# Caching resource sharing in radio access networks: a game theoretic approach

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## Abstract

Deployment of caching in wireless networks has been considered an effective method to cope with the challenge brought on by the explosive wireless traffic. Although some research has been conducted on caching in cellular networks, most of the previous works have focused on performance optimization for content caching. To the best of our knowledge, the problem of caching resource sharing for multiple service provider servers (SPSs) has been largely ignored. In this paper, by assuming that the caching capability is deployed in the base station of a radio access network, we consider the problem of caching resource sharing for multiple SPSs competing for the caching space. We formulate this problem as an oligopoly market model and use a dynamic non-cooperative game to obtain the optimal amount of caching space needed by the SPSs. In the dynamic game, the SPSs gradually and iteratively adjust their strategies based on their previous strategies and the information given by the base station. Then through rigorous mathematical analysis, the Nash equilibrium and stability condition of the dynamic game are proven. Finally, simulation results are presented to show the performance of the proposed dynamic caching resource allocation scheme.

## Key words

Video caching Oligopoly market Game theory Nash equilibrium Stability analysis## CLC number

TP393## 1 Introduction

According to a recent report from Cisco (http://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/mobile-white-paper-c11-520862.html).

Deploying caching in a RAN is a promising technique to cope with these challenges and satisfy the quality-of-experience (QoE) and quality-of-service (QoS) (Pedersen and Dey, 2016), which has attracted a lot of attention. By storing popular video objects closer to the mobile users, most video requests can be served from the RAN caches, thereby decreasing the load of original servers and RAN backhaul, as well as the delivery delay (Wang *et al.*, 2014).

One direction is caching popular content in devices (Malak and Al-Shalash, 2014). Pedersen and Dey (2014) introduced a reactive mobile device caching (rMDC) framework, using which a mobile device could cache requested videos reactively and share these video contents with its neighbors using device-to-device communication. In Golrezaei *et al.* (2014), a novel scheme was presented to improve the throughput of video transmission in cellular communication systems by exploiting the large storage resources on modern smartphones. Nam and Chung (2015) proposed a cluster-based cooperative content caching scheme to reduce the content delivery cost of mobile devices. However, these techniques often face great challenges in motivating users to share their batteries, apart from limited bandwidth.

Another direction is caching within wireless networks. Recently, there have been some efforts conducting research on caching in cellular networks. Erman *et al.* (2011) showed the potential benefits of caching data in the carrier CN based on a study of Hypertext Transfer Protocol (HTTP) traffic collected in a cellular network. Hamidouche *et al.* (2014) proposed a many-to-many matching game theory to address the caching problem between small base stations (BSs) and service provider servers (SPSs). The idea of caching content objects at an evolving node B (eNodeB) and leveraging the information from in-network caches to improve caching performance has also been considered by Ming *et al.* (2014). Zhang *et al.* (2014) proposed a cooperative caching scheme based on the network coding technique to improve the cache hit rate and decrease the query processing time. A novel multiple-input multiple-output (MIMO) cooperation framework has been proposed by Liu and Lau (2015), to improve the video streaming performance by jointly optimizing cache control and playback buffer management. In addition, Gu *et al.* (2014) and Pingyod and Somchit (2014a; 2014b) have investigated the cache replacement strategy for cache-enabled nodes in cellular networks. Mavromoustakis (2008) has proposed a stream-oriented modeled scheme to provide optimized and guaranteed QoS, and later introduced a reliable file-sharing scheme for vehicular peer-to-peer (P2P) devices to increase end-to-end availability for delay sensitive streams (Mavromoustakis, 2013). Kryftis *et al.* (2014) presented a novel multimedia service delivery architecture to satisfy users’ requests efficiently by exploiting a resource prediction system.

Caching video contents at eNodeB could reduce transport energy but it needs additional caching energy. The problem of energy consumption in the eNodeB caching mechanism has also been studied. Xu *et al.* (2014) built a theoretical model to formulate the energy consumption, with the purpose of minimizing the total network energy consumption at eNodeB caches. Yang *et al.* (2014) have studied the energy efficiency problem in wireless cooperative caching networks and a suboptimal caching strategy has been proposed. An effective algorithm has been proposed by Arai *et al.* (2014) to minimize the total energy consumption of user equipments (UEs) by reassigning UEs to eNodeBs.

*et al.*, 2011; Chaudhry

*et al.*, 2015; Ding

*et al.*, 2015; Liu YX

*et al.*, 2015). To the best of our knowledge, the problem of caching space sharing for SPSs has been largely ignored. However, the issue is very important because to maximize the revenue of SPSs, each SPS always hopes to get enough caching space to cache its contents to improve the users’ experience, which may lead the SPSs to compete for the caching space. Therefore, our work is different from previous works. In this study, by assuming that caching capability is deployed in the BS in a RAN, we consider the problem of caching resource sharing for multiple SPSs to compete for the caching space. Some distinct features are listed as follows:

- 1.
We focus on the caching resource sharing problem in RANs. The system is modeled as an oligopoly market, in which the SPSs compete for the caching resources provided by the BS and the cost of the caching resources is defined by a price function.

