Delay Minimization in Sliced Multi-Cell Mobile Edge Computing (MEC) Systems
11 Delay Minimization in Sliced Multi-Cell MobileEdge Computing (MEC) Systems
Sheyda Zarandi and Hina Tabassum,
Senior Member, IEEE
Abstract —Here, we consider the problem of jointly optimizingusers’ offloading decisions, communication and computing re-source allocation in a sliced multi-cell mobile edge computing(MEC) network. We minimize the weighted sum of the gapbetween the observed delay at each slice and its correspondingdelay requirement, where weights set the priority of each slice.Fractional form of the objective function, discrete subchannelallocation, considered partial offloading, and the interferenceincorporated in the rate function, make the considered problem acomplex mixed integer non-linear programming problem. Thus,we decompose the original problem into two sub-problems: (i)offloading decision-making and (ii) joint computation resource,subchannel, and power allocation. We solve the first sub-problemoptimally and for the second sub-problem, leveraging on noveltools from fractional programming and Augmented Lagrangianmethod, we propose an efficient algorithm whose computationalcomplexity is proved to be polynomial. Using alternating op-timization, we solve these two sub-problems iteratively untilconvergence is obtained. Simulation results demonstrate theconvergence of our proposed algorithm and its effectivenesscompared to existing schemes.
Index Terms —Network slicing, partial offloading, interference,MEC, resource allocation.
I. I
NTRODUCTION
Network slicing is an indispensable technique to supportheterogeneous services in fifth generation (5G) networks [1].Using network slicing, multiple logical network slices canbe created on a common physical infrastructure. Each slicecan be tailored to a specific application with distinct Quality-of-Service (QoS) requirement. On another note, resource-intensive and latency sensitive services necessitate mobile edgecomputing (MEC) that brings computational resources to theRadio Access Network (RAN) edge. Thus, users would useboth RAN and computation resources to offload and processtheir tasks at the MEC servers. On the other hand, in a slicednetwork, resources are restricted for each slice based on aservice level agreement (SLA) with infrastructure provider(InP). Subsequently, joint optimization of RAN resources (e.g.,subchannel and power) and computation resources (e.g., CPUcycles of MEC servers) with optimal computation offloadingin a sliced network becomes imperative.Recently, the problem of delay minimization in a multi-cell MEC network was solved through communication andcomputation resource allocations (RAs) without network slic-ing [2]–[4]. However, in all these works, the interference waseither ignored [2], [3] or simplified [4]. Also, in [2], offloadingdecisions were not optimized, [3] did not consider RAN RA,and [4] considered a binary offloading scheme.A handful of research studies considered RA in slicedcellular networks [1], [5]–[9]. In [1], the authors minimized
S. Zarandi and H. Tabassum are with the Lassonde School of Engineeringat York University, Canada (e-mail: [email protected], [email protected]). Thiswork is supported by the Discovery Grant from the Natural Sciences andEngineering Research Council of Canada. a weighted combination of energy consumption and delaythrough subchannel and computation RA. This work con-sidered two slices on a single base station (BS) with nointerference. In [5], the authors minimized delay through com-putation RA, considering multiple BSs, and in [6], the authorsmaximized the offloaded workload that can be supported ina given time at each fog node through energy optimizationand server allocation. However, in both [5] and [6], the inter-cell interference was ignored and offloading decisions andRAN RA were not considered. The authors in [7] optimizedthe traffic allocation in a multi-tier sliced architecture, whilepreventing over-provisioning. However RAN and computationRA were considered abstractly, i.e., neither subchannel, power,and computation RA were considered, nor offloading decisionswere optimized. Similarly, in [8], an abstract view of ’resource’was adopted to minimize the weighted system delay, i.e., RANand computation RA were not addressed.Recently, using stochastic optimization, joint subchannel,power and