I Introduction
To achieve the goals of the fifth generation (5G) mobile wireless systems [1] and alleviate the existing challenges in cloud radio access networks (CRANs)[2][4], the fog radio access network (FRAN) has been proposed as a new network architecture by incorporating of fog computing, edge storage and centralized cloud computing into radio access networks [5][6]. Fog computing, which is similar to edge computing, is first proposed by Cisco [7]. It extends cloud computing based services to the edge of the network. In FRANs, services cannot only be executed in a centralized unit such as the base band unit (BBU) pool in CRAN, but also can be hosted at smart terminal devices which are closer to the users. Meanwhile, through the usercentric adaptive techniques such as devicetodevice (D2D), distributed coordination, and largescale centralized cooperation, users do not have to connect to the centralized cloud computing unit to complete the data transmission, which will improve both spectrum and energy efficiencies as well as relieve the load of fronthaul and alleviate the burden of BBU pool. In order to execute the above, the traditional access point (AP) is evolved to the fog access point (FAP) through equipped with a certain caching and sufficient computing capabilities to execute the local cooperative signal processing in the physical layer.
Extensive studies have been done to study the user access mode schemes in CRANs and heterogeneous cloud radio access networks (HCRANs). The performance of uplink transmissions under different user access modes in CRANs have been investigated in [8], and the impact of the remote radio heads (RRHs) intensity, the number of antennas and the pathloss exponent on the performance is characterized. By using the stochastic geometry tool, the authors in [9] have provided the probabilistic characterization of the signal interference ratio (SIR) distribution and the ergordic rate at a randomly located user in a downlink HCRAN system. In [10]
, the successful access probability has been derived for CRAN RRH cluster mode, and based on the analytical result, two coalition game based cluster algorithms have been designed. And in our previous study
[11], the ergodic rates of two tier FRANs under different access modes are characterized, and a mode selection mechanism is presented.However, most of literatures focused on the mode selection schemes based on the user received SIR and distance, whereas little attention has been paid to study the impact of cache and fronthaul delay. The fronthaul limitations between RRHs and cloud are known to impose a formidable bottleneck to the system performances and a remarkable challenge to block the commercial practices [12]. Consequently, taking advantage of the fogcomputing to switch the content cache or data process to network edge devices is the key to increase both of the spectral and energy efficiency, as well as relieve the burden of fronthaul in cloud computing based network architectures. D2D communications as one of the network edge communication technologies which has the ability of support content delivery and sharing is more and more paid extensive attention. Existing research on D2D networks mode selection mainly focused on underlaid cellular networks with fixed location model [13] [15]. Different from the traditional underlaid cellular networks, the main challenge to design a user access mode selection algorithm in FRANs is that the FAPs and D2D users are often deployed randomly and the network performance of FRANs may drastically deteriorate with the increasing number of users who select the access mode with more fronthaul load.
In this paper, we present an adaptive user access mode selection algorithm for downlink FRAN by taking into account the different nodes locations, cache sizes as well as the fronthaul delay cost. The main contributions of this paper are threefolds.

We characterize the ergodic rate and the coverage probability of both D2D and FAP modes in FRANs. We investigate the impacts of the user node density, and the quality of service (QoS) constrains and the intratier interference. The closedform expressions are presented in some special cases, which can make the analysis not only tractable, but also flexible.

Based on the above results obtained from the stochastic geometry analysis, an evolutionary game theoretic approach is presented to solve the problem of user access mode selection, while taking into account different access service requirements and fronthaul delay cost.

