Let's Share VMs: Optimal Placement and Pricing across Base Stations in MEC Systems
Marie Siew, Kun Guo, Desmond Cai, Lingxiang Li, Tony Q.S. Quek
LLet’s Share VMs: Optimal Placement and Pricingacross Base Stations in MEC Systems
Marie Siew † , Kun Guo † , Desmond Cai § , Lingxiang Li ∗ , Tony Q.S. Quek † † Information Systems Technology and Design Pillar, Singapore University of Technology and Design § Institute of High Performance Computing, Singapore ∗ University of Electronic Science and Technology of China, Chinamarie [email protected]; [email protected]; [email protected]; [email protected];[email protected]
Abstract —In mobile edge computing (MEC) systems, usersoffload computationally intensive tasks to edge servers at basestations. However, with unequal demand across the network,there might be excess demand at some locations and underutilizedresources at other locations. To address such load-unbalancedproblem in MEC systems, in this paper we propose virtualmachines (VMs) sharing across base stations. Specifically, weconsider the joint VM placement and pricing problem across basestations to match demand and supply and maximize revenue atthe network level. To make this problem tractable, we decomposeit into master and slave problems. For the placement masterproblem, we propose a Markov approximation algorithm MAPon the design of a continuous time Markov chain. As forthe pricing slave problem, we propose OPA - an optimal VMpricing auction, where all users are truthful. Furthermore, givenusers’ potential untruthful behaviors, we propose an incentivecompatible auction iCAT along with a partitioning mechanismPUFF, for which we prove incentive compatibility and revenueguarantees. Finally, we combine MAP and OPA or PUFF to solvethe original problem, and analyze the optimality gap. Simulationresults show that collaborative base stations increases revenue byup to 50 % . Index Terms —Edge Computing, Network Economics
I. I
NTRODUCTION
Mobile Edge Computing (MEC) is an enabler of excit-ing new technologies and applications like deep learning ondevices, virtual and augmented reality, and smart city dataanalytics. These exciting new technologies and applicationshave high computation requirements. MEC enables them byallowing users to offload computationally intensive tasks tothe network edge (e.g., base stations in cellular networks andaccess points in WiFi networks), which are equipped withcomputing capability by connecting to the edge servers [1].With servers placed at the network edge near the end users,Wide-area-network (WAN) delay is avoided, allowing it tomeet the stringent latency requirements of delay sensitivetasks, that cloud computing is unable to [2].Unlike cloud computing, the computational resources at theedge server are limited. Hence optimizing resource allocation
This work was supported in part by the National Natural Science Foundationof China under Grants 61901528, 62001254 and 61771263, and in part by theHunan Natural Science Foundation under Grant 2020JJ5769. (Correspondingauthor: Kun Guo). in MEC is an important research question. In particular, de-mand for computation is uneven across the network. Leadingto excess demand at some coverage areas, and underutilizedresources at others. In this load-unbalanced scenario, thereare users not being served, and from the network operator’sperspective, resources are not efficiently utilized and revenueis not maximized. This prompts a global optimization andorganization of resources over the network, to place resourcesmore effectively in light of the network’s demand pattern.Virtual machine (VM) migration is perceived as a promisingway to solve the load-unbalanced scenario [3]. There havebeen works on VM migration in MEC [4]–[8]. These worksinvestigate at the level of a single user, in response to usermobility. In contrast, there has been a lack of work fromthe global perspective. To this end, we propose the idea of“
Collaborative Base Stations ”, where base stations share theirVMs with each other. This involves the migration of VMs, inaccordance with the relative demand across base stations. Inparticular, we consider a joint optimization of VM placementand pricing at base stations to match the demand and supplyfrom the network level. A joint formulation is used because onone hand, the price at one base station has an impact on users’demand, which affects the VM placements. On the other hand,VM placement determines the resource supply at one basestation. This way, users’ demand will be satisfied as much aspossible and the revenue across the network is maximized.However, some difficulties arise when solving the formu-lated joint VM Migration and Pricing for Profit maximizationproblem (
MPP ). Firstly, there is a sophisticated coupling ofthe price and VM placement variables, making it difficultto solve
MPP directly. Secondly,
MPP is a combinatorialoptimization problem, with the number of VMs deployed ateach base station being integers. It could be intractable, whenthe number of base stations increases and the total numberof VMs deployed at the edge increases. Thirdly, the pricing atone base station is affected by the demand and bid informationreported by the user. Users’ potential untruthful behaviorsmake pricing at base stations challenging.To tackle these difficulties, we first use primal decom-position to decouple the variables, decomposing
MPP intothe slave problem NP - Normalized Pricing problem, and a r X i v : . [ c s . I T ] J a n aster problem VP - VM Placement problem. Next, wepropose an online Markov approximation enabled algorithmwhich solves the combinatorial VP in a distributed manner.This helps to deal with the potential intractability when theproblem size gets large. It does so by modelling the differentVM configurations as states of a Continuous Time MarkovChain (CTMC). The VM migrations happen according to thetransition rate of the CTMC, which is in turn dependent on theperformance level (revenue) of the placement configurations.How is the revenue of the VM placement configurationsobtained? We solve NP to obtain the optimal revenue for eachplacement configuration. Specifically, at each base station weconduct either OPA - the Optimal Pricing Auction, or iCAT -an incentive CompAtible Truthful auction, which ensures usersare truthful. iCAT guarantees the revenue R , when R is lessthan or equal to the optimal. To successfully estimate R , wefurther present a user partitioning mechanism. The results ofthe auction will be fed back to the base station and networkoperator, directly influencing the transition rates of the CTMC.Our contributions are summarized as follows: • To deal with unequal demand across the MEC coverageareas, we formulate a joint VM migration and pricingproblem across base stations to match demand and supplyat the network level. This works towards ensuring thatuser demand is met, resource placement is optimizedglobally, and the operator’s revenue is maximized. • Due to 1) the combinatorial nature of the problem, 2) thecoupling of price and placement variables, and 3) usershaving the incentive to hide their true valuations, we useprimal decomposition to decompose the problem into amaster and slave problem. For the master VM placementproblem, we present
