (P1 ′ ) into anequivalent parameterized subtractive-form problem via the following theorem. Theorem 1: If ( X ∗ , y ∗ ) is the optimal solution to Problem (P1 ′ ), then there exist α ∗ = { α ∗ k,n } , β ∗ = { β ∗ k,n } , and γ ∗ = { γ ∗ k,n } such that ( X ∗ , y ∗ ) is the optimal solution to the followingparameterized problem with ( α , β , γ ) = ( α ∗ , β ∗ , γ ∗ ) : ( P2 ) : max ( X , y ) ∈F X k ∈K X n ∈N α k,n h x k,n ω k − β k,n (cid:16) r k,n + (1+ d n ) yn − f k,n (cid:17)i + X k ∈K X n ∈N ( x k,n z k,n − γ k,n ) , (14)where F denotes the feasible solution set satisfying the constraints (10)-(13). Moreover, ( X ∗ , y ∗ ) also satisfies the following conditions when ( α , β , γ ) = ( α ∗ , β ∗ , γ ∗ ) , for all k and n : α k,n (cid:16) r k,n + (1+ d n ) yn − f k,n (cid:17) − , (15) β k,n (cid:16) r k,n + (1+ d n ) yn − f k,n (cid:17) − x k,n ω k = 0 , (16) γ k,n − x k,n z k,n = 0 . (17) Proof:
See Appendix A.Theorem 1 reveals that the sum-of-ratios maximization Problem (P1 ′ ) shares the same optimalsolution with the parameterized subtractive-form Problem (P2) when ( α , β , γ ) = ( α ∗ , β ∗ , γ ∗ ) .Here, ( α ∗ , β ∗ , γ ∗ ) denotes the optimal tuple of parameters that meets the system equations (15)-(17) together with its corresponding solution ( X , y ) to Problem (P2). Based on Theorem 1,we can solve Problem (P1 ′ ) by a two-layer iterative approach: In the inner layer, we solve thesubtractive-form Problem (P2) with given ( α , β , γ ) . Then, in the outer layer, we find the optimal ( α ∗ , β ∗ , γ ∗ ) satisfying (15)-(17). B. Solving Problem (P2) Given ( α , β , γ ) In this subsection, we solve Problem (P2) for given parameters ( α , β , γ ) (cid:23) , which can befurther re-expressed as max ( X , y ) ∈F X k ∈K X n ∈N x k,n ( α k,n ω k + z k,n ) − X n ∈N h(cid:16) X k ∈K α k,n β k,n f k,n (cid:17) (1 + d n ) y n − i (18)where the objective function (18) is derived from (14) by omitting the constant terms P k P n γ k,n and P k P n α k,n β k,n r k,n .It can be readily proved that Problem (18) is convex because the objective function is concaveand the constraints are linear. Then, the convex optimization methods can be used to solve thisproblem optimally. By introducing a set of Lagrangian multipliers µ = { µ n } (cid:23) associatedwith the constraints (11), the dual problem of Problem (18) can be expressed as min µ (cid:23) θ ( µ ) = X k ∈K φ k ( µ ) + X n ∈N ξ n ( µ n ) , (19)where φ k ( µ ) = max { x k,n } n ∈N X n ∈N x k,n ( α k,n ω k + z k,n − µ n )s . t . X n ∈N x k,n = 1 , ≤ x k,n ≤ , ∀ n ∈ N , (20) ξ n ( µ n ) = max ≤ y n ≤ M n µ n y n − (cid:16) X k ∈K α k,n β k,n f k,n (cid:17) (1 + d n ) y n − . (21) Since all the constraints in the convex Problem (18) are linear, the Slater’s condition is satisfiedand the strong duality holds [30]. The primal Problem (18) can therefore be equivalently solvedby the dual Problem (19).
1) Optimal JMH Policy in Dual Domain:
We can observe that the dual function θ ( µ ) has adecomposable structure. Specifically, given µ , θ ( µ ) can be determined by solving K independentsubproblems (20), where each user k individually makes its own JMH decision { x k,n } n ∈N overthe BSs, and at the same time N independent subproblems (21), where each BS n optimizes itsown computational load y n .To solve the JMH subproblem (20) for each user, we have the following observation: Remark 1 (JMH Revenue):
The value of ( α k,n ω k + z k,n − µ n ) in subproblem (20) can beinterpreted as the revenue of user k when its service is migrated to BS n . Specifically, with α k,n referred as to the offloading rate in each iteration [see (15)], ( α k,n ω k + z k,n ) represents theprofit obtained from BS n , consisting of the weighted offloading rate ω k α k,n and the modifiedcost z k,n . On the other hand, the Lagrangian multiplier µ n is the price of BS n to provideservice. Therefore, the difference between the profit and the payment, ( ω k α k,n + z k,n − µ n ) , canbe measured as the revenue of user k obtained from BS n .Based on Remark 1, the objective of subproblem (20) can be interpreted as maximizing therevenue of user k over all the BSs. Through a direct observation, each subproblem (20) alwayshas an optimal binary solution { x ∗ k,n } : x ∗ k,n = , if n = n ∗ k = argmax n ′ ∈N { α k,n ′ ω k + z k,n ′ − µ n ′ } , , otherwise , (22)i.e., each user selects one BS with the highest revenue. Note that when there are multiplemaximizers n ∗ k ’s, user can choose any one of them without affecting the value of dual function.For subproblem (21), the optimal amount of load y ∗ n at BS n can be obtained via differentiating ξ n ( µ n ) in (21) with respect to y n and letting the result be zero: y ∗ n = min n ln µ n − ln q n ln(1+ d n ) + 1 , M n o if µ n ≥ µ min n , q n d n , , if ≤ µ < µ min n , (23)where q n = ( P k ∈K α k,n β k,n f k,n ) ln(1 + d n ) . For the dual Problem (19), we use the subgradient method to find the optimal Lagrangianmultipliers µ ∗ , in which each µ n is updated as µ t +1 n = h µ tn − ǫ t (cid:16) y tn − X k ∈K x tk,n (cid:17)i + , (24)for t = 1 , , ... , where [ · ] + , max {· , } and ǫ t is the step size chosen in iteration t . In thispaper, we adopt a harmonic series step size ǫ t = ǫ/ ( t + 1) , t = 1 , , · · · . ǫ > is a properlydesigned constant. Since the primal problem is a convex problem satisfying Slater’s condition,the subgradient method in (24) operated with the above step-size rule guarantees the convergenceto an optimal dual solution µ ∗ and the primal optimal value [31].
