Pricing and Resource Allocation via Game Theory for a Small-Cell Video Caching System
Jun Li, He Chen, Youjia Chen, Zihuai Lin, Branka Vucetic, Lajos Hanzo
IIEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1
Pricing and Resource Allocation via Game Theoryfor a Small-Cell Video Caching System
Jun Li,
Member, IEEE , He (Henry) Chen,
Member, IEEE , Youjia Chen,Zihuai Lin,
Senior Member, IEEE , Branka Vucetic,
Fellow, IEEE , and Lajos Hanzo,
Fellow, IEEE
Abstract —Evidence indicates that downloading on-demandvideos accounts for a dramatic increase in data traffic overcellular networks. Caching popular videos in the storage ofsmall-cell base stations (SBS), namely, small-cell caching, is anefficient technology for reducing the transmission latency whilstmitigating the redundant transmissions of popular videos overback-haul channels. In this paper, we consider a commercializedsmall-cell caching system consisting of a network service provider(NSP), several video retailers (VR), and mobile users (MU). TheNSP leases its SBSs to the VRs for the purpose of makingprofits, and the VRs, after storing popular videos in the rentedSBSs, can provide faster local video transmissions to the MUs,thereby gaining more profits. We conceive this system withinthe framework of Stackelberg game by treating the SBSs as aspecific type of resources. We first model the MUs and SBSsas two independent Poisson point processes, and develop, viastochastic geometry theory, the probability of the specific eventthat an MU obtains the video of its choice directly from thememory of an SBS. Then, based on the probability derived, weformulate a Stackelberg game to jointly maximize the averageprofit of both the NSP and the VRs. Also, we investigate theStackelberg equilibrium by solving a non-convex optimizationproblem. With the aid of this game theoretic framework, weshed light on the relationship between four important factors: theoptimal pricing of leasing an SBS, the SBSs allocation among theVRs, the storage size of the SBSs, and the popularity distributionof the VRs. Monte-Carlo simulations show that our stochasticgeometry-based analytical results closely match the empiricalones. Numerical results are also provided for quantifying theproposed game-theoretic framework by showing its efficiency onpricing and resource allocation.
Index Terms —Small-cell caching, cellular networks, stochasticgeometry, Stackelberg game
Jun Li is with the School of Electronic and Optical Engineering, NanjingUniversity of Science and Technology, Nanjing, CHINA, 210094. E-mail:[email protected] Chen, Youjia Chen, Zihuai Lin, and Branka Vucetic are withthe School of Electrical and Information Engineering, The Univer-sity of Sydney, AUSTRALIA. E-mail: { he.chen, youjia.chen, zihuai.lin,branka.vucetic } @sydney.edu.au.Lajos Hanzo is with the Department of Electronics and Computer Science,University of Southampton, U.K. E-mail: [email protected] work is partially supported by the National Natural Science Foundationof China (No. 61501238, No. 61271230, No. 61472190), by the JiangsuProvincial Science Foundation Project BK20150786, by the Specially Ap-pointed Professor Program in Jiangsu Province, 2015, by the open researchfund of National Key Laboratory of Electromagnetic Environment (No.201500013), by the open research fund of National Mobile CommunicationsResearch Laboratory, Southeast University (No. 2013D02), by the AustralianResearch Council (No. DP120100405 and No. DP150104019), and by theFaculty of Engineering and IT Early Career Researcher Scheme 2016, TheUniversity of Sydney. I. I
NTRODUCTION
Wireless data traffic is expected to increase exponentiallyin the next few years driven by a staggering proliferationof mobile users (MU) and their bandwidth-hungry mobileapplications. There is evidence that streaming of on-demandvideos by the MUs is the major reason for boosting the tele-traffic over cellular networks [1]. According to the predictionof mobile data traffic by Cisco, mobile video streaming willaccount for of the overall mobile data traffic by .The on-demand video downloading involves repeated wirelesstransmission of videos that are requested multiple times bydifferent users in a completely asynchronous manner, whichis different from the transmission style of live video streaming.Often, there are numerous repetitive requests of popularvideos from the MUs, such as online blockbusters, leadingto redundant video transmissions. The redundancy of datatransmissions can be reduced by locally storing popular videos,known as caching, into the storage of intermediate networknodes, effectively forming a local caching system [1, 2]. Thelocal caching brings video content closer to the MUs andalleviates redundant data transmissions via redirecting thedownloading requests to the intermediate nodes.Generally, wireless data caching consists of two stages: dataplacement and data delivery [3]. In the data placement stage,popular videos are cached into local storages during off-peakperiods, while during the data delivery stage, videos requestedare delivered from the local caching system to the MUs.Recent works advanced the caching solutions of both device-to-device (D2D) networks and wireless sensor networks [4–6]. Specifically, in [4] a caching scheme was proposed for aD2D based cellular network relaying on the MUs’ cachingof popular video content. In this scheme, the D2D clustersize was optimized for reducing the downloading delay. In [5,6], the authors proposed novel caching schemes for wirelesssensor networks, where the protocol model of [7] was adopted.Since small-cell embedded architectures will dominate infuture cellular networks, known as heterogeneous networks(HetNet) [8–13], caching relying on small-cell base stations(SBS), namely, small-cell caching, constitutes a promising so-lution for HetNets. The advantages brought about by small-cellcaching are threefold. Firstly, popular videos are placed closerto the MUs when they are cached in SBSs, hence reducing thetransmission latency. Secondly, redundant transmissions overSBSs’ back-haul channels, which are usually expensive [14],can be mitigated. Thirdly, the majority of video traffic isoffloaded from macro-cell base stations to SBSs. a r X i v : . [ c s . I T ] F e b EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 2
In [15], a small-cell caching scheme, named ‘Femto-caching’, is proposed for a cellular network having embeddedSBSs, where the data placement at the SBSs is optimized in acentralized manner for the sake of reducing the transmissiondelay imposed. However, [15] considers an idealized system,where neither the interference nor the impact of wirelesschannels is taken into account. The associations between theMUs and the SBSs are pre-determined without consideringthe specific channel conditions encountered. In [16], small-cell caching is investigated in the context of stochastic net-works. The average performance is quantified with the aid ofstochastic geometry [17, 18], where the distribution of networknodes is modeled by Poisson point process (PPP). However,the caching strategy of [16] assumes that the SBSs cache thesame content, hence leading to a sub-optimal solution.As detailed above, current research on wireless cachingmainly considers the data placement issue optimized for reduc-ing the downloading delay. However, the entire caching systemdesign involves numerous issues apart from data placement.From a commercial perspective, it will be more interesting toconsider the topics of pricing for video streaming, the rentalof local storage, and so on. A commercialized caching systemmay consist of video retailers (VR), network service providers(NSP) and MUs. The VRs, e.g., Youtube, purchase copyrightsfrom video producers and publish the videos on their web-sites. The NSPs are typically operators of cellular networks,who are in charge of network facilities, such as macro-cellbase stations and SBSs.In such a commercial small-cell caching system, the VRs’revenue is acquired from providing video streaming for theMUs. As the central servers of the VRs, which store the pop-ular videos, are usually located in the backbone networks andfar away from the MUs, an efficient solution is to locally cachethese videos, thereby gaining more profits from providingfaster local transmissions. In turn, these local caching demandsraised by the VRs offer the NSPs profitable opportunities fromleasing their SBSs. Additionally, the NSPs can save consid-erable costs due to reduced redundant video transmissionsover SBSs’ back-haul channels. In this sense, both the VRsand NSPs are the beneficiaries of the local caching system.However, each entity is selfish and wishes to maximize itsown