MAP , a Markov approximation-enabled algorithm which solves the combinatorial prob-lem in a distributed manner at individual base stations. • To solve the pricing problem, we present an optimal pric-ing auction
OPA , and prove that it is optimal. Besides, asusers might have an incentive to hide their true valuations,we present an incentive compatible auction iCAT , provethat it is dominant strategy incentive compatible andthat its revenue is R , when R is less than or equalto the optimal. To estimate the target R , we presenta user partitioning algorithm PUFF , and prove that itscompetitive ratio is 4. • We present the combined algorithm cMAP which solvesour original joint VM placement and pricing problem,with an optimality gap of β log | V | . Following which,we conduct a perturbation analysis and show that theoptimality gap of the stationary distribution caused bypotential perturbations is bounded by − exp( − βψ max ) ,where ψ max is the perturbation error. • Finally, we provide simulation results which show thatour proposed solution cMAP : MAP + OPA convergeto optimality, and analyze the impact of β . While theperformance of cMAP : MAP + PUFF is not optimal, ithas a competitive ratio of , as we have proved. Results show that our mechanism cMAP increases revenue by upto 50 % , compared to the baseline where base stations donot collaborate and VMs are not migrated.The rest of this paper is organized as follows. In Section II,we introduce related works. The system model and problemformulation are given in Section III, which is followed bythe optimal VM placement algorithm and the auction pricingalgorithms in Sections IV and V. In Section VI, we givethe complete implementation and analysis. In Section VII wediscuss simulations results and in Section VIII we conclude.II. R ELATED W ORKS
There are two mainstream ways to address the load-unbalanced problems for efficient resource utilization in MECsystems. On this basis we introduce the related works.The first way is to optimize users’ task offloading decisions,i.e. whether or not to offload, and which base station the useroffloads to [1], [3]. In this way, the computing resources atbase stations are fixed and the users are handovered amongbase stations. For instance, [9]–[11] have optimized taskoffloading to strike a balance between energy consumptionand delay from the perspective of users. [12] studied the staticedge server placement problem. [13]–[15] aimed to maximizethe network revenue through task offloading.This paper considers an alternative way, in which thecomputing resources are migrated among base stations to servethe associated users. Particularly, VM migration in MEC drawsattention in industry and academic fields [3], [16], [17]. (Notethat while there has been work on VM placement or migrationfor revenue maximization in clouds [18], [19], these works arespecific with respect to data center topologies.) Most of thework on VM or service migration in MEC focus on improvinguser experience (e.g. reducing delay), in light of user mobility[4]–[8]. For example in [4] Plachy et al. proposed a dynamicVM placement and communication path selection algorithm.In [5] Taleb et al. optimized a policy on the service migrationdecision, given the user’s distance. In [6], Ouyang et al. used Lyapunov optimization to optimize the placements overdifferent timeslots. Another line of research regarding VMmigration looks at how it can maximize network profit orrevenue. In [20], Sun et al. optimized the tradeoff betweenmaximizing the migration gain and minimizing the migrationcost. In this work, we investigate from a novel perspective. Welook at VM migration in MEC at a global level, in light of thenetwork’s demand patterns, for revenue maximization. And weformulate a joint VM migration and pricing problem becausethe price and migration decisions have a coupled impact onrevenue. To the best of our knowledge, we do not know ofmany other works which take this approach.Our proposed incentive compatible auctions and their proofsborrow from, but are different from the Profit Extractor andRandom Sampling Auction in [21], [22]. Profit Extractor andRandom Sampling Auction cater to fully digital goods, withzero marginal cost of producing the next good, and hence aninfinite supply. Unlike this, our network has a limited supplyof VMs, resulting in unique novel algorithms and proofs.II. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
Consider an MEC system with K base stations withheterogeneous computing capability. Each base station k isequipped with an edge server containing v k VMs. These arevirtualised computing resources which users can offload theircomputationally intensive tasks to, at a price of p k . Since thebase stations are controlled by the same network operator,these VMs can be migrated from one base station to another, tooptimize the utilization of resources. This global coordinationof resources will help to deal with load-unbalanced scenarioswhere there are excess demands in one coverage area, andunderutilized resources in another part of the network.Each base station k has a set of users [1 , ..., i, ..., n k ] ∈ U k which are associated with it. Each user i offloads its com-putationally intensive tasks to the edge server for auxiliaryprocessing. Different users require different number of VMs,with user i at base station k requiring r k,i VMs. At base station k , different users respond differently to the price p k .A user i at base