- 2.
The caching resource sharing problem is formulated as a dynamic non-cooperative Cournot game. In addition, a Newton-Raphson method based iterative algorithm is proposed to obtain the optimal amount of caching space needed by SPSs (i.e., the Cournot equilibrium solution).

- 3.
We evaluate the performance of the proposed caching resource sharing scheme under different system parameters. The stability characteristics of the scheme are also analyzed.

## 2 System description

In this section, we first briefly depict the system model. Then the problem formulation is presented.

### 2.1 System model

Under this framework, we consider that there are *N* users in the set \({\mathcal N} = \{ {U_1},\;{U_2},\; \ldots ,\;{U_N}\} \) belonging to the BS and *M* SPSs in the set \({\mathcal M} = \{ {S_1},\;{S_2},\; \ldots ,\;{S_M}\} \) wanting to buy storage resources from the BS to cache their videos. The BS is ready to allocate some portion of the resources with SPS_{ i }, denoted as *b*_{ i }. The BS charges the SPSs for the resources at a rate of *c*(*b*) (per MB), where *b* is the total amount of storage resource demand of all SPSs. Besides, the cost of RAN backhaul resources paid by SPSs for transmitting 1 MB data is *C* (i.e., the cost of SPS_{ i } is *C*_{ i }).

### 2.2 Problem formulation

In this subsection, we formulate the caching resource sharing problem as an oligopoly market model. In an oligopoly market, a few players compete with each other to achieve the highest profit based on the amount of supplied commodity in the market. In the caching resource sharing problem considered here, all SPSs compete with each other to share the caching resources provided by the BS, and the goal of all SPSs is to achieve their highest revenue. We can model this situation as a Cournot game.

In a Cournot game, players are the SPSs. The commodity is the caching resource owned by the BS. Each player’s strategy corresponds to the amount of allocated caching resources, denoted as *b*_{ i } for SPS_{ i }. The payoff for each player is the revenue of each SPS (denoted as *π*_{ i } for SPS_{ i }) by using the caching resources. In the following part, we first present the price function used by the BS. Then based on the delivery cost without/with caching, the utility function of SPSs is given.

#### 2.2.1 Price function used by the BS

*x*and

*y*are non-negative constants, and \({\mathcal B} = \{ {b_1},\;{b_2},\; \ldots ,\;{b_M}\} \) is the set of strategies of all SPSs. To meet the demand price theory in economics (i.e., when the supply is fixed and the demand increases, the price will go up), we define

*π*≥ l;in this case, the price function is convex. Caching content objects at the BS would consume additional costs (e.g., central processing unit/system bus usage and storage cost), which depend mainly on the caching hardware technology, such as dynamic random access memory (DRAM), high-speed solid state disk (SSD), ternary content-addressable memory (TCAM), and static random access memory (SRAM). In this study, let

*w*denote the average cost (per MB) of these object-caching technologies. Then we assume \(c({\mathcal B}) > w\); otherwise, the BS is not willing to sell its storage resources to the SPSs.

#### 2.2.2 Delivery cost without caching

*et al.*, 2014). The total costs of SPS

_{ i }for processing the requests can be computed as follows:

*n*

_{ i }denotes the total number of requests arriving at SPS

_{ i },

*q*

_{ i }is the mean size (MB) of a requested video object, and

*R*

_{ i },

*G*

_{ i },

*T*

_{ i }are the cost components denoting the cost (per MB) from the users’ devices to the BS, the cost of the CN link (per MB) from theBSto the PGW, andthe cost (per MB)fromthe PGW to the source servers of SPS

_{ i }, respectively.

#### 2.2.3 Delivery cost with caching

*et al.*(2014), the costs for SPS

_{ i }to fulfill the requests can be computed as follows:

*c*(

*b*) is the cost (per MB) of caching objects paid to the BS, and

*N*

_{ i }is the number of video objects cached at the BS. Therefore, \(({n_i} - n_i^{{\rm{cache}}}) \cdot ({R_i} + {G_i} + {T_i}) \cdot {q_i}\)represents the cost incurred when the requested objects need to be obtained from source servers, \(n_i^{{\rm{cache}}} \cdot {R_i} \cdot {q_i}\) is the cost incurred on the RAN path from the BS to the users’ devices, and

*N*

_{ i }·

*c*(

*b*) ·

*q*

_{ i }is the additional cost of SPS

_{ i }for caching video objects at the BS.

