Mean-Field Game-Theoretic Edge Caching
Hyesung Kim, Jihong Park, Mehdi Bennis, Seong-Lyun Kim, Mérouane Debbah
CChapter 1
Mean-Field Game-TheoreticEdge Caching
Hyesung Kim , Jihong Park ,Mehdi Bennis , Seong-Lyun Kim , and Mérouane Debbah Mobile networks are envisaged to be extremely densified in 5G and beyond tocope with the ever-growing user demand [1, 2, 3, 4]. Edge caching is a keyenabler of such an ultra-dense network (UDN), through which popular con-tent is prefetched at each small base station (SBS) and downloaded with lowlatency [5, 6] while alleviating the significant backhaul congestion between adata server and a large number of SBSs [7]. Focusing on this, in this chapterwe study the content caching strategy of an ultra-dense edge caching network(UDCN) . Optimizing the content caching of a UDCN is a crucial yet challeng-ing problem. Due to the sheer amount of SBSs, even a small misprediction ofuser demand may result in a large amount of useless data cached in capacity-limited storages. Furthermore, the variance of interference is high due to shortinter-SBS distances [8], making it difficult to evaluate cached data downloadingrates, which is essential in optimizing the caching file sizes. To resolve these H. Kim was with the School of Electrical and Electronic Engineering, Yonsei University,and is currently with Samsung Research, 135-090 Seoul, Korea (email: [email protected]). J. Park is with the School of Information Technology, Deakin University, Geelong, VIC3220, Australia (email: [email protected]). M. Bennis is with the Centre of Wireless Communications, University of Oulu, 90014Oulu, Finland (email: mehdi.bennis@oulu.fi). S.-L. Kim is with the School of Electrical and Electronic Engineering, Yonsei University,120-749 Seoul, Korea (email: [email protected]). M. Debbah is with Université Paris-Saclay, CNRS, CentraleSupélec, 91190 Gif-sur-Yvette,France (e-mail: [email protected]) and the Lagrange Mathematical andComputing Research Center, 75007 Paris, France. a r X i v : . [ c s . I T ] J a n CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING problems, we first present a spatio-temporal user demand model in continuoustime, in which the long-term and short-term content popularity variations at aspecific location are modeled using the Chinese restaurant process (CRP) andthe Ornstein-Uhlenbeck (OU) process, respectively. Based on this, we aim todevelop a scalable and distributed edge caching algorithm by leveraging themean-field game (MFG) theory [9, 10].To this end, at first the problem of optimizing distributed edge cachingstrategies in continuous time is cast as a non-cooperative stochastic differentialgame (SDG). As the game player, each SBS decides how much portion of eachcontent file is prefetched by minimizing its long run average (LRA) cost thatdepends on the prefetching overhead, cached file downloading rates under inter-SBS interference, and overlapping cached files among neighboring SBSs, i.e.,content overlap. This minimization problem is tantamount to solving a partialdifferential equation (PDE) called the Hamilton-Jacobi-Bellman equation (HJB)[11]. The major difficulty is that the HJB solution of an SBS is intertwined withthe HJB solutions of other SBSs, as they interact with each other through theinter-SBS interference and content overlap. The complexity of this problem isthus increasing exponentially with the number of SBSs, which is unfit for aUDCN. Alternatively, exploiting MFG, we decouple the SBS interactions in away that each SBS interacts only with a virtual agent whose behaviors followthe state distribution of the entire SBS population, known as mean-field (MF)distribution. For the given SBS population, the MF distribution is uniquely andderived by locally solving a PDE, called the Fokker-Planck-Kolmogorov equation(FPK). Consequently, the optimal caching problem at each SBS boils down tosolving a single pair of HJB and FPK, regardless of the number of SBSs. Suchan MF approximation guarantees achieving the epsilon Nash equilibrium [9, 12],when the number of SBSs is very large (theoretically approaching infinity) whiletheir aggregate interactions are bounded. Both conditions are satisfied in aUDCN [13, 14], mandating the use of MFG.To describe the MFG-theoretic caching framework and show its effectivenessfor a UDCN, this chapter is structured as follows. Related works on UDCNanalysis and MFG-theoretic approaches are briefly reviewed in chapter 1.2. Thenetwork, channel, and caching models as well as the spatio-temporal dynamicsof user demand, caching, and interference are described in chapter 1.3. Forthe SBS supporting a reference user, its optimal caching problem is formulatedand solved using MFG in chapter 1.4. The performance of the MFG-theoreticedge caching is numerically evaluated in terms of LRA and the content overlapamount in chapter 1.5, followed by concluding remarks in chapter 1.6.
Edge caching in cellular networks has received significant attention in 5G andbeyond [6, 7, 15]. In the context of MFG-theoretic edge caching in a UDN,we briefly review its preceding user demand models, interference analysis, andMFG-based applications as follows. .2. RELATED WORKS User Demand Model and Interference Analysis.
The effectivenessof edge caching is affected significantly by user demand according to contentpopularity. The user demand model in most of the works on edge cachingrelies commonly on the Zipf’s law. Under this model, the content popularityin the entire network region is static and follows a truncated power law [16],which is too coarse to reflect spatio-temporal content popularity dynamics in aUDCN. A time-varying user demand model has been considered in [17, 18] whileignoring spatial characteristics, which motivated us to seek for a more detaileduser demand model reflecting spatio-temporal content popularity variations.The spatial characteristics of interference dynamics has been analyzed in[19, 20] using stochastic geometry. These works however rely on a globallystatic user demand model, and thus ignore the temporal and local dynamics ofinterference [21]. By contrast, in this chapter we consider the spatio-temporalcontent popularity dynamics, and analyze their impact on interference.The impacts of SBS densification on interference in a UDN have been investi-gated in [8, 22, 23, 24, 25, 26, 27], in which the interference dynamics is governedby the spatial dynamics of user demand, i.e., locations [8]. While interesting,these works neglect temporal user demand variations. It is worth noting thata recent study [13] has considered spatio-temporal user demand fluctuations.However, it does not take into account temporal content popularity correlation.The gap has been filled by its follow-up work [28] that models the correlatedcontent popularity using the CRP, which is addressed in this chapter.
MFG Applications.