computation RA was considered in a multi-cellsliced network to minimize system energy consumption in[9], while ignoring offloading decisions. It should be notedthat energy consumption can be modeled as a convex functionof transmit power and subchannel allocation variables, and isdifferent from delay, which at its simplest form, is a function ofinverse of non-convex data rate. Also, when all users offload,as in [9], the delay can be easily restated in terms of the users’data rate. However, with offloading decision optimization, suchsimplifications are not applicable.To our best knowledge, the problem of delay minimizationwith joint offloading, computation, and communication RA ina cooperative multi-cell MEC network with or without slicing is not investigated in the literature. Our contributions are: • We jointly optimize users’ offloading decisions, RAN andcomputing RA in a multi-cell MEC network to minimize theweighted sum of the difference between the delay observed ateach slice and its corresponding desired delay. The fractionalform of the objective function, discrete subchannel allocation,the partial offloading scheme, and the interference incorpo-rated in the rate function, turns this problem into a mixedinteger non-linear programming problem (MINLP) for whichwe proposed an efficient and novel algorithm. • We decouple the original problem into two sub-problems:(i) offloading decision-making and (ii) joint computationresource, subchannel, and power allocation. We solve thefirst sub-problem optimally. For the second sub-problem, wepropose an efficient algorithm with polynomial computationalcomplexity, leveraging on tools from fractional programmingand Augmented Lagrangian method (ALM). Using alternatingoptimization, we solve these two sub-problems iteratively untilconvergence. Complexity analysis is also presented. • Simulation results demonstrate the efficacy of our pro- a r X i v : . [ c s . I T ] J a n lice Coordinator X2-Interface
Backhaul Links
Offloading SignalIntercell Interference
Local offloading
Available Resources In Use
Slice 1
Slice K
Fig. 1: System Modelposed algorithm compared to existing schemes and provideinsights related to the impact of interference, slice prioritiza-tion, and cooperative MEC offloading, while demonstratingthe convergence in a few iterations.II. S
YSTEM M ODEL AND A SSUMPTIONS
We consider a MEC network with M edge points (or BSs)with co-located servers . The set of MEC servers is denotedas M = { , , · · · , M } . The available spectrum at each cell isdivided into N subchannels each with bandwidth B . Networkresources are sliced to accommodate K = { , , · · · , K } tenants each of which provide one specific type of service.Furthermore, the set of users for each tenant k is denoted by U k and the set of all users is U = { , , · · · , U } . Each tenant k has a SLA with InP in which the proportion of computationcapacity, β Ek , and available bandwidth, α k , reserved for itsusers is determined. The task of each user u is represented bythe tuple ( L u , C u ), with L u as the size of the task and C u asthe computational demand (CPU cycles) to process each bit.To facilitate slice resource management, we consider asoftware-defined network (SDN) controller referred to as slicecoordinator (SC). The SC keeps track of resource utilizationin each slice and ensures that service providers (SPs) followresource constraints in SLA and do not exceed their share ofresources. This network architecture is given in Fig. 1.We denote y u,j as the proportion of the task of user u exe-cuted on the MEC server j . Thus, we have (cid:80) j ∈{M∪ } y u,j =1 , ∀ u ∈ U , where index denotes local computation.
1) Communication Model:
We consider that if a user of-floads its task, it first sends it to its assigned server denoted by m u , and then the remaining communication (possible hand-offs between servers) would be done over the high speedbackhaul links. Denoting ˆ U j as the set of users associated toserver j , the data rate of each user u over subchannel n is: r u,n = B log (cid:18) x u,n p u,n h u,m u ,n σ + I u,n (cid:19) (1)where p u,n , I u,n , and σ represent the transmit power of user u over subchannel n , its inter-cell interference calculated as I u,n = (cid:80) j ∈{M\ m u } (cid:80) u (cid:48) ∈ ˆ U j x u (cid:48) ,n p u (cid:48) ,n h u (cid:48) ,m u ,n , and receivernoise power, respectively. Also, h u,j,n is the path-gain betweenuser u and BS j over subchannel n , and x u,n denotes the The edge nodes can connect to each other using any type of topology suchas full-mesh or star topology. binary subchannel allocation variable which is equal to one ifsubcarrier n is assigned to user u , and zero otherwise.Now, we can calculate the total data rate of each user u as R u ( X , P ) = (cid:80) n ∈N r u,n , where N , X , P , denote the set of N subchannels, subchannel allocation matrix, and transmit powerallocation matrix, respectively. Denoting Y as the matrix ofoffloading decisions, the communication delay of user u is: T comm u,m u ( X , P , Y ) = (cid:80) j ∈M y u,j L u R u ( X , P ) . (2)
2) Computing Model:
As a partial offloading scheme isadopted here, users’ task may be partly processed locally.Denoting the computation capability of local device for user u as f Lu (CPU cycles per second), the local computation delaywould be: T L u ( Y ) = y u, L u C u f Lu . (3)With F representing the matrix of all computation resource al-location variables, since the task of user u might be processedby servers other than its assigned server, the computation delayof user u is: T Eu ( X , P , F , Y ) = T comm u,m u ( X , P , Y )+ (cid:88) j ∈{M\ m u } y u,j T ho m u ,j + T comp u ( F , Y ) . (4)where T ho m u ,j denotes the hand-off delay, including the timefor communicating with SC and the average round trip timefor task transfer between m u and j th server. Moreover, T comp u denotes the offloading computation delay of user u . If tasks’fragments are processed sequentially (one after the other), T comp u would be the summation of delays of user u ineach server j as in (5). In case of parallel processing, thecomputation delay of user would be equal to the delay in theslowest server. However, in order to retain a tractable formfor our objective function, we consider an upper-bound andcalculate the computation delay in both cases as follows: T comp u ( F , Y ) = (cid:88) j ∈M y u,j L u C u f u,j , (5)where f u,j represents the computation resource that is allo-cated to user u in server j (CPU cycles per second). Note thateven when parallel computation of the tasks is possible , dueto 1) positivity of computation delay and 2) the independencebetween f u,j for different servers, this upper bound wouldnot significantly effect the optimized value of computationresource allocation in the slowest server, as minimizing thesum translates into minimizing each component separately.Due to the typically small size of response, we ignore thedownlink transmission delay. Thus, the total delay of eachuser u is: T u ( X , P , F , Y ) = T Lu ( Y ) + T Eu ( X , P , F , Y ) . (6)III. P ROBLEM F ORMULATION
In this section, we formulate the problem of minimizing theweighted sum of the difference between the delay observedat a given slice and its corresponding delay requirement ( orweighted sum of the delay deviation at each slice ), throughointly optimizing users’ offloading decisions, RAN and com-puting RA in a cooperative multi-cell MEC network. Thisproblem offers SPs a valuable insight into the adequacy of theirleased resources to meet the service quality requirement oftheir subscribers and the average delay they would experienceunder the existing SLA. Analysing the results obtained, SPscan better plan their future strategies to whether maintaintheir current SLA, invest more on leasing resources, or tomodify their subscription policy to either increase or decreasethe number of users they accept. Now, we formally state theoptimization problem as follows: P : min X , P , F , Y (cid:88) k ∈K (cid:88) u ∈U k λ k ( T u ( X , P , F , Y ) − ¯ T k ) Subject to: C : (cid:88) u ∈ ˆ U j x u,n ≤ , ∀ n ∈ N , ∀ j ∈ M ,C : x u,n ∈ { , } , ∀ n ∈ N , ∀ j ∈ M , ∀ u ∈ ˆ U j ,C : 0 ≤ (cid:88) n ∈N x u,n p u,n ≤ P max ,u , ∀ u ∈ U ,C : y u, L u C u ≤ F Lu , ∀ u ∈ U ,C : (cid:88) u ∈U y u,j L u C u ≤ F Ej , ∀ j ∈ M ,C : (cid:88) u ∈U k (cid:88) n ∈N x u,n ≤ α k M N, ∀ k ∈ K ,C : (cid:88) u ∈U k (cid:88) j ∈M f u,j ≤ β Ek S E , ∀ k ∈ K ,C : (cid:88) j ∈{M∪ } y u,j = 1 , ∀ u ∈ U ,C : y u,j ∈ [0 , , ∀ u ∈ U , ∀ j ∈ M . (7)In the above optimization problem, ¯ T k denotes the desireddelay threshold of each slice k and λ k is the weighting factorwhose value is defined in SLA and handles the precedenceof slices over each