#### 2.2.4 Utility function of SPSs

_{ i }as follows:

*C*

_{ i }is composed of the CN link cost (per MB) from the BS to the PGW and the cost (per MB) from the PGW to the source servers (i.e.,

*C*

_{ i }=

*G*

_{ i }+

*T*

_{ i }),

*b*

_{ i }denotes the portion of caching resources that SPS

_{ i }wants to buy from the BS (i.e.,

*b*

_{ i }=

*N*

_{ i }·

*q*

_{ i }), and \({{{n_i^{{\rm{cache}}}} \over {{n_i}}}}\) denotes the ratio of the requested cached objects to the total requested objects (related to the specific caching policies). There are many caching policies, such as least recently used (LRU), least frequently used (LFU), and user preference profile (UPP). The LRU (Sleator and Tarjan, 1985) replaces videos in the caching space that have been least used recently if the caching space is full. The LFU (Lee

*et al.*, 2001) caches the ‘most popular videos’ and discards the least frequently used ones by keeping track of the number of requests to each video. The UPP (Ahlehagh and Dey, 2014) caching policy uses the UPPs of active users in a cell to cache video objects. Although there are many caching policies, they have a common goal, which is caching the videos with the highest requested probability. The larger the requested probability of a video is, the higher the priority of caching by the SPSs.

*et al.*, 1999; Cha

*et al.*, 2009); the probability of accessing a video at rank

*k*out of \(N_i^{{\rm{total}}}\) available video objects can be expressed as follows:

_{ i }, \({\Omega _i} = \sum\nolimits_{j = 1}^{N_i^{{\rm{total}}}} {{j^{ - \beta }}} \)is a constant used to normalize the requested probability. A larger

*β*means that more requests are concentrated on a few hot video objects. Typically, the value of

*β*ranges between zero and one (i.e.,

*β*∈ (0, 1))(Breslau

*et al.*, 1999). In this case, if SPS

_{ i }wants to cache

*N*

_{ i }most popular video objects at the BS, \({{n_i^{{\rm{cache}}}} \over {{n_i}}}\) could be computed as follows:

*et al.*(1999), we can obtain \(\sum\nolimits_{k = 1}^{{N_i}} {{k^{ - \beta }}} \approx {{N_i^{1 - \beta }} \over {1 - \beta }}\). Thus, for Zipf-like distributions, the cumulative probability that one of the top

*N*

_{ i }video objects is accessed could be given approximately as follows:

_{ i }can be computed as follows:

The Cournot game model has been built, and the equilibrium solution will be described in the next section.

## 3 Equilibrium solution of Cournot game and stability analysis

As mentioned in Section 2, we can model the problem of caching space sharing as a Cournot game. If we assume that each SPS can completely observe the strategies and the profits gained by the other SPSs, a simple static Cournot game can be modeled to solve the problem. However, this assumption is not practical; in other words, an SPS may not be able to observe the payoff of another SPS. Moreover, the current strategy adopted by another SPS may be unknown, and only the pricing information from the BS could be observed by SPSs. Based on the above situation, in this section, we discuss a dynamic Cournot game model. In this scenario, an SPS adjusts its strategy according to the variations in payoff because of the differential price charging by the BS. The goal of this caching resource sharing problem is to maximize the profit of all SPSs by using the equilibrium concept. In this section, we first introduce the dynamic Cournot game. Then we give the equilibrium solution and stability analysis of the dynamic Cournot game.

### 3.1 Dynamic Cournot game

*b*

_{ i }by using the marginal profit function as shown in Eq. (10). Let

*b*

_{ i }(

*t*) denote the strategy of SPS

_{ i }at iteration

*t*, and

*b*

_{ i }(

*t*+ 1) is defined similarly. Therefore, the relationship between the strategies in the current and the future iterations can be expressed as follows (Niyato and Hossain, 2007; 2008):

*α*

_{ i }is the learning rate of SPS

_{ i }and

Therefore, the dynamic Cournot game can be represented as shown in Eq. (12).

### 3.2 Equilibrium solution of dynamic Cournot game

An element in this set corresponds to the strategy of an SPS, and a strategy profile must include one and only one strategy for each SPS. For ease of analysis, we can also write \({\mathcal B}\) as (*b*_{ i }, \({{\mathcal B}_{ - i}}\)), where *b*_{ i } is the strategy of SPS_{ i }, and \({{\mathcal B}_{ - i}}\) denotes the set of strategies of all SPSs except SPS_{ i } (i.e., \({{\mathcal B}_{ - i}} = \left\{ {{b_j}|j = 1,\;2,\; \ldots ,\;M;j \ne i} \right\}\), *M*; *j* ≠ *i*}. Now, we analyze the Cournot game model to obtain the stable solutions called ‘Nash equilibrium’.