The MFG theory is built upon an asymptoticallylarge number of agents in a non-cooperative game. This fits naturally with aUDN within which assuming an infinite number of SBSs becomes reasonable[1, 2, 8, 29]. In this respect, SBS transmit power control and the user mobilitymanagement in a UDN have been studied in [22, 23]. For a massive number ofdrones, their rate-maximizing altitude control and collision-avoid path planninghave been investigated in [30] and [31, 32, 33, 34], respectively. In a similar vein,in this chapter we utilize the MFG theory to simplify the spatio-temporal anal-ysis on interference and content overlap in a UDCN. One major limitation ofMFG-based methods is that solving a pair of HJB-FPK PDEs may still be chal-lenging when the agent’s state dimension is large. In fact, existing PDE solversrely mostly on the Euler discretization method such that the derivatives in aPDE are approximated using finite differences. To guarantee the convergenceof a numerical PDE solution, the larger state dimension is considered, the finerdiscretization step size is required under the Courant-Friedrichs-Lewy (CFL)condition [35], increasing the computing complexity. To avoid this problem,in this chapter we describe the state of each SBS separately for each contentfile, reducing the dimension of each PDE. Alternatively, machine learning meth-ods have been applied in recent works [32, 33, 36] by which solving a PDE isrecast as a regression learning problem. By leveraging this method, incorporat-ing large-sized edge caching states (e.g., joint optimization of transmit power,caching strategy, and mobility management) could be an interesting topic forfuture research.
CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING
In this section, we describe a downlink UDN under study, followed by its com-munication channel and caching models.
Network Model.
SBSs and their users are independently and uniformlydistributed in a two-dimensional plane R with finite densities, forming twoindependent Poisson point processes (PPPs) [8, 37]. Following [8], the networkis assumed to be a UDN such that SBS density λ b is much higher than userdensity λ u , i.e., λ b (cid:29) λ u . In this UDN, the i -th user is located at the coordinates y i ∈ R , and receives signals from multiple SBSs located within a reception ball b ( y i , R ) centered at y i with radius R > , as depicted in Fig. 1. The radius R can be determined based on the noise floor so that the average received signalpower should be larger than the noise floor. When R → ∞ , the receptionball model becomes identical to a conventional PPP based network model instochastic geometric analysis [37, 38]. Channel and Antenna Pattern Models.
The transmitted signals fromSBSs experience path-loss attenuation and multi-path fading. Specifically, thepath loss l k,i from the k -th SBS located at z k ∈ R to the i -th user at y i ∈ R is given as l k,i = min(1 , || z k − y i || − α ) , where α > is the path-loss exponent.The transmitted signals experience an independent and identically distributedfading with the coefficient g k,i ( t ) . We assume that the fading coefficient is nottemporally correlated. Consequently, the received signal power at the i -th useris given as S ( t ) = P | h k,i ( t ) | , where P denotes the transmit power of every SBS,and h k,i is the channel gain determined by | h k,i ( t ) | = l k,i | g k,i ( t ) | . Next, thetransmission of each SBS is directional using N a antennas. Following [39], thebeam pattern follows a sectored uniform linear array model, in which the centerof the mainlobe beam points at the receiving user. The mainlobe gain is givenas N a with the beam width θ N a = 2 π/ √ N a while ignoring side lobes. Caching Model.
Consider a set M of M content files in total, each ofwhich is encoded using the maximum distance separable dateless code [40]. Attime t , a fraction p k,j ( t ) ∈ [0 , of the j -th content file with the size L j isprefetched to the k -th SBS from a remote server through a capacity-limitedbackhaul link as shown in Fig. 1.1. The SBS is equipped with a data storage ofsize C k,j assigned for the content file j , and therefore we have p k,j ( t ) L j ≤ C k,j .Each user in the network requests the j -th content file with probability x j .Within the user’s content request range R c > , there exists a set N of N SBSs [8]. If multiple SBSs cached the requested file (i.e., cache hitting), then theuser downloads the file from a randomly selected SBS. If there is no SBS cachedthe requested file (i.e., cache missing), then the file is downloaded to a randomlyselected SBS from the remote server via the backhaul, which is then deliveredto the user from the SBS. At time t , the goal of the k -th SBS is to determineits file caching fraction vector p k ( t ) = { p k, ( t ) , · · · , p k,j ( t ) , · · · p k,M ( t ) } . .3. SYSTEM MODEL The effectiveness of caching strategies is affected by spatio-temporal dynamicsof content popularity among users, backhaul and storage capacities in SBSs,and interference across SBSs, as we elaborate next.
User Demand Dynamics.
The user demand on content files is oftenmodeled as a Zipf distribution [16]. Such a long-term user demand patternin a wide area is too coarse to capture the spatial demand and its temporalvariations [21], calling for a detailed spatio-temporal user demand model for aUDCN. To this end, we consider that each SBS regularly probes the contentpopularity within the distance R s , and the content popularity for each SBSfollows an independent stochastic process. For the content popularity of eachSBS, its temporal dynamics is described by the long-term fluctuations acrosstime t = T, T, · · · , κT and short-term fluctuations over t ∈ [( κ − T, κT ] asconsidered in [41]. These long-term and short-term content popularity dynamicsare modeled using the Chinese restaurant process (CRP) and the Ornstein-Uhlenbeck (OU) process, respectively, as detailed next.Following the CRP [5, 42], the long-term content popularity variations aredescribed by the analogy of the table selection process in a Chinese restaurant.Here, a UDCN becomes a Chinese restaurant, wherein the content files andusers are the tables and customers in the restaurant, respectively. Treating therestaurant table seating problem as a long-term content popularity updatingmodel, we categorize content files into two groups: the set U rk ( κT ) of the filesthat have been requested by N k ( κT ) users at least once at SBS k until time κT ; and the set U uk ( κT ) of the files that have not yet been requested until then.For these two groups, the mean popularity µ k,j ( κT ) of the j -th content file atSBS k during t ∈ [( κ − T, κT ] is given by: µ k,j ( κT ) = (cid:40) n k,j ( κT ) − νN k ( κT )+ θ for j ∈ U rk ( κT ) ν | U rk ( κT ) | + θN k ( κT )+ θ for j ∈ U uk ( κT ) , (1.1)where n k,j ( κT ) is the number of accumulated downloading requests for the j -thcontent file at SBS k until time κT , and θ and ν are positive constants. Notethat each content file popularity depends on the popularity of other files and thenumber of other files. Consequently, more popular files are more often requested,proportionally to the previous request history n k,j ( κT ) , while unpopular filescan also be requested with a probability proportional to θ and ν . For simplicitywithout loss of generality, we omit the index κT of µ k,j ( κT ) , and focus only onthe case when κ = 1 hereafter.Next, for a given mean content popularity µ k,j , at time t during a short-termperiod t ∈ [0 ≤ t ≤ T ] , the content request probability x k,j ( t ) of the j -th fileat SBS k is described by the OU process [41], a stochastic differential equation(SDE) given as follows:d x k,j ( t ) = r ( µ k,j − x k,j ( t )) d t + η d W k,j ( t ) , (1.2) CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING where W k,j ( t ) is the Wiener process, and r and η are positive constants. Itdescribes that the short-term content popularity is drifted from the long-termmean content popularity µ k,j by x k,j ( t ) , and is randomly fluctuated by W k,j ( t ) .Fig. 1.1 illustrates the user demand pattern generated from the aforemen-tioned long-term and short-term content popularity dynamics. As observed bySBS 1 and SBS 2, for the same content files A, B, and C, these two spatiallyseparated SBSs have different popularity dynamics, while at the same SBS eachcontent popularity is updated according to a given temporal correlation. Fur-thermore, as shown in SBS 2 at around t = T , the previously unpopular file C can emerge as an up-do-date popular file. Caching Dynamics.