benefit, raising a competition and optimization problemamong these entities, which can be effectively solved withinthe framework of game theory.We note that game theory has been successfully appliedto wireless communications for solving resource allocationproblems. In [19], the authors propose a dynamic spectrumleasing mechanism via power control games. In [20], a price-based power allocation scheme is proposed for spectrumsharing in Femto-cell networks based on Stackelberg game.Game theoretical power control strategies for maximizing theutility in spectrum sharing networks are studied in [21, 22].In this paper, we propose a commercial small-cell cachingsystem consisting of an NSP, multiple VRs and MUs. Weoptimize such a system within the framework of Stackelberggame by viewing the SBSs as a specific type of resources forthe purpose of video caching. Generally speaking, Stackelberggame is a strategic game that consists of a leader and several followers competing with each other for certain resources [23].The leader moves first and the followers move subsequently.Correspondingly, in our game theoretic caching system, weconsider the NSP to be the leader and the VRs as the followers.The NSP sets the price of leasing an SBS, while the VRscompete with each other for renting a fraction of the SBSs.To the best of the authors’ knowledge, our work is the firstof its kind that optimizes a caching system with the aid ofgame theory. Compared to many other game theory basedresource allocation schemes, where the power, bandwidth andtime slots are treated as the resources, our work has a totallydifferent profit model, established based on our coveragederivations. In particular, our contributions are as follows.1) By following the stochastic geometry framework of [17,18], we model the MUs and SBSs in the network astwo different ties of a Poisson point process (PPP) [24].Under this network model, we define the concept ofa successful video downloading event when an MUobtains the requested video directly from the storage ofan SBS. Then we quantify the probability of this eventbased on stochastic geometry theory.2) Based on the probability derived, we develop a profitmodel of our caching system and formulate the profitsgained by the NSP and the VRs from SBSs leasing andrenting.3) A Stackelberg game is proposed for jointly maximizingthe average profit of the NSP and the VRs. Given thisgame theoretic framework, we investigate a non-uniformpricing scheme, where the price charged to different VRsvaries.4) Then we investigate the Stackelberg equilibrium of thisscheme via solving a non-convex optimization problem.It is interesting to observe that the optimal solution isrelated both to the storage size of each SBS and to thepopularity distribution of the VRs.5) Furthermore, we consider an uniform pricing scheme.We find that although the uniform pricing scheme isinferior to the non-uniform one in terms of maximizingthe NSP’s profit, it is capable of reducing more back-haul costs compared with the latter and achieves themaximum sum profit of the NSP and the VRs.The rest of this paper is organized as follows. We describethe system model in Section II and establish the relatedprofit model in Section III. We then formulate Stackelberggame for our small-cell caching system in Section IV. InSection V, we investigate Stackelberg equilibrium for the non-uniform pricing scheme by solving a non-convex optimizationproblem, while in Section VI, we further consider the uniformpricing scheme. Our simulations and numerical results aredetailed in Section VII, while our conclusions are providedin Section VIII. II. S
YSTEM M ODEL
We consider a commercial small-cell caching system con-sisting of an NSP, V VRs, and a number of MUs. Let usdenote by L the NSP, by V = {V , V , · · · , V V } the set ofthe VRs, and by M one of the MUs. Fig. 1 shows an example EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 3 !" $ %& ! ' %& ! ( %& ! ) %& ! Fig. 1. An example of the small-cell caching system with four VRs. of our caching system relying on four VRs. In such a system,the VRs wish to rent the SBSs from L for placing their videos.Both the NSP and each VR aim for maximizing their profits.There are three stages in our system. In the first stage, theVRs purchase the copyrights of popular videos from videoproducers and publish them on their web-sites. In the secondstage, the VRs negotiate with the NSP on the rent of SBSsfor caching these popular videos. In the third stage, the MUsconnect to the SBSs for downloading the desired videos. Wewill particulary focus our attention on the second and thirdstages within this game theoretic framework. A. Network Model
Let us consider a small-cell based caching network com-posed of the MUs and the SBSs owned by L , where eachSBS is deployed with a fixed transmit power P and the storageof Q video files. Let us assume that the SBSs transmit overthe channels that are orthogonal to those of the macro-cellbase stations, and thus there is no interference incurred by themacro-cell base stations. Also, assume that these SBSs arespatially distributed according to a homogeneous PPP (HPPP) Φ of intensity λ . Here, the intensity λ represents the number ofthe SBSs per unit area. Furthermore, we model the distributionof the MUs as an independent HPPP Ψ of intensity ζ .The wireless down-link channels spanning from the SBSsto the MUs are independent and identically distributed ( i.i.d. ),and modeled as the combination of path-loss and Rayleighfading. Without loss of generality, we carry out our analysisfor a typical MU located at the origin. The path-loss betweenan SBS located at x and the typical MU is denoted by (cid:107) x (cid:107) − α ,where α is the path-loss exponent. The channel power of theRayleigh fading between them is denoted by h x , where h x ∼ exp(1) . The noise at an MU is Gaussian distributed with avariance σ .We consider the steady-state of a saturated network, whereall the SBSs keep on transmitting data in the entire frequencyband allocated. This modeling approach for saturated networkscharacterizes the worst-case scenario of the real systems,which has been adopted by numerous studies on PPP analysis,such as [18]. Hence, the received signal-to-interference-plus-noise ratio (SINR) at the typical MU from an SBS located at x can be expressed as ρ ( x ) = P h x (cid:107) x (cid:107) − α (cid:80) x (cid:48) ∈ Φ \ x P h x (cid:48) (cid:107) x (cid:48) (cid:107) − α + σ . (1) The typical MU is considered to be “covered” by an SBSlocated at x as long as ρ ( x ) is no lower than a pre-set SINRthreshold δ , i.e., ρ ( x ) ≥ δ. (2)Generally, an MU can be covered by multiple SBSs. Note thatthe SINR threshold δ defines the highest delay of downloadinga video file. Since the quality and code rate of a video cliphave been specified within the video file, the download delaywill be the major factor predetermining the QoS perceived bythe mobile users. Therefore, we focus our attention on thecoverage and SINR in the following derivations. B. Popularity and Preferences
We now model the popularity distribution, i.e., the distri-bution of request probabilities, among the popular videos tobe cached. Let us denote by F = {F , F , · · · , F N } the fileset consisting of N video files, where each video file containsan individual movie or video clip that is frequently requestedby MUs. The popularity distribution of F is represented by avector t = [ t , t , · · · , t N ] . That is, the MUs make independentrequests of the n -th video F n , n = 1 , · · · , N , with theprobability of t n . Generally, t can be modeled by the Zipfdistribution [25] as t n = 1 /n β (cid:80) Nj =1 /j β , ∀ n, (3)where the exponent β is a positive value, characterizing thevideo popularity. A higher β corresponds to a higher contentreuse, where the most popular files account for the majorityof download requests. From Eq. (3), the file with a smaller n corresponds to a higher popularity.Note that each SBS can cache at most Q video files, andusually Q is no higher than the number of videos in F , i.e., wehave Q ≤ N . Without loss of generality, we assume that N/Q is an integer. The N files in F are divided into F = N/Q filegroups (FG), with each FG containing Q video files. The n -thvideo, ∀ n ∈ { ( f − Q + 1 , · · · , f Q } , is included in the f -thFG, f = 1 , · · · , F . Denote by G f the f -th FG, and by p f theprobability of the MUs’ requesting a file in G f , and we have p f = fQ (cid:88) n =( f − Q +1 t n , ∀ f. (4)File caching is then carried out on the basis of FGs, whereeach SBS caches one of the F FGs.At the same time, the MUs have unbalanced preferenceswith regard to the V VRs, i.e., some VRs are more popularthan others. For example, the majority of the MUs may tend toaccess Youtube for video streaming. The preference distribu-tion among the VRs is denoted by q = [ q , q , · · · , q V ] , where q v , v = 1 , · · · , V , represents the probability that the MUsprefer to download videos from V v . The preference distribution q can also be modeled by the Zipf distribution. Hence, we have q v = 1 /v γ (cid:80) Vj =1 /j γ , ∀ v, (5) EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 4 where γ is a positive value, characterizing the preference ofthe VRs. A higher γ corresponds to a higher probability ofaccessing the most popular VRs. C. Video Placement and Download