station k has willingness to pay u k,i . Thewillingness to pay can be viewed as the utility a user gets fromjob computation using the VM. Different users have differentwillingness to pay. For example, a user with a more urgent jobwould have a higher willingness to pay than a user who is notas urgent. A user who will execute the job no matter what, withless regard of the price would have a higher willingness to pay(e.g., IoT sensors’ periodic data analytics). A user will decideto execute its job if its payoff π k,i = u k,i − p k is non-negative,i.e. if utility minus payment is greater than 0 ( π k,i ≥ .Therefore, the demand (total number of VM requests) at basestation k will be (cid:80) i ∈ U k r k,i { u k,i >p k } , where { u k,i >p k } isthe indicator function representing whether user i ’s willingnessto pay is higher than p k .The demand for VMs at each base station k could be higheror lower than the supply v k . Hence, the network operatorwould perform a global optimization of VMs, shifting them tolocations with higher demand, to achieve a higher utilizationof resources and to optimize its profit. At the same time, thenetwork operator sets prices p k differently across coverageareas, to obtain the highest possible revenue, in light of thevarying demand across the network. The joint Migration andPricing for Profit maximization problem ( MPP ) is as follows:
MPP : max p , v K (cid:88) k =1 p k min (cid:40) (cid:88) i ∈ U k r k,i { u k,i ≥ p k } , v k (cid:41) s.t. v k ∈ Z +0 , k = 1 , ..., K K (cid:88) k =1 v k = V, (1)where V is the total number of VMs, placed by the net-work operator across K base stations. Besides, Z +0 indi-cates the set of non-negative integers. In MPP , the deci-sion variables are the prices across the various base stations p = [ p , ..., p k , ..., p K ] , in which each element is normalized(i.e., p k ∈ [0 , ) without loss of generality, and the VMplacements across the network v = [ v , ..., v k , ..., v K ] . The objective function is the sum of the revenue obtained acrossbase stations. In particular, it is the price multiplied by thenumber of units of demand which is met with supply.Some difficulties arise when solving MPP . Firstly,
MPP isa combinatorial optimization problem, with v k being integers.It could be intractable, when the number of base stationsincreases and the total number of VMs increases. Even if werelax v k to continuous values, the problem is still non-convex.Secondly, there is a coupling of p and v in the objectivefunction, making it difficult to solve MPP directly.To tackle the difficulties in solving
MPP , firstly we useprimal decomposition [23], such that
MPP is decomposed intoslave problem NP - Normalized Pricing problem, and masterproblem VP - VM Placement problem. Specifically, fixing v ,the slave problem is as follows: NP : max p K (cid:88) k =1 p k min (cid:40) (cid:88) i ∈ U k r k,i { u k,i ≥ p k } , v k (cid:41) . (2)Given the optimal solution from the slave problem, the masterproblem updates the VM migration decisions: VP : max v Φ ∗ v s.t. v ∈ V , (3)where Φ ∗ v is the optimal value of NP for the given v and V = { v | (cid:80) Kk =1 v k = V (cid:84) v k ∈ Z +0 , k = 1 , ..., K } is the set ofall possible VM placements across the network, with size | V | .Following this, we propose a distributed Markov Approxi-mation implementation to solve VP . And finally, we proposeboth optimal and incentive compatible auction mechanisms tosolve NP . We discuss the details in the following sections.IV. T HE OPTIMAL VM PLACEMENT ALGORITHM
In this section, we will show how we solve the master prob-lem VP . Particularly, we first reformulate and approximate VP and then, propose a Markov approximation-enabled algorithm,named MAP - Markov Approx VM Placement algorithm.
A. Reformulating and Approximating VP The master problem VP can be rewritten as VP-EQ : max π v (cid:88) v ∈ V π v Φ ∗ v s.t. ≤ π v ≤ , ∀ v ∈ V (cid:88) v ∈ V π v = 1 , (4)where π v could be seen as the proportion of time spent inconfiguration v . VP is an NP hard combinatorial optimization problem,and hence challenging to solve, even for a centralized im-plementation. Even if we relax v k to continuous values, theproblem is still non-convex. Therefore, we use the log-sum-exp approximation f (Φ ∗ v ) = β log( (cid:80) v ∈ V exp( β Φ ∗ v )) to ap-proximate VP-EQ . This approximation allows for a distributedimplementation at individual base stations. This is useful whenthe system dynamics change - when new users enter, or whensers move from coverage area to area. This approximation isupper bounded by β log | V | , following Proposition 1 [24]: Proposition 1.
For β > , we have max v Φ ∗ v ≤ β log( (cid:88) v ∈ V exp( β Φ ∗ v )) ≤ max v Φ ∗ v + 1 β log | V | . (5)Therefore, max v Φ ∗ v = lim β →∞ β log( (cid:80) v ∈ V exp( β Φ ∗ v )) ,i.e., the approximation tends towards VP-EQ for large β . Asthe log-sum-exp function is a closed and convex function,the conjugate of its conjugate is itself, and hence we have β log( (cid:80) v ∈ V exp( β Φ ∗ v )) = (cid:80) v π v Φ ∗ v − β (cid:80) v π v log π v , accord-ing to [24], [25]. Therefore the log-sum-exp approximation of VP-EQ is equivalent to the following problem
VP-approx : max π v (cid:88) v π v Φ ∗ v − β (cid:88) v π v log π v s.t. ≤ π v ≤ , ∀ v ∈ V (cid:88) v ∈ V π v = 1 . (6)By solving the KKT conditions of VP-approx , the optimalsolution is achieved in Theorem 1.
Theorem 1.
The optimal solution to
VP-approx is π ∗ v = exp( β Φ ∗ v ) (cid:80) v ∈ V exp( β Φ ∗ v ) . (7) Proof.
Let λ be the Lagrange multiplier associated with theconstraint (cid:80) v ∈ V π v = 1 . The Lagrangian of VP-approx willthen be L ( π v , λ ) = (cid:88) v ∈ V π v Φ ∗ v − β (cid:88) v ∈ V π v log π v − λ ( (cid:88) v ∈ V π v − . (8)Therefore, the KKT conditions will be: Φ ∗ v − β (log π ∗ v + 1) − λ = 0 , ∀ v ∈ V , (9) (cid:88) v ∈ V π v = 1 , (10) λ ≥ . (11)Solving the KKT conditions for the primal and dual op-timal points π ∗ v and λ ∗ , we obtain π ∗ v = exp( β (Φ ∗ v − λ ) − . Using the constraint (cid:80) v ∈ V π v = 1 , we obtain λ ∗ = β log (cid:80) v exp( β Φ ∗ v − . Finally, substituting λ ∗ into π ∗ v = exp( β (Φ ∗ v − λ ) − , we obtain (7).Therefore, by time-sharing among VM placement config-urations according to the probability distribution π ∗ v , we areable to solve VP-approx , and hence
VP-EQ , VP , and MPP approximately.
