Computing Resource Allocation of Mobile Edge Computing Networks Based on Potential Game Theory
Heng Liu, Haoming Jia, Jiaqi Chen, Xiaohu Ge, Yonghui Li, Lin Tian, Jinglin Shi
aa r X i v : . [ c s . N I] J a n Computing Resource Allocation of Mobile EdgeComputing Networks Based on Potential GameTheory
Heng Liu † , Haoming Jia † , Jiaqi Chen † , Xiaohu Ge †∗ , Yonghui Li ‡ , Lin Tian § , Jinglin Shi §¶† School of Electronic Information and CommunicationsHuazhong University of Science and Technology, Wuhan, China ‡ School of Electrical and Information EngineeringUniversity of Sydney, Sydney, NSW 2006, Australia § Beijing Key Laboratory of Mobile Computing and Pervasive DevicesInstitute of Computing Technology, Chinese Academy of Sciences, China ¶ University of Chinese Academy of Sciences, ChinaContact Email: [email protected]
Abstract —Mobile edge computing (MEC) networks are oneof the key technologies for ultra-reliability and low-latencycommunications. The computing resource allocation solutionneeds to be carefully designed to guarantee the computingresource efficiency of MEC networks. Based on the potentialgame theory, a computing resource allocation solution is proposedto reduce energy consumption and improve computing resourceefficiency in MEC networks. The computing resource allocationsolution includes two parts: the first part is the power controlscheme based on the potential game theory and the second partis the computing resource allocation scheme based on linearprogramming. The power control scheme is to find a set of thetransmission powers of base stations (BSs) that maximizes thepotential function of MEC networks. The computing resourceallocation scheme is to maximize the average computing resourceallocation coefficient of the MEC networks based on the results ofthe power control scheme. Compared with traditional solutions,simulation results indicate the computing resource utilization andenergy efficiency of the proposed computing resource allocationsolution are significantly improved.
Index Terms —mobile edge computing, potential game, powercontrol, computing resource allocation, PSO algorithm
I. I
NTRODUCTION
The exponential increment of data traffic, the sustainedgrowth of terminals, as well as more and more diverse servicescenarios, increase the pressure of the fourth generation (4G)cellular networks, which has led to the advent of the fifth gen-eration (5G) cellular networks [1] [2]. The MEC technologyis a promising solution for 5G networks, which can providethe complex computing capability at the radio access network(RAN) [3] [4]. The function of the cloud data center is sunkto the edge of cellular networks by MEC technologies, whichprovide users with some functions of the core network suchas computing, storage and communication resources in basestations (BSs) at the edge of wireless networks. However, theinterference among adjacent BSs not only influences wirelesstraffic transmissions but also affects the resource allocation in MEC networks. It is an important challenge for resourceallocation optimization in MEC networks.In the literature about resource allocation in MEC networks,a joint caching and offloading mechanism was proposed toupload uncached computation results as well as downloadcomputation result at BSs [5]. However, this mechanism isnot suitable for applications involving with large compu-tational demands and time-critical requirements as well aslarge-scale computational results, such as augmented reality,interactive online gaming, and multimedia conversions. Anenergy-efficient resource allocation scheme was presented fora multi-user MEC system with inelastic computation tasksand non-negligible task execution durations [6]. However, thisscheme only focuses on reducing energy consumption andneglects resource allocation coefficient in MEC systems. Afair resource-allocation scheme was proposed to maximizethe total throughput of a wireless network when each users’transmission rate is constrained with the minimum transmis-sion rate [7]. Based on the double-sided auction game, anefficient resource allocation scheme with limited resourcesbetween suppliers and consumers was proposed in [8]. Anew distributed resources block (RB) and power allocation(PA) algorithm based on non-cooperative game theory waspresented to improve the energy efficiency of MEC networks[9]. However, the computing resource allocation utilization andenergy efficiency of MEC networks have not been simultane-ously investigated in existing studies.This paper focuses on reducing the energy consumptionand improving the computing resource allocation utilizationin MEC networks. The main contributions of this paper issummarized as follows:1) To improve the computing resources utilization and en-ergy efficiency of MEC networks, a computing resourceallocation solution are proposed in this paper. Simulationresults show that the proposed solution can save energyconsumption and improve resource utilizations.) To reduces the energy consumption of MEC networks,a power control scheme is developed based on thepotential game theory.3) To improve the average computing resource allocationcoefficient of MEC networks, a new computing resourceallocation scheme is developed based on the linearprogramming.The rest of this paper is arranged as follows. Section IIgives the system model. Section III describes the proposedcomputing resource allocation solution. Simulation results andanalysis is presented in Section IV. Section V draws theconclusions. II. SYSTEM MODEL
Fig. 1. Mobile edge computing system model.