The remaining storage capacity varies according tothe instantaneous caching strategy. Let us assume that SBSs have finite storagesize and discard content files at a rate of e k,j from the storage unit in orderto make space for caching other contents. Considering the discarding rate, wemodel the evolution law of the storage unit at SBS k as follows:d Q k,j ( t ) = ( e k,j − L j p k,j ( t )) d t, (1.3)where Q k,j ( t ) denotes the remaining storage size dedicated to content j of SBS k at time t , and L j is data size of content j . Note that L j p k,j ( t ) representsthe data size of content j downloaded by SBS k at time t . Since each user candownload the requested file from one of multiple SBSs within its reception ball,for the given limited storage size, it is crucial to minimize overlapping contentcaching while maximizing the cache hitting rates, by determining the file cachingfraction p k,j ( t ) at SBS k . This problem is intertwined with other SBSs’ cachingdecisions, and the difficulty is aggravated under ultra-dense SBS deployment,seeking a novel solution with low-complexity using MFG to be elaborated inSec. 1.4. Interference Dynamics . In a UDN, there is a considerable number ofSBSs with no associated user within its coverage. These SBSs become idle anddoes not transmit any signal according to the definition of UDN ( λ b (cid:29) λ u ) [8].Hence, this dormant SBS does not cause interference to neighbor SBSs. Thisleads to a spatially dynamic distribution of interference characterized by users’locations. We assume that active SBSs have always data to transmit. Let usdenote the SBS active probability by p a . The aggregate interference is imposedby the active SBSs with probability p a . Assuming that p a is homogeneous overSBSs yields p a ≈ − [1 + λ u / (3 . λ b )] − . [44]. It provides that the densityof interfering SBSs is equal to p a λ b . Then, at the typical user selected uni-formly at random, the signal-to-interference-plus-noise ( SINR ) with N a numberof transmit antennas is given as: SINR ( t ) = N a P | h ( t ) | σ + θ Na π N a I f ( t ) . (1.4)where the aggregate interference I f ( t ) depends on the set Φ R ( p a λ b ) of ac-tive SBS coordinates within the reception ball of radius R , given by I f ( t ) = .3. SYSTEM MODEL (a) Spatial popularity dynamics of the network at t = 0 .(b) Temporal popularity dynamics at SBSs 1 and 2 during T . Figure 1.1:
An illustration of a UDCN and its intrinsic spatio-temporal popularitydynamics. (a) Spatially dynamics of popularity (b) Temporal dynamics where thecontent popularity changes for long-term and short-term duration. The long-termdynamics are captured by the Chinese restaurant process, which determines the meanpopularity for a certain time period of T . During this period, the instantaneouspopularity is captured by the mean reversion model following the OU process [43]. CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING (cid:80) | Φ R ( p a λ b ) | k P | h k,i ( t ) | . The term θ Na π N a in (1.4) is given by the directionalbeam pattern. Given uniformly distributed (i.e., isotropically distributed) users,an SBS becomes an interferer with probability θ N a / (2 π ) with the main lobe gain N a and the beam width θ N a = 2 π/ √ N a . Note that the interference term I f ( t ) depends on the spatial locations of SBSs and users (through p a ). This becomesa major bottleneck in anayzing UDCN, calling for a tractable way of handling I f ( t ) using MFG to be discussed in Sec. 1.4. We utilize the framework of non-cooperative games to devise a fully distributedalgorithm. The goal of each SBS k is to determine its own caching amount p ∗ k,j ( t ) for content j in order to minimize an LRA cost. The LRA cost is determined byspatio-temporally varying content request probability, network dynamics, con-tent overlap, and aggregate inter-SBS interference. As the SBSs’ states andcontent popularity evolves, the caching strategies of the SBSs must adapt ac-cordingly. Minimizing the LRA cost under the spatio-temporal dynamics canbe modeled as a dynamic stochastic differential game (SDG) [45]. In the fol-lowing subsection, we specify the impact of other SBSs’ caching strategies andinter-SBS interference in the SDG by defining the LRA cost. An instantaneous cost function J k,j ( t ) defines the LRA cost. It is affected bybackhaul capacity, remaining storage size, average rate per unit bandwidth,and overlapping contents among SBSs. SBS k cannot download more than B k,j ( t ) , defined as the allocated backhaul capacity for downloading content j at time t . In the proposed LRA cost, the download rate L j p k,j ( t ) is preventedfrom exceeding the backhaul capacity constraint B k,j ( t ) by the backhaul costfunction φ k,j as φ k,j ( p k,j ( t )) = − log( B k,j ( t ) − L j p k,j ( t )) . If L j p k,j ( t ) ≥ B k,j ( t ) ,the value of the cost function φ k,j goes to infinity. This form of cost functionis widely used to model barrier or constraint of available resources as in [18].As cached content files occupy the storage, it causes processing latency [46] ordelay to search requested files by users. This overhead cost is proportional to thecached data size in the storage unit. To incorporate this, a storage cost functionis proposed baed on the occupation ratio of the storage unit normalized by thestorage size as follows: ψ k,j ( Q k,j ( t )) = γ ( C k,j − Q k,j ( t )) /C k,j , (1.5)where Q k,j ( t ) is storage cost function at time t , and γ is a constant storage costparameter. Then, the global instantaneous cost is given by: J k,j ( p k,j ( t ) , p − k,j ( t )) = φ k,j ( p k,j ( t ))(1+ I rk,j ( p − k,j ( t ))) R k ( t, ˆ I f ( t )) x j ( t ) + ψ k,j ( Q k,j ( t )) , (1.6) .4. GAME THEORETIC FORMULATION FOR EDGE CACHING I rk,j ( p − k,j ( t )) denotes the expected amount of overlapping content perunit storage size, C k,j , p − k,j ( t ) is a vector of caching control variable of all theother SBSs except SBS k , ˆ I f ( t ) denotes the normalized aggregate interferencefrom other SBSs with respect to the SBS density and the number of antennas,and R k ( t ) is the average downlink rate per unit bandwidth. The cost increaseswith the amount of overlapping contents and aggregate interference, which aredescribed in the next subsection. From the global cost function (1.6), the LRAcaching cost is given by: J k,j = E (cid:34)(cid:90) Tt J k,j ( p k,j ( t ) , p − k,j ( t )) d t (cid:35) . (1.7) The caching strategy of an SBS inherently makes an impact on the caching con-trol strategies of other SBSs. These interactions can be defined and quantifiedby the amount of overlapping contents and interference. These represent majorbottlenecks for optimizing distributed caching for two reasons: first of all, theyundergo changes with respect to the before-mentioned spatio-temporal dynam-ics, and it is hard to acquire the knowledge of other SBSs’s caching strategiesdirectly. In this context, our purpose is to estimate these interactions in adistributed fashion without full knowledge of other SBSs’ states or actions.