B. Solving VP: Algorithm design
The idea consists of designing a Markov Chain, in whichthe state space is the space of possible VM placement config-urations | V | , and the stationary distribution is π ∗ v , the optimalsolution to VP-approx . This would allow us to solve the joint VM placement and pricing problem
MPP with an optimalitygap of β log | V | . To help us in the construction of the Markovchain, we use the following result from [25]: Lemma 1.
For any distribution of the form π ∗ v in (7), thereexists at least one continuous-time time-reversible ergodicMarkov chain whose stationary distribution is π ∗ v . A continuous time-reversible markov chain (CTMC) iscompletely defined by its state space and transition rate. Welet the state space be the space of possible VM placementconfigurations V . The transition rate q vv (cid:48) indicates the rateat which the CTMC shifts from placement configuration v to v (cid:48) . According to [25], for the CTMC to converge tostationary distribution π ∗ v , it needs to satisfy the following twoconditions: 1) Irreducibility, meaning that any two states of theCTMC are reachable from each other. 2) Satisfaction of thedetailed balanced equation: for any v , v (cid:48) ∈ V , π ∗ v q vv (cid:48) = π ∗ v (cid:48) q v (cid:48) v .In other words, exp( β Φ ∗ v ) q vv (cid:48) = exp( β Φ ∗ v (cid:48) ) q v (cid:48) v based on (7).Condition 1 can be satisfied because any two states (place-ment configurations) are reachable from each other. For Con-dition 2, let us set q vv (cid:48) = 0 for any two states which involvethe migration of more than one VM from one base stationto another. This is done to reduce the computation required,especially when the network is large. For states which involvethe migration of only one VM, we have q vv (cid:48) = exp( 12 β (Φ ∗ v (cid:48) − Φ ∗ v )) . (12)The detailed balance equation will be satisfied. The transitionrate q vv (cid:48) is exponentially proportional to the performanceof the target minus current VM placement configuration.Therefore, when the performance (optimal revenue) of thetarget configuration is relatively higher than the current, therewill be a higher transition rate, and vice versa.The performance of each configuration v is equivalent to itsrevenue obtained. In the next section, we show how to obtainthe optimal revenue given a VM placement configuration v .In particular, we propose auction mechanisms to solve theslave problem NP . Following which, we will show how thealgorithms solving the master problem VP and slave problem NP are combined to solve the original problem MPP .V. T HE A UCTION P RICING M ECHANISMS
In this section, we show how the slave problem NP can besolved. Specifically, NP defined in (2), can be decomposedinto individual pricing problems for each base station, whereeach base station k solves the following problem: NP-k : max p k p k min (cid:40) (cid:88) i ∈ U k r k,i { u k,i ≥ p k } , v k (cid:41) . (13) NP-k can be solved by an auction. We provide two solutions,firstly
OPA - Optimal Pricing Auction, which assumes theusers are truthful, submitting bids b k,i equal to their truevaluations u k,i , and then PUFF - Partitioning Users For truth-Fulness mechanism, which includes an incentive CompAtibleTruthful auction iCAT . Our auction mechanisms are priorree, since they can be carried out without knowledge on thedistribution of users’ valuations u k,i . A. The Optimal Pricing Auction (OPA)
The mechanics behind
OPA are as follows: users submittuple ( r k,i , b k,i ) to base station k , where r k,i is the amountof VMs requested by user i at base station k , and b k,i isthe bid indicating the user’s willingness to pay for one VM.Since all users are truthful, the bid reported by the user isequal to its valuation (i.e., b k,i = u k,i ). At price p k , all userswith valuation u k,i ≥ p k will be willing to participate in theauction. Then, we prove the optimal price will be p ∗ k ∈ B k = U k in Theorem 2, where B k and U k are the set of bids andvaluations for users at base station k , respectively. Theorem 2.
When all users are truthful, the optimal price of
NP-k , termed as p ∗ k , is found in B k = U k .Proof. When all users are truthful, we have B k = U k . Then,we prove that p ∗ k is found in B k .For the case with p k > max i ∈ U k b k,i = max i ∈ U k u k,i , { b k,i ≥ p k } = { u k,i ≥ p k } = 0 holds, such that all userswould reject to rent the VMs at base station k . There-fore, the revenue attained at base station k is Rev ( p k ) = p k min { (cid:80) i ∈ U k r k,i { u k,i ≥ p k } , v k } = 0 .Then, we analyse the case with p k < max i ∈ U k b k,i . Rear-range B k in descending order and denote the set of orderedbids by { b , b , ..., b n k } , where b i represents the i -th highestbid. Using the fact that b k,i = u k,i , we have Rev ( p k = b i − (cid:15) ) = ( b i − (cid:15) ) min (cid:40)(cid:88) i ∈ U k r k,i { b k,i ≥ ( b i − (cid:15) ) } , v k (cid:41) < b i min (cid:40) (cid:88) i ∈ U k r k,i { b k,i ≥ b i } , v k (cid:41) = Rev ( p k = b i ) , (14)where (cid:15) < b i − b i − , no new users rent the VMs atbase station k by changing the price from p k = b i to p k = b i − (cid:15) , that is, min (cid:8)(cid:80) i ∈ U k r k,i { b k,i ≥ ( b i − (cid:15) ) } , v k (cid:9) =min (cid:8)(cid:80) i ∈ U k r k,i { b k,i ≥ b i } , v k (cid:9) hold. Based on (14), we thusconclude that p ∗ k lies in B k .Using this insight that the optimal price belongs to theset of bids, the structure of our proposed OPA is summa-rized in Algorithm 1. In detail, after receiving the tuple ( r k,i , b k,i ) from all the users, base station k will sort theminto descending order with respect to b k,i . For each uniquebid b k,i , the platform will set ¯ p k = b k,i , and calculatethe revenue Rev (¯ p k ) = ¯ p k min { (cid:80) i ∈ U k r k,i { u k,i ≥ ¯ p k } , v k } .Following which, it will optimize over ¯ p k and achieve p ∗ k = argmax ¯ p k = b k,i , ∀ i ∈ U k Rev (¯ p k ) . B. The Incentive CompAtible Truthful Auction (iCAT)