2) Optimal Primal Solution Recovery for Problem (P2):
Although the optimal µ ∗ is obtainedby the above subgradient method, its associated solution ( X ( µ ∗ ) , y ( µ ∗ )) by (22) and (23) maynot be optimal and can even be infeasible for the primal Problem (P2). This is because the dualsubgradient method does not guarantee to find an optimal primal solution even for the convexproblem satisfying strong duality, unless the dual function θ ( µ ) is differentiable at µ ∗ [31], [32].In our Problem (P2), it arises from the fact that the dual subproblem (20) is a linear programming(LP) problem. When there exists a user that has more than one BS with the same maximumrevenue at µ ∗ , the binary-form solution in (22) is not a unique solution to the dual Problem (19)such that it may not be primal optimal to Problem (P2) (see [32, Proposition 7]). However, theoptimal solution to the inner-layer Problem (P2) with given ( α , β , γ ) is required for ensuringthe convergence of sum-of-ratios algorithm [29].To address this issue, we adopt the average procedure [32] to recover the primal solution.The idea behind is to reconstruct an approximate primal feasible solution by a weighted convexcombination of the previous primal iterates ( { x tk,n } , { y tn } ) obtained by (22) and (23), which isshown to converge an optimal primal solution. Theorem 2 (Primal Convergence):
Consider the primal-and-dual iteration scheme [(22), (23),and (24)] for Problem (P2) and that we recursively average the primal iterates ( { x tk,n } , { y tn } ) obtained by (22) and (23) as follows: ¯ x tk,n = (cid:18) − t ν P ts =1 s ν (cid:19) ¯ x t − k,n + t ν P ts =1 s ν x tk,n , (25) ¯ y tn = (cid:18) − t ν P ts =1 s ν (cid:19) ¯ y t − n + t ν P ts =1 s ν y tn , (26)for t = 1 , , ... , where ν > is a proper constant for controlling weights. Then, ¯ x tk,n → x ∗ k,n , ∀ k, n , and ¯ y tn → y ∗ n , ∀ n , i.e., converge to the optimal solution of Problem (P2). Algorithm 1
Optimal Algorithm for Solving Problem (P2)
Input: ( α , β , γ ) . Initialize { µ n ≥ } . repeat Compute { x k,n } and { y n } for given µ according to (22) and (23), respectively. Update µ based on subgradient method in (24). Update primal variables { ¯ x k,n } and { ¯ y n } according to (25) and (26), respectively. until µ converges. return x ∗ k,n = ¯ x k,n , ∀ k, n and y ∗ n = ¯ y n , ∀ n . Output: ( X ∗ , y ∗ ) for given ( α , β , γ ) . Proof:
See [32, Theorem 2].We summarize the detailed procedures of solving the inner-layer Problem (P2) in Algorithm 1.
C. Finding Optimal Parameters ( α ∗ , β ∗ , γ ∗ ) After obtaining the optimal ( X ∗ , y ∗ ) for given ( α , β , γ ) in above subsection, we develop analgorithm to find the optimal ( α ∗ , β ∗ , γ ∗ ) for solving Problem (P1 ′ ). For notational brevity, wedenote q k,n , r k,n + (1+ d n ) yn − f k,n and define some functions (for all k and n ) as follows: ψ k,n ( α k,n ) = α k,n q k,n − , (27) ψ k,n ( β k,n ) = β k,n q k,n − x k,n ω k , (28) ψ k,n ( γ k,n ) = γ k,n − x k,n z k,n , (29)where ( { x k,n } , { y n } ) is the inner-layer optimal solution obtained by Algorithm 1.According to [29, Theorem 3.1], the unique optimal solution of ( α ∗ , β ∗ , γ ∗ ) is achieved if andonly if ψ ik,n = 0 , ∀ k, n and ∀ i ∈ { , , } , as in conditions (15)-(17). We employ the modifiedNewton method [29] to update α k,n , β k,n and γ k,n to meet above conditions. Specifically, theparameters (for all k and n ) are point-wisely updated as α l +1 k,n = (cid:0) − ζ l (cid:1) α lk,n + ζ l q k,n , (30) β l +1 k,n = (cid:0) − ζ l (cid:1) β lk,n + ζ l x k,n ω k q k,n , (31) γ l +1 k,n = (cid:0) − ζ l (cid:1) γ lk,n + ζ l x k,n z k,n , (32) where l is the iteration index for the sum-of-ratios algorithm. ζ l is the step size at iteration l chosen via the following manner. Let m l be the smallest integer among m ∈ { , , ... } satisfying X k ∈K X n ∈N (cid:26)(cid:12)(cid:12)(cid:12) ψ k,n (cid:16) (1 − ρ m ) α lk,n + ρ m q k,n (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ k,n (cid:16) (1 − ρ m ) β lk,n + ρ m x k,n ω k q k,n (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ψ k,n (cid:16) (1 − ρ m ) γ lk,n + ρ m x k,n z k,n (cid:17)(cid:12)(cid:12)(cid:12) (cid:27) ≤ (1 − ερ m ) X k ∈K X n ∈N (cid:16) | ψ k,n ( α lk,n ) | + | ψ k,n ( β lk,n ) | + | ψ k,n ( γ lk,n ) | (cid:17) , (33)where ε ∈ (0 , and ρ ∈ (0 , . We set ζ l = ρ m l at the l -th iteration.As indicated in [29], the sum-of-ratios iterative algorithm can converge to the global optimumof Problem (P1 ′ ) if the inner-layer Problem (P2) for given ( α , β , γ ) is optimally solved andthe outer-layer update of ( α , β , γ ) is via the modified Newton method (30)-(32). Evidently, theglobal optimum of Problem (P2) can be guaranteed by Algorithm 1 due to its convexity. Thus,our proposed sum-of-ratios algorithm can achieve the global optimal solution to Problem (P1 ′ ). D. Integer Recovery for Problem (P1)
Let ( X ′ , y ′ ) denote the optimal solution to Problem (P1 ′ ). As explained in Section III-B, ( X ′ , y ′ )may be fractional due to the possibility that the binary-form solution in (22) is not the primaloptimal to Problem (P2). Therefore, in this subsection, we discuss the integer recovery on JMHdecisions X ′ to finalize solving Problem (P1). There are two major challenges in recovery forour problem instance. First of all, the recovery operation needs to guarantee the obtained resultstill meeting the hard constraints (6) and (8) of Problem (P1). Second, since the user’s offloadingrate R k,n is a function of the sum of users’ decisions P k x k,n [see in (4)], rounding x k,n withoutconsidering this correlation may accumulate a significant variance in P k x k,n , which in turnaffects { R k,n } greatly in the objective of Problem (P1) and incurs high performance loss. Inorder to recover a feasible decision solution with less rounding loss, we propose an effectiverounding method that captures the problem structure. The key idea is to utilize an importantproperty of Problem (P1) given any feasible integer y , which is described as follows. Theorem 3:
Define Y , { y ∈ Z N | P n ∈N y n = K, and ≤ y n ≤ M n , ∀ n ∈ N } as thefeasible integer set of y , where Z N denotes the integer set. For any given y ∈ Y , Problem (P1)is reduced into an integer linear programming (ILP) problem, expressed as ( P3 ) : max X X k ∈K X n ∈N x k,n u k,n ( y n ) (34) s . t . X n ∈N x k,n = 1 , ∀ k ∈ K , (35) X k ∈K x k,n = y n , ∀ n ∈ N , (36) x k,n ∈ { , } , ∀ k ∈ K , ∀ n ∈ N , (37)where u k,n ( y n ) , ω k rk,n + (1+ dn ) yn − fk,n − λ P j ∈N x k,j c k,j,n is pre-calculated for the given y . AndProblem (P3) is equivalent to the linear assignment problem (LAP). Proof:
See [33, Theorem 1].According to Theorem 3, we can map Problem (P1) with any given y ∈ Y into an equivalentLAP problem. It is well-known that the LAP problem is a special linear integer programmingproblem with a nice combinatorial property that its integer-relaxed problem always has an integeroptimal solution, i.e., LAP is equivalent to its continuous relaxation. Also, the famous Hungarianalgorithm [34] can provide an optimal solution to LAP in a polynomial complexity of O ( K ) .As a result, the optimal X to Problem (P1) can be efficiently obtained once y ∈ Y is determined.Next, we turn to construct an effective y ∈ Y by rounding the fractional-optimal y ′ . Notethat y ′ satisfies P n ∈N y ′ n = K because of the necessarily optimal condition P k ∈K x ′ k,n = y ′ n in(11), and ⌈ y ′ n ⌉ ≤ M n , ∀ n , since y ′ n ≤ M n by (13) and M n is integral, where ⌊·⌋ / ⌈·⌉ denotes thefloor/ceil operation. Let s , K − P n ∈N ⌊ y ′ n ⌋ and ˆ y , { ˆ y n } be the recovered integer solution. Weround y ′ to construct ˆ y by setting ˆ y n = ⌈ y ′ n ⌉ for s BSs with the maximal value of ( y ′ n − ⌊ y ′ n ⌋ ) and setting ˆ y n = ⌊ y ′ n ⌋ for the rest of BSs. Mathematically, the recovered ˆ y n is given by ˆ y n = ⌈ y ′ n ⌉ , if y ′ n − ⌊ y ′ n ⌋ is one of the s largest , ⌊ y ′ n ⌋ , otherwise . (38) Proposition 1: ˆ y constructed by rule (38) satisfies:a) ˆ y ∈ Y , i.e., ˆ y is an integer vector meeting P n ∈N y n = K and ≤ y n ≤ K , ∀ n ∈ N ;b) ˆ y ∈ arg min y ∈Y k y − y ′ k q , with q ≥ , i.e., ˆ y is one of the closest integer vectors in set Y to the fractional-optimal y ′ , for any norm q ≥ . Proof:
See Appendix B.Compared with the method that directly rounds x k,n and incurs a unstable rounding error on | ˆ y n − y ′ n | , the rounding rule (38) generates a feasible integer y ∈ Y with the smallest roundingerror k y − y ′ k q . Moreover, as the optimal X with given y ∈ Y can be optimally solved byHungarian algorithm, it can be perceived that our recovery method has lower performance lossthan that of the direct rounding. Algorithm 2
Whole Algorithm for Solving Problem (P1) Initialize ( α , β , γ ) (cid:23) . repeat Given ( α , β , γ ) , obtain the optimal solution ( X ′ , y ′ ) to Problem (P2) by Algorithm 1. Update ( α , β , γ ) using (30), (31), and (32). until P i =1 P k ∈K P n ∈N | ψ ik,n | < ǫ , where ǫ controls accuracy. Round y ′ → ˆ y by rule (38). Given ˆ y , obtain the optimal solution ˆ X by solving Problem (P3). Output: ˆ X .Based on the discussions above, we present the whole algorithm procedures of solving Problem(P1) in Algorithm 2. Its computational complexity is dominated by the sum-of-ratios algorithmin Steps 2-5 and solving the LAP problem at the rounding stage in Step 7. The sum-of-ratiosalgorithm is an iterative method that repeatedly solves the parameterized Problem (P2) byAlgorithm 1 and updates auxiliary parameters until convergence. The complexity of Algorithm 1is O ( N K/δ ) , where the complexity of computing ( X , y ) per iteration is O ( N K ) and thesubgradient method iterates O (1 /δ ) to converge, given a solution accuracy of δ > [30]. Thus,the total complexity of sum-of-ratios algorithm is O ( T N K/δ ) , where T is the number of sum-of-ratio iterations and is independent of the amount of variables and fractional functions [29].Solving the LAP Problem (P3) using Hungarian algorithm is of complexity O ( K ) . Therefore,Algorithm 2 has the total complexity of O ( T N K/δ + K ) . E. Extension: JMH with Radio Resource Allocation
In the previous sections, we consider the full frequency reuse scheme in multiuser’s offloadingto reduce the complexity of analysis. Allocating users with different sub-bands is howevernecessary in large systems to mitigate their mutual interference. To this end, in this subsectionwe consider radio resource allocation into the JMH design.Consider that the spectrum of BSs do not overlap each other and each BS allocates its time andfrequency radio resources, which is known as physical resource blocks (RBs), to the associatedusers in an orthogonal manner. We denote η k,n as the spectral efficiency in uplink transmissionbetween user k and BS n , and b k,n as the amount of RBs allocated by BS n to user k . Based on above assumptions, the achievable uplink transmission rate of user k to BS n is rewritten by r k,n = b k,n η k,n and the JMH Problem (P1) with radio resource allocation can be formulated as max X , B X k ∈K X n ∈N x k,n ω k b k,n η k,n + F k,n − λ X k ∈K X n ∈N X j ∈N x k,j x k,n c k,j,n (39) s . t . X k ∈K x k,n b k,n ≤ B n , ∀ n ∈ N , (40) X n ∈N x k,n = 1 , ∀ n ∈ K , (41) X k ∈K x k,n ≤ M n , ∀ n ∈ N , (42) x k,n ∈ { , } , b k,n ≥ , ∀ k ∈ K , ∀ n ∈ N , (43)where B = { b k,n } . Note that if b k,n = 0 , the offloading rate R k,n in (39) is equal to zero. (40) isthe radio resource capacity constraint on each BS, with B n denoting the total amount of RBs atBS n . It is easy to check that constraint (40) can be equivalently re-written as P k ∈K b k,n ≤ B n since x k,n = 0 in (39) would enforce b k,n = 0 .We can extend the proposed Algorithm 2 to solve Problem (39). Specifically, in the stage ofsum-of-ratios algorithm design, the integer-relaxed Problem (39) can be addressed by solvinga sequence of subtractive-form problems with given auxiliary parameters ( α , β , γ ) , which areconvex problems expressed as max ( X , y ) ∈F B ∈S R = X k ∈K X n ∈N α k,n (cid:18) x k,n ω k − β k,n (cid:16) b k,n η k,n + (1 + d n ) y n − f k,n (cid:17)(cid:19) + X k ∈K X n ∈N ( x k,n z k,n − γ k,n ) , = X k ∈K X n ∈N x k,n ( α k,n ω k + z k,n ) − X n ∈N h(cid:16) X k ∈K α k,n β k,n f k,n (cid:17) (1 + d n ) y n − i − X k ∈K X k ∈N α k,n β k,n b k,n η k,n − X k ∈K X k ∈N γ k,n (44)where S , { B | P k ∈K b k,n ≤ B n , b k,n ≥ } . y n , z k,n , and F , as specified in Section III-A,are respectively the auxiliary variable, the modified JMH cost, and the set