Without loss of generality, one MEC server and K BSs areconfigured for a MEC network. P k denotes the transmissionpower of the k − th BS ( k ∈ B = { , , ..., K } ). Themaximum transmission power of each BS is denoted as P max . All users are assumed to be governed by a uniformlydistribution with the density ρ . The coverage of the k − th BSis configured as a circle with the radius of r k . The maximumcoverage radius is denoted as r M when the transmission poweris P max . The distance between the m − th BS and the n − thBS is denoted by R mn . The required computing resourcesfor the k − th BS is denoted by f BSk and the computingresources actually allocated to the k − th BS is denoted by s BSk . Assumed that the required computing resources of allusers are configured as a constant f UE . The total number ofactual computing resources for the MEC server is denoted by S . The system model of MEC networks is illustrated in Fig.1. The coverage probability of BSs is expressed as [10] p c ( T, α ) = P [ SIN R > T ] , (1)where T is the threshold of the signal-to-interference-and-noise ratio (SINR), α is the path loss exponent.Based on the results in [11]–[13], the SINR of the k − thBS is expressed as SIN R k = hr − αk P k σ + I k , (2a) I k = X m ∈ B,m = k P m R − αmk , (2b)where h is the channel fading assumed to be an exponentiallydistributed random variable with parameter µ ( µ > [14], σ is the noise power in wireless channels.As a consequence, the coverage radius r k is derived as SIN R k = hr − αk P k σ + I k = T ⇒ r k = ( hP k T ( σ + I k ) ) α . (3)Furthermore, the cumulative distribution function (CDF) ofthe coverage radius r k is expressed as F r k ( r ) = P ( r k < r ) = P ( h < T ( σ + I k ) r α P k )= 1 − e − µT ( σ Ik ) rαPk . (4)Moreover, the probability density function (PDF) of thecoverage radius r k is expressed as f r k ( r ) = dF r k ( r ) dr = αµT ( σ + I k ) r α − P k e − µT ( σ Ik ) rαPk . (5)In the end, the required computing resources of the k − thBS is derived as f BSk = Z r M f UE ρf r k ( r )2 πrdr = 2 f UE ρπ αµT ( σ + I k ) P k Z r M r α e − µT ( σ Ik ) Pk r α dr. (6) III. A
LLOCATION SOLUTION OF C OMPUTING R ESOURCE
Based on the potential game theory, a computing resourceallocation solution is proposed in this section to reduce energyconsumption and improve computing resource efficiency inMEC networks. The computing resource allocation solutionincludes two parts: the first part is the power control schemebased on the potential game theory and the second part isthe computing resource allocation scheme based on linearprogramming.