In reality, users may have an incentive to submit bidsunequal to their true valuations (i.e. b k,i (cid:54) = u k,i ), hopingto achieve a higher payoff. Therefore, we present incentive Algorithm 1
OPA: Optimal Pricing Auction Input:
Tuple ( r k,i , b k,i ) , ∀ i ∈ U k Sort ( r k,i , b k,i ) according to descending order with respectto b k,i . for all unique b k,i do Set ¯ p k = b k,i Rev (¯ p k ) ← ¯ p k min { (cid:80) i ∈ U k r k,i { u k,i ≥ ¯ p k } , v k } (cid:46) ByEq. (13) end for Output: p ∗ k ← argmax ¯ p k = b k,i , ∀ i ∈ U k Rev (¯ p k ) end compatible auction mechanism iCAT , by which the user’sdominant strategy is to be truthful.Given a target revenue R , the auction mechanism willpost price p k = R min { (cid:80) i ∈ Uk r k,i ,v k } , where (cid:80) i ∈ U k r k,i is thetotal demand of the users currently in the auction. Userswill decide whether or not to accept the offer by weighingif their payoff p k − u k,i is not lesser than (individualrationality met). If any user i rejects the offer, he is removedfrom future rounds of the auction. Then, the set of usersin the auction is updated as U k ← U k \ { i } . The processrepeats: base station k obtains the new demand (cid:80) i ∈ U k r k,i of users currently in the auction, and broadcasts the newprice p k = R min { (cid:80) i ∈ Uk r k,i ,v k } . If all users remaining in theauction accept the offer, they will be the winners, payingthe last offer price p k . Therefore, base station k would rent min { (cid:80) i r k,i { u k,i ≥ p k } , v k } units of VMs to users with bidsin the set U k at price p k = R min { (cid:80) i ∈ Uk r k,i { uk,i ≥ pk } ,v k } .The complete iCAT is summarized in Algorithm 2. Themain idea behind this mechanism is that it prunes the setof auction users until it obtains a set U k where: the usersin U k are willing to pay p k = R min { (cid:80) i ∈ Uk r k,i ,v k } , the priceat which the base station obtains revenue R given demand (cid:80) i ∈ U k r k,i . Note that our auction mechanism does not involvethe users submitting any bids b k,i . Truthfulness is ensuredvia the structure of the mechanism, as proved in Theorem3. In particular, we prove that iCAT is dominant strategyincentive compatible, meaning that being truthful gives theusers a higher payoff compared to any other strategy. Theorem 3.
Mechanism iCAT is dominant strategy incentivecompatible.Proof.
If a user rejects an offer, he will be out of the auctionand unable to participate in the next round, hence getting apayoff of . Therefore rejecting p k , when p k < u k,i , is adominated strategy.Likewise, accepting p k > u k,i is a dominated strategy, sinceprices will rise the next round. Therefore the dominant strategyfor every user i is to report his true value u k,i .The following theorem provides an optimality guaran-tee for iCAT. It uses the benchmark OptRev ≥ ( U allk ) = max p k p k min { (cid:80) i ∈ U allk r k,i { u k,i ≥ p k } , v k } , which has a re-quirement of at least two users being in the market. This is lgorithm 2 iCAT: incentive CompAtible Truthful Auction Input:
Initialize U k , the number of VMs required by user i ( r k,i ), and target revenue R at base station k . while U k is not empty do Base station k posts price p k = R min { (cid:80) i ∈ Uk r k,i ,v k } ; if u k,i < p k for any user i ∈ U k then User i rejects to join in the auction; Base station k updates U k ← U k \ { i } ; else All users in U k would join in the auction; Exit while loop; end if end while
Output: p k ← R min { (cid:80) i ∈ Uk r k,i , v k } and Rev ( p k ) ← R with U k not empty, otherwise, p k ← and Rev ( p k ) ← . end not a serious constraint in light of the number of users at onebase station. Besides, we use U allk to indicate the initial U k in iCAT , that is, the total number of users at base station k . Theorem 4.
The mechanism iCAT achieves a revenue of R ifOptRev ≥ ( U allk ) ≥ R , and a revenue of otherwise.Proof. According to Theorem 2, we haveOptRev ≥ ( U allk ) = u ∗ k,x min (cid:88) i ∈ U ∗ k,x r k,i , v k , (15)for some u ∗ k,x and U ∗ k,x = { i | u k,i ≥ u ∗ k,x } .If OptRev ≥ ( U allk ) > R , then some u k,x not equal to u ∗ k,x could be found to obtain a revenue Rev ( u k,x ) equal to R . Onthe contrary, if OptRev ≥ ( U allk ) < R , by (15) we will not beable to find any u k,x satisfying u k,x ≥ R min { (cid:80) i ∈ Uallk r k,i ,v k } .According to line 12 in Algorithm 2, a revenue of 0 is obtainedin this case. Besides, for the case with OptRev ≥ ( U allk ) = R ,the revenue of R is achieved naturally.Intuitively, the target revenue R plays a key role in iCAT .How shall the base station estimate R ? For truthfulness, wewant R to be estimated independently of the bidders we runauction iCAT on. Hence, we further propose a partitioningmechanism PUFF - Partitioning Users For truthFulness, forthe base station to estimate R while preserving truthfulness. C. Partitioning Users For Truthfulness (PUFF)
The operations of
PUFF are as follows: We partition the setof all users into two sets. Following which, we calculate theoptimal revenues R and R for each set. Next, we use theoptimal revenues as ’estimates of R ’ for the opposing set andrun iCAT in each set. Note that when the total supply is lessthan the total demand, we will run the separate auctions using (cid:98) v k / (cid:99) and (cid:100) v k / (cid:101) number of VMs. The complete PUFF issummarized in Algorithm 3.In the following theorem, we show that PUFF is truthful.