Nash equilibrium is a combination of optimal strategies, that is, \({{\mathcal B}^*} = \left\{ {b_1^*,\;b_2^*,\; \ldots ,\;b_M^*} \right\}\), where \(b_i^*\) denotes the best response of SPS_{ i }, whereby this participant cannot increase its payoff by choosing a different action given the other participants’ actions. It means that \({\pi _i}\left( {{b_i},\;{\mathcal B}_{ - i}^*} \right) \le {\pi _i}\left( {b_i^*,\;{\mathcal B}_{ - i}^*} \right)\), where \({{\mathcal B}_{ - i}^*}\)denotes the set of optimal strategies of all SPSs except SPS_{ i }. So, the Nash equilibrium in a non-cooperative game is a situation in which all of the players use the optimal strategies and none of them wish to deviate from the equilibrium (Dufwenberg, 2011).

In an actual system, the SPSs can estimate the value of \({{\partial {\pi _i}\left( {\mathcal B} \right)} \over {\partial {b_i}}}\) (Niyato and Hossain, 2008). In a particular case, at time *t*, each SPS submits the storage resource size *b*_{ i }(*t*)±*δ* to the BS, where *δ* is a small number (e.g., *δ* = 0.001). Then the BS computes the price *c*^{−}(·) and *c*^{+}(·) for *b*_{ i }(*t*) − *δ* and *b*_{ i } (*t*) + *δ* respectively, according to the price function. Next, the BS sends the information to these SPSs. With this information, each SPS can compute the profits \(\pi _i^ - \left( \cdot \right)\) and \(\pi _i^ + \left( \cdot \right)\) locally and then estimate the marginal profit from \({{\partial {\pi _i}\left( \cdot \right)} \over {\partial {b_i}\left( t \right)}} \approx {{\pi _i^ + \left( \cdot \right) - \pi _i^ - \left( \cdot \right)} \over {2\delta }}\)

*b*

_{ i }(

*t*+1) =

*b*

_{ i }(

*t*) =

*b*

_{ i }(

*i*= 1, 2, …

*M*) (Agiza

*et al.*, 1999). With a linear price function (i.e.,

*τ*=1; note that this linear price function is a common assumption for an oligopoly market), the optimal response can be obtained by solving the following set of equations (

*i*=1, 2, …

*M*):

*M*= 2); Eq. (13) can be rewritten as follows (note that to facilitate the analysis, we let

*x*= 0):

*b*

_{1}and

*b*

_{2}as follows:

_{1}, i.e.,

*b*

_{1}. Then according to the relationship between

*b*

_{1}and

*b*

_{2}(i.e., Eq. (15)), we can obtain the optimal solution of SPS

_{2}, i.e.,

*b*

_{2}.

Now, we consider the situation with more than two SPSs. Because in a dynamic Cournot game, SPSs adjust their demanded caching space *b*_{ i } dynamically, the caching resource sharing problem can be expressed as a system of nonlinear equations, as in Eq. (17). Inspired by Kelley (2003), we can use the following iterative algorithm based on the Newton-Raphson method to obtain the Cournot game equilibrium solution. The basic idea of the Newton-Raphson method is to linearize nonlinear equations successively, each step solving linear equations, consequently forming an iterative algorithm. The iterative algorithm is given in Algorithm .

*b*

_{ i }(

*k*+ 1) =

*b*

_{ i }(

*k*) −

*μ*

_{ i }(

*k*)

*d*

_{ i }(

*k*) (

*i*=1, 2,…,

*M*), to guarantee that

*b*

_{ i }is a positive value, and

When the caching space demands converge (i.e., ∥*f*_{ i }(* b*(

*k*)) −

*f*

_{ i }(

*(*

**b***k*− 1))∥ <

*ε*)where

*= (*

**b***b*

_{1},

*b*

_{2},…

*b*

_{ M })

^{T}and

*i*= 1, 2, …,

*M*, the algorithm terminates. In this case, the equilibrium solutions of the Cournot game are obtained (i.e.,

*(*

**b***k*)). The performance of the proposed algorithm will be shown later through simulations.

### 3.3 Stability analysis of the dynamic Cournot game

Now we give the stability analysis of the caching space allocation scheme by considering the eigenvalues of the Jacobian matrix (Eq. (19)) and applying the Routh-Hurwitz stability condition (Sonis, 1996). By definition, the equilibrium point is stable if, and only if, all the eigenvalues of the Jacobian matrix are inside the unit circle of the complex plane (i.e., |λ_{ i }| < 1) (Agiza *et al.*, 1999; Niyato and Hossain, 2008).

When we consider the scenario with two SPSs, the Jacobian matrix can be expressed as Eq. (20).

*is the identity matrix. The solution is*

**I**By definition, if the evolutionary equilibrium of the dynamic Cournot game with two SPSs is stable, |λ_{1}| < 1 and |λ_{2}| < 1.