Next, we introduce the small-cell caching system with itsdetailed parameters. In the first stage, each VR purchases the N popular videos in F from the producers and publishes thesevideos on its web-site. In the second stage, upon obtainingthese videos, the VRs negotiate with the NSP L for rentingits SBSs. As L leases its SBSs to multiple VRs, we denote by τ = [ τ , τ , · · · , τ V ] the fraction vector, where τ v representsthe fraction of the SBSs that are assigned to V v , ∀ v . We assumethat the SBSs rented by each VR are uniformly distributed.Hence, the SBSs that are allocated to V v can be modeled asa “thinned” HPPP Φ v with intensity τ v λ .The data placements of the second stage commence duringnetwork off-peak time after the VRs obtain access to the SBSs.During the placements, each SBS will be allocated with oneof the F FGs. Generally, we assume that the VRs do not havethe a priori information regarding the popularity distributionof F . This is because the popularity of videos is changingperiodically, and can only be obtained statistically after thesevideos quit the market. It is clear that each VR may havemore or less some statistical information on the popularitydistribution of videos based on the MUs’ downloading history.However, this information will be biased due to limitedsampling. In this case, the VRs will uniformly assign the F FGs to the SBSs with equal probability of F for simplicity. Weare interested in investigating the uniform assignment of videofiles for drawing a bottom line of the system performance. Asthe FGs are randomly assigned, the SBSs in Φ v that cachethe FG G f can be further modeled as a “more thinned” HPPP Φ v,f with an intensity of F τ v λ .In the third stage, the MUs start to download videos. Whenan MU M requires a video of G f from V v , it searches theSBSs in Φ v,f and tries to connect to the nearest SBS thatcovers M . Provided that such an SBS exists, the MU M will obtain this video directly from this SBS, and we therebydefine this event by E v,f . By contrast, if such an SBS doesnot exist, M will be redirected to the central servers of V v for downloading the requested file. Since the servers of V v are located at the backbone network, this redirection of thedemand will trigger a transmission via the back-haul channelsof the NSP L , hence leading to an extra cost.III. P ROFIT M ODELING
We now focus on modeling the profit of the NSP andthe VRs obtained from the small-cell caching system. Theaverage profit is developed based on stochastically geometricaldistributions of the network nodes in terms of per unit areatimes unit period ( /U AP ), e.g., /month · km . A. Average Profit of the NSP
For the NSP L , the revenue gained from the caching systemconsists of two parts: 1) the income gleaned from leasing SBSs to the VRs and 2) the cost reduction due to reduced usage ofthe SBSs’ back-haul channels. First, the leasing income /U AP of L can be calculated as S RT = V (cid:88) j =1 τ j λs j , (6)where s j is the price per unit period charged to V j forrenting an SBS. Then we formulate the saved cost /U AP dueto reduced back-haul channel transmissions. When an MUdemands a video in G f from V v , we derive the probability Pr( E v,f ) as follows. Theorem 1:
The probability of the event E v,f , ∀ v, f , can beexpressed as Pr( E v,f ) = τ v C ( δ, α )( F − τ v ) + A ( δ, α ) τ v + τ v , (7)where we have A ( δ, α ) (cid:44) δα − F (cid:0) , − α ; 2 − α ; − δ (cid:1) and C ( δ, α ) (cid:44) α δ α B (cid:0) α , − α (cid:1) . Furthermore, F ( · ) inthe function A ( δ, α ) is the hypergeometric function, whilethe Beta function in C ( δ, α ) is formulated as B ( x, y ) = (cid:82) t x − (1 − t ) y − d t . Proof:
Please refer to Appendix A. (cid:4)
Remark 1:
From
Theorem 1 , it is interesting to observe thatthe probability
Pr( E v,f ) is independent of both the transmitpower P and the intensity λ of the SBSs. Furthermore, since Q is inversely proportional to F , we can enhance Pr( E v,f ) byincreasing the storage size Q .We assume that there are on average K video requests fromeach MU within unit period, and that the average back-haulcost for a video transmission is s bh . Based on Pr( E v,f ) inEq. (7), we obtain the cost reduction /U AP for the back-haulchannels of L as S BH = F (cid:88) j =1 V (cid:88) j =1 p j q j ζK Pr( E j ,j ) s bh . (8)By combining the above two items, the overall profit /U AP for L can be expressed as S NSP = S RT + S BH . (9) B. Average Profit of the VRs
Note that the MUs can download the videos either from thememories of the SBSs directly or from the servers of the VRsat backbone networks via back-haul channels. In the first case,the MUs will be levied by the VRs an extra amount of moneyin addition to the videos’ prices because of the higher-ratelocal streaming, namely, local downloading surcharge (LDS).We assume that the LDS of each video is set as s ld . Thenthe revenue /U AP for a VR V v gained from the LDS can becalculated as S LDv = F (cid:88) j =1 p j q v ζK Pr( E v,j ) s ld . (10)Additionally, V v pays for renting the SBSs from L . The relatedcost /U AP can be written as S RTv = τ v λs v . (11) EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 5
Upon combining the two items, the profit /U AP for V v , ∀ v ,can be expressed as S V Rv = S LDv − S RTv . (12)IV. P ROBLEM F ORMULATION
In this section, we first present the Stackelberg game for-mulation for our price-based SBS allocation scheme. Then theequilibrium of the proposed game is investigated.
A. Stackelberg Game Formulation
Again, Stackelberg game is a strategic game that consists ofa leader and several followers competing with each other forcertain resources [23]. The leader moves first and the followersmove subsequently. In our small-cell caching system, wemodel the NSP L as the leader, and the V VRs as the fol-lowers. The NSP imposes a price vector s = [ s , s , · · · , s V ] for the lease of its SBSs, where s v , ∀ v , has been defined inthe previous section as the price per unit period charged on V v for renting an SBS. After the price vector s is set, the VRsupdate the fraction τ v , ∀ v , that they tend to rent from L .
1) Optimization Formulation of the Leader:
Observe fromthe above game model that the NSP’s objective is to maximizeits profit S NSP formulated in Eq. (9). Note that for ∀ v , thefraction τ v is a function of the price s v under the Stackelberggame formulation. This means that the fraction of the SBSsthat each VR is willing to rent depends on the specific pricecharged to them for renting an SBS. Consequently, the NSPhas to find the optimal price vector s for maximizing its profit.This optimization problem can be summarized as follows. Problem 1:
The optimization problem of maximizing L ’sprofit can be formulated as max s (cid:23) S NSP ( s , τ ) , s.t. V (cid:88) j =1 τ j ≤ . (13)
2) Optimization Formulation of the Followers:
The profitgained by the VR V v in Eq. (12) can be further written as S V Rv ( τ v , s v ) = F (cid:88) j =1 p j q v ζK Pr( E v,j ) s ld − τ v λs v = F (cid:88) j =1 p j q v ζKs ld τ v ( A ( δ, α ) − C ( δ, α ) + 1) τ v + C ( δ, α ) F − λs v τ v . (14)We can see from Eq. (14) that once the price s v is fixed,the profit of V v depends on τ v , i.e., the fraction of SBSs thatare rented by V v . If V v increases the fraction τ v , it will gainmore revenue by levying surcharges from more MUs, whileat the same time, V v will have to pay for renting more SBSs.Therefore, τ v has to be optimized for maximizing the profitof V v . This optimization can be formulated as follows. Problem 2:
The optimization problem of maximizing V v ’sprofit can be written as max τ v ≥ S V Rv ( τ v , s v ) . (15) Problem 1 and
Problem 2 together form a Stackelberggame. The objective of this game is to find the StackelbergEquilibrium (SE) points from which neither the leader (NSP)nor the followers (VRs) have incentives to deviate. In thefollowing, we investigate the SE points for the proposed game.
B. Stackelberg Equilibrium
For our Stackelberg game, the SE is defined as follows.