Theorem 5.
Mechanism
PUFF is dominant strategy truthful.
Algorithm 3
PUFF: Partitioning Users For truthFulness Mech-anism Input:
Initialize U k and the number of VMs required byuser i ( r k,i ). Randomly partition U k into two sets S and S of equalsize. if (cid:80) i ∈ U k r k,i > v k then Calculate R = optimal revenue of S given (cid:98) v k / (cid:99) VMs, and R = optimal revenue of S given (cid:100) v k / (cid:101) VMs; Run auction iCAT ( S , (cid:98) v k / (cid:99) , R ) on set S , and iCAT ( S , (cid:100) v k / (cid:101) , R ) on set S . else Calculate R = optimal revenue of S given v k VMs,and R = optimal revenue of S given v k VMs. Run auction iCAT ( S , v k , R ) on set S , and iCAT ( S , v k , R ) on set S . end if end Proof.
Auction iCAT is truthful when implemented with an R estimated independently of the users it is run on.Next, we state a lemma which helps us towards provinglower bounds on the performance of PUFF . Lemma 2.
The revenue of
PUFF is at least min( R , R ) .Proof. Either R > R , R > R , or R = R holds in the PUFF . Therefore, at least one auction out of iCAT ( S , R )and iCAT ( S , R ) succeeds, i.e. gets a revenue of above 0,giving a revenue of min( R , R , R + R ) .Following which, we prove bounds on the optimality gapof PUFF , proving that its competitive ratio is , in a specialcase where all users i request one VM, i.e., r k,i = 1 . Theorem 6.
Assume r k,i = 1 for all users. Let Rev be theexpected revenue of
PUFF . We will have RevOptRev ≥ ( U allk ) ≥ .Proof. We know from Theorem 2 that OptRev ≥ ( U allk ) = u ∗ k,x min { (cid:80) i ∈ U ∗ k,x r k,i , v k } for some u ∗ k,x and U ∗ k,x = { i | u k,i ≥ u ∗ k,x } . Let D = (cid:80) i ∈ U allk r k,i and S = v k .Further, we first analyse the case where D ≥ S . Given this u ∗ k,x , we will have R ≥ u k,x min { (cid:80) i ∈ U ∗ k,x ∩ S r k,i , (cid:98) v k / (cid:99)} and R ≥ u k,x min { (cid:80) i ∈ U ∗ k,x ∩ S r k,i , (cid:100) v k / (cid:101)} . Therefore, wededuce that Rev
OptRev ≥ ( U allk ) ( a ) ≥ E [min( R , R )] u ∗ k,x min { (cid:80) i ∈ U ∗ k,x r k,i , v k } ( b ) ≥ E [min( u ∗ k,x min { A, (cid:98) v k / (cid:99)} , u ∗ k,x min { B, (cid:100) v k / (cid:101)} )] u ∗ k,x min { (cid:80) i ∈ U ∗ k,x r k,i , v k } ( c ) ≥ min( (cid:98) v k / (cid:99) , E [min { A, B } ]min { (cid:80) i ∈ U ∗ k,x r k,i , v k } ( d ) ≥ min {(cid:98) v k / (cid:99) , / (cid:80) i ∈ U ∗ k,x r k,i } min { (cid:80) i ∈ U ∗ k,x r k,i , v k } ≥ . (16)n inequality ( b ) , we have A = (cid:80) i ∈ U ∗ k,x ∩ S r k,i and B = (cid:80) i ∈ U ∗ k,x ∩ S r k,i . Note that the transition from inequality ( c ) to ( d ) is due to the fact that if we flip k ≥ coins (correspondingto partitioning the winners into the 2 sets), E [min( H, T )] ≥ [22], Chapter 13.Likewise, for the case where D ≤ S , following the samelogic we have Rev
OptRev ≥ ( U allk ) ≥ min { v k , / (cid:80) i ∈ U ∗ k,x r k,i } min { (cid:80) i ∈ U ∗ k,x r k,i , v k } ≥ . (17)It is emphasized that, iCAT , PUFF and their proofs borrowfrom, but are different from the Profit Extractor and RandomSampling Auction in [21], [22]. Profit Extractor and RandomSampling Auction cater to fully digital goods, with 0 marginalcost of producing the next good, and hence an infinite supply.Unlike this, our network has a limited supply of VMs, resultingin unique novel algorithms and proofs.VI. C
OMBINED A LGORITHM AND A NALYSIS
In this section, we present the combined VM placementand pricing mechanism, describing its implementation. Next,we analyse its performance, termed cMAP , and prove boundson the optimality gap caused by potential perturbations on Φ ∗ v . A. Algorithm Implementation
The distributed and combined Markov Approx VM Place-ment and Pricing Algorithm ( cMAP ) is summarized in Algo-rithm 4 and works as follows: Each round, we randomly selecta base station. The base station k considers potential config-urations v (cid:48) in which it has gained one VM, or sent one VMto elsewhere. The network operator obtains the target revenue Φ v (cid:48) using OPA , PUFF , or via historical data. The base stationthen starts exponential clocks for each of these configurations,following the transition rate q vv (cid:48) ← exp(0 . β (Φ ∗ v (cid:48) − Φ ∗ v )) .When the performance of the target configuration is relativelyhigher (or lower) than the current, there will be a higher (orlower) rate of switching. The process repeats until convergenceto the stationary distribution, the optimal point of VP-approx .This point approximates the optimal point of
MPP withan optimality gap of β log | V | , according to Proposition 1.Note that due to its distributed nature, our algorithm is ableto handle the dynamic scenarios when new users enter thesystem, or when users shift from region to region. B. Algorithm Analysis