Content Overlap.
As shown in Fig. 1.1a, in UDNs, there may be overlap-ping contents downloaded by multiple SBSs located within radius R c from therandomly selected typical user. For example, let us consider that these neigh-boring SBSs cache the most popular contents with the intention of increasingthe local caching gain (i.e., cache hit). Since only one of the SBS candidates isassociated with the user to transmit the cached content file, caching the identicalcontent of other SBSs becomes a waste of storage and backhaul usage. In thiscontext, overlapping contents increase redundant cost due to inefficient resourceutilization [47]. The amount of overlapping contents is determined by otherSBSs’ caching strategies. We define the content overlap function I rk,j ( p − k,j ( t )) as the expected amount of overlapping content per unit storage size C k,j , whichis given by: I rk,j ( p − k,j ( t )) = 1 C k,j N r ( j ) |N | (cid:88) i (cid:54) = k p i,j ( t ) , (1.8)where N r ( j ) denotes the number of contents whose request probability is asymp-totically equal to x j . It can be defined as cardinality of the following set: { m | m ∈ M s.t. | x m − x j | ≤ (cid:15) } . When the value of (cid:15) is sufficiently small, N r ( j ) becomes the number of contents whose request probability is equal to that ofcontent j . If there is a large number of contents with equal request probabilities,a given content is randomly selected and cached. Hence, the occurrence proba-bility of content overlap decreases with a higher diversity of content caching.0 CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING
Inter-SBS Interference.
In a UDN, user location determines the loca-tion of interferers, or the density of the user determines the density of interferingSBSs. It is because there are SBSs that have no users in their own coverageand become dormant without imposing interference to their neighboring SBSs.These spatial dynamics of interference in UDN is a bottleneck for optimizingdistributed caching such that an SBS in a high interference environment cannotdeliver the cached content to its own users. To incorporate this spatial interac-tion, following the interference analysis in UDNs [22], interference normalizedby SBS density and the number of antennas is given by: ˆ I f ( t ) = ( λ u πR ) N − a λ − α b (cid:18)
1+ 1 − R − α α − (cid:19) P E g [ | g ( t ) | ] , (1.9)where ˆ I f ( t ) denotes the normalized interference with respect to SBS densityand the number of antennas. It gives us an average downlink rate per unitbandwidth R k ( t ) and its upper bound in UDN as follows: R k ( t ) = E S,I f [log(1 + SINR ( t ))] (1.10) ≤ E S log S k ( t ) σ N a λ α/ b + E I f [ ˆ I f ( t )] , (1.11)where σ is the noise power. Note that inequation (1.11) shows the effect ofinterference on the upper bound of an average SE. It is because we considerthat only the SBSs within the pre-defined reception ball cause interference to atypical user. Hence, the equality in (1.11) holds, when the size of reception ball R goes to infinity, including all the SBSs in the networks as interferers. As the SBSs’ states and content popularity evolves according to the dynamics(1.2) and (1.3), an individual SBS’s caching strategy must adapt accordingly.Hence, minimizing the LRA cost under the spatio-temporal dynamics can bemodeled as a dynamic stochastic differential game (SDG), where the goal of eachSBS k is to determine its own caching amount p ∗ k,j ( t ) for content j to minimizethe LRA cost J k,j ( t ) (1.7): (P1) v k,j ( t ) = inf p k,j ( t ) J k,j ( t ) (1.12)subject to d x j ( t ) = r ( µ − x j ( t )) d t + η d W j ( t ) , (1.13)d Q k,j ( t ) = ( e k,j − L j p k,j ( t )) d t. (1.14)In the problem P1 , the state of SBS k and content j at time t is defined as s k,j ( t ) = { x j ( t ) , R k ( t ) , Q k,j ( t ) } , ∀ k ∈ N , ∀ j ∈ M . The stochastic differentialgame (SDG) for edge caching is defined by ( N , S k,j , A k,j , J k,j ) where S k,j isthe state space of SBS k and content j , A k is the set of all caching controls { p k,j ( t ) , ≤ t ≤ T } admissible for the state dynamics. .4. GAME THEORETIC FORMULATION FOR EDGE CACHING P1 , the long-term average of content request proba-bility µ is necessary for the dynamics of content request probability (1.13). Todetermine the value of µ , the mean value m k ( t ) of the cardinality of the set U rk ( t ) needs to be obtained. Although the period { ≤ t ≤ T } is not infinite, weassume that the inter-arrival time of the content request is sufficiently smallerthan T and that numerous content requests arrive during that period. Then,the long-term average of content request probability µ becomes an asymptoticmean value ( t → ∞ ) . Noting that (cid:80) j n k,j ( t ) = N k ( t ) , the mean value of m k ( t ) is asymptotically given by [48] as follows: (cid:104)| U rk ( t ) |(cid:105) (cid:39) (cid:40) Γ( θ +1) α Γ( θ + α ) N k ( t ) α for α > θ log( N k ( t ) + θ ) for α = 0 (1.15)where the expression (cid:104) . (cid:105) is the average value, and Γ( . ) is the Gamma function.The problem P1 can be solved by using a backward induction method wherethe minimized LRA cost v k,j ( t ) is determined in advance through solving thefollowing N coupled HJB equations. ∂ t v k,j ( t )+ inf p k,j ( t ) (cid:20) J k,j ( p k,j ( t ) , p − k,j ( t ))+ η ∂ xx v k,j ( t )+( e k,j − L j p k,j ( t )) (cid:124) (cid:123)(cid:122) (cid:125) ( A ) ∂ Q k v k,j ( t ) + r ( µ − x j ( t )) (cid:124) (cid:123)(cid:122) (cid:125) ( B ) ∂ x v k,j ( t ) (cid:21) (1.16)The HJB equations (1.16) for k = 1 , ..., N have a unique joint solution if thedrift functions defining temporal dynamics (A) and (B) and the cost function(1.6) are smooth [11]. Since the smoothness of them is satisfied, we can assurethat a unique solution of equation (1.16) exists. The optimal joint solution ofHJB equations achieves Nash equilibrium (NE) as the problem P1 is a non-cooperative game wherein players do not share their state or strategy [11, 12].The unique minimized cost v ∗ k,j ( t ) of the problem P1 and its corresponding NEcan be defined as follows: Definition 1 : The set of SBSs’ caching strategies p ∗ = { p ∗ ,j ( t ) , ..., p ∗ N,j ( t ) } ,where p ∗ k,j ( t ) ∈ A k,j for all k ∈ N , is a Nash equilibrium, if for all SBS k andfor all admissible caching strategy set { p ,j ( t ) , ..., p N,j ( t ) } , where p k,j ( t ) ∈ A k,j for all k ∈ N , it is satisfied that J k,j ( p ∗ k,j ( t ) , p ∗− k,j ( t )) ≤ J k,j ( p k,j ( t ) , p ∗− k,j ( t )) , (1.17)under the temporal dynamics (1.13) and (1.14) for common initial states x j (0) and Q k,j (0) .Unfortunately, this approach is accompanied with high computational com-plexity in achieving the NE (1.17), when N is larger than two because an in-dividual SBS should take into account other SBSs’ caching strategies p − k,j ( t ) to solve the inter-weaved system of N HJB equation (1.16). Furthermore, it2
CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING requires collecting the control information of all other SBSs including their ownstates, which brings about a huge amount of information exchange among SBSs.This is not feasible and impractical for UDNs. For a sufficiently large numberof SBSs, this problem can be transformed to a mean-field game (MFG), whichcan achieve the (cid:15) -Nash equilibrium [45].
To reduce the aforementioned complexity in solving the SDG P1 , the followingfeatures are utilized. When the number of SBSs becomes large, the influence ofevery individual SBS can be modeled with the effect of the collective (aggregate)behavior of the SBSs. MFG theory enables us to transform these multipleinteractions into a single aggregate interaction, called MF interaction, via MFapproximation. According to [43], this approximation holds under the followingconditions: (i) a large number of players, (ii) the exchangeability of players underthe caching control strategy, and (iii) finite MF interaction. If these conditionsare satisfied, the MF approximation can provide the optimal solution which theoriginal SDG achieves.The first condition (i) corresponds to the definition of UDNs. For condition(ii), players (i.e., SBSs) in the formulated SDG are said to be exchangeable orindistinguishable under the control p k,j ( t ) and the states of players and contentsif the player’s control is invariant by their indices and decided by only their ownstates. In other words, permuting players’ indices cannot change their owncontrol strategies. Under this exchangeability, it is sufficient to investigate andre-formulate the problem for a generic SBS by dropping its index k .The MF interactions (1.8) and (1.9) should asymptotically converge to a fi-nite value under the above conditions. The content overlap (1.8) in MF regime,called MF overlap, goes to zero when the number of contents per SBS is ex-tremely large, i.e. M (cid:29) N . Such a condition implies that the cardinality ofthe set consisting of asymptotically equal content popularity goes to infinity. Inother words, N r ( j ) goes to infinity yielding that the expected amount of over-lapping content per unit storage size I rk,j ( p − k,j ( t )) becomes zero. In terms ofinterference, the MF interference converges as the ratio of SBS density to userdensity goes to infinity, i.e. N a λ αb / ( λ u R ) → ∞ [22]. This condition corre-sponds to the notion of UDN [8] or massive MIMO ( N a → ∞ ). Thus, the MFapproximation can be utilized as the conditions inherently hold for UDCNs .To approximate the interactions from other SBSs, we need a state distribu-tion of SBSs and contents at time t , called MF distribution m t ( x ( t ) , Q ( t )) . TheMF distribution is derived from the following empirical distribution. M ( N × M ) t ( x ( t ) , Q ( t )) = 1 N M M (cid:88) j =1 N (cid:88) k =1 δ { x j ( t ) ,Q k ( t ) } (1.18)When the number of SBSs increases, the empirical distribution M ( N × M ) t ( x j ( t ) , Q ( t )) converges to m t ( x j ( t ) , Q ( t )) , which is the density of contents and SBSs in state .4. GAME THEORETIC FORMULATION FOR EDGE CACHING ( x j ( t ) , Q ( t )) . Note that we omit the SE R ( t ) from the density measure to onlyconsider temporally correlated state without loss of generality.To this end, we derive a Fokker-Planck-Kolmogorov (FPK) equation [45]that is a partial differential equation capturing the time evolution of the MFdistribution m t ( x j ( t ) , Q ( t )) under dynamics of the popularity x j ( t ) and theavailable storage size Q ( t ) . The FPK equation for m t ( x j ( t ) , Q ( t )) subject tothe temporal dynamics (1.2) and (1.3) are given as follows: ∂ t m t ( x j ( t ) , Q ( t )) + r ( µ j − x j ( t )) ∂ x m t ( x j ( t ) , Q ( t ))+ ( e j − L j p j ( t )) ∂ Q m t ( x j ( t ) , Q ( t )) − η ∂ xx m t ( x j ( t ) , Q ( t )) . (1.19)Let us denote the solution of the FPK equation (1.19) as m ∗ t ( x j ( t ) , Q ( t )) . Ex-changeability and existence of the MF distribution allow us to approximate theinteraction I rk,j ( p − k,j ( t )) as a function of m ∗ t ( x j ( t ) , Q ( t )) as follows: I rj ( t, m ∗ t ( x j ( t ) , Q ( t ))) = (cid:90) Q (cid:90) x m ∗ t ( x j , Q ) p j ( t,x ( t ) ,Q ( t )) C k,j N r ( j ) d x d Q. (1.20)This interaction from (1.20) can be estimated without observing other SBSs’caching strategies. Thus, it is not necessary for an SBS to have full knowledgeof the states or the caching control policies of other SBSs. An SBS needs tosolve only a pair of equations, namely the FPK equation (1.19) and the followingmodified HJB one obtained by applying the MF approximation (1.20) to (1.16): ∂ t v j ( t )+ inf p j ( t ) (cid:20) J j ( p j ( t ) , I j ( t, m ∗ t ( x j ( t ) , Q ( t )))+ η ∂ xx v j ( t )+ ( e j − L j p j ( t )) ∂ Q v j ( t ) + r ( µ − x j ( t )) ∂ x v j ( t ) (cid:21) . (1.21)FPK equation (1.19) and HJB equation (1.21) are intertwined with eachother for the MF distribution and the optimal caching amount, which dependson the optimal trajectory of the LRA cost v ∗ j ( t ) . The optimal LRA cost v ∗ j ( t ) is found by applying backward induction to the single HJB equation (1.21).Also, its corresponding MF distribution (state