In the scenario that there are more than two SPSs in the system, we can obtain all eigenvalues by solving the characteristic equation of the Jacobian matrix (i.e., φ(λ) = det(λ* I* −

*)= 0). Then based on the Routh-Hurwitz stability condition, we can confirm whether the Nash equilibrium point is stable.*

**J**## 4 Simulation results and discussion

In this section, we use simulations to evaluate the performance of the distributed dynamic caching resource sharing scheme. For illustration purposes, we first consider a simple network environment with two SPSs and one BS. The simulation parameters are set as follows. For the price function, we set *x* =0, *y* = 1, and *τ* =1 (Niyato and Hossain, 2007; 2008). We assume that each video object of each SPS has the same size, which is 25 MB (i.e., *q*_{ i } = 25), and that the content popularity follows the Zipf-like distribution with the skewness factor *β* = 0.8 (Ahlehagh and Dey, 2014). Besides, the total number of requests arriving at each SPS is 10 000 (i.e., *n*_{ i } = 10 000). The backhaul cost of transmitting 1 MB data is 10.0 (i.e., *C*_{ i } = 10). The total number of video contents owned by each SPS is 1000 (i.e., \(N_i^{{\rm{total}}} = 1000\)). Note that some of these parameters have to be adjusted based on the evaluation scenarios.

_{1}increases with the increase of its total number of requests (

*n*

_{1}) and decreases with the increase of the total number of requests arriving at SPS

_{2}(

*n*

_{2}). This is not hard to understand. When there are more requests, SPSs need larger bandwidth to deliver video objects, which will result in the increase of backhaul costs. In this case, SPSs prefer to buy more caching space. In Figs. 3c and 3d, we analyze the relationship between the caching space demand from the SPS and the backhaul cost of transmitting 1 MB data. Relationships with mean size of a video content are evaluated in Figs. 3e and 3f. In Figs. 3g and 3h, we investigate the relationship between the caching space demand from the SPS and the total number of videos owned by each SPS. Obviously, caching space demand depends largely on the price given by the BS to sell its storage resources and backhaul costs used by SPSs to transmit requested video objects to UEs. While caching space demand is a decreasing function of resource price, it is an increasing function of backhaul costs. Specifically, when the resource price increases the SPS needs to pay the BS more fees for a certain amount of storage resources, resulting in the situation in which the SPS tends to buy less caching space. However, with the increase of backhaul costs, the SPS is willing to buy more caching space to reduce expenditure in backhaul bandwidth. When there are more requests arriving at SPS

_{2}, SPS

_{2}tends to demand more storage resources, and therefore the BS will charge higher price for the same size of resources (as shown in Eq. (1), the price function is convex). As a result, the profit of SPS

_{1}decreases and the size of the demanded storage resources from SPS

_{1}becomes smaller. This analysis method can be used for other evaluation results in Fig. 3.

_{1}first increases with the increase of its caching space. Then when the caching space size reaches a certain value, the revenue of SPS

_{1}begins to decrease. This is because when the storage resources demanded by either SPS (i.e., SPS

_{1}or SPS

_{2}) increases, the price given by the BS will become larger (Eq. (1) with

*τ*=1). In this case, the cost of buying storage resources will increase. As a result, the SPS first benefits from buying caching space. However, at a particular point (e.g.,

*b*

_{1}= 245), the revenue gained by the SPS decreases because the storage resource price becomes too high.

*k*= 1, 2) to denote the variations of several parameters (e.g.,

*n*

_{ k }and

*C*

_{ k }). From the figure, we note that the best response of the SPSs has a demand for a larger caching space when

*g*

_{ k }is higher. This is because the SPSs will reduce more backhaul cost when the value of

*g*

_{ k }is larger. So, in this case, the SPSs expect to get more storage resources to maximize their revenue by caching more popular video contents.

*β*increases,

*g*

_{1}and

*g*

_{2}become larger, resulting in both the SPSs needing more caching space. In this case, the BS will achieve benefits from charging higher price from the SPSs (as shown in Eq. (1), the price function is convex). However, at a particular point (e.g.,

*β*= 0.75), the caching space demand of SPS

_{1}decreases because the price of the storage resources becomes too high and the cost of sharing storage resources increases at a rate larger than the revenue of SPS

_{1}.

*b*

_{1}(0) =

*b*

_{2}(0) = 300. With the proposed dynamic caching space sharing scheme in which an SPS is not aware of others, and it adjusts its demand based on the marginal profit for the SPS, we illustrate the variations in caching space demand for both SPSs in Fig. 7 when

*g*

_{1}= 60 000 and

*g*

_{2}= 50 000. Obviously, with these parameters, the Nash equilibrium point is located at

*b*

_{1}= 250.95 and

*b*

_{2}= 217.39. As shown in Fig. 7, when the learning rate is set properly (e.g.,

*α*

_{1}= 0.0015 and

*α*

_{2}=0.0015), the caching space demand will converge gradually and finally reach the Nash equilibrium. However, if the learning rate is large (e.g.,

*α*

_{1}=0.0023 and

*α*

_{2}= 0.0023), the strategies of both SPSs will swing drastically and may never converge to the Nash equilibrium. This is because if the learning rate is too large, the determination of an SPS strategy greatly depends on the latest information obtained from the BS, such as the pricing information. Therefore, the strategies of SPSs fluctuate too much to converge to the equilibrium.