other. Furthermore, constraint C indicatesthat each subchannel can be allocated to at most one user ineach cell and C shows the binary nature of the subchannelallocation variable. In constraint C , users’ transmit poweris restricted between zero and a maximum threshold denotedby P max ,u . In constraints C and C , the limitation of localand edge computation resources are specified for each userand server, respectively, with F Lu and F Ej denoting the totalcomputation capacity of user u and server j (both in CPUcycles per second), in that order. Constraints C and C ensurethat resource consumption at each slice follows SLA. Thatis, C limits the spectrum usage for each slice. Since thereare M cells in the system and each cell has access to N subchannels, then in total we have N M subchannels, fromwhich only α k percent can be used by users of slice k .Similar to communication resources, the proportion of the totalcomputation capacity S E ( S E = (cid:80) j ∈M F Ej ) that is allocatedto each slice k is limited to β Ek as given in constraint C .Constraints C and C clarify the partial offloading decisionscheme adopted in this work.As the result of interference included in the rate function,the binary subchannel allocation variables, and the objective function which is in the form of summation of ratios, opti-mization problem (7) is MINLP and thus difficult to tackle. Inthe what follows we present our resource allocation algorithm.IV. P ROPOSED R ESOURCE A LLOCATION F RAMEWORK
To tackle the difficulties of solving problem (7), we first takeadvantage of the problem structure and decompose it into thefollowing two subproblems: P1 : min Y (cid:88) k ∈K (cid:88) u ∈U k λ k ( T u ( Y ) − ¯ T k ) Subject to: C , C , C , C . (8) P2 : min X , P , F (cid:88) k ∈K (cid:88) u ∈U k λ k ( T u ( X , P , F ) − ¯ T k ) Subject to: C − C , C , C . (9)In problem (8), both the objective function and constraint setare affine with respect to the variable Y . As such, it can besolved using standard optimization tools such as CVX toolbox.The first challenge in (9) is the multiplication of subchanneland power allocation variables in (1) as well as in constraint C . To tackle this challenge, we first replace all x u,n p u,n termswith p u,n and then add the following constraint to (9): C , : 0 ≤ p u,n ≤ x u,n P max,u (10)By using the above modification, users’ transmit power wouldbe automatically set to zero over subchannels they do notown. By adding this constraint, data rate function R u ( X , P ) would become a function of trasmit power only ( R u ( P ) ). Thisstep solves the variable multiplication issue, however discretesubchannel allocation variable is still challenging. To deal withthis issue we replace C with the following two constraints: C , : 0 ≤ x u,n ≤ , C , : x u,n − x u,n ≤ . (11) Remark 2:
Although we relax x u,n to a continuous variablein C , , since the only two values in [0,1] that fit C , are 0and 1, the binary nature of this variable would be preserved.The fractional form of users’ delay, T u , is the next issue wefocus on. After offloading decision is obtained through solvingsubproblem P1 , edge computation delay, T comp u ( F , Y ) , in theobjective function of P2 would turn into a convex function andhand-off delay would be a constant. This leaves us with thesummation of users’ transmission delay, whose non-convexitycan be easily proved. Lemma 1.
Using tools from fractional programming, problem(9) can be restated as:min X , P , F T ( P , F ) = (cid:88) k ∈K (cid:88) u ∈U k λ k (cid:104) y u,j L u R u ( P ) + (cid:88) j ∈M ,j (cid:54) = m u y u,j T ho m u ,j + (cid:88) j ∈M y u,j L u C u f u,j − ¯ T k (cid:105) Subject to: C − C , C , C . (12) Proof.
An optimization problem with the form min x ∈ C x (cid:80) Ii =1 B i ( X ) A i ( X ) , can be restated equivalently as [10]: min x ∈ C x ,t ∈ R + I (cid:88) i =1 t i B i ( X ) + I (cid:88) i =1 t i A i ( X ) , (13)here t i = B i ( X ) A i ( X ) . Using (13) and by setting B i = 1 and A i = R u ( P ) , we restate problem (9) as given in Lemma 1.Due to the presence of interference, R i ( P ) is still a non-convex function of transmit power. Lemma 2.
We can obtain an equal but convex representationof communication delay function by restating the rate as: ˆ r u,n ( P , z u,n ) =log (cid:16) z u,n (cid:112) h u,m u ,n p u,n − z u,n ( I u,n + σ ) (cid:17) , (14) Proof.