A distributed algorithm is proposed to reach the evolutionary equilibrium, and its performance is compared with that of the max rate based user access mode selection scheme. The accuracy of the stochastic geometry analytical results are validated by Monte Carlo simulation results.
The remainder of this paper is organized as follows. Section II describes the system model, cache model, delay model as well as the user access modes in a downlink FRAN. In Section III, the dynamic behaviors of potential FAP users are modeled as an evolutionary game. And we present the results of the coverage probability and ergodic rate in FRAN system for both D2D and FAP modes. While in Section IV, we propose a distributed protocol to converge the evolutionary equilibrium, and the evolutionary stable strategy (ESS) is analyzed. Simulation results provided in Section V prove that the performance comparisons of different user access modes with different system parameters and the robustness of our protocol. Finally, Section VI concludes the paper.
Ii System Model
Iia Radio Access Network Model
As shown in Fig.1, a FRAN downlink system is considered in this paper, where a group of FAPs are deployed according to a twodimensional PPP with density of in a disc plane . Each FAP is assumed as single antenna configuration with a fixed transmit power , and interfaced to the cloud computing layer through a wired fronthaul network which is composed of links connecting FAPs with several gateways. Then, the signals over gateways will be largescale processed in the cloud network server via dedicated fiber [16]. The locations of gateways are distributed according to a homogeneous PPP with intensity .
The spatial distribution of users is modeled as an independent PPP distribution with constant intensity . We set as the probability that a user can direct transmit its local cache content to its nearby content require user (i.e.support D2D mode). Then, the location distribution of the D2D transmit users can be denoted as a thinning homogeneous with the density of . Meanwhile according to Marking Theorem, the distribution of content require users follows a stationary PPP with the density of .
In this paper, we focus on the usercentric access mode selection for content require users via the above FRAN system within two phases, namely cache prefetching and access mode selection. The total number of content require users can be estimated as
, where is the area of the disc plane .IiB Cache PreFetching Phase
In cache prefetching phase, the FAPs will cache content from the central cloud server file library which stored contents with uncoded cache strategies that may be requested by users . Then, the D2D transmit users will cache part of the content from FAPs via wireless link. The prefetching content is limited by the local caching storage space size of FAPs and D2D transmit users, which denote by and (), respectively, and the prefetching content in and is assumed to be constant across access mode selection phase.
Prior studies have been found that users are mostly interested in downloading the most popular video contents [17], which leads to only a small portion of the contents are frequently accessed by the majority of users. Therefore, in this paper we assume the FAPs and D2D transmit users only cache the most popularity content, and the demand probability can be modeled as the following Zipf distribution
(1) 
where , and the video content with a smaller index has a larger probability of being requested by users, i.e. , if . Zipf exponent controls the relative popularity of files, and with the larger the caching storage has a fewer of popular video contents accounting for the majority of the requests.
Next, we define the content caching probability as the probability of an event that a user can find the requested file in its accessed node, i.e., . By setting the caching storages in FAPs and D2D transmit users to only cache the most popularly requested video contents, the content caching probabilities of each FAP and D2D transmit user can be respectively obtained as
(2) 
(3) 
IiC Access Mode Selection Phase
In access mode selection phase, a content require user can select a communication mode independently according to the perceived performance, distance, local cache and the video streaming traffic delay cost. Let signify that a desired content require user is associated to a node located at , and denotes the distance between and . And we can categorize the content require users into two types: potential D2D mode users and potential FAP mode users, the definitions of which are given as follows:

Potential D2D mode Users: A content require user is a potential D2D mode user if it can successfully obtain the requested contents from another D2D transmit user in a known location within a distance threshold .

Potential FAP mode Users: A content require user is a potential FAP mode user if the user cannot find a D2D transmit user within , or the content require user can find a D2D transmit user near it but the requested content is not cached in that D2D transmit user. Thus, will try to access its nearest FAP node.
It’s worth noting that all potential D2D mode users can select both the D2D mode and FAP mode, while each of the potential FAP users only can select the FAP mode to access its nearest FAP. More specifically, if a potential D2D mode user selects D2D mode, a communication link is established between the tagged content require user and its service D2D transmit user which has a known location. On the other hand, when FAP mode is selected and the requested content is cached in the FAP node, the potential D2D mode user or potential FAP user can download data directly from FAP. If user cannot find the requested content in its nearby FAP, then user will download its data through the fronthaul in the same way as traditional CRANs.
Let and denote the number of potential D2D mode and FAP mode users, respectively, where denotes the probability of at least one D2D transmit user located in distance threshold meanwhile has the requested content , which can be given as
(4) 
Proof.
By using the property of 2D Poisson process, the probability distribution of the nodes number
in a circle area with radius limit can be derived as(5) 
Let , and . Then, we have the probability of none D2D transmit user has the the requested video content within the distance limit . Therefore, (4) can be given as the probability of complementary events. ∎
During the access mode selection phase, we consider two kinds of delay, cache processing delay and average FAP fronthaul packet delay. The cache processing delay mainly comes from the cache data processing time, and in this work we assume that the processing delay in D2D transmit users and FAPs as constants and , respectively.
On the other hand, the FAP fronthaul packet delay is defined as the time needed for a packet to be transmitted from fronthaul to a FAP. It depends on the size of the packet and the number of FAPs associated with the gateway [19]
. In wired scenario, Gamma distribution is usually used to model the distribution of the delay in routers or switches
[20][21]. Then, the fronthual packet delay is expressed as(6) 
where the first and second terms of Gamma function represent the effect of number of connecting nodes and packet size on delay, respectively. , and are constants, representing the processing capability of the nodes, and is the packet size.
Therefore, we can directly obtain the mean FAP fronthaul packet delay from the expectation of Gamma distribution as
(7) 
And the total delay in access mode can be expressed as follows
(8) 
Iii EVOLUTIONARY GAME APPROACH for USER ACCESS MODE SELECTION
In this section, the evolutionary game formulation for user access mode selection is given, and the replicator dynamics is used to model the strategy adaptation process.
Iiia Formulation of the Evolutionary Game
We define the evolutionary game as follows:

Players: Each user who can choose among multiple access mode is a player of the game. In this paper, the players are all of the potential D2D mode users. Note that the potential FAP mode users are not involved in the game since FAP mode is the only access mode available to these users.

Strategies: The set of strategies is the selection of a communication mode. The set of strategies for the potential D2D mode players is , and there is obvious only one strategy possible for the potential FAP players .