A. Power Control Scheme Based on Potential Game1) Game Formulation:
In this paper the power controlproblem is modeled as an exact potential game model. In thisexact potential game model, related utility function of BSs andpotential function of the MEC networks are shown below.The proposed game model is denoted as
Γ = { B, P = { P k } k ∈ B , { u k } k ∈ B } , where B is the set of players, P is thestrategy vector of which the element P k denotes the transmitpower of the player k . For the MEC networks, the powercontrol problem can be described as P ∗ = arg max P (Φ( P )) , (7)where Φ( P ) is the potential function of the MEC networks,which represents the attainable maximum required computingresources considering interfering BSs.otential game is a common game in communication net-works. A game can be regarded as the potential game if theinfluence on the global utility caused by the change in players’strategy is modeled as a single global function. Such singleglobal function is regarded as the potential function.The expression of an exact potential game in [15] is givenas Φ( t ′ k , t − k ) − Φ( t k , t − k ) = u ( t ′ k , t − k ) − u ( t k , t − k ) , (8)where t k is the strategy of player k , t − k is the strategy ofall players except k . Φ( t k , t − k ) is the potential function ofMEC networks, which denotes the overall benefit of the MECnetworks. As the player’s individual utility function, u ( t k , t − k ) denotes individual benefit of each BS. According to (8), theincrement of the potential function of the MEC networks isequal to the increment of the individual utility function of oneplayer, caused by the change in the strategy of the player.Based on [16], the individual utility function represent thedifference between the benefit and cost of the MEC networks.The benefit of the k − th BS in the MEC networks is denotedas the attainable maximum required computing resources withno interference, which is expressed as f BSk, max = 2 f UE ρπ αµT σ P k Z r M r α e − µTσ Pk r α dr. (9)The cost of the k − th BS consists of two parts. One isthe reduction of the required computing resources caused byintroducing the interference from the m − th BS ( m = k ),denoted as I k,m . The other part is the reduction of the requiredcomputing resources considering the interference received bythe m − th BS from the k − th BS, denoted as I m,k . Basedon (6), I k,m is expressed as I k,m = 2 f UE ρπ αµTP k ( σ Z r M r α e − µTσ Pk r α dr − ( σ + P m R − αmk ) Z r M r α e − µT ( σ PmR − αmk ) Pk r α dr ) , (10)where P m is the transmit power of the m − th BS, and R mk isthe distance between the m − th BS with the k − th BS. Andthe estimated reduction of the required computing resourcesof the m − th BS caused by the k − th BS is expressed as I m,k = 2 f UE ρπ αµTP m ( σ Z r M r α e − µTσ Pm r α dr − ( σ + P k R − αkm ) Z r M r α e − µT ( σ PkR − αkm ) Pm r α dr ) , (11)where R km is the distance between the k − th BS and the m − th BS. And R km = R mk .Furthermore, the utility function of the k − th BS can beexpressed as u ( t k , t − k ) = 2 f UE ρπ αµT σ P k Z r M r α e − µTσ Pk r α dr − εK − X m ∈ B,m = k ( I k,m + I m,k ) , (12) where ε is a constant for balancing required computing re-sources with interference.The potential function (i.e. overall benefit of the MECnetworks) the weighted sum of individual utility functions ofall BSs, expressed as Φ( P ) = X k ∈ B (2 f UE ρπ αµT σ P k Z r M r α e − µTσ Pk r α dr − εK − b X m ∈ B,m = k I k,m +(1 − b ) X m ∈ B,m = k I m,k )) , (13) where b is a constant for building exact potential game.Based on (12) and (13), the game is proved to be anexact potential game in the appendix. Monderer and Shapleydemonstrated the theorem that each finite potential game hasat least one pure strategy NE [17]. The theorem guarantee theexistence of NE for the exact potential games [18].