Definition 1:
Let s (cid:63) (cid:44) [ s (cid:63) , s (cid:63) , · · · , s (cid:63)V ] be a solution for Problem 1 , and τ (cid:63)v be a solution for Problem 2 , ∀ v . Define τ (cid:63) (cid:44) [ τ (cid:63) , τ (cid:63) , · · · , τ (cid:63)V ] . Then the point ( s (cid:63) , τ (cid:63) ) is an SE forthe proposed Stackelberg game if for any ( s , τ ) with s (cid:23) and τ (cid:23) , the following conditions are satisfied: S NSP ( s (cid:63) , τ (cid:63) ) ≥ S NSP ( s , τ (cid:63) ) ,S V Rv ( s (cid:63)v , τ (cid:63)v ) ≥ S V Rv ( s (cid:63)v , τ v ) , ∀ v. (16)Generally speaking, the SE of a Stackelberg game can beobtained by finding its perfect Nash Equilibrium (NE). In ourproposed game, we can see that the VRs strictly competein a non-cooperative fashion. Therefore, a non-cooperativesubgame on controlling the fractions of rented SBSs is for-mulated at the VRs’ side. For a non-cooperative game, theNE is defined as the operating points at which no players canimprove utility by changing its strategy unilaterally. At theNSP’s side, since there is only one player, the best responseof the NSP is to solve Problem 1 . To achieve this, we need tofirst find the best response functions of the followers, basedon which, we solve the best response function for the leader.Therefore, in our game, we first solve
Problem 2 given aprice vector s . Then with the obtained best response function τ (cid:63) of the VRs, we solve Problem 1 for the optimal price s (cid:63) .In the following, we will have an in-depth investigation onthis game theoretic optimization.V. G AME T HEORETIC O PTIMIZATION
In this section, we will solve the optimization problem inour game under the non-uniform pricing scheme, where theNSP L charges the VRs with different prices s , · · · , s V forrenting an SBS. In this scheme, we first solve Problem 2 atthe VRs, and rewrite Eq. (14) as S V Rv ( τ v , s v ) = Γ v s ld τ v Θ τ v + Λ − λs v τ v . (17)where Γ v (cid:44) (cid:80) Fj =1 p j q v ζK , Θ (cid:44) A ( δ, α ) − C ( δ, α ) + 1 , and Λ (cid:44) C ( δ, α ) F . We observe that Eq. (17) is a concave functionover the variable τ v . Thus, we can obtain the optimal solutionby solving the Karush-Kuhn-Tucker (KKT) conditions, and wehave the following lemma. Lemma 1:
For a given price s v , the optimal solution of Problem 2 is τ (cid:63)v = (cid:32)(cid:114) Γ v Λ s ld Θ λ (cid:114) s v − ΛΘ (cid:33) + , (18)where ( · ) + (cid:44) max( · , . Proof:
The optimal solution τ (cid:63)v of V v can be obtained byderiving S V Rv with respect to τ v and solving d S V Rv d τ v = 0 underthe constraint that τ v ≥ . (cid:4) EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 6
We can see from
Lemma 1 that if the price s v is set toohigh, i.e., s v ≥ Γ v s ld Λ λ , the VR V v will opt out for renting anySBS from L due the high price charged. Consequently, theVR V v will not participate in the game.In the following derivations, we assume that the LDS oneach video s ld is set by the VRs to be the cost of a video trans-mission via back-haul channels s bh . The rational behind thisassumption is as follows. Since a local downloading reduce aback-haul transmission, this saved back-haul transmission canbe potentially utilized to provide extra services (equivalent tothe value of s bh ) for the MUs. In addition, the MUs enjoy thebenefit from faster local video transmissions. In light of this,it is reasonable to assume that the MUs are willing to acceptthe price s bh for a local video transmission.Substituting the optimal τ (cid:63)v of Eq. (18) into Eq. (9) andcarry out some further manipulations, we arrive at S NSP = V (cid:88) j =1 λs j (cid:32)(cid:114) Γ j Λ s bh Θ λ (cid:115) s j − ΛΘ (cid:33) + + (cid:80) Fi =1 p i q j ζKs bh (cid:18)(cid:113) Γ j Λ s bh Θ λ (cid:113) s j − ΛΘ (cid:19) + Θ (cid:18)(cid:113) Γ j Λ s bh Θ λ (cid:113) s j − ΛΘ (cid:19) + + Λ= V (cid:88) j =1 ξ i Θ (cid:18) − Λ λs j + (cid:18) √ s bh − s bh √ s bh (cid:19) (cid:112) Γ j Λ λs j + Γ j s bh (cid:19) = V (cid:88) j =1 ξ i Θ (cid:0) − Λ λs j + Γ j s bh (cid:1) , (19)where ξ j is the indicator function, with ξ j = 1 if s j < Γ j s bh Λ λ and ξ j = 0 otherwise. Upon defining the binary vector ξ (cid:44) [ ξ , ξ , · · · , ξ V ] , we can rewrite Problem 1 as follows.
Problem 3:
Given the optimal solutions τ (cid:63)v , ∀ v , gleanedfrom the followers, we can rewrite Problem 1 as min ξ , s (cid:23) V (cid:88) j =1 ξ j (cid:0) Λ λs j − Γ j s bh (cid:1) , s.t. V (cid:88) j =1 ξ j (cid:32)(cid:115) Γ j Λ s bh λs j − Λ (cid:33) ≤ Θ . (20)Observe from Eq. (20) that Problem 3 is non-convex dueto ξ . However, for a given ξ , this problem can be solved bysatisfying the KKT conditions. In the following, we commencewith the assumption that ξ = , i.e., ξ v = 1 , ∀ v , and then weextend this result to the general case. A. Special Case: ξ v = 1 , ∀ v In this case, all the VRs are participating in the game, andwe have the following optimization problem.
Problem 4:
Assuming ξ v = 1 , ∀ v , we rewrite Problem 3 as min s (cid:23) V (cid:88) j =1 s j , s.t. V (cid:88) j =1 (cid:115) Γ j s j ≤ ( V Λ + Θ) (cid:114) λ Λ s bh . (21) The optimal solution of Problem 4 is derived and given inthe following lemma.
Lemma 2:
The optimal solution to
Problem 4 can be de-rived as ˆ s (cid:44) [ˆ s , · · · , ˆ s V ] , where ˆ s v = Λ s bh (cid:16)(cid:80) Vj =1 (cid:112) Γ j (cid:17) √ Γ v λ ( V Λ + Θ) , ∀ v. (22) Proof:
Please refer to Appendix B. (cid:4)
Note that the solution given in
Lemma 2 is found underthe assumption that ξ v = 1 , ∀ v . That is, ˆ s v given in Eq. (22)should ensure that τ (cid:63)v > , ∀ v , in Eq. (18), i.e., Λ s bh (cid:16)(cid:80) Vj =1 (cid:112) Γ j (cid:17) √ Γ v λ ( V Λ + Θ) < Γ v s bh Λ λ . (23)Given the definitions of Γ v , Λ , and Θ , it is interesting to findthat the inequality (23) can be finally converted to a constrainton the storage size Q of each SBS, which is formulated as Q > max
N C ( δ, α ) (cid:16)(cid:80) Vj =1 (cid:113) q j q v − V (cid:17) A ( δ, α ) − C ( δ, α ) + 1 , ∀ v . (24)The constraint imposed on Q can be expressed in a concisemanner in the following theorem. Theorem 2:
To make sure that ˆ s v in Eq. (22) does becomethe optimal solution of Problem 4 when ξ v = 1 , ∀ v , thesufficient and necessary condition to be satisfied is Q > Q min (cid:44)
N C ( δ, α ) (cid:16)(cid:80) Vj =1 (cid:113) q j q V − V (cid:17) A ( δ, α ) − C ( δ, α ) + 1 , (25)where q V is the minimum value in q according to Eq. (5). Proof:
Please refer to Appendix C. (cid:4)
Remark 2:
Observe from Eq. (25) that since q j q V increasesexponentially with γ according to Eq. (5), the value of Q min ensuring ξ v = 1 , ∀ v , will increase exponentially with γ/ .Note that we have Q ≤ N . In the case that Q min in Eq. (25)is larger than N for a high VR popularity exponent γ , someVRs with the least popularity will be excluded from the game. B. Further Discussion on Q We define a series of variables U v , ∀ v , as follows: U v (cid:44) N C ( δ, α ) (cid:16)(cid:80) vj =1 (cid:113) q j q v − v (cid:17) A ( δ, α ) − C ( δ, α ) + 1 , (26)and formulate the following lemma. Lemma 3: U v is a strictly monotonically-increasing func-tion of v , i.e., we have U V > U V − > · · · > U . Proof:
Please refer to Appendix D. (cid:4)
For the special case of the previous subsection, the optimalsolution for ξ v = 1 , ∀ v , is found under the condition thatthe storage size obeys Q > U V . In other words, Q should belarge enough such that every VR can participate in the game.However, when Q reduces, some VRs have to leave the gameas a result of the increased competition. Then we have thefollowing lemma. EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 7
Lemma 4:
When U v < Q ≤ U v +1 , the NSP can only retainat most the v VRs of V , V , · · · , V v in the game for achievingits optimal solution. Proof:
Please refer to Appendix E. (cid:4)
From
Lemma 4 , when we have U v < Q ≤ U v +1 , and giventhat there are u VRs, u ≤ v , in the game, we can have anoptimal solution for s . Problem 5: when U v < Q ≤ U v +1 is satisfied, and giventhat there are u , u ≤ v , VRs in the game, we can formulatethe following optimization problem as min s (cid:23) u (cid:88) j =1 s j , s.t. u (cid:88) j =1 (cid:115) Γ j s j ≤ ( u Λ + Θ) (cid:114) λ Λ s bh . (27)Similar to the solution of Problem 4 , we arrive atthe optimal solution for the above problem as ˆ s u (cid:44) [ˆ s ,u , · · · , ˆ s i,u , · · · , ˆ s V,u ] , where ˆ s i,u = Λ s bh ( (cid:80) uj =1 3 √ Γ j ) √ Γ i λ ( u Λ+Θ) , i = 1 , · · · , u, ∞ , i = u + 1 , · · · , V. (28) C. General Case
Let us now focus our attention on the general solution ofthe original optimization problem, i.e., of
Problem 3 . Withoutloss of generality, we consider the case of U v < Q ≤ U v +1 .Then Problem 3 is equivalent to the following problem.