of ( X , y ) satisfyingconstraints (10)-(13).It can be seen that for given ( α , β , γ ) , Problem (44) can be solved optimally by solving twoseparate problems. The first problem of optimizing ( X , y ) is identical to Problem (18) and can be solved by Algorithm 1. For the second problem of optimizing B , the optimal RB allocation b ∗ k,n can be easily obtained as b ∗ k,n = B n ( α k,n β k,n /η k,n ) / P k ∈K ( α k,n β k,n /η k,n ) / . (45)In outer layer of updating ( α , β , γ ) , we can use the same modified Newton method in (30)-(32) to find the optimal ( α ∗ , β ∗ , γ ∗ ) . Therefore, the sum-of-ratios algorithm design for solvingresource allocation included Problem (39) is almost the same as the original one except the extracomputation of b ∗ k,n in each iteration.In the stage of integer recovery of X as well as finding its corresponding optimal B ∗ , we firstapply the rounding rule (38) to recover an y ∈ Y . Then, given the recovered y , the residualProblem (39) is expressed as max X , B X k ∈K X n ∈N x k,n V k,n ( b k,n ) (46) s . t . X k ∈K x k,n b k,n ≤ B n , ∀ n ∈ N , (47) X n ∈N x k,n = y n , ∀ k ∈ K , (48) X k ∈K x k,n = 1 ∀ n ∈ N , (49) x k,n ∈ { , } , b k,n ≥ , ∀ k ∈ K , ∀ n ∈ N , (50)where V k,n ( b k,n ) , ω k bk,nηk,n + (1+ dn ) yn − fk,n − λ P j ∈N x k,j c k,j,n is a concave function of b k,n . Comparedwith Problem (P3), Problem (47) is coupled with RB allocation and more challenging to solve.Fortunately, we can leverage the analysis of Problem (P3) and the Lagrangian relaxation methodto offer an effective solution for Problem (47). Specifically, let ν = { ν n } (cid:23) be the Lagrangianmultipliers associated with constraint (47). For given ν , we consider the relaxed problem Z ( ν ) , max X , B X k ∈K X n ∈N x k,n [ V k,n ( b k,n ) − ν n b k,n ] + X n ∈N ν n B n , s . t . (48) − (50) . (51)The optimal b ∗ k,n ( ν n ) in Problem (51) can be determined by b ∗ k,n ( ν n ) , argmax b k,n ≥ { V k,n ( b k,n ) − ν n b k,n } = f k,n (1 + d n ) y n − (cid:20)r ω k ν n − η k,n (cid:21) + . (52)Let U k,n ( ν n ) , V k,n ( b ∗ k,n ( ν n )) − ν n b ∗ k,n ( ν n ) . By plugging b ∗ k,n ( ν n ) into (51), we have max X X k ∈K X n ∈N x k,n U k,n ( ν n ) + X n ∈N ν n B n , (53) VMs
VMs
Initial association
HandoverVM migration
Macro BS
Fig. 2: A hotspot mitigation scenario, where an overloaded macro-BS migrates some users’ services to small-BSsby JMH. s . t . (48) − (49) , x k,n ∈ { , } , ∀ k ∈ K , ∀ n ∈ N , which is a LAP problem like Problem (P3) (see Theorem 3) and similarly can be solved byHungarian algorithm. The optimal Lagrangian multiplier ν ∗ to the dual problem min ν (cid:23) Z ( ν ) can be found using the subgradient method. Note that due to the non-convexity of Problem (46),the optimal ( X ∗ , B ∗ ) obtained in dual domain may not be the primal optimum, meaning that theduality gap exists. However, the proposed dual-based algorithm is of low complexity and yieldsto a good solution to the primal Problem (46) in some sense.The complexity of the modified Algorithm 2 for solving Problem (39) is O ( T N K/δ +( N K + K ) /δ ) , where O ( N K + K ) /δ ) is the complexity of the lagrangian relaxationmethod, including O ( N K ) and O ( K ) for determining B and X in each iteration and O ( δ ) for subgradient method convergence.IV. H OTSPOT M ITIGATION C ASE
In this section, we consider the JMH design for a hotspot mitigation scenario as depicted inFig 2, where a macro-BS distributes its load among N idle small-BSs in a small cell. Specifically,the macro-BS, denoted by BS , initially hosts all the K users’ services and attempts to migratesome of them to N idle small-BSs for alleviating its load. Let N + = N ∪{ } denote the set of allthe BSs. To facilitate exposition, we assume the users associated with the same BS n ∈ N + havethe average transmission rates and computation rates, i.e., r k,n = r n and f k,n = f n , ∀ k ∈ K .Also, the JMH cost from BS to BS n ∈ N is assumed to be identical for each user, i.e., c k, ,n = c n , ∀ k ∈ K . Under the assumptions, Problem (P1) reduces to a problem of determiningthe number of services (or users) allocated to each BS n ∈ N + , which can be formulated as ( P4 ) : max y ∈ Z N +1 R = X n ∈N " y n r n + (1+ d n ) yn − f n − λy n c n (54) s . t . X n ∈N + y n = K, (55) ≤ y n ≤ M n , ∀ n ∈ N + , (56)where y = ( y , · · · , y N ) and c = 0 , i.e., no cost incurs if a service is hosted by the macro-BS.Like Problem (P1), Problem (P4) is also an integer nonlinear programming problem, whichin general has no efficient method to solve it optimally. Nevertheless, we show in the followingthat Problem (P4) can be optimally solved, provided that the total number of users K ≤ K ∗ ,where K ∗ is the optimal number of the total users that yields the maximum network utility R .For the other case that K > K ∗ , the proposed algorithm in the preceding section can be adoptedto find a suboptimal solution in an efficient manner. A. Optimal Load Distribution for K ≤ K ∗ In this subsection, we develop an optimal relaxation-and-rounding based algorithm to solveProblem (P4), conditioned on K ≤ K ∗ . The key idea is to verify that the integer-relaxed Problem(P4) is a convex problem given K ≤ K ∗ and design the optimal rounding method in the sequel.We first relax the integer y into real numbers and solve the relaxed Problem (P4). To proceed,we define the one-sided optimal load of BS n as J n = argmax ≤ y n ≤ M n ( y n r n + (1+ d n ) yn − f n − λy n c n ) . (57)Clearly, J n is the amount of load achieving the maximum utility at BS n . By taking the firstderivative with respect to y n , we derive a general solution of J n as J n = min { J ′ n , M n } , if λc n ≤ .h r n + f n (1+ d n ) i , , otherwise , (58)where J ′ n ≥ is the root of the following equation: r n + (1 + d n ) y n − f n [1 − y n ln(1 + d n )] = λc n (cid:20) r n + (1 + d n ) y n − f n (cid:21) . (59)It can be checked that the LHS of (59) is monotonically decreasing while the RHS is monoton-ically increasing over y n ≥ . Thus, J ′ n can be obtained by the simple bisection search.Now, we make a key observation of the relaxed Problem (P4) under different values of K : Proposition 2:
Define K ∗ , P n ∈N + J n and R ( K ) as the optimal objective value of therelaxed Problem (P4) in terms of K . The following properties hold: • Property 1: If K = K ∗ , the optimal load distribution is y ∗ n = J n , ∀ n ∈ N + . • Property 2: If
K < K ∗ , y ∗ n ≤ J n and R is strictly concave in ≤ y n ≤ J n , ∀ n ∈ N + . • Property 3: If
K > K ∗ , y ∗ n ≥ J n , ∀ n ∈ N + . • Property 4: R ( K ) monotonically increases in ≤ K ≤ K ∗ and R ( K ∗ ) > R ( K ) , ∀ K > K ∗ . Proof:
See Appendix B.Proposition 2 reveals that K ∗ is the optimal number of users that the network can accommodateto achieve the maximum network utility. Seen from Property , when K < K ∗ , provisioningmore users’ services into the macro-BS can help increase the network utility, mainly becausethe resources on each BS are under-utilized (i.e., y ∗ n ≤ J n by Property 2) after the optimal JMH.On the contrary, when K > K ∗ , there are too many users hosted by the macro-BS such that eachBS is overloaded (i.e., y ∗ n ≥ J n by Property 3) even after the optimal JMH. In this case, moresmall-BSs are needed to increase the network capacity and address the overloaded condition.Using Property 2 in Proposition 2, for an under-utilized system (i.e., K ≤ K ∗ ), we can safelyimpose constraints y n ≤ J n , ∀ n , into the relaxed Problem (P4) without loss of optimality: max y ∈ R N +1 R = X n ∈N + " y n r n + (1+ d n ) yn − f n − λy n c n (60) s . t . X n ∈N + y n = K, ≤ y n ≤ J n , ∀ n ∈ N + . With the objective function R being concave over the feasible region of y , Problem (60) is aconvex problem and can be readily solved and the details are omitted here due to space limitation.After solving Problem (60), we next propose a rounding method to recover the optimal integersolution to Problem (P4). Proposition 3:
Any solution y ∗ = ( y ∗ , · · · , y ∗ N ) ∈ Z N +1+ to Problem (P4) satisfies y ∗ n ∈ {⌊ y ′ n ⌋ , ⌈ y ′ n ⌉} , (61)where y ′ = ( y ′ , · · · , y ′ N ) ∈ R N +1+ denotes the unique solution of Problem (60). Proof:
See Appendix C.Thanks to Proposition 3, we can dramatically reduce the range of numerical searching theoptimal integer y ∗ n . Moreover, since the recovered y n has to satisfy the sum constraint (55), wecan further derive the optimal rounding rule as follows. Proposition 4 (Optimal Rounding Rule):
Let R n ( y n ) , y n rn + (1+ dn ) yn − fn − λy n c n , ∀ n , and s , K − P n ∈N + ⌊ y ′ n ⌋ . The optimal y ∗ that solves Problem (P4) is given by y ∗ n = ⌈ y ′ n ⌉ , if R n ( ⌈ y ′ n ⌉ ) − R n ( ⌊ y ′ n ⌋ ) is one of the s largest , ⌊ y ′ n ⌋ , otherwise . (62) Proof:
Based on Proposition 3 and constraint (55), the optimal integer y ∗ should meet thecombinatorial condition that s of the y ∗ n ’s satisfy y ∗ n = ⌈ y ′ n ⌉ and the rest N + 1 − s satisfy y ∗ n = ⌊ y ′ n ⌋ . Among all the possible combinations, (62) is the one with the maximum R and thusthe optimal solution to Problem (P4), which completes the proof. B. Modified Algorithm for
K > K ∗ It is worth noting that we can simplify the proposed Algorithm 2 to suboptimally solveProblem (P4) for any K . Recall that Algorithm 2 consists of the optimal sum-of-ratios algorithmfor solving the relaxed problem and the suboptimal rounding method for recovering a feasibleinteger solution. For Problem (P4), the same rounding rule (38) can be applied in the integerrecovery while the sum-of-ratios algorithm design can be simplified as follows.We start by transforming the relaxed Problem (P4) into a standard sum-of-ratios problem: max y ∈G X n ∈N + " y n r n + (1+ d n ) yn − f n + y n z n (63)where G , { y ∈ R N +1 | P n ∈N + y n = K, ≤ y n ≤ M n , ∀ n } and z n = max n { λc n } − λc n ≥ , ∀ n , is the modified cost coefficient for reshaping the objective function into a form of sum ofnon-negative functions.The sum-of-ratios Problem (63) can be solved by two-layer optimization. The inner layer isto find the optimal solution to the subtractive-form problem with given auxiliary parameters ( α n , β n , γ n ) , which is a convex problem expressed as max y ∈F X n ∈N + α n h y n − β n (cid:16) r n + (1+ d n ) yn − f n (cid:17)i + X n ∈N + [ y n z n − γ n ] . (64)The subtractive-form Problem (64) can be easily solved through the following proposition. Proposition 5:
The optimal load distribution that solves Problem (64) is y ∗ n = (cid:20) (cid:18) ( α n + z n − ν ) f n α n β n ln(1 + d n ) (cid:19)(cid:21) M n , ∀ n ∈ N + , (65)where [ · ] ba = max { a, min {· , b }} . ν satisfying P n ∈N + y ∗ n = K can be obtained by bisection search. TABLE I: System Parameters
Parameter ValueNumber of BSs, N Number of users, K System bandwidth, B MHzPath loss from user to BS . . l [ km ] dBUser transmit power, p k . WExpected computation rate, f k,n [0 . × , × ] bits/secDegradation factor, d n . Weight of user k ’s offloading rate, ω k Weight of JMH cost, λ , . Number of Monte Carlo simulations
Maximum number of VM, M n TABLE II: Sum Utility [ × ] v.s. Number of Users Number of Users 6 8 10 60Upper Bound 0.814994 1.110704 1.306949 6.408008Optimal via Exhaustive Search 0.814991 1.110697 1.306941 –Proposed 0.814991 1.110697 1.306941 6.407563
In the outer layer, we use the modified Newton method to find the optimal ( α ∗ n , β ∗ n , γ ∗ n ) likeAlgorithm 2. Thus, we omit the detailed description of the sum-of-ratios algorithm for Problem(P4) when K > K ∗ . V. S IMULATION R ESULTS
In this section, we perform simulation to evaluate the performance of our proposed algorithms.We consider N = 7 BSs deployed in a square area of km with a regular hexagonal-lattice layout(see [35, Fig. 2]). All the users are randomly distributed within the area at the beginning and theirBS associations are initialized using the conventional max-SINR association scheme. We adoptthe Random Waypoint Mobility model [36] to generate the new user’s locations for the consideredtime slot, with the parameters taken as: the static probability and pause time p s = t p = 0 , and theuser velocities chosen uniformly at random within the interval [ v min , v max ] = [0 , m/s. We set theJMH cost c k,j,n = W + W k if j = n , and c k,j,n = 0 otherwise, where W = 10 is the handovercost while W k denotes the VM migration cost, which is chosen from the set { , , } × according to user k ’s subscribed service. Unless mentioned otherwise, the main communicationand computation parameters used in the simulations are summarized in Table I.