distribution) m ∗ t ( x j ( t ) , Q ( t )) isobtained by forward solving the FPK equation (1.19). These solutions of HJBand FPK equations [ m ∗ t ( x j ( t ) , Q ( t )) , v ∗ j ( t )] define the mean-field equilibrium(MFE), defined as follows: Definition 2 : The generic caching strategies p ∗ j ( t ) achieves an MFE if for alladmissible caching strategy set { p ,j ( t ) , ..., p N,j ( t ) } where p k,j ( t ) ∈ A k,j for all k ∈ N it is satisfied that J j ( p ∗ j ( t ) , m ∗ t ( x j ( t ) , Q ( t )) ≤ J j ( p j ( t ) , m ∗ t ( x j ( t ) , Q ( t )) , (1.22)under the temporal dynamics (1.13) and (1.14) for an initial MF distribution m . The MFE corresponds to the (cid:15) -Nash equilibrium: J k,j ( p ∗ k,j ( t ) , p ∗− k,j ( t )) ≤ J j ( p ∗ j ( t ) , m ∗ t ( x j ( t ) , Q ( t )) − (cid:15), (1.23)4 CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING where (cid:15) asymptotically becomes to zero for a sufficiently large number of SBSs.Let us define p ∗ j ( t ) as an optimal caching control strategy which achieves theMFE yielded by the optimal caching cost trajectory v ∗ j ( t ) and MF distribution m ∗ t ( x j ( t ) , Q ( t )) . The solution p ∗ j ( t ) is given by the following Proposition. Proposition 1.
The optimal caching amount is given by: p ∗ j ( t ) = 1 L j (cid:34) B j ( t ) − I rj ( t, m ∗ t ( x j ( t ) , Q ( t ))) R ( t, I f ( t )) x j ( t ) ∂ Q v ∗ j (cid:35) + , (1.24) where m ∗ t ( x ( t ) , Q ( t )) and v ∗ j ( t ) are the solutions of (1.19) and (1.21) , respec-tively.Proof : The optimal control control of the differential game with HJB equationsis the argument of the infimum term (1.21) [11]. p ∗ j ( t ) = arg inf p j ( t ) (cid:20) J j ( p j ( t ) ,I j ( t, m ∗ t ( x j ( t ) , Q ( t )))+ η ∂ xx v j ( t )+( e j − L j p j ( t )) ∂ Q v j ( t ) + r ( µ − x j ( t )) ∂ x v j ( t ) (cid:21) (1.25)The infimum term (1.25) is a convex function of p j ( t ) for all time t , since its firstand second-order derivative are lower than zero. Hence, we can apply Karush-Khun-Tucker (KKT) conditions and get a sufficient condition for the uniqueoptimal control p ∗ j ( t ) by finding a critical point given by: ∂∂p j ( t ) (cid:2) J j ( p j ( t ) , I j ( t, m ∗ t ( x j ( t ) , Q ( t )))+( e j − L j p j ( t )) ∂ Q v j ( t ) (cid:3) = 0 . (1.26)Due to the convexity, the solution of equation (1.26) is the unique optimalsolution described as follows: p ∗ j ( t ) = 1 L j (cid:34) B j ( t ) − I rj ( t, m ∗ t ( x j ( t ) , Q ( t ))) R ( t, I f ( t )) x j ( t ) ∂ Q v ∗ j (cid:35) + . (1.27)Remark that p ∗ j ( t ) is a function of m ∗ t ( x j ( t ) , Q ( t )) and v ∗ j , which are solutionsof the equations (1.19) and (1.21), respectively. The expression of p ∗ j ( t ) (1.26)provides the final versions of the HJB and FPK equations as follows: .4. GAME THEORETIC FORMULATION FOR EDGE CACHING ∂ t v j ( t ) − log (cid:16) B j ( t ) − (cid:104) B j ( t ) − I rj ( t,m ∗ t ( x j ( t ) ,Q ( t ))) R ( t,I f ( t )) x j ( t ) ∂ Q v j (cid:105) + (cid:17) R ( t, I f ( t )) x j ( t ) × (1+ I rj ( t,m ∗ t ( x j ( t ) ,Q ( t )))) + α ( C − Q ( t )) C + r ( µ j − x j ( t )) ∂ x v j ( t )+ (cid:32) e j − (cid:20) B j ( t ) − I rj ( t, m ∗ t ( x,Q )) R ( t, I f ( t )) x j ( t ) ∂ Q v j (cid:21) + (cid:33) ∂ Q v j ( t )+ η ∂ xx v j ( t ) , ∂ t m t ( x j ( t ) , Q ( t )) + r ( µ j − x j ( t )) ∂ x m t ( x j ( t ) , Q ( t )) − η ∂ xx m t ( x j ( t ) , Q ( t ))+ (cid:32) e j − (cid:34) B j ( t ) − I rj ( t, m t ( x j ( t ) , Q ( t ))) R ( t, I f ( t )) x j ( t ) ∂ Q v ∗ j (cid:35) + (cid:33) ∂ Q m t ( x j ( t ) ,Q ( t )) . From these equations, we can find the values of v ∗ j ( t ) and m ∗ t ( x ( t ) , Q ( t )) . Notethat the smoothness of the drift functions and in the dynamic equation and thecost function (1.6) assures the uniqueness of the solution [11]. (cid:4) Proposition 1 provides the optimal caching amount of p ∗ j ( t ) is in a water-filling fashion of which water level is determined by the backhaul capacity B j ( t ) .Noting that the average rate per unit bandwidth R ( t ) increases with the numberof antenne N a and SBS density λ b , SBSs cache more contents from the serverwhen they can deliver content to users with high wireless capacity. Also, SBSsdiminish the caching amount of content j , when the estimated amount of contentoverlap I rj ( t, m ∗ t ( x j ( t ) , Q ( t ))) is large.Remark that the existence and uniqueness of the optimal caching controlstrategy are guaranteed. The optimal caching algorithm converges to a uniqueMFE, when the initial conditions m , x j (0) , and Q (0) are given. The specificprocedure of this MF caching algorithm is described in the following Algo-rithm 1.The respective processes of solving P1 in ways of SDG and MFG are depictedin Fig. 1.2. Remark that the solution of the MFG becomes equivalent to thatof the N -player SDG P1 as N increases. The complexity of the proposedmethod is much lower compared to solving the original N -player SDG P1 . Thenumber of PDEs to solve for one content is reduced to two from the number ofSBSs N . Thus, the complexity is consistent even though the number of players N becomes large. This feature is verified via simulations as shown in Fig.1.3, which represents the number of iterations required to solve the HJB-FPKequations (1.21) and (1.19) as a function for different SBS densities λ b . Here,it is observed that the caching problem P1 is numerically solved by withina few iterations for highly dense networks. It means that the computationalcomplexity remains consistent regardless of the SBS density λ b , or the number6 CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING
Figure 1.2:
Ultra-dense edge caching flow charts according to the approaches of SDGand MFG, respectively. (a) In the framework of SDG, we solve the game of N SBSs(players) interacting with each individual SBS. (b) By incorporating MFG theory andSG into the framework, we can estimate the collective interaction of other SBSs. Thisrelaxes the N -SBS caching game to a two-SBS caching game. .4. GAME THEORETIC FORMULATION FOR EDGE CACHING Algorithm 1