2) PSO Based Potential Game:
The particle swarm opti-mization (PSO) algorithm is proposed to solve the potentialgame in Algorithm 1.Each particle in PSO algorithm represents a solution to aspecific problem. In other words, a particle is a point in amulti-dimensional search space in which we are attempting tofind an optimal location with respect to a fitness function [19].Parameters of PSO algorithm include group number N , max-imum iteration number Ger , inertia weight ω , self-learningfactor c , group learning factor c and search dimension d which is equal to the total number of BSs. The notations inthe PSO algorithm are shown in Table I [19]. TABLE IN
OTATION
Symbols Meanings x A set of positions (states) of N particles x i The position of particle i , which is a set of BSs’s transmit power U i The current fitness of particle ippm
The historical optimal position of particle iUppm
The historical optimal fitness of particle igpm
The historical optimal position of group
Ugpm
The historical optimal fitness of group
B. Computing Resource Allocation Scheme Based on LinearProgramming
In this section, we discuss the classification of the resultsobtained in the proposed PSO algorithm. The actually allo-cated computing resources at the k − th BS is denoted as s k .The computing resources required by the k − th BS f k can becalculated based on (6) and the optimal power control scheme. S indicates the total number of computing resources ownedby the MEC server. The classification is discussed as follows.1) P B f BSk ≤ S , indicates that the total number of comput-ing resources required by all BSs is less than or equalto the total number of actual computing resources.Then s BSk = f BSk , ∀ k ∈ B , the required computingresources of each BS can be satisfied.2) P B f BSk > S , indicates that the total number of comput-ing resources required by all BSs is more than the totalnumber of actual computing resources.o solve this problem, the resource allocation coefficientis proposed in this paper, which denotes the ratio of theactually allocated computing resources to the requiredcomputing resources. The resource allocation coefficientis denoted as
Sat = K P k ∈ B s
BSk f BSk . In order to maximizethe average computing resource allocation coefficient, aspecific linear programming problem is formulated as max
Sat = 1 K X k ∈ B s BSk f BSk s.t. ( P k ∈ B s BSk ≤ S ≤ s BSk ≤ f BSk , ∀ k ∈ B . (14)
Algorithm 1
PSO algorithm based on potential game Input:
Total number of base stations K ; maximum trans-mit power of BSs P max ; users density ρ ; unit user requiredcomputing resources f UE ; maximum coverage radius ofBSs r M ; total number of computing resources S . Initialize the swarm randomly for each particle i in the search space do
1) Initialize feasible position and velocity
2) Set position information ppm , U ppm , gpm and U gpm
3) Set the lower bound and upper bound of eachparameter end for while maximum iterations is not attained do for each particle i do
1) Calculate fitness value U i
2) Update
U ppm , ppm , U gpm and gpm if U i > U ppm then U ppm = U i ppm = x i if U i > U gpm then U gpm = U i gpm = x i end if end if end for for each particle i do
1) Calculate particle velocity: v = ω ∗ v + c ∗ rand () ∗ ( ppm − x ) + c ∗ rand () ∗ ( gpm − x )
2) Update particle position: x = x + v end for end while Output: gpm and
U gpm
IV. S
IMULATIONS A ND R ESULT A NALYSIS
In this section, the proposed computing resource allocationsolution is simulated in a grid-based system. The simulationresults of the proposed solution are analyzed and comparedwith the reference solution. Reference solution 1 is the clas-sical equal allocation solution [20], reference solution 2 is s BSk = min( f BSk , S/K ) . TABLE IIS
IMULATION P ARAMETERS
Parameter Value Parameter Value f UE ε ρ − users/m b µ N T Ger σ − W ω P max c r M c S The m × m square zone as the simulation scenario.And the main simulation parameters are shown in Table II[21] [22] [23] [24]. BS density(BSs/m ) -3 A v e r age u t ili t y f un c t i on =2.5 Proposed solution=2.5 Reference solution 1=3.5 Proposed solution=3.5 Reference solution 1 Fig. 2. Average utility function with respect to the BS density consideringdifferent path loss exponent.
BS density(BSs/m ) -3 A v e r age c o m pu t i ng r e s ou r c e s e ff i c i en cy ( C P U cyc l e s / ( b i t· W ) =2.5 Proposed solution=2.5 Reference solution 1=3.5 Proposed solution=3.5 Reference solution 1 Fig. 3. Average computing resource efficiency with respect to the BS densityconsidering different path loss exponent.