*g*

_{ k }. We observe that the location of the Nash equilibrium depends on the value of

*g*

_{ k }. From Fig. 9, we can see that as the value of

*g*

_{3}increases, SPS

_{3}prefers to demand a larger caching space size from the BS. Moreover, the value of

*g*

_{3}affects the caching space demands of other SPSs. From the figure, we can see that with the increase of the value of

*g*

_{3},the demanded caching space and revenue of other SPSs (i.e., SPS

_{1}and SPS

_{2}) decrease. Furthermore, the revenue and the demanded caching space of other SPSs decrease at a rate smaller than the increasing rate of SPS

_{3}. In addition, the revenue of the BS increases at a low rate. Similar results are expected for scenarios with a larger number of SPSs.

## 5 Conclusions and future work

In this paper, we have discussed the issues of the caching resource sharing strategy in RANs to save the bandwidth of RAN backhaul networks. The caching resource sharing problem is formulated as a dynamic Cournot game. Then we computed the equilibrium solution of a specific system model with two SPSs and proposed an iteration algorithm based on the Newton-Raphson method to obtain the equilibrium solution with more than two SPSs. Next, the stability analysis of the dynamic Cournot game has been presented. Simulation results have been presented to illustrate the performance of the proposed scheme and demonstrate the instability effects due to changes in the learning rate.

In this paper, we consider the scenario in which there is only one BS to cache video objects of several SPSs. In our future work, we will consider inter-domain collaborative caching scenarios, wherein multiple BSs cooperate in caching. The storage resource sharing problem in these cooperative scenarios should consider the imperfect and dynamic radio channel condition (Xie *et al.*, 2012a). Moreover, this research is based on the LTE network. However, in recent years, the fifth-generation (5G) network has become an advanced research hotspot in academia and industry (Liang *et al.*, 2015). So, the potential caching techniques that might be used in 5G mobile networks will be considered in our future work. Future works also include exploring new relevant opportunities and challenges of caching contents in 5G systems.