Payoff: The payoff quantifies the performance satisfaction level of a potential player which depends on the ergodic rate rate as well as the delay cost. The payoff function for a mode potential player selected access mode is defined as follows:
(9) 
where x
denotes the vector of the proportion of users choosing different access modes.
denotes the utility function measuring the achieved performance in mode choosing mode , which will be further discussed in Section IV. denotes the bandwidth in mode . is the proportion of individuals of potential users in choosing strategy . is the number of potential users in mode choosing mode , is the set of subareas in mode , for the potential D2D mode users, this set can be defined as since both D2D and FAP modes are available for these users. is the coefficient of linear pricing function used by mode to charge a user.Moreover, since the FAP fronthaul bandwidth is limited, it will lead to negative impact on their payoff if too many users select FAP mode to receive their data through fronthaul link. On the other hand, the processing delay cost of both of D2D and FAP modes will also increase with the growth of the users at the same time. In order to evaluate these impacts, we define the cost price coefficient and to reflect the influence of the proportion of mode selection.
(10) 
where and are positive constants.
In order to measure the users’ performance utility functions presented in the evolutionary game formulation (9), in the next subsection, we use stochastic geometry tool to derive the coverage probability and ergodic rate for both D2D and FAP modes by taking into account the different nodes locations, the SIR QoS constrains and the interference.
IiiB Performance Analysis for Different User Access Modes
In this subsection, we derive the coverage probability and ergodic rate for FRAN with two different user access modes. The ergodic rate in FRAN is defined as , where the unit of the ergodic rate is in terms of nats/s/Hz, is the QoS constraint of access mode .
In this paper, we limit our attention to the interferencelimited scenario since the interference is much larger than the noise. Path loss is represented by a standard power law path loss function
, where is the path loss exponent.Then, if is served by a D2D transmit user at a fixed distance of to , the received SIR at the desired user is given by
(11) 
where characterizes the flat Rayleigh channel fading between the potential D2D mode user and its associated D2D transmit user, and denotes the path loss. is the intratier interference from other D2D transmit users, and
denote the exponentially distributed fading power over the Rayleigh fading channel and path loss from other D2D transmit users to
, respectively. denotes intertier interference from FAPs, the definition of and are similar to that in .Next, if is served by a single nearest FAP in FAP mode, the SIR is given by
(12) 
where , and denote the intratier interference from other FAPs and intertier interference from D2D transmit users, respectively, and the definitions of , and are given in (11).
1.D2D mode
The coverage probability of a user in D2D mode can be denoted as
(13) 
where .
Proof.
See Appendix A. ∎
Similarly, the achievable ergodic rate can be derived as
(14) 
where follows in the high SIR conditions , Ei is the exponential integral function.
Proof.
See Appendix B. ∎
2.FAP mode
In FAP mode, the desired user will try to access its nearest FAP to download the requested content
. The probability density function (PDF) of the distance between
and can be derived by using a similar way as (4)(15) 
Thus, the coverage probability of FAP mode can be calculated as
(16) 
where .
Proof.
See Appendix C. ∎
And the ergodic rate for FAP mode can be given as
(17) 
Special Case: Path loss exponent for FAP to user link is 4 (), and SIR threshold
An approximated closedform approximate expression can be derived in this special case, and we give the ergodic rate in the following lemma 1.
1.
The ergodic rate for nearest FAP mode with = 4 and can be expressed as
(18) 
Proof.
See Appendix D. ∎
IiiC Replicator Dynamics
Evolutionary game theory differs from classical game theory by focusing more on the dynamics of strategy change as influenced not solely by the quality of the various competing strategies, but by the effect of the frequency with which those various competing strategies are found in the population [25][26]. In a dynamic evolutionary game, the replicator dynamic is a simple model of evolution and prestigebiased learning in games [27].
We consider an evolutionary game of user access mode selection where the group of potential users in can choose among the available wireless access modes . The game is repeated, and the user adopts a strategy that gives a higher payoff which is determined by the payoff matrix and the proportion of each strategy in the population. The speed of the user in observing and adapting the access mode selection is controlled by parameter .
For a small period of time, the rate of strategy is governed by the replicator dynamics, which is defined as follows:
(19) 
where is the average payoff of the users in potential group which is computed as
(20) 
Based on this replicator dynamics of the users in potential group , the number of users choosing access mode increases if their payoff is above the average payoff. It is impossible for a user to choose access mode , which provides a lower payoff than the current payoff. This replicator dynamics satisfies the condition of . In the next section, we will use above expressions to analyze the stability of the access mode selection game.
Iv EVOLUTIONARY EQUILIBRIUM and STABILITY ANALYSIS
In this section, the evolutionary equilibrium is considered to be the solution of the formulated access mode selection game and we also analyzed its stability. An evolutionary equilibrium is a fixed point where there is no change in proportion of players choosing different strategies. In other words, since the rate of strategy adaptation is zero (i.e., ), there is no user who deviates to gain a