Mean-Field Caching Control
Require: x j ( t ) , m , B ( t ) and Q (0) Find the optimal trajectory of caching cost and state distribution [ v ∗ j ( t ) , m ∗ t ( x j ( t ) , Q ( t ))] by solving HJB (1.21) and FPK (1.19) equations Calculate I rj ( t, m ∗ t ( x j ( t ) ,Q ( t ))) , I f ( t ) and ∂ Q v ∗ j Compute the instantaneous caching amount p ∗ j ( t ) : p ∗ j ( t ) = L j (cid:104) B j ( t ) − I rj ( t,m ∗ t ( x j ( t ) ,Q ( t ))) R ( t,I f ( t )) x j ( t ) ∂ Q v ∗ j (cid:105) + Get values of [ x j ( t ) , Q ( t )] according to the dynamics Go line 2Figure 1.3:
The number of iterations required to solve the coupled HJB and FPKequations for different densities of SBSs. CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING of players N . Fig. 1.3 also shows that this consistency holds for different initialstorage state distribution of SBSs. The number of iterations to reach the optimalcaching strategy is bounded within tens of iterations even for low SBS density.The proposed algorithm provides the solution faster for more densified networks. Numerical results are provided for evaluating the proposed algorithm underspatio-temporal content popularity and network dynamics illustrated in Fig. 1.1.Let us assume that the initial distribution of the SBSs m is given as normaldistribution and that the storage size Q ( t ) belongs to a set [0 , for all time t .Considering Rayleigh fading with mean one, the parameters are configured asshown in Table I. To solve the coupled PDEs (the first step of the Algorithm 1)using a finite element method, we used the MATLAB PDE solver.Table 1.1: Key simulation parameters Parameter ValueSBS density λ b )User density λ u − , . × − (users/m )Transmit power P
23 dBmNoise floor -70 dBmNumber of contents 20
CRP parameters θ, ν θ = 1 , ν = 0 . Reception ball radius R / √ π kmNetwork size 20 km ×
20 kmFile discarding rate e j To demonstrate that the proposed MF caching algorithm achieves the MFE, itis assumed that SBSs have full knowledge of contents request probability, whichimplies perfect popularity information is available at SBSs. The trajectory ofthe proposed caching algorithm and MF distribution is numerically analyzedwhen the content request probability is static. In this case, the caching con-trol strategies do not depend on the evolution law of the content popularity.Specifically, in HJB (1.21) and FPK (1.19) equations, the derivative terms withrespect to content request probability x become zero.Fig. 1.4 shows the evolution of the optimal caching amount p ∗ ( t ) with respectto the storage state and time. The value of p ∗ ( t ) is maintained lower than thecontent request probability to reduce the content overlap and prevent redundantbackhaul and storage usage.Fig. 1.5 shows heat-maps representing the instantaneous density of SBSshaving the remaining storage size Q ( t ) in terms of the MF distribution m ∗ t ( Q ( t )) .5. NUMERICAL RESULTS The optimal caching amount p ∗ ( t ) at the MF equilibrium under twodifferent content popularities 0.4 and 0.7, assuming that the content popularity isstatic. The initial MF distribution m ( Q (0)) is given as N (0 . , . ) . CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING
Figure 1.5:
A heat map illustration of the MF distribution m ∗ t ( Q ( t )) that representsthe instantaneous density of SBSs having the remaining storage space Q ( t ) for anarbitrary content during a long-term period { ≤ t ≤ T } , when the proposed MFcaching algorithm is applied. A bright-colored point means there are many SBSswith the unoccupied storage size corresponding to the point. It shows the temporalevolution of the density of SBSs with respect to different content popularity x j , andinitial distribution m ( Q (0)) ( B ( t ) = 1 , N r ( j ) = 20 , λ u = 0 . , λ b = 0 . . for a content during a period { ≤ t ≤ T } , where T = 1 . A bright-colored pointmeans there are many SBSs with the unoccupied storage size corresponding tothe point. It is observed that the unoccupied storage space of SBSs does notdiverge from each other as the proposed algorithm brings SBSs’ state in theMFE. At this equilibrium, the amount of cached content file decreases when thecontent popularity x becomes low. This tendency corresponds to the trajectoryof the optimal caching probability in Fig. 1.4. Almost every SBS has cachedthe content over time, but not used its entire storage. The remaining storagesaturates even though the content popularity is equal to . . This implies thatSBSs adjust the caching amount of popular content in consideration of thecontent overlap expected to possibly increase the cost. This section evaluates the performance of the proposed MF caching algorithmunder the spatio-temporal content popularity dynamics. Additionally, we evalu-ate the robustness of our scheme to imperfect popularity information in terms ofthe LRA caching cost. To this end, we compare the performance of the proposedMF caching algorithm with the following caching algorithms. .5. NUMERICAL RESULTS
Long run average costs of the caching strategies with respect to differentuser density λ u . ( Q (0) = 0 . , x (0) = 0 . , η = 0 . ). Figure 1.7:
Long run average costs of different caching strategies with perfect andimperfect popularity information. ( Q (0) = 0 . , x (0) = 0 . , η = 0 . ). CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING
Figure 1.8:
LRA cost Increment due to imperfect popularity information. For dif-ferent SBS density λ b , the proposed MF caching and the baseline caching withoutconsidering the content overlap are compared ( Q (0) = 0 . , x (0) = 0 . , η = 0 . ). • Baseline caching algorithm that does not consider the amount of contentoverlap but determines the instantaneous caching amount ˆ p j ( t ) propor-tionally to the instantaneous request probability x j ( t ) subject to currentbackhaul, storage state, and interference described as follows: ˆ p j ( t ) = L j (cid:104) B j ( t ) − R ( t,I f ( t )) x j ( t ) (cid:105) + .• Uniformly random caching that randomly determines the caching amountfollowing the uniform distribution.