30 40 50 60 70 80 90 100 110 120
Number of users (K) S u m u t ili t y × Upper BoundProposedMTRANo Migration × Fig. 3: Sum utility vs. K . Degradation factor (d) S u m u t ili t y × ProposedMTRANo Migration
Fig. 4: Sum utility vs. d . A. JMH in a General Multi-cell MEC System
In Table II, we evaluate the sum utility of the proposed Algorithm 2 in comparison withthe globally optimal solution by exhaustive search, and the upper-bound result, referring to theoptimal solution of Problem (P1 ′ ). Note that we only provide the performance of the exhaustivesearch in a small network size due to its exponential complexity. It can be observed that, theperformance gap between the upper bound and the exhaustive search does exist. Meanwhile,we can see that the proposed algorithm achieves the optimal performance, indicating that theproposed rounding method in Algorithm can efficiently recover the optimal integer solutionsfrom the fractional results of the relaxation stage.Next, we introduce two benchmark schemes for performance comparison: 1) No migration:
All the users continue the associations with their original BSs; 2)
Radio-oriented migration:
Eachuser connects to the BS with the highest value of r k,n − λc k,n , which represents the traditionalBS handover without considering the system dynamics on the computation side.In Fig. 3, we compare the sum utility performance of different algorithms versus the numberof users K . First, we can observe that the performance of the proposed algorithm approaches tothe upper bound, indicating its close-to-optimal performance. The proposed algorithm and radio-oriented migration scheme have large utility gain against the no-migration scheme, since thesetwo schemes jointly manage computation-and-radio resources according to the system dynamics.The radio-oriented migration performs well when ≤ K ≤ ; however it begins to degradewhen K ≥ . This is because when K is small, each BS is lightly loaded and wireless channelcondition dominates system performance. When K becomes large, the load of each BS becomesvaried, leading to notable computation-rate variations among BSs caused by I/O interference. User's maximum velocity ( v max ) S u m u t ili t y × ProposedMTRANo Migration (a)
User's maximum velocity ( v max ) P e r c en t age o f m i g r a t ed u s e r s ( % ) ProposedMTRANo Migration (b)
Fig. 5: (a) Sum utility vs. user’s maximum velocity v max . (b) Percentage of migrated users vs. v max . In this case, the radio-oriented migration scheme without considering the computation rate ofBSs will suffer severe performance degradation. In contrast, our proposed JMH framework canefficiently mitigate the I/O interference and thus further improve system performance especiallywhen K is large. For instance, when K = 90 , the proposed algorithm obtains about utilityimprovement over the radio-oriented migration scheme.In Fig. 4, we evaluate the impact of degradation factor on the sum utility performance,where the factor of each BS is set to be identical, i.e., d n = d , ∀ n . As expected, the proposedalgorithm has the slowest descending rate among all the algorithms, showing that our proposedalgorithm has the best performance resistance against the I/O interference. We also observe thatthe performance of radio-oriented migration is close to that of the proposed algorithm when d issmall, however, it dramatically decreases when d increases. This is aligned with the discussionin Fig. 3 that the radio-oriented migration performs well when the channel condition is dominantwhile it has poor performance when the I/O interference becomes a key factor.Fig. 5(a) shows the impact of user’s mobility on the sum utility, where v max denotes theuser’s maximum velocity, with a larger v max indicating more dramatic location changes and inturn higher channel variations. As expected, the performance of no-migration scheme drasticallydecreases as v max increases due to the channel deterioration of the initial BSs. In contrast, theproposed algorithm and radio-oriented migration scheme can efficiently resist the impact of v max , thanks to their flexible user-BS association. On the other hand, when v max = 0 , i.e., user’slocation remains static, there are still performance gains achieved by the proposed algorithm andradio-oriented migration compared with no migration. This is because besides user’s movement,wireless fading is time-varying, which affects channel condition and thus the JMH policies. -1.5 -1 -0.5 0 0.5 1 1.5 log λ S u m u s e r ' s o ff l oad i ng r a t e × T o t a l J M H c o s t × No migration
Rate v.s. λ Cost v.s. λ Fig. 6: Sum offloading rate/total JMH cost vs. λ .
10 20 30 40 50 60 70 80
Number of users (K) S u m u t ili t y × ProposedNo migration
Fig. 7: Sum utility vs. K in radio-resource allocation case. Fig. 5(b) shows the percentage of migrated users among the total number of users versus v max .As we observe, the percentage of migrated users increases with v max in both proposed algorithmand radio-oriented migration scheme, which fits our intuition that the user’s migration demandgrows as the level of mobility increases. Compared with the radio-oriented migration scheme,the proposed algorithm has a lower migration percentage and slower ascending rate against themobility level; combining with the sum utility behaviors of these two schemes shown in Fig.5(a), these demonstrate that our proposed algorithm can reduce unnecessary migrations and makemore accurate migration decisions to improve the sum utility.Fig. 6 shows the impacts of λ on the sum user’s offloading rate and total JMH cost ofthe proposed algorithm. It can be observed that when the price of JMH cost λ is small, theproposed algorithm triggers more migrations to improve sum user’s offloading rate at the cost ofhigh JMH cost consumption. However, as λ increases, the price of doing migration operationsincreases and our proposed algorithm avoids more worthless migrations (i.e., those with littleoffloading-rate improvement but at high JMH cost). Therefore, there exists a tradeoff between thesum offloading rate improvement and the JMH cost consumption. We also observe that setting λ ∈ [0 . , can achieve over offloading-rate improvement compared to the no-migrationscheme while maintaining the JMH cost less than half of the maximum JMH cost consumption,which is a desirable interval to balance the performance of these two metrics.Fig. 7 shows the sum utility versus the number of users K under the radio-resource allocationscenario, where the bandwidth of each BS is set as B n = B/N and the user-BS associations areinitialized by choosing the BS with the highest value of spectral efficiency η k,n . We can observethat the proposed algorithm has a much larger and more stable performance than the no-migration scheme, thanks to its high spectrum efficiency achieved by radio-resource allocation among users.The performance of the proposed algorithm increases with K when K is small while decreasesslowly when K ≥ . This is because when K is small, increasing the number of user at eachBS can help leverage the VM-multiplexing gain to further improve the system performance.However, when K is large, the I/O interference becomes the dominant issue of degrading thesystem performance. In this case, our algorithm can efficiently mitigate the interference so thatthe system performance decreases at a slower rate than that of the no-migration scheme. B. Hotspot Mitigation Case