As mathematically proven in [11] and since in
Lemma 1 , we set A i = R u ( P ) , and R u ( P ) = (cid:80) n ∈N r u,n , r u,n can be equally restated as (14). This modification, notonly makes r u,n a concave function of P , R u ( P ) would alsobecome a convex function.In (14), z u,n is a slack variable that will be updatediteratively. Using Lemma 2 , we convexify the complex non-convex function T comm u,m u ( P ) , also we redefine R u ( P ) = (cid:80) n ∈N ˆ r u,n ( P , z u,n ) . For optimizing P , X , and F we adoptALM. For a given z u,n , the augmented Lagrangian functionis given in (15).In the augmented Lagrangian function, Ψ is a positiveconstant that plays the role of an adjustable penalty coefficientand Γ is the vector of all Lagrangian multipliers Θ , ∆ , Φ , ξ , and Ξ . Solving problem (9) or, equivalently (12) can bedone in three steps. In the first step, we consider Lagrangianmultipliers to be fixed and minimize L ( X , P , F , Z ) given in(15). In the second step, Lagrangian multipliers would beupdated as: θ t +1 u = (cid:34) θ tu + Ψ (cid:32) (cid:88) n ∈N p u,n − P max,u (cid:33)(cid:35) + , (16) δ t +1 k = (cid:34) δ tk + Ψ (cid:0) (cid:88) u ∈U k (cid:88) j ∈M f u,j − β Ek S E (cid:1)(cid:35) + , (17) φ t +1 n,j = (cid:34) φ tn,j + Ψ (cid:0) (cid:88) u ∈U x u,n − (cid:1)(cid:35) + , (18) ξ t +1 u,n,j = (cid:34)(cid:88) n ∈N (cid:88) j ∈M (cid:88) u ∈U ξ tu,n,j + Ψ (cid:18) (cid:88) n ∈N (cid:88) j ∈M (cid:88) u ∈ ˆ U j x u,n − x u,n (cid:19)(cid:35) + (19) Ξ t +1 u,n = (cid:34) (cid:88) u ∈U (cid:88) n ∈N Ξ tu,n + Ψ( (cid:88) u ∈U (cid:88) n ∈N p u,n − x u,n P max,u ) (cid:35) + . (20)The third step is executed after a solution is obtained for(15). In this last step, using the values obtained for P and X , we update slack variable z u,n as z u,n = √ p ∗ u,n h u,mu,n ( I u,n + σ ) .Our proposed algorithm is given in Algorithm 1 .V. C
OMPUTATION C OMPLEXITY A NALYSIS
Our proposed algorithm is divided into two sub-problems,i.e., (i) offloading decision optimization and (ii) joint computa-tion and RAN RA. For the first sub-problem, we use interiorpoint method in CVX whose complexity is in the order of O (log( C/t ξ(cid:15) )) , where C , t , ξ , and (cid:15) denote the total numberof constraints, the initial point for interior point method, the Algorithm 1
Proposed Algorithm Obtain the solution of problem (8) and initialize Z . Repeat Initialize
Γ = [Θ , ∆ , Φ , ξ, Ξ] with small numbers. Repeat Solve problem (15) considering Γ to be fixed, Update Γ using (16), (17), (18), (19), and (20). Until convergence. Set z u,n = √ p ∗ u,n h u,mu,n ( I u,n + σ ) for all users and subchannels. Until
Convergencestopping criterion, and a representation of the accuracy of themethod, respectively. For the second sub-problem based onALM the order of complexity at each iteration is O ( KU M ) which is polynomial [12].VI. S IMULATION R ESULTS AND D ISCUSSIONS
We consider a network with two cells each having 6 usersand 16 available subchannels, unless stated otherwise. Similarto [8], we consider three slices/services as: elastic services with flexible latency constraints, inelastic services that requireultra-low latency, and background services with low latencyrequirement. The weighting parameter λ is set to [3 , , forinelastic, elastic, and background services, with 50ms, 100ms,and 5s desired delay threshold, respectively. The value of L u is 1 MB and the CPU cycle, C u , is randomly chosen from [1500 , , . As a convex problem, initial point doesnot effect the solution of (8), however, to avoid increasing thecomplexity, the initial point of the problem (9) is obtainedby checking various values and selecting the best values thatminimizes our objective function.Fig. 2 depicts the effect of number of users in each cell onthe sum of weighted delay deviation at each slice. We havecompared our algorithm with 1) Joint Offloading and Compu-tation RA (JOCRA): where only offloading and computing RAis considered (with interference and server cooperation, thisscenario is in fact an improvement on [5]), 2)