LRA Cost Comparison.
Fig. 1.6 shows the LRA cost evaluation of theproposed MF caching algorithm, uniformly random caching, and the baselinecaching algorithm, which disregards the content overlap among neighboringSBSs. The LRA costs over time for different user density λ u are numericallyevaluated. The proposed caching control algorithm reduces about of theLRA cost as compared to the caching algorithm without considering the contentoverlap. This performance gain is due to avoiding redundant content overlapand having an SBS under lower interference environment to cache more con-tents. As the user density λ u becomes higher for a fixed SBS density λ b , thefinal values of the LRA cost increase for all the three caching schemes. WhenUDCNs are populated by numerous users, the fluctuation of spatial dynamics ofpopularity increases and the number of SBSs having associated users increases.Hence, both the aggregate interference imposed by the SBSs and the contentpopularity severely change over the spatial domain. In this environment, the ad-vantage of the proposed algorithm compared to the popularity based algorithmbecomes larger, yielding a higher gap between the final values of the producedLRA cost. Demand Misprediction Impact.
Accurate content popularity informa- .5. NUMERICAL RESULTS
The amount of overlapping contents per storage usage ( Q (0) = 0 . , η =0 . ). tion may not be available at SBSs due to misprediction or estimation error ofcontent popularity. It is thus assessed how the proposed algorithm is robustagainst imperfect popularity information (IPI) given as follows: ˆ x ( t ) = x ( t ) + ∆( t ) , (1.28)where ˆ x ( t ) denotes a content request probability estimated by an SBS, and ∆( t ) represents an observation error for the request probability x ( t ) at time t . AnSBS has perfect popularity information (PPI) if ∆( t ) is equal to zero for all t (i.e. ˆ x ( t ) = x ( t ) ). The magnitude of ∆ determines the accuracy of the popularity.For numerical evaluations, an observation error ∆ is assumed to follow a nor-mal distribution N (0 . , . ) . SBSs respectively determine their own cachingcontrol strategies based on imperfect content request probability ˆ x ( t ) (1.28) in-stead of PPI x ( t ) . With this IPI, the LRA caching cost over time is evaluatedas shown in Fig. 1.7. The impact of IPI increases with the number of SBSsbecause redundant caching occurs at several SBSs. Also, the LRA incrementdue to IPI is evaluated for our MF caching algorithm and the popularity basedone for different SBS density, i.e., the number of neighboring SBSs as shownin Fig. 1.8. The numerical results corroborate that the proposed algorithmis more robust against imperfect information of content popularity in compar-ison with the popularity-based benchmark scheme. In particular, our cachingstrategy reduces about of the LRA cost increment as compared to thepopularity-based baseline method.Fig. 1.9 shows the amount of overlapping contents per storage usage as afunction of the initial content probability x (0) . The proposed MF caching al-gorithm reduces caching content overlap averagely 42% compared to popularity4 CHAPTER 1. MEAN-FIELD GAME-THEORETIC EDGE CACHING based caching. However, MF caching algorithm yields a higher amount of con-tent overlap than random caching does when the content request probabilitybecomes high. The reason is that the random policy downloads contents re-gardless of their popularity, so the amount of content overlap remains steady.On the other hand, MF caching increases the downloaded volume of popularcontent.
In this chapter, scalable and distributed edge caching in a UDCN has been in-vestigated. To accurately reflect time-varying local content popularity, spatio-temporal content popularity modeling and interference analysis have been ap-plied in optimizing the edge caching strategy. Finally, by leveraging MFG, thecomputing complexity of optimizing the caching strategy has been reduced toa constant overhead from the cost exponentially increasing with the numberof SBSs in conventional methods. Numerical simulations corroborate that theproposed MFG-theoretic edge caching yields lower LRA costs while achievingmore robustness against imperfect content popularity information, compared toseveral benchmark schemes ignoring content popularity fluctuations or cachedcontent overlap among neighboring SBSs. ibliography [1] M Kamel, W Hamouda, and A Youssef. Ultra-Dense Networks: A Survey.IEEE Commun Surveys Tuts. 2016 Fourth quarter;18(4):2252–2545.[2] J Zender. Beyond the Ultra-Dense Barrier: Paradigm Shifts on theRoad Beyond 1000x Wireless Capacity. IEEE Wireless Commun. 2017Jan2017;24(3):96–102.[3] Shim T, Park J, Ko S, et al. Traffic convexity aware cellular networks:a vehicular heavy user perspective. IEEE Wireless Communications.2016;23(1):88–94.[4] P Popovski, J J Nielsen, C Stefanovic, E de Carvalho, E G Ström, KF Trillingsgaard, A Bana, D Kim, R Kotaba, J Park, and R B Sørensen.Wireless Access for Ultra-Reliable Low-Latency Communication (URLLC):Principles and Building Blocks. IEEE Netw. 2018 Mar;32(2):16–23.[5] E Baştuˇg, M Bennis, and M Debbah. Living on the Edge: The Role ofProactive Caching in 5G Wireless Networks. IEEE Commun Mag. 2014Aug;52(8):82–89.[6] S Tamoor-ul-Hassan, M Bennis, P H J Nardelli, and M Latva-aho. Cachingin Wireless Small Cell Networks: A Storage-Bandwidth Tradeoff. IEEECommun Lett. 2016 Jun;20(6):1175–1178.[7] X Wang, M Chen, T Taleb, A Ksentini, and V Leungi. Cache in the Air:Exploiting Content Caching and Delivery Techniques for 5G Systems. IEEECommun Mag. 2014 Feb;52(2):131–139.[8] J Park, S -L Kim, and J Zander. Tractable Resource Management with Up-link Decoupled Millimeter-Wave Overlay in Ultra-Dense Cellular Networks.IEEE Trans Wireless Commun. 2016 Jun;15(6):4362–4379.[9] P E Caines. Mean Field Games. Encyclopedia of Systems and ControlSpringer London. 2014;p. 1–6.[10] N Şen, and P E Caines. Mean Field Games with Partial Observation. SIAMJournal on Control and Optimization. 2019;57(3):2064–2091.256
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