In this subsection, we turn our attention to the special case of hotspot-mitigation scenario.We consider a macro BS, denoted by BS , with the assistance of N = 3 BSs. For BS , weset [ r , f , d ] = [5 Mbps , × bit/s , . . For each BS n = 1 , , , we consider a homogenoussetting of [ r n , f n , c n , d n ] = [2 Mbps , × bit/s , × , . for the ease of graphic illustration.Fig. 8(a) shows the utility performance of our proposed algorithm versus K , where theproposed algorithm includes the relaxation-and-rounding based algorithm to resolve the case K ≤ K ∗ and the modified Algorithm 2 towards the case K > K ∗ . For comparison, we alsopresent the optimal performance obtained by exhaustive search and the performance of no-migration scheme mentioned in the preceding section. As can be seen in Fig. 8(a), the proposedalgorithm can achieve the optimal performance for all K , which verifies its optimality behaviorwhen K ≤ K ∗ and the effectiveness of finding the optimal solution when K > K ∗ . We alsoobserve that, for both proposed algorithm and no-migration scheme, the utility monotonicallyincreases with K when K is small and it begins to decrease when K exceeds some thresholdsdue to the computation-rate degradation caused by I/O interference. Nevertheless, compared withthe no-migration scheme, our proposed algorithm not only greatly prolongs the utility growthuntil K > K ∗ but also keeps the utility reduction in a much slower rate afterwards.To illustrate the mechanism behind the optimal JMH scheme, we further analyze the loaddistribution among BSs shown in Fig. 8(b), along with the results shown in Fig. 8(a). Specifically,by varying K from to , the optimal utility goes through the following four stages:1) Stage I ( ≤ K ≤ ): The utility increases with K and no JMH occurs since BS is stillunder-utilized. The representative load distribution is K = 12 in Fig. 8(b).2) Stage II ( < K ≤ ): Unlike no-migration scheme, the utility of the optimal JMH schemekeeps increasing in this stage thanks to migrating the load to the helper BSs. The load
10 20 30 40 50 60 70
Total number of users (K) S u m u t ili t y × Exhaustive SearchProposedNo Migration
Begin JMH K* Begin load imbalance (a)
12 24 32 44 48
Total number of users (K) y n BS 0BS 1BS 2BS 3
12 24 32 44 48
Total number of users (K) y n BS 0BS 1BS 2BS 3 J J ∼ J (b) Fig. 8: (a) Sum utility vs. K . (b) Load distributions among BSs sampled from Line “Proposed” in Fig. 8(a). of each BS (i.e., y n ) gradually increases as K grows up and all of them are below theirone-side optimal load levels [i.e., J n defined in (57)]. When K = 32 , the utility achievesits maximum by the scheme that y n ≈ J n , ∀ n, shown in K = 32 . All the observationsin this stage verify the results in Proposition 2 that when K ≤ K ∗ , R ( K ) monotonicallyincreases and y ∗ n ≤ J n , ∀ n ; when K = K ∗ , R ( K ) achieves its optimum by y ∗ n = J n , ∀ n .3) Stage III ( < K ≤ ): The utility begins to decrease. Nevertheless, as shown in K = 44 ,the corresponding optimal JMH scheme is implemented in a load-balance manner, whereeach BS is lightly overloaded (i.e., slightly above J n ) to share the total load, withoutsacrificing the performance of any one of the BSs.4) Stage IV (
K > ): The utility decreases slowly in an approximately linear rate. However,contrary to Stage III, it is realized by load imbalance that allocates all the unwanted loadinto one of the BSs while maintaining the others at their optimal load levels.To summarize, in the considered hotspot-mitigation scenario, our proposed algorithm canachieve higher utility than no-migration scheme. It performs well especially when the totalnumber of users K is in Stage II or Stage III, which is conducted in efficient resource utilizationand load balance among BSs. However, when K is in Stage IV, the system still remains loadimbalance after the optimal JMH, implying that there are too many services accommodated atthe system and the number of helper BSs is not enough. In this case, adding more BSs is neededto address the overloaded issue. VI. C ONCLUSIONS
In this paper, we studied the JMH optimization problem in a multi-user multi-cell MEC system,where the I/O interference is considered. We proposed a novel efficient algorithm to solve the combinatorial problem, which achieves the close-to-optimal performance. In addition, we alsoconsidered the JMH design for a special hotspot-mitigation scenario. We obtained the followinguseful insights for practical multi-user multi-cell/server MEC design: First, communication aspectdominates the system performance when the number of users is small, and computation is thekey factor when the number of users is large due to the I/O interference. Second, there existsa threshold on the number of users, such that load balance among BSs can be captured withinthe threshold while load imbalance happens beyond the threshold.A PPENDIX
A. Proof of Theorem 1
By introducing the auxiliary variables β = { β k,n } and γ = { γ k,n } , Problem (P1 ′ ) can beequivalently transformed as max ( X , y ) ∈F β , γ X k ∈K X n ∈N β k,n + X k ∈K X n ∈N γ k,n (66) s . t . x k,n ω k ≥ β k,n (cid:16) r k,n + (1+ d n ) yn − f k,n (cid:17) , ∀ k, n (67) x k,n z k,n ≥ γ k,n , ∀ k, n (68)where F denotes the set of ( X , y ) satisfying the convex constraints (10)-(12). Clearly, the optimal ( X ∗ , y ∗ , β ∗ , γ ∗ ) to Problem (66) satisfies the following conditions: β ∗ k,n = x ∗ k,n ω k r k,n + (1+ d n ) y ∗ n − f k,n , γ ∗ k,n = x ∗ k,n z k,n , ∀ k, n. (69)Let α = { α k,n } and ν = { ν k,n } be the Lagrangian multipliers associated with constraints (67)and (68), respectively. If ( X ∗ , y ∗ , β ∗ , γ ∗ ) is the optimal solution to Problem (66), there mustexist α ∗ and ν ∗ such that α ∗ k,n = 1 r k,n + (1+ d n ) y ∗ n − f k,n , ν ∗ k,n = 1 , ∀ k, n. (70)(70) is derived from Fritz-John optimality condition. Its detailed derivation can be found in theproof of Lemma 2.1 in [29]. We see that the conditions of α ∗ k,n , β ∗ k,n and γ ∗ k,n in (69) and (70)are correspondingly equivalent to (15)-(17).On the other hand, it can be checked that the optimal ( X ∗ , y ∗ ) of Problem (66) satisfies theKarush-Kuhn-Tucker (KKT) conditions for Problem (14) for given ( α , β , γ ) = ( α ∗ , β ∗ , γ ∗ ) .Here ν is omitted in Problem (14) since ν ∗ k,n = 1 . As Problem (14) is convex programming for α ≻ and β , γ (cid:23) , the KKT condition is also sufficient optimality condition. Thus, ( X ∗ , y ∗ ) is also the solution of Problem (14) for ( α , β , γ ) = ( α ∗ , β ∗ , γ ∗ ) , which completes the proof. B. Proof of Proposition 1