Joint offloading,Subchannel, Power RA (JSPRA): in which only RAN RA isaddressed and computation resource is equally allocated tousers, and 3) our proposed scheme without server cooperation.We can clearly observe the significance of joint computationand RAN RA in the delay that users experience. In fact, ifwe ignore computation RA we would have and if weoverlook communication RA we will have increase innetwork delay deviation on average. In Fig. 1, the impactof cooperation among cells is also illustrated. At first, whennumber of users is not too high, there is almost no need forcooperation. However, as the number of users increases, weobserve that the effect of cooperation becomes noteworthy(i.e., reduction on average). The positive delay deviationoccur when network becomes infeasible (i.e., insufficient re-sources in at least one slice) and satisfying the QoS of highpriority services takes precedence in the network. Thus, wecan preserve the QoS of slices by increasing their weight ( λ k )for prioritization of the slice or the quota of reserved resources( β and α ) to avoid infeasibility. However, such modificationsare often a function of the cost SPs are willing to pay. in L ( X , P , F , Z , Γ ) = T ( X , P , F , Z ) + 12Ψ (cid:34)(cid:32)(cid:34) (cid:88) u ∈U θ u + Ψ( (cid:88) n ∈N p u,n − P max,u ) (cid:35) + (cid:33) − (cid:88) u ∈U θ u + (cid:32)(cid:34) (cid:88) k ∈K δ k + Ψ (cid:0) (cid:88) u ∈U k (cid:88) j ∈M f u,j − β Ek S E (cid:1)(cid:35) + (cid:33) − (cid:88) k ∈K δ k + (cid:32)(cid:34) (cid:88) n ∈N (cid:88) j ∈M φ n,j + Ψ (cid:0) (cid:88) u ∈U x u,n − (cid:1)(cid:35) + (cid:33) − (cid:88) n ∈N (cid:88) j ∈J φ n,j + (cid:16)(cid:34) (cid:88) n ∈N (cid:88) j ∈M (cid:88) u ∈U ξ u,n,j + Ψ (cid:18) (cid:88) n ∈N (cid:88) j ∈M (cid:88) u ∈ ˆ U j x u,n − x u,n (cid:19)(cid:35) + (cid:17) − (cid:88) n ∈N (cid:88) j ∈M (cid:88) u ∈U ξ u,n,j + (cid:32)(cid:34) (cid:88) u ∈U (cid:88) n ∈N Ξ u,n + Ψ( (cid:88) u ∈U (cid:88) n ∈N p u,n − x u,n P max,u ) (cid:35) + (cid:33) − (cid:88) u ∈U (cid:88) n ∈N Ξ u,n (cid:35) . (15) Number of Users -505101520253035 S u m o f W e i gh t ed D e l a y D e v i a t i on Proposed Algorithm Proposed Algorithm [Inelastic Slice] JOCRA (Enhanced version of [5])JSPRA Proposed Algorithm w/o Server Cooperation
Fig. 2: Weighted delay deviation Vs.Number of users
Number of Cells A v e r age W e i gh t ed D e l a y D e v i a t i on P e r U s e r Proposed Algorithm JOCRA (Enhanced version of [5]) Proposed Algorithm w/o Inelastic Slice JSPRA
Fig. 3: Weighted delay deviation Vs.Number of cells
Number of Iterations S u m o f W e i gh t ed D e l a y D e v i a t i on = [0.35,0.35,0.3] = [0.5,0.3,0.2] Fig. 4: Weighted delay deviation Vs.Number of iterationsIn Fig. 3, we examine how increasing the number of cellsimpacts the delay of users. We again compare our proposedalgorithm with
JOCRA and
JSPRA . As the number of usersper cell remains constant here, we depict the average delaydeviation per user. Increasing the number of cells notablyincreases the delay of users, however this increase is moresignificant when communication RA is overlooked. Because,while the average amount of resources available for usersremains almost the same (since the number of users in eachcell is constant), more cells means intensified interference inthe network. To deal with the negative effect of this intensifiedinterference, precise RAN RA becomes imperative.The convergence of our proposed algorithm and the impor-tance of slice resource management is numerically demon-strated in Fig. 4. Here, we observe that: i) our algorithmconverges to its final solution after a few iterations, and ii)careful resource reservation plays a significant role in the QoSusers of each slice achieve.VII. C
ONCLUSION
In this work we propose a framework to minimize the delayin cooperative MEC network by optimizing both
RAN andcomputation resources and offloading decisions , using toolsfrom fractional programming, convexification of rate function,and ALM. The problem of routing between edge servers is avenue for future works, especially with wireless backhauling.R
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