Problem 6:
When U v < Q ≤ U v +1 , there are at most v VRs in the game. Then
Problem 3 can be converted to min ξ , s (cid:23) v (cid:88) j =1 ξ j (cid:0) Λ λs j − Γ j s bh (cid:1) , s.t. v (cid:88) j =1 ξ j (cid:32)(cid:115) Γ j Λ s bh λs j − Λ (cid:33) ≤ Θ . (29)The problem in Eq. (29) is again non-convex due to theuncertainty of ξ u , u = 1 , · · · , v . We have to consider the cases,where there are u , ∀ u , most popular VRs in the game. Weobserve that for a given u , Problem 6 converts to
Problem 5 .Therefore, to solve
Problem 6 , we first solve
Problem 5 with agiven u and obtain ˆ s u according to Eq. (28). Then we choosethe optimal solution, denoted by s (cid:63)v , among ˆ s , · · · , ˆ s v as thesolution to Problem 6 , which is formulated as s (cid:63)v =arg min ˆ s u min u (cid:88) j =1 (cid:0) Λ λs j − Γ j s bh (cid:1) , u = 1 , · · · , v . (30)Based on the above discussions, we can see that the optimalsolution s (cid:63) of Problem 3 is a piece-wise function of Q , i.e., s (cid:63) = s (cid:63)v when U v < Q ≤ U v +1 . Now, we formulate the solution s (cid:63) = [ s (cid:63) , · · · , s (cid:63)V ] to Problem 3 in a general manneras follows. s (cid:63)v = Λ s bh ( (cid:80) ˆ uj =1 3 √ Γ j ) √ Γ v λ (ˆ u Λ+Θ) , v = 1 , · · · , ˆ u, ∞ , v = ˆ u + 1 , · · · , V, (31)where regarding ˆ u , we have ˆ u = arg min u { S u : u = 1 , , · · · , T } , (32)with S u formulated as S u = u (cid:88) j =1 Λ s bh (cid:16)(cid:80) uj =1 (cid:112) Γ j (cid:17) (cid:112) Γ j ( u Λ + Θ) − Γ j s bh ,T = , U < Q ≤ U , · · · ,v, U v < Q ≤ U v +1 , · · · ,V, U V < Q. (33)To gain a better understanding of the optimal solution inEq. (31), we propose a centralized algorithm at L in Table Ifor obtaining s (cid:63) . Remark 3:
The optimal solution s (cid:63) in Eq. (31), combinedwith the solution of τ (cid:63) given by Eq. (18) in Lemma 1 ,constitutes the SE for the Stackelberg game.
Algorithm 1 : Input:
Storage size Q , number of videos N , VRs’ preference distribution q , channel exponent α , and pre-set threshold δ . Output:
Optimal pricing vector s (cid:63) . Steps:
1: Based on N , q , α , and δ , the NSP calculates U v , ∀ v , according toEq. (26);2: By comparing Q to U v , the NSP obtains the value of the integer T inEq. (33);3: Calculate S u , u = 1 , , · · · , T , according to Eq. (33);4: Compare among S , · · · , S T for finding the index ˆ u of the minimum S ˆ u ;5: Based on ˆ u , N , q , α , and δ , the NSP obtains the optimal solution s (cid:63) according to Eq. (31). TABLE IT HE CENTRALIZED ALGORITHM AT THE
NSP
FOR OBTAINING THEOPTIMAL SOLUTION S (cid:63) . Furthermore, by substituting the optimal s (cid:63) into the expres-sion of S NSP in Eq. (19), we get S NSP ( s (cid:63) , τ (cid:63) ) =1Θ ˆ u (cid:88) j =1 Γ j s bh − Λ s bh (cid:16)(cid:80) ˆ uj =1 (cid:112) Γ j (cid:17) (cid:112) Γ j (ˆ u Λ + Θ) . (34) Remark 4:
Since we have Γ v ∝ q v , ∀ v , and q v increasesexponentially with the VR preference parameter γ accordingto Eq. (5), S NSP ( s (cid:63) , τ (cid:63) ) also increases exponentially with γ . EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 8
VI. D
ISCUSSIONS OF O THER S CHEMES
Let us now consider two other schemes, namely, an uniformpricing scheme and a global optimization scheme.
A. Uniform Pricing Scheme
In contrast to the non-uniform pricing scheme of the previ-ous section, the uniform pricing scheme deliberately imposesthe same price on the VRs in the game. We denote the fixedprice by s . In this case, similar to Lemma 1 , Problem 2 canbe solved by τ (cid:63)v = (cid:32)(cid:114) Γ v Λ s bh Θ λ (cid:114) s − ΛΘ (cid:33) + . (35)We first focus our attention on the special case of ξ v = 1 , ∀ v . Then Problem 4 can be converted to that of minimizing s subject to the constraint (cid:80) Vj =1 (cid:113) Γ j s ≤ ( V Λ + Θ) (cid:113) λ Λ s bh . Wethen obtain the optimal ˆ s for this special case as ˆ s = Λ s bh (cid:16)(cid:80) Vj =1 (cid:112) Γ j (cid:17) λ ( V Λ + Θ) . (36)To guarantee that all the VRs are capable of participating inthe game, i.e., ξ v = 1 , ∀ v , with the optimal price ˆ s , we let ˆ s < Γ v s bh Λ λ . Then we have the following constraint on thestorage Q as Q > Q (cid:48) min (cid:44)
N C ( δ, α ) (cid:16)(cid:80) Vj =1 (cid:113) q j q V − V (cid:17) A ( δ, α ) − C ( δ, α ) + 1 . (37)We can see that the we require a larger storage size Q in Eq. (37) than that in Eq. (25) under the non-uniformpricing scheme to accommodate all the VRs, since we have (cid:80) Vj =1 (cid:113) q j q V > (cid:80) Vj =1 (cid:113) q j q V . Following Remark 2 , we con-clude that Q (cid:48) min of the uniform pricing scheme will increaseexponentially with γ/ .Then based on this special case, the optimal s (cid:63) =[ s (cid:63) , · · · , s (cid:63)V ] in the uniform pricing scheme can be readilyobtained by following a similar method to that in the previoussection. That is, s (cid:63)v = Λ s bh ( (cid:80) ˆ uj =1 √ Γ j ) λ (ˆ u Λ+Θ) , v = 1 , · · · , ˆ u, ∞ , v = ˆ u + 1 , · · · , V, (38)where regarding ˆ u , we have ˆ u = arg min u { S u : u = 1 , , · · · , T } , (39)with S u = u Λ s bh (cid:16)(cid:80) uj =1 (cid:112) Γ j (cid:17) ( u Λ + Θ) − u (cid:88) j =1 Γ j s bh ,T = , ¯ U < Q ≤ ¯ U , · · · ,v, ¯ U v < Q ≤ ¯ U v +1 , · · · ,V, ¯ U V < Q. (40) Note that ¯ U v in Eq. (40) is defined as ¯ U v (cid:44) N C ( δ, α ) (cid:16)(cid:80) vj =1 (cid:113) q j q v − v (cid:17) A ( δ, α ) − C ( δ, α ) + 1 . (41)It is clear that the uniform pricing scheme is inferiorto the non-uniform pricing scheme in terms of maximizing S NSP . However, we will show in the following problem thatthe uniform pricing scheme offers the optimal solution tomaximizing the back-haul cost reduction S BH at the NSP inconjunction with τ (cid:63)v , ∀ v , from the followers. Problem 7:
With the aid of the optimal solutions τ (cid:63)v , ∀ v ,from the followers, the maximization on S BH is achieved bysolving the following problem: min ξ , s (cid:23) V (cid:88) j =1 ξ j (cid:16) √ s bh (cid:112) Γ j Λ λ √ s j − Γ j s bh (cid:17) , s.t. V (cid:88) j =1 ξ j (cid:32)(cid:115) Γ j Λ s bh λs j − Λ (cid:33) ≤ Θ . (42)The optimal solution to Problem 7 can be readily shownto be s (cid:63) given in Eq. (38). This proof follows the similarprocedure of the optimization method presented in the pre-vious section. Thus it is skipped for brevity. In this sense,the uniform pricing scheme is superior to the non-uniformscheme in terms of reducing more cost on back-haul channeltransmissions. B. Global Optimization Scheme
In the global optimization scheme, we are interested in thesum profit of the NSP and VRs, which can be expressed as S GLB = S NSP + V (cid:88) j =1 S V Rj = V (cid:88) j =1 F (cid:88) j =1 p j q j ζKs bh τ j ( A ( δ, α ) − C ( δ, α ) + 1) τ j + C ( δ, α ) F = 2 S BH . (43)Observe from Eq. (43), we can see that the sum profit S GLB is twice the back-haul cost reduction S BH , where the vector τ is the only variable of this maximization problem. Problem 8:
The optimization of the sum profit S GLB canbe formulated as max τ (cid:23) V (cid:88) j =1 τ j (cid:80) Fj =1 p j q j ζKs bh ( A ( δ, α ) − C ( δ, α ) + 1) τ j + C ( δ, α ) F , s.t. V (cid:88) j =1 τ j ≤ . (44) Problem 8 is a typical water-filling optimization problem.By relying on the classic Lagrangian multiplier, we arrive atthe optimal solution as ˆ τ v = (cid:32) √ q v η − C ( δ, α ) F A ( δ, α ) − C ( δ, α ) + 1 (cid:33) + , ∀ v, (45)where we have η = (cid:80) ¯ vj =1 √ q j ¯ vC ( δ,α ) F + A ( δ,α ) − C ( δ,α )+1 , and ¯ v satisfiesthe constraint of ˆ τ v > . EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 9
C. Comparisons
Let us now compare the optimal SBS allocation variable τ v in the context of the above two schemes. First, we investigate τ (cid:63)v in the uniform pricing scheme. By substituting Eq. (38)into Eq. (35), we have τ (cid:63)v = (cid:32)(cid:114) Γ v Λ s bh Θ λ (cid:115) s (cid:63)v − ΛΘ (cid:33) + = √ qvη (cid:48) − C ( δ,α ) F A ( δ,α ) − C ( δ,α )+1 , v = 1 , · · · , ˆ u , v = ˆ u + 1 , · · · , V, (46)where η (cid:48) = (cid:80) ˆ uj =1 √ q j ˆ uC ( δ,α ) F + A ( δ,α ) − C ( δ,α )+1 , and ˆ u ensures τ (cid:63)v > .Then, comparing τ (cid:63)v given in Eq. (46) to the optimal solution ˆ τ of the global optimization scheme given by Eq. (45), we cansee that these two solutions are the same. In other words,the uniform pricing scheme in fact represents the globaloptimization scheme in terms of maximizing the sum profit S GLB and maximizing the back-haul cost reduction S BH .VII. N UMERICAL R ESULTS
In this section, we provide both numerical as well as Monte-Carlo simulation results for evaluating the performance ofthe proposed schemes. The physical layer parameters of oursimulations, such as the path-loss exponent α , transmit power P of the SBSs and the noise power σ are similar to thoseof the 3GPP standards. The unit of noise power and transmitpower is Watt, while the SBS and MU intensities are expressedin terms of the numbers of the nodes per square kilometer.Explicitly, we set the path-loss exponent to α = 4 , the SBStransmit power to P = 2 Watt, the noise power to σ =10 − Watt, and the pre-set SINR threshold to δ = 0 . . Forthe file caching system, we set the number of files in F to N = 500 and set the number of VRs to V = 15 . For thenetwork deployments, we set the intensity of the MUs to ζ =50 /km , and investigate three cases of the SBS deploymentsas λ = 10 /km , /km and /km .For the pricing system, the profit /U AP is considered tobe the profit gained per month within an area of one squarekilometer, i.e., /month · km . We note that the profits gainedby the NSP and by the VRs are proportional to the cost s bh of back-haul channels for transmitting a video. Hence, withoutloss of generality, we set s bh = 1 for simplicity. Additionally,we set K = 10 /month , which is the average number of videorequests from an MU per month.We first verify our derivation of Pr( E v,f ) by comparing theanalytical results of Theorem 1 to the Monte-Carlo simulationresults. Upon verifying
Pr( E v,f ) , we will investigate theoptimization results within the framework of the proposedStackelberg game by providing numerical results. A. Performance Evaluation on
Pr( E v,f ) For the Monte-Carlo simulations of this subsection, all theaverage performances are evaluated over a thousand networkscenarios, where the distributions of the SBSs and the MUschange from case to case according the PPPs characterized by Φ and Ψ , respectively. τ v P r o b a b ili t y o f D i rec t D o w n l oa d i n g λ =10, Simulation λ =20, Simulation λ =30, SimulationNumerical Result Q=10 Q=500Q=100 Q=50 Fig. 2. Comparisons between the simulations and analytical results on
Pr( E v,f ) . We consider four kinds of storage size Q in each SBS, i.e., Q = 10 , , , , and three kinds of SBS intensity, i.e., λ = 10 , , . γ M i n i m u m Q Non−Uniform PricingUniform Pricing 0.66 0.98Number of Videos N=500
Fig. 3. The minimum number of Q that allows all the VRs to participate inthe game under different preference parameter γ . In the case that the minimum Q is larger than N , it means that some VRs will be inevitable excluded fromthe game. Note that
Pr( E v,f ) in Theorem 1 is the probability that anMU can obtain its requested video directly from the memoryof an SBS rented by V v . We can see from the expression of Pr( E v,f ) in Eq. (7) that it is a function of the fraction τ v of the SBSs that are rented by V v . Although τ v should beoptimized according to the price charged by the NSP, herewe investigate a variety of τ v values, varying from to , toverify the derivation of Pr( E v,f ) .Fig. 2 shows our comparisons between the simulationsand analytical results on Pr( E v,f ) . We consider four differentstorage sizes Q in each SBS by setting Q = 10 , , , .Correspondingly, we have four values for the number of file EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 10 γ N u m b er o f P a r t i c i p a n t s Non−Uniform PricingUniform PricingQ=50Q=10 Q=500Q=100
Fig. 4. Number of participants, i.e., the VRs that are in the game, vs.the preference parameter γ , under the two schemes. We also consider fourdifferent values of the storage size Q , i.e., , , , . γ R e v e nu e Non−Uniform PricingUniform Pricing S GLB =2 × S BH S NSP
Fig. 5. Various revenues, including S NSP and S GLB , vs. the preferenceparameter γ , under the two schemes. groups, i.e., F = 50 , , , . Furthermore, we consider theSBS intensities of λ = 10 , , . From Fig. 2, we cansee that the simulations results closely match the analyticalresults derived in Theorem 1 . Our simulations show that theintensity λ does not affect Pr( E v,f ) , which is consistent withour analytical results. Furthermore, a larger Q leads to a highervalue of Pr( E v,f ) . Hence, enlarging the storage size is helpfulfor achieving a higher probability of direct downloading. B. Impact of the VR Preference Parameter γ The preference distribution q of the VRs defined in Eq. (5)is an important factor in predetermining the system perfor-mance. Indeed, we can see from Eq. (5) that this distributiondepends on the parameter γ . Generally, we have < γ ≤ , N u m b er o f P a r t i c i p a n t s Non−Uniform PricingUniform Pricing γ =0.3 γ =1 Fig. 6. Number of participants vs. the storage size Q , under the two schemes.We also consider two different values of γ , i.e., γ = 0 . , . R e v e nu e Non−Uniform PricingUniform Pricing S NSP S GLB =2 × S BH Fig. 7. Various revenues, including S NSP and S GLB , vs. the storage size Q , under the two schemes. with a larger γ representing a more uneven popularity amongthe VRs. First, we find the minimum Q that can keep allthe VRs in the game. This minimum Q for the non-uniformpricing scheme (NUPS) is given by Eq. (25), while theminimum Q for the uniform-pricing scheme (UPS) is given byEq. (37). From the two equations, this minimum Q increasesexponentially with γ/ in the NUPS, while it also increasesexponentially with a higher exponent of γ/ in the UPS.Fig. 3 shows this minimum Q for different values of the VRpreference parameter γ .We can see that the UPS needs a larger Q than the NUPSfor keeping all the VRs. This gap increases rapidly with thegrowth of γ . For example, for γ = 0 . , the uniform pricingscheme requires almost more storages, while for γ = 0 . ,it needs more. We can also observe in Fig. 3 that for EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 11 P r i ce o n I nd i v i du a l VR s Non−Uniform PricingUniform Pricing