## References

- Agiza, H.N., Bischi, G.I., Kopel, M., 1999. Multistability in a dynamic Cournot game with three oligopolists.
*Math. Comput. Simul.*,**51**(1–2):63–90. http://dx.doi.org/10.1016/S0378-4754(99)00106-8MathSciNetCrossRefGoogle Scholar - Ahlehagh, H., Dey, S., 2014. Video-aware scheduling and caching in the radio access network.
*IEEE/ACM Trans. Netw.*,**22**(5):1444–1462. http://dx.doi.org/10.1109/TNET.2013.2294111CrossRefGoogle Scholar - Arai, S., Fadlullah, Z.M., Ngo, T.,
*et al.*, 2014. An efficient method for minimizing energy consumption of user equipment in storage-embedded heterogeneous networks.*IEEE Wirel. Commun.*,**21**(4):70–76. http://dx.doi.org/10.1109/MWC.2014.6882298CrossRefGoogle Scholar - Bastug, E., Bennis, M., Debbah, M., 2014. Living on the edge: the role of proactive caching in 5G wireless networks.
*IEEE Commun. Mag.*,**52**(8):82–89. http://dx.doi.org/10.1109/MCOM.2014.6871674CrossRefGoogle Scholar - Beyranvand, H., Lim, W., Maier, M.,
*et al.*, 2015. Backhaulaware user association in FiWi enhanced LTE-A heterogeneous networks.*IEEE Trans. Wirel. Commun.*,**14**(6):2992–3003. http://dx.doi.org/10.1109/TWC.2015.2399308CrossRefGoogle Scholar - Breslau, L., Cao, P., Fan, L.,
*et al.*, 1999. Web caching and Zipf-like distributions: evidence and implications. Proc. IEEE INFOCOM Jointly with the 18th Annual Conf. of the IEEE Computer and Communications Societies, p.126–134. http://dx.doi.org/10.1109/INFCOM.1999.749260Google Scholar - Cha, M., Kwak, H., Rodriguez, P.,
*et al.*, 2009. Analyzing the video popularity characteristics of large-scale user generated content systems.*IEEE/ACM Trans. Netw.*,**17**(5):1357–1370. http://dx.doi.org/10.1109/TNET.2008.2011358CrossRefGoogle Scholar - Chaudhry, M.T., Ling, T.C., Hussain, S.A.,
*et al.*, 2015. Thermal-aware relocation of servers in green data centers.*Front. Inform. Technol. Electron. Eng.*,**16**(2):119–134. http://dx.doi.org/10.1631/FITEE.1400174CrossRefGoogle Scholar - Chuang, M.C., Chen, M.C., 2015. A mobile proxy architecture for video services over high-speed rail environments in LTE-A networks.
*IEEE Syst. J.*,**9**(4):1264–1272. http://dx.doi.org/10.1109/JSYST.2014.2354435CrossRefGoogle Scholar - Ding, J., Huang, T., Liu, J.,
*et al.*, 2015. Virtual network embedding based on real-time topological attributes.*Front. Inform. Technol. Electron. Eng.*,**16**(2):109–118. http://dx.doi.org/10.1631/FITEE.1400147CrossRefGoogle Scholar - Dufwenberg, M., 2011. Game theory.
*Wiley Interdiscip. Rev. Cogn. Sci.*,**2**(2):167–173. http://dx.doi.org/10.1002/wcs.119MathSciNetCrossRefGoogle Scholar - Erman, J., Gerber, A., Hajiaghayi, M.,
*et al.*, 2011. To cache or not to cache: the 3G case.*IEEE Internet Comput.*,**15**(2):27–34. http://dx.doi.org/10.1109/MIC.2010.154CrossRefGoogle Scholar - Golrezaei, N., Mansourifard, P., Molisch, A.F.,
*et al.*, 2014. Base-station assisted device-to-device communications for high-throughput wireless video networks.*IEEE Trans. Wirel. Commun.*,**13**(7):3665–3676. http://dx.doi.org/10.1109/TWC.2014.2316817CrossRefGoogle Scholar - Gu, J.X., Wang, W., Huang, A.P.,
*et al.*, 2014. Distributed cache replacement for caching-enable base stations in cellular networks. Proc. IEEE Int. Conf. on Communications, p.2648–2653. http://dx.doi.org/10.1109/ICC.2014.6883723Google Scholar - Hamidouche, K., Saad, W., Debbah, M., 2014. Many-tomany matching games for proactive social-caching in wireless small cell networks. Proc. 12th Int. Symp. on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks, p.569–574. http://dx.doi.org/10.1109/WIOPT.2014.6850348Google Scholar
- Kelley, C., 2003. Solving Nonlinear Equations with Newton’s Method. Society for Industrial and Applied Mathematics. http://dx.doi.org/10.1137/1.9780898718898Google Scholar
- Kryftis, Y., Mavromoustakis, C.X., Mastorakis, G.,
*et al.*, 2014. Resource usage prediction for optimal and balanced provision of multimedia services. Proc. IEEE 19th Int. Workshop on Computer Aided Modeling and Design of Communication Links and Networks, p.255–259. http://dx.doi.org/10.1109/CAMAD.2014.7033245Google Scholar - Lee, D., Choi, J., Kim, J.H.,
*et al.*, 2001. LRFU: a spectrum of policies that subsumes the least recently used and least frequently used policies.*IEEE Trans. Comput.*,**50**(12):1352–1361. http://dx.doi.org/10.1109/TC.2001.970573MathSciNetCrossRefGoogle Scholar - Liang, C., Yu, F.R., Zhang, X., 2015. Information-centric network function virtualization over 5G mobile wireless networks.
*IEEE Netw.*,**29**(3):68–74. http://dx.doi.org/10.1109/MNET.2015.7113228CrossRefGoogle Scholar - Liu, A., Lau, V., 2015. Exploiting base station caching in MIMO cellular networks: opportunistic cooperation for video streaming.
*IEEE Trans. Signal Process.*,**63**(1):57–69. http://dx.doi.org/10.1109/TSP.2014.2367473MathSciNetCrossRefGoogle Scholar - Liu, J., Huang, T., Chen, J.Y.,