Fig. 2 shows the average utility function with respect tothe BS density considering different path loss exponents. Theaverage utility function of the proposed solution is comparedwith that of the reference solution 1. When the BS density isfixed, the average utility function increases with the increaseof the path loss exponent. When the path loss exponent isfixed, the average utility function decreases with the increaseof the BS density. The average utility function of the proposedsolution is larger than that of the reference solution 1. .8 1 1.2 1.4 1.6 1.8 2 2.2 2.4
BS density(BSs/m ) -3 A v e r age c o m pu t i ng r e s ou r c e a ll o c a t i on c oe ff i c i en t =2.5 Proposed solution=2.5 Reference solution 2=3.5 Proposed solution=3.5 Reference solution 2 Fig. 4. Average computing resource allocation coefficient with respect to theBS density considering different path loss exponent.
Fig. 3 shows the average computing resource efficiency(average required computing resources/transmit power of BSs)with respect to the BS density considering different path lossexponents. The average computing resource efficiency of theproposed solution is compared with that of the reference solu-tion 1. When the BS density is fixed, the average computingresource efficiency increases with the increase of the path lossexponent. When the path loss exponent is fixed, the averagecomputing resource efficiency decreases with the increase ofthe BS density. The average computing resource efficiencyof the proposed solution is larger than that of the referencesolution 1.Fig. 4 shows the computing resource allocation coefficientwith respect to the BS density considering different pathloss exponents. The computing resource allocation coefficientdecreases with the increase of the BS density.The performance of the proposed solution simulated in Fig.2, Fig. 3 and Fig. 4 is better than the performance of thereference solutions for the MEC networks.V. C
ONCLUSION
A computing resource allocation solution for MEC networksis proposed in this paper. The optimization model is con-structed, and the potential game theory is explored to guaranteethe convergence of utility function. This solution included thepower control scheme based on potential game theory andthe resource allocation scheme based on linear programming.Finally, the proposed computing resource allocation solutionis evaluated by using a grid-based system. Simulation resultsshow that compare with traditional solutions, the computingresource utilization and energy efficiency of the proposed com-puting resource allocation solution are significantly improved.So the proposed computing resource allocation solution isapplicable under energy saving scene. We hope that thecomputing resource allocation solution proposed in this papercan promote the development of saving energy and computingresource allocation in MEC networks in the future. ACKNOWLEDGMENTThe authors would like to acknowledge the support fromNational Key R & D Program of China (2016YFE0133000):EU-China study on IoT and 5G (EXICITING-723227)A
PPENDIX : D