Fig. 8. Price charged on each VR for renting an SBS per month. γ > . in the UPS and for γ > . in the NUPS, theminimum Q becomes larger than the overall number of videos N . In both cases, since we have Q ≤ N ( Q > N results inthe same performance as Q = N ), some unpopular VRs willbe excluded from the game.Next, we study the number of VR participants that stayin the game for the two schemes upon increasing γ . Wecan see from Fig. 4 that the number of VR participantskeeps going down upon increasing γ in the both schemes.The NUPS always keeps more VRs in the game than theUPS under the same γ . At the same time, by considering Q = 10 , , , , it is shown that for a given γ , a higher Q will keep more VRs in the game.Fig. 5 shows two kinds of revenues gained by the twoschemes for a given storage of Q = 500 , namely, the globalprofit S GLB defined in Eq. (43) and the profit of the NSP S NSP defined in Eq. (9). Recall that we have S GLB = 2 S BH according to Eq. (43). We can see that the revenues of bothschemes increase exponentially upon increasing γ , as statedin Remark 4 . As our analytical result shows, the profit S NSP gained by the NUPS is optimal and thus it is higher than thatgained by the UPS, while the UPS maximizes both S GLB and S BH . Fig. 5 verifies the accuracy of our derivations. C. Impact of the Storage Size Q Since γ is a network parameter that is relatively fixed, theNSP can adapt the storage size Q for controlling its perfor-mance. In this subsection, we investigate the performance asa function of Q . Fig. 6 shows the number of participants inthe game versus Q , where γ = 0 . and are considered. It isshown that for a larger Q , more VRs are able to participatein the game. Again, the NUPS outperforms the UPS owing toits capability of accommodating more VRs for a given Q . Bycomparing the scenarios of γ = 0 . and , we find that for γ = 0 . , a given increase of Q can accommodate more VRsin the game than γ = 1 . F r a c t i o n o f R e n t e d S B S s Non−Uniform PricingUniform Pricing
Fig. 9. The fraction of SBSs that are rented by each VR.
Fig. 7 shows both S NSP and S GLB versus Q for the twoschemes for a given γ = 1 . We can see that the revenues ofboth schemes increase with the growth of Q . It is shown thatthe profit S NSP gained by the NUPS is higher than the onegained by the UPS, while the UPS outperforms the NUPS interms of both S GLB and S BH . D. Individual VR Performance
In this subsection, we investigate the performance of eachindividual VR, including the price charged to them for rentingan SBS per month, and the fractions of the SBSs they rent fromthe NSP. We fix γ = 0 . and choose a large storage size of Q = 500 for ensuring that all the VRs can be included. Fig. 8shows the price charged to each VR for renting an SBS. TheVRs are arranged according to their popularity order, rangingfrom V to V , with V having the highest popularity and V the lowest one. We can see from the figure that in the NUPS,the price for renting an SBS is higher for the VRs havinga higher popularity than those with a lower popularity. Bycontrast, in the UPS, this price is fixed for all the VRs. Fig. 9shows the specific fraction of the rented SBSs at each VR. Inboth schemes, the VRs associated with a high popularity tendto rent more SBSs. The UPS in fact represents an instance ofthe water-filling algorithm. Furthermore, the UPS seems moreaggressive than the NUPS, since the less popular VRs of theUPS are more difficult to rent an SBS, and thus these VRs arelikely to be excluded from the game with a higher probability.VIII. C ONCLUSIONS
In this paper, we considered a commercial small-cellcaching system consisting of an NSP and multiple VRs, wherethe NSP leases its SBSs to the VRs for gaining profits and forreducing the costs of back-haul channel transmissions, whilethe VRs, after storing popular videos to the rented SBSs, canprovide faster transmissions to the MUs, hence gaining moreprofits. We proposed a Stackelberg game theoretic framework
EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 12 by viewing the SBSs as a type of resources. We first modeledthe MUs and SBSs using two independent PPPs with the aid ofstochastic geometry, and developed the probability expressionof direct downloading. Then, based on the probability derived,we formulated a Stackelberg game for maximizing the averageprofit of the NSP as well as individual VRs. Next, we investi-gate the Stackelberg equilibrium by solving the associated non-convex optimization problem. We considered a non-uniformpricing scheme and an uniform pricing scheme. In the formerscheme, the prices charged to each VR for renting an SBSare different, while the latter imposes the same price foreach VR. We proved that the non-uniform pricing schemecan effectively maximize the profit of the NSP, while theuniform one maximizes the sum profit of the NSP and the VRs.Furthermore, we derived a relationship between the optimalpricing of renting an SBS, the fraction of SBSs rented by eachVR, the storage size of each SBS and the popularity of theVRs. We verified by Monte-Carlo simulations that the directdownloading probability under our PPP model is consistentwith our derived results. Then we provided several numericalresults for showing that the proposed schemes are effective inboth pricing and SBSs allocation.A
PPENDIX AP ROOF OF T HEOREM V v and cache G f are modeled as a “thinned” HPPP Φ v,f having the intensity of F τ v λ . We consider a typical MU M who wishes to connectto the nearest SBS B in Φ v,f . The event E v,f represents thatthis SBS can support M with an SINR no lower than δ , andthus M can obtain the desired file from the cache of B .We carry out the analysis on Pr( E v,f ) for the typical MU M located at the origin. Since the network is interferencedominant, we neglect the noise in the following. We denoteby z the distance between M and B , by x Z the location of B , and by ρ ( x Z ) the received SINR at M from B . Then theaverage probability that M can download the desired videofrom B is Pr( ρ ( x Z ) ≥ δ )= (cid:90) ∞ Pr h x Z z − α (cid:80) x ∈ Φ \{ x Z } h x (cid:107) x (cid:107) − α ≥ δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z f Z ( z ) d z = (cid:90) ∞ Pr h x Z ≥ δ (cid:32) (cid:80) x ∈ Φ \{ x Z } h x (cid:107) x (cid:107) − α (cid:33) z − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z π F τ v λz exp (cid:18) − π F τ v λz (cid:19) d z (47) = (cid:90) ∞ E I (exp ( − z α δI )) 2 π F τ v λz exp (cid:18) − π F τ v λz (cid:19) d z, where we have I (cid:44) (cid:80) x ∈ Φ \{ x Z } h x (cid:107) x (cid:107) − α , and the PDF of z , i.e., f Z ( z ) , is derived by the null probability of the HPPP Φ v,f with the intensity of F τ v λ . More specifically in Φ v,f , since the number of the SBSs k in an area of A follows thePoisson distribution, the probability of the event that there isno SBS in the area with the radius of z can be calculatedas [17] Pr( k = 0 | A = πz ) = e − A F τ v λ ( A F τ v λ ) k k ! = e − πz F τ v λ . (48)By using the above expression, we arrive at f Z ( z ) =2 π F τ v λz exp (cid:0) − π F τ v λz (cid:1) . Note that the interference I con-sists of I and I , where I emanates from the SBSs in Φ excluding Φ v,f , while I is from the SBSs in Φ v,f excluding B . The SBSs contributing to I , denoted by Φ v,f , have theintensity of (cid:0) − F τ v (cid:1) λ , while those contributing to I havethe intensity of F τ v λ .Correspondingly, the calculation of E I (exp ( − z α δI )) willbe split into the product of two expectations over I and I .The expectation over I is calculated as E I (exp ( − z α δI )) ( a ) = E Φ v,f (cid:32) (cid:89) x ∈ Φ v (cid:90) ∞ exp (cid:16) − z α δh x (cid:107) x (cid:107) − α (cid:17) exp( − h x ) d h x (cid:33) ( b ) = exp (cid:18) − (cid:18) − F τ v (cid:19) λ (cid:90) R (cid:18) −
11 + z α δ (cid:107) x k (cid:107) − α (cid:19) d x k (cid:19) = exp (cid:18) − π (cid:18) − F τ v (cid:19) λ α z δ α B (cid:18) α , − α (cid:19)(cid:19) , = exp (cid:18) − π (cid:18) − F τ v (cid:19) λC ( δ, α ) z (cid:19) , (49)where ( a ) is based on the independence of chan-nel fading, while ( b ) follows from E (cid:18)(cid:81) x u ( x ) (cid:19) =exp (cid:0) − λ (cid:82) R (1 − u ( x )) d x (cid:1) , where x ∈ Φ and Φ is an PPPin R with the intensity λ [24], and C ( δ, α ) has been definedas α δ α B (cid:0) α , − α (cid:1) .The expectation over I has to take into account z as thedistance from the nearest interfering SBS. Then we have E I (exp( − z α δI ))= exp (cid:18) − F τ v λ π (cid:90) ∞ z (cid:18) −
11 + z α δr − α (cid:19) r d r (cid:19) ( a ) = exp (cid:32) − F τ v λπδ α z α (cid:90) ∞ δ − κ α − κ d x (cid:33) (50) ( b ) = exp (cid:18) − F τ v λπδz α − F (cid:18) , − α ; 2 − α ; − δ (cid:19)(cid:19) , where ( a ) defines κ (cid:44) δ − z − α r α , and F ( · ) in ( b ) is the hypergeometric function. As we defined A ( δ, α ) = δα − F (cid:0) , − α ; 2 − α ; − δ (cid:1) , by substituting (49) and (50)into (47), we have Pr( ρ ( x Z ) ≥ δ ) = (cid:90) ∞ exp (cid:18) − π (cid:18) − F τ v (cid:19) λC ( δ, α ) z (cid:19) exp (cid:18) − π F τ v λz A ( δ, α ) (cid:19) π F τ v λz exp (cid:18) − π F τ v λz (cid:19) d z = F τ v C ( δ, α )(1 − F τ v ) + A ( δ, α ) F τ v + F τ v . (51)This completes the proof. (cid:4) EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 13 A PPENDIX BP ROOF OF L EMMA L ( s , µ, ν ) = V (cid:88) j =1 s j + µ V (cid:88) j =1 (cid:115) Γ j s j − ( V Λ + Θ) (cid:114) λ Λ s bh − V (cid:88) j =1 ν j s j , (52)where µ and ν j are non-negative multipliers associated withthe constraints (cid:80) Vj =1 (cid:113) Γ j s j − ( V Λ+Θ) (cid:113) λ Λ s bh ≤ and s j ≥ ,respectively. Then the KKT conditions can be written as ∂L ( s , µ, ν ) ∂s j = 0 , ∀ j = 1 , · · · , V,µ V (cid:88) j =1 (cid:115) Γ j s j − ( V Λ + Θ) (cid:114) λ Λ s bh = 0 , and ν j s j = 0 , ∀ j. (53)From the first line of Eq. (53), we have s j = (cid:115) µ Γ j − ν j ) . (54)Obviously, we have s j (cid:54) = 0 , ∀ j , otherwise the constraint (cid:80) Vj =1 (cid:113) Γ j s j − ( V Λ + Θ) (cid:113) λ Λ s bh ≤ cannot be satisfied.Thus, we have ν j = 0 , ∀ j . Furthermore, we have µ (cid:54) = 0 according to Eq. (54) since s j is non-zero. This means that (cid:80) Vj =1 (cid:113) Γ j s j − ( V Λ + Θ) (cid:113) λ Λ s bh = 0 .By substituting Eq. (54) into this constraint, we have √ µ = √ Λ s bh (cid:80) Vj =1 (cid:112) j √ λ ( V Λ + Θ) . (55)Then it follows that s j = Λ s bh (cid:16)(cid:80) Vv =1 √ Γ v (cid:17) (cid:112) Γ j λ ( V Λ + Θ) . (56)This completes the proof. (cid:4) A PPENDIX CP ROOF OF T HEOREM
Q > NC ( δ,α ) (cid:16)(cid:80) Vj =1 3 (cid:113) qjqV − V (cid:17) A ( δ,α ) − C ( δ,α )+1 is a sufficient condition for theoptimal solution in Eq. (22). In other words, as long as Q issatisfied, we have the conclusion that the solution in Eq. (22)is optimal and ξ v = 1 , ∀ v .Next, we prove the necessary aspect. Without loss ofgenerality, we assume that N C ( δ, α ) (cid:16)(cid:80) V − j =1 (cid:113) q j q V − − V + 1 (cid:17) A ( δ, α ) − C ( δ, α ) + 1 < Q ≤ N C ( δ, α ) (cid:16)(cid:80) Vj =1 (cid:113) q j q V − V (cid:17) A ( δ, α ) − C ( δ, α ) + 1 . (57) This leads to s V ≥ Γ v s bh Λ λ , and the VR V V will be excludedfrom the game. In this case, we have ξ j = 1 , j = 1 , · · · , V − ,and Problem 4 will be rewritten as follows.
Problem 9:
We rewrite
Problem 4 as min s (cid:23) V − (cid:88) j =1 s j , s.t. V − (cid:88) j =1 (cid:115) Γ j s j ≤ (( V − (cid:114) λ Λ s bh . (58)Similar to the proof of Lemma 2 , and combined with theconstraint of Q in Eq. (57), the optimal solution of Problem 9 is given by ˆ s v = Λ s bh ( (cid:80) V − j =1 3 √ Γ j ) √ Γ v λ (( V − , v = 1 , · · · , V − , ∞ , v = V. (59)We can see that the optimal solution given in Eq. (59)contradicts to the optimal solution of Problem 4 given inEq. (22). Hence,
Q > NC ( δ,α ) (cid:16)(cid:80) Vj =1 3 (cid:113) qjqV − V (cid:17) A ( δ,α ) − C ( δ,α )+1 is a necessarycondition for finding the optimal solution in Eq. (22). Thiscompletes the proof. (cid:4) A PPENDIX DP ROOF OF L EMMA v , v = 1 , · · · , V and v = v +1 . Then we provethat U v > U v . We have U v = N C ( δ, α ) (cid:16)(cid:80) v j =1 (cid:113) q j q v − v (cid:17) A ( δ, α ) − C ( δ, α ) + 1 = N C ( δ, α ) (cid:16)(cid:80) v j =1 (cid:113) q j q v − v + (cid:80) v j = v +1 (cid:113) q j q v − ( v − v ) (cid:17) A ( δ, α ) − C ( δ, α ) + 1= N C ( δ, α ) (cid:16)(cid:80) v j =1 (cid:113) q j q v − v (cid:17) A ( δ, α ) − C ( δ, α ) + 1 ( a ) > N C ( δ, α ) (cid:16)(cid:80) v j =1 (cid:113) q j q v − v (cid:17) A ( δ, α ) − C ( δ, α ) + 1 = U v , (60)where ( a ) comes from the fact that q v < q v . This completesthe proof. (cid:4) A PPENDIX EP ROOF OF L EMMA L can only keep at most v VRs, it hasto retain the v most popular VRs to maximize its profit. Letus now prove that if L keeps ( v + w ) VRs, w = 1 , · · · , V − v ,in the game, it cannot achieve the optimal solution for U v In the case that L keeps ( v + w ) VRs, we havethe optimization problem of min s (cid:23) v + w (cid:88) j =1 s j , s.t. v + w (cid:88) j =1 (cid:115) Γ j s j ≤ (( v + w )Λ + Θ) (cid:114) λ Λ s bh . (61) EEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, ACCEPTED FOR PUBLICATION 14 Similar to the proof of Theorem 2 , we obtain that Q > NC ( δ,α ) (cid:18)(cid:80) v + wj =1 3 (cid:114) qjqv + w − ( v + w ) (cid:19) A ( δ,α ) − C ( δ,α )+1 = U v + w is the necessarycondition for the ( v + w ) VRs to participate in the game.This contradicts to the premise U v < Q ≤ U v +1 , since wehave Q > U v +1 according to Lemma 3 . Let us now considerthe cases of w (cid:48) = 0 , − , · · · , − v . To ensure there are ( v + w (cid:48) ) VRs in the game, Q has to satisfy the conditionthat Q > U v + w (cid:48) . 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