*et al.*, 2011. A new algorithm based on the proximity principle for the virtual network embedding problem.*J. Zhejiang Univ.-Sci. C (Comput. & Electron.)*,**12**(11):910–918. http://dx.doi.org/10.1631/jzus.C1100003CrossRefGoogle Scholar - Liu, Y.X., Li, K.L., Tang, Z.,
*et al.*, 2015. Energy-aware scheduling with reconstruction and frequency equalization on heterogeneous systems.*Front. Inform. Technol. Electron. Eng.*,**16**(7):519–531. http://dx.doi.org/10.1631/FITEE.1400399CrossRefGoogle Scholar - Malak, D., Al-Shalash, M., 2014. Optimal caching for deviceto-device content distribution in 5G networks. Proc. Globecom Workshops, p.863–868. http://dx.doi.org/10.1109/GLOCOMW.2014.7063541Google Scholar
- Mavromoustakis, C.X., 2008. On the impact of caching and a model for storage-capacity measurements for energy conservation in asymmetrical wireless devices. Proc. 16th Int. Conf. on Software, Telecommunications and Computer Networks, p.243–247. http://dx.doi.org/10.1109/SOFTCOM.2008.4669488Google Scholar
- Mavromoustakis, C.X., 2013. Mitigating file-sharing misbehavior with movement synchronization to increase endto-end availability for delay sensitive streams in vehicular P2P devices.
*Int. J. Commun. Syst.*,**26**(12):1599–1616. http://dx.doi.org/10.1002/dac.2335CrossRefGoogle Scholar - Ming, Z.X., Xu, M.W., Wang, D., 2014. InCan: in-network cache assisted eNodeB caching mechanism in 4G LTE networks.
*Comput. Netw.*,**75**(A):367–380. http://dx.doi.org/10.1016/j.comnet.2014.10.021CrossRefGoogle Scholar - Nam, Y., Chung, J.M., 2015. Cooperative content delivery for cost minimization in wireless networks. Proc. 17th Asia-Pacific Network Operations and Management Symp., p.566–568. http://dx.doi.org/10.1109/APNOMS.2015.7275393Google Scholar
- Niyato, D., Hossain, E., 2007. A game-theoretic approach to competitive spectrum sharing in cognitive radio networks. Proc. IEEE Wireless Communications and Networking Conf., p.16–20. http://dx.doi.org/10.1109/WCNC.2007.9Google Scholar
- Niyato, D., Hossain, E., 2008. Competitive spectrum sharing in cognitive radio networks: a dynamic game approach.
*IEEE Trans. Wirel. Commun.*,**7**(7):2651–2660. http://dx.doi.org/10.1109/TWC.2008.070073CrossRefGoogle Scholar - Pedersen, H.A., Dey, S., 2014. Mobile device video caching to improve video QoE and cellular network capacity. Proc. 17th ACM Int. Conf. on Modeling, Analysis and Simulation of Wireless and Mobile Systems, p.103–107. http://dx.doi.org/10.1145/2641798.2641820Google Scholar
- Pedersen, H.A., Dey, S., 2016. Enhancing mobile video capacity and quality using rate adaptation, RAN caching and processing.
*IEEE/ACM Trans. Netw.*,**24**(2):996–1010. http://dx.doi.org/10.1109/TNET.2015.2410298CrossRefGoogle Scholar - Pingyod, A., Somchit, Y., 2014a. Content updating method in FemtoCaching. Proc. 11th Int. Joint Conf. on Computer Science and Software Engineering, p.123–127. http://dx.doi.org/10.1109/JCSSE.2014.6841854Google Scholar
- Pingyod, A., Somchit, Y., 2014b. Rank-based content updating method in FemtoCaching. Proc. IEEE Region 10 Conf., p.1–6. http://dx.doi.org/10.1109/TENCON.2014.7022376Google Scholar
- Sleator, D.D., Tarjan, R.E., 1985. Amortized efficiency of list update and paging rules.
*ACM Commun.*,**28**(2):202–208. http://dx.doi.org/10.1145/2786.2793MathSciNetCrossRefGoogle Scholar - Sonis, M., 1996. Once more on Hénon map: analysis of bifurcations.
*Chaos Solit. Fract.*,**7**(12):2215–2234. http://dx.doi.org/10.1016/S0960-0779(96)00081-1MathSciNetCrossRefGoogle Scholar - Wang, X.F., Chen, M., Taleb, T.,
*et al.*, 2014. Cache in the air: exploiting content caching and delivery techniques for 5G systems.*IEEE Commun. Mag.*,**52**(2):131–139. http://dx.doi.org/10.1109/MCOM.2014.6736753CrossRefGoogle Scholar - Xie, R., Yu, F.R., Ji, H., 2012a. Dynamic resource allocation for heterogeneous services in cognitive radio networks with imperfect channel sensing.
*IEEE Trans. Veh. Technol.*,**61**(2):770–780. http://dx.doi.org/10.1109/TVT.2011.2181966CrossRefGoogle Scholar - Xie, R., Yu, F.R., Ji, H.,
*et al.*, 2012b. Energy-efficient resource allocation for heterogeneous cognitive radio networks with femtocells.*IEEE Trans. Wirel. Commun.*,**11**(11):3910–3920. http://dx.doi.org/10.1109/TWC.2012.092112.111510CrossRefGoogle Scholar - Xu, Y.M., Li, Y., Wang, Z.H.,
*et al.*, 2014. Coordinated caching model for minimizing energy consumption in radio access network. Proc. IEEE Int. Conf. on Communications, p.2406–2411. http://dx.doi.org/10.1109/ICC.2014.6883683Google Scholar - Yang, C.C., Chen, Z.Y., Yao, Y.,
*et al.*, 2014. Energy efficiency in wireless cooperative caching networks. Proc. IEEE Int. Conf. on Communications, p.4975–4980. http://dx.doi.org/10.1109/ICC.2014.6884109Google Scholar - Zhang, Y.D., Xu, X.J., Wang, X.L.,
*et al.*, 2014. NC-COCA: network coding-based cooperative caching scheme. Proc. IEEE 17th Int. Conf. on Computational Science and Engineering, p.976–980. http://dx.doi.org/10.1109/CSE.2014.195Google Scholar