EMONSTRATION OF V ALIDITY FOR THE P ROPOSED P OTENTIAL G AME M ODEL
Now let us prove that the game is the exact potential game,which is equivalent to prove the equation in (8) holds. Firstly,the proposed potential function in (13) will be decomposed: Φ( t k , t − k )= 2 f UE ρπαµT X B ( σ P k Z r M r α e − µTσ Pk r α dr − ε bK − X m ∈ B,m = k ( σ P k Z r M r α e − µTσ Pk r α dr − σ + P m R − αmk P k Z r M r α e − µT ( σ PmR − αmk ) Pk r α dr ) − ε − bK − X m = k ( σ P m Z r M r α e − µTσ Pm r α dr − σ + P k R − αkm P m Z r M r α e − µT ( σ PkR − αkm ) Pm r α dr ))= 2 f UE ρπαµT ( σ P k Z r M r α e − µTσ Pk r α dr + ε bK − X m ∈ B,m = k σ + P m R − αmk P k Z r M r α e − µT ( σ PmR − αmk ) Pk r α dr + ε − bK − X m ∈ B,m = k σ + P k R − αkm P k Z r M r α e − µT ( σ PkR − αkm ) Pm r α dr − ε bK − X m = k σ P k Z r M r α e − µTσ Pk r α dr − ε − bK − X m = k σ P m Z r M r α e − µTσ Pm r α dr )+2 f UE ρπαµT X n ∈ B,n = k ( σ + P k R − αkn P n Z r M r α e − µT ( σ PkR − αkn ) Pn r α dr + ε − bK − σ + P n R − αnk P k Z r M r α e − µT ( σ PnR − αnk ) Pk r α dr − ε bK − σ P n Z r M r α e − µTσ Pn r α dr − ε − bK − σ P m Z r M r α e − µTσ Pm r α dr )+2 f UE ρπαµT X n ∈ B,n = k ( σ P n Z r M r α e − µTσ Pn r α dr − ε bK − X m = n,m = k ( σ P n Z r M r α e − µTσ Pn r α dr − σ + P m R − αmn P n Z r M r α e − µT ( σ PmR − αmn ) Pn r α dr ) − ε − bK − X m = n,m = k ( σ P m Z r M r α e − µTσ Pm r α dr − σ + P n R − αnm P m Z r M r α e − µT ( σ PnR − αnm ) Pm r α dr )) 2 f UE ρπαµT ( σ P k Z r M r α e − µTσ Pk r α dr + ε K − X m ∈ B,m = k σ + P m R − αmk P k Z r M r α e − µT ( σ PmR − αmk ) Pk r α dr + ε K − X m ∈ B,m = k σ + P k R − αkm P k Z r M r α e − µT ( σ PkR − αkm ) Pm r α dr − ε K − X m ∈ B,m = k σ P k Z r M r α e − µT σ Pk r α dr − ε K − X m ∈ B,m = k σ P m Z r M r α e − µTσ Pm r α dr )+2 f UE ρπαµT X n = k ( σ P n Z r M r α e − µTσ Pn r α dr − ε bK − X m = n,m = k ( σ P n Z r M r α e − µTσ Pn r α dr − σ + P m R − αmn P n Z r M r α e − µT ( σ PmR − αmn ) Pn r α dr ) − ε − bK − X m = n,m = k ( σ P m Z r M r α e − µTσ Pm r α dr − σ + P n R − αnm P m Z r M r α e − µT ( σ PnR − αnm ) Pm r α dr )) Since the items other than the first four items are not relatedto k , we can ignore them in the next calculation. Φ( t ′ k , t − k ) − Φ( t k , t − k )= 2 f UE ρπαµT ( σ P ′ k Z r M r α e − µTσ P ′ k r α dr + ε K − X m ∈ B,m = k σ + P m R − αmk P ′ k Z r M r α e − µT ( σ PmR − αmk ) P ′ k r α dr + ε K − X m ∈ B,m = k σ + P ′ k R − αkm P ′ k Z r M r α e − µT ( σ P ′ kR − αkm ) Pm r α dr − ε K − X m ∈ B,m = k σ P ′ k Z r M r α e − µTσ P ′ k r α dr − ε K − X m = k σ P m Z r M r α e − µTσ Pm r α dr ) − (2 f UE ρπαµT ( σ P k Z r M r α e − µTσ Pk r α dr + ε K − X m ∈ B,m = k σ + P m R − αmk P k Z r M r α e − µT ( σ PmR − αmk ) Pk r α dr + ε K − X m ∈ B,m = k σ + P k R − αkm P k Z r M r α e − µT ( σ PkR − αkm ) Pm r α dr − ε K − X m = k σ P k Z r M r α e − µTσ Pk r α dr − ε K − X m = k σ P m Z r M r α e − µTσ Pm r α dr ))= u ( t ′ k , t − k ) − u ( t k , t − k ) R EFERENCES[1] X. Ge, S. Tu, G. Mao, C. Wang and T. Han, ”5G Ultra-Dense CellularNetworks,” in
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