Proof of Part a): To show ˆ y ∈ Y is equivalent to verifying that y n ≤ M n , ∀ n ∈ N and the sumconstraint P n ∈N y n = K are met by ˆ y simultaneously. The first condition is obvious becausewith y ′ n ≤ M n and a non-negative integer M n , ˆ y n ≤ ⌈ y ′ n ⌉ ≤ M n holds. For the second one, firstit is easy to prove that s is is an integer among [0 , N − , and for a given s ∈ [0 , N − , ˆ y by rule (38) can meet the sum constraint as long as there exist at least s BSs with a fractional y ′ n for taking the ceil operation. Note that a fractional y ′ n naturally implies ⌈ y ′ n ⌉ ≤ M n is met.Consider the non-trivial case s ≥ . Since y ′ n − ⌊ y ′ n ⌋ < , y ′ satisfying the sum constraint mustcontain at least s + 1 fractional entries to generate s ≥ , i.e., s + 1 fractional y ′ n ’s exist when s ≥ and thus the second condition holds. Summarizing above conditions yields the result.Proof of Part b): We first prove that ˆ y by (38) is an optimal solution to the following problem: min y ∈ Z N k y − y ′ k q , s . t . X n ∈N y n = K, (71)for a given positive integer K and non-negative real numbers y ′ with P n ∈N y ′ n = K , and forevery ≤ q ≤ ∞ norm. Let y ∗ = { y ∗ n } be an optimal solution to Problem (71). The proof isprovided as follows:1) Show y ∗ n ∈ {⌊ y ′ n ⌋ , ⌈ y ′ n ⌉} , ∀ n ∈ N : This can be proved by contradiction. Suppose that y ∗ contains y ∗ i < ⌊ y ′ i ⌋ for some i ∈ N and y ∗ j > ⌈ y ′ j ⌉ for some j ∈ N with j = i . Then, werepeatedly construct a better y = { y n } by letting y n = y ∗ n for n / ∈ { i, j } , y i = y ∗ i + 1 ≤ ⌊ y ′ i ⌋ , y j = y ∗ j − ≥ ⌈ y ′ j ⌉ , (72)such that P y n = P y ∗ n = K , and k y − y ′ k qq = X n ( | y ∗ n − y ′ n | ) q + X i ( | y ∗ i − y ′ i | − q + X j (cid:0) | y ∗ j − y ′ j | − (cid:1) q < X n ( | y ∗ n − y ′ n | ) q + X i ( | y ∗ i − y ′ i | ) q + X j (cid:0) | y ∗ j − y ′ j | (cid:1) q = k y ∗ − y ′ k qq , (73)until either set { i } or set { j } is empty. This contradicts to the optimality assumption andthus the above case does not hold. Suppose that y ∗ contains y ∗ i < ⌊ y ′ i ⌋ for some i ∈ N while y ∗ n ∈ {⌊ y ′ n ⌋ , ⌈ y ′ n ⌉} is satisfied for n = i . For this case, we also can construct a y by letting y i = y ∗ i + 1 and y k = y ∗ k − , with y ∗ k = ⌈ y ′ k ⌉ 6 = y ′ k , such that P y n = K and k y − y ′ k qq < k y ∗ − y ′ k qq ( by |⌈ y ′ k ⌉ − y ′ k | −|⌊ y ′ k ⌋ − y ′ k | < ), until set { i } is empty. Note that y k always exists if set { i } 6 = ∅ due to P n ∈N y ∗ n = K . Thus, the above case does not hold for optimality. The opposite case y ∗ contains y ∗ j > ⌈ y ′ j ⌉ for some j is also not optimal,proved by the similar approach. Summarizing the above discussion yields the result.2) Show ˆ y is the optimal among set H , { y ∈ Z N | y n ∈ {⌊ y ′ n ⌋ , ⌈ y ′ n ⌉} , P n ∈N y n = K } : As y n ∈ {⌊ y ′ n ⌋ , ⌈ y ′ n ⌉} for all n , y ∈ Z N must contain s entries to be y n = ⌈ y ′ n ⌉ and the restto be y n = ⌊ y ′ n ⌋ to preserve P n ∈N y n = K . Evidently, ˆ y by rule (38) can obtain smallestvalue of k y − y ′ k qq among all y ∈ H since rounding upwards s entries with the largest y ′ n − ⌊ y ′ n ⌋ achieves the maximum rounding-error reduction from recovering y n = ⌊ y ′ n ⌋ , ∀ n to y n ∈ {⌊ y ′ n ⌋ , ⌈ y ′ n ⌉} and P n ∈N y n = K .Since ˆ y is an optimal solution to Problem (71) and satisfies ˆ y n ≤ M n for all n according tothe result of Part i), ˆ y is also an optimal solution to Problem (71) with the additional constraints y n ≤ M n , ∀ n ∈ N , i.e., ˆ y ∈ arg min y ∈Y k y − y ′ k q , ending the proof. C. Proof of Proposition 2
To prove this proposition, we need the following property of R , which can be easily verifiedby taking the first derivative of R with respect to y n . Lemma 1:
The objective value R is monotonically increasing with y n in [0 , J ′ n ] and mono-tonically decreasing when y n > J ′ n , for all n ∈ N + . Property 1:
It is due to the facts that i) y n = J n , ∀ n ∈ N + can achieve the maximal servicemigration utility at each BS and thus maximize the sum utility R ; and ii) P n ∈N + J n = K ∗ meets the constraint (55). Property 2:
Given
K < K ∗ , y ∗ n ≤ J n , ∀ n ∈ N + can be verified by contradiction as follows.Since K < K ∗ , it follows that P n ∈N + y ∗ n < P n ∈N + J n and there always exists an non-emptysubset e N ⊆ N + such that y ∗ n < J n , ∀ n ∈ e N . Suppose that there exist y ∗ i > J i for some i ∈ N + .By Lemma 1, we can always find a larger R by decreasing y ∗ i and meanwhile increasing some y ∗ n ’s with n ∈ e N to rein in the constraint (55), which contradicts the definition of the optimal y ∗ n . Thus, the optimal workload distribution satisfies y ∗ n ≤ J n , ∀ n ∈ N + .We first prove the concavity of R in ≤ y n ≤ J ′ n , ∀ n ∈ N + . Take the second derivative of R with respect to y n : d Rdy n = − (cid:16) r n + (1+ d n ) yn − f n [1 − y n ln(1+ d n )] (cid:17) (1+ d n ) y n − ln(1+ d n ) f n h r n + (1+ d n ) yn − f n i | {z } A + − y n (1+ d n ) y n − ln (1+ d n ) f n h r n + (1+ d n ) yn − f n i | {z } B . (74)As B ≤ and the denominator of A is positive for y n ≥ , to prove d Rdy n ≤ in ≤ y n ≤ J ′ n ,it is sufficient to show the term g ( y n ) , r n + (1+ d n ) yn − f n [1 − y n ln(1+ d n )] in A is positive, ∀ y n ∈ [0 , J ′ n ] . Consider the non-trivial case J ′ n > . According to the definition of J ′ n in (59),we have g ( J ′ n ) > . Also, it it easily proved that g ( y n ) monotonically decreases with y n ≥ .Thus, g ( y n ) ≥ g ( J ′ n ) > , ∀ y n ∈ [0 , J ′ n ] and the strong concavity of R holds in ≤ y n ≤ J ′ n .Note that J n ≤ [ J ′ n ] + by (58). Hence the strong concavity of R is also valid in ≤ y n ≤ J n . Property 3:
Similar to the proof of Property 2 and thus omitted.
Property 4:
Let K , K ∈ [0 , K ∗ ] and K < K . Define { y (1) n } n ∈N + as the optimal solutionto the integer-relaxed Problem (P4) for given K . Since K = P n ∈N + y (1) n < K ≤ P n ∈N + J n and y (1) n ≤ J n , ∀ n ∈ N + by Property 2, there always exists an increment { δ n ≥ } that meets P n ∈N + ( y (1) n + δ n ) = K and y (1) n + δ n ≤ J n , ∀ n ∈ N + . Then, for any K , K , we have R ( K ) ( a ) < X n ∈N + y (1) n + δ n r n + (1+ d n ) y (1) n + δn − f n − λ ( y (1) n + δ n ) c n ( b ) ≤ R ( K ) , (75)where ( a ) is derived by the monotonically increasing property of R when y n ∈ [0 , J n ] in Lemma1 and ( b ) is because { y (1) n + δ n } n ∈N + is a feasible solution to the continuous relaxation of Problem(P4) given K . Hence, R ( K ) is monotonically increasing with K in [0 , K ∗ ] .As R ( K ∗ ) is the optimal objective value of the continuous relaxation of Problem (P4) without the constraint (55) by Lemma 1, R ( K ∗ ) is the upper bound of R ( K ) , ∀ K , ending the proof. D. Proof of Proposition 3
The results of this proposition is established by employing the theorems in [37], which ispresented as follows for convenience.
Lemma 2:
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