Our combined mechanism cMAP attains an optimality gapof β log | V | for the original problem MPP . In practice, thesystem may obtain an inaccurate value of Φ ∗ v , the optimalrevenue under configuration v . This may occur when weimplement the incentive compatible auction mechanism PUFF and estimate R .In light of this we analyse the impact of the perturbations,by bounding the optimality gap caused by the perturbations, onproblem VP-approx . To this end, we construct a new CTMC
Algorithm 4 cMAP: Combined Markov Approx VM Place-ment and Pricing Algorithm Input: V , the total number of VMs across the network, { U k } , the set of users across all base stations, and { r k,i } ,the number of VMs required by all users. Initialise a configuration v . Network operator calculates Φ ∗ v ← OPA ( v , { U k } , { r k,i } )or PUFF ( v , { U k } , { r k,i } ); while True do Randomly select a base station k . Consider configurations v (cid:48) with v k ± VMs at k . for all configurations v (cid:48) do Network operator obtains the target revenue Φ ∗ v (cid:48) ← OPA ( v (cid:48) , { U k } , { r k,i } ) or PUFF ( v (cid:48) , { U k } , { r k,i } ); Set clocks with transition rate q vv (cid:48) ← exp(0 . β (Φ ∗ v (cid:48) − Φ ∗ v )) ; end for The CTMC transits to the next state according to q vv (cid:48) ; end while which takes into account the perturbations, and characterizeits stationary distribution, in the following.For each state v with optimal revenue Φ ∗ v , we let Φ v beits corresponding perturbed inaccurate revenue. The pertur-bation error (cid:15) v = Φ v − Φ ∗ v lies in the range [ − ψ v , ψ v ] . Foreach state v , we quantize the error into a v + 1 potentialvalues [ − ψ v , ..., − ψ v /a v , , ..., ψ v /a v , ..., ψ v ] , where the error (cid:15) v = na v ψ v with probability ρ v n , n = 0 , ± , .. ± a v , and (cid:80) a v n = − a v ρ v n = 1 . This means that we have constructed a newCTMC in which each state v of the original CTMC is nowexpanded into a v + 1 states. The transition rate follows thefollowing equation: q v n v (cid:48) n (cid:48) = exp(0 . β (Φ ∗ v (cid:48) n (cid:48) − Φ ∗ v n )) ρ v (cid:48) n (cid:48) . (18)Based on the detailed balanced equation π v n q v n v (cid:48) n (cid:48) = π v (cid:48) n (cid:48) q v (cid:48) n (cid:48) v n , we have π v n exp( 12 β (Φ ∗ v (cid:48) n (cid:48) − Φ ∗ v n )) ρ v (cid:48) n (cid:48) = π v (cid:48) n (cid:48) exp( 12 β (Φ ∗ v n − Φ ∗ v (cid:48) n (cid:48) )) ρ v n , (19)which results in π v n exp( β Φ ∗ v (cid:48) n (cid:48) ) ρ v (cid:48) n (cid:48) = π v (cid:48) n (cid:48) exp( β Φ ∗ v n ) ρ v n . (20)Because (cid:80) v (cid:48) ∈ V (cid:80) a v (cid:48) n (cid:48) = − a v (cid:48) π v (cid:48) n (cid:48) = 1 , we obtain π v n = exp( β Φ ∗ v n ) ρ v n (cid:80) v (cid:48) ∈ V (cid:80) a v (cid:48) n (cid:48) = − a v (cid:48) exp( β Φ ∗ v (cid:48) n (cid:48) ) ρ v (cid:48) n (cid:48) . (21)Letting σ v (cid:48) = (cid:80) a v (cid:48) n (cid:48) = − a v (cid:48) ρ v (cid:48) n (cid:48) exp( β n (cid:48) a v (cid:48) ψ v (cid:48) ) , the distribution ofthe new perturbed CTMC will be π v = a v (cid:88) n = − a v π v n = σ v exp( β Φ ∗ v ) (cid:80) v (cid:48) ∈ V σ v (cid:48) exp( β Φ ∗ v (cid:48) ) . (22)We use the Total Variation Distance [26], [27] as a metric toquantify the optimality gap between the stationary distributionf the perturbed CTMC π v and π ∗ v , the optimal solution of VP-approx , as follows: d T V ( π ∗ v , π v ) = 12 (cid:88) v ∈ V | π ∗ v − π v | . (23)With the stationary distribution of the perturbed CTMC π v ,we use a result in [27], which proved that the total variationdistance is bounded as follows d T V ( π ∗ v , π v ) ≤ − exp( − βψ max ) , (24)where ψ max = max v ψ v , the largest perturbation error amongstates v . The revenue gap is hence bounded as follows: | π ∗ v Φ ∗ v − π v Φ v |≤ max (1 − exp( − βψ max )) , (25)where Φ max = max v Φ v .The upper bound on both the Total Variation Distancebetween the two distributions d T V ( π ∗ v , π v ) and the optimalitygap | π ∗ v Φ ∗ v − π v Φ v | is independent with respect to ρ v n , thedistribution of perturbed revenues, and is independent with re-spect to | V | , the total number of configurations. This indicatesthat the optimality gap does not increase with the networksize and number of configurations | V | . Besides this, usingMarkov Approximation enables us to perform a distributedimplementation on this large combinatorial problem.VII. S IMULATION R ESULTS
In this section, we evaluate the performance of our com-bined mechanisms cMAP : MAP (which solves the VM place-ment problem) along with either
OPA or PUFF (which solvethe normalized pricing problem), and provide some insights.
A. Convergence, and insights on pricing
Firstly, we consider a network in which there are 5 BSs, and10 VMs being distributed amongst these 5 BSs. The 5 basestations have (2 , , , , users respectively. We set r k,i , thenumber of VMs required by user i at BS k , to be between1 to 3 VMs. u k,i , the willingness to pay of user i at BS k ,follows uniform distribution U [0 , . Fig. 1. Convergence of the cMAP . Under this setup, we run cMAP (the combined MarkovApproximation VM Placement Algorithm) along with auction
OPA . for different values of β . We plot the running averageover a window size of jumps, in comparison with theoptimal value, as seen in Fig 1. The optimal value is obtainedby exhaustive search, evaluating the solution to MPP overall combinations of v . As seen, for β = 50 , we are ableto achieve optimality. For β = 10 , the converged stationarydistribution over configurations of v is near optimal. Under β = 10 , the top 5 most common states are v = (2 , , , , , (2 , , , , , (2 , , , , , (3 , , , , , (2 , , , , , whichare best able to meet the total demand of (2 , , , , . Noticethat as β increases, performance improves: the running averageis closer to the optimal point, and fluctuations decrease. Thefluctuations occur because under our Markov Approximation-inspired algorithm, we converge not to a specific state ofthe CTMC, but to a stationary distribution over the states ofthe CTMC. Recall that the converged stationary distributionhas an optimality gap of β log | V | from the optimal point ofthe original problem VP . This shows that as β → ∞ , theperformance converges to the optimal value of VP . A potentialtradeoff in having a higher β is: if Φ ∗ v > Φ ∗ v (cid:48) , according to(12) there will be a lower rate of switching, and a higherprobability of staying in the current state. As β increases, thenetwork is more likely to stay in the current state. This maylead to a longer time spent in local minimums, due to the lackof exploration, and hence a longer convergence time.Next, with the current setup we compare the performanceof our proposed mechanisms MAP + OPA and
MAP + PUFF to the following baselines:1) Cooperative BS + Uniform Pricing: Under this scenario,the base stations are cooperative. They share the VMs witheach other, where the VMs are transferred within the networkvia our proposed
MAP . Unlike our proposed combined so-lution, here we use uniform pricing: a common price is setthroughout the network, regardless of the demand pattern. Abenefit of uniform pricing is that it is faster to implement.2) Non-cooperative BS + Auction: Under this scenario,the base stations are no longer cooperative - they do notshare the VMs with each other. We obtain the average resultunder the non-cooperative scenario, by averaging over all thepossible combinations of v . For each configuration v , we usethe optimal auction OPA to obtain Φ ∗ v .3) Non-cooperative BS + Uniform Pricing: Under thisscenario, the base stations not only do not share the VMswith each other, but also do not consider the demand pattern,using a common price throughput the network.We plot the revenue obtained under the various methods,and show how the performance varies when different pricesare set as the uniform price in Fig. 2. As seen, our proposedalgorithms cMAP outperforms the baselines, especially when OPA is used as the pricing mechanism. While
MAP in com-bination with
PUFF is not near-optimal, we have proved that
PUFF has a competitive ratio of . The baselines involvinguniform price perform best when the price is ”neutral” - neithertoo low nor too high. If the price is too high, the users (likelyhaving a lower willingness to pay) would not choose to use theVMs. If the price is too low, the revenue the network operator ig. 2. The effect of different uniform prices on revenue. obtains will be low. Fig. 2 also shows that resource sharingamong base stations increases the revenue. B. A larger setup, with insights on willingness to pay and thedemand-supply ratio
Next, we enlarge our setup and compare the performance ofour proposed mechanisms with the different baselines. In thissetup, there are 20 VMs shared amongst the 5 base stations.The number of users at each base station are randomized,along with r k,i , the number of VM units each user requests.We let the users’ willingness to pay u k,i follow a uniformdistribution U [ a, b ] . Fig. 3. The impact of willingness to pay on revenue.
In Fig. 3, we show the impact of users’ willingness to payon the revenue. The range of u k,i is adjusted, from the uniformdistribution U [0 , . (low willingness to pay), to U [0 . , . , U [0 . , . and U [0 . , (high willingess to pay). Our propsedsolution MAP + OPA (with β = 10 ) outperforms thebaselines, obtaining a near-optimal revenue. Our results showthat on average, having base station cooperation increases therevenue by up to percent. As seen in Fig. 3, when theusers have a higher willingness to pay, the revenue increases.Notice that uniform pricing ( p = 0 . ) does not perform well,when the users have low willingness to pay.Fig. 4 illustrates the impact of revenue when the DemandSupplyratio is varied. Supply is fixed at VMs, while demand isincreased, from D = 7 (low demand), to D = 21 (near equal Fig. 4. The impact of the Demand/Supply ratio on revenue. demand and supply) and high demand D = 38 . Our solution cMAP outperforms the baselines, especially when demandincreases, as the supply of VMs is shifted around the networkto meet demand more effectively, and an optimal auction isused to extract the highest revenue possible. Our results showthat on average, having base station cooperation increases therevenue by up to . As seen in Fig. 4, as the DemandSupply ratioincreases, revenue increases because more units of demand arebeing met. Once the DemandSupply ratio hits 1, revenue no longerincreases much due to the lack of global supply in the system.VIII. C ONCLUSIONS
In this paper, we have addressed the load-unbalanced prob-lem in MEC systems, by jointly optimizing the VM placementand pricing across base stations. Specifically, we have formu-lated a revenue maximization problem from the network oper-ator’s perspective, which was decomposed to a VM placementmaster problem and a normalized pricing slave problem. Theobjective function of the master problem is the optimal valueof the slave problem. Then, we solved the master problemby designing a CTMC and solved the slave problem byproposing auctions considering users’ truthful and untruthfulbehaviors, respectively. By combining the algorithms proposedfor the master and slave problem, cMAP is implementedfor VM placement and pricing decision making across basestations. Through theoretical analysis, we give the optimalgap of cMAP , which is nested with
OPA (auction mechanismwith users’ truthful behaviors) and
PUFF (auction machanismwith users’ untruthful behaviors), respectively. Finally, wedemonstrated the convergence and efficiency of cMAP . Forfuture work we will analyse the impact of factors like havinga heterogeneous cost of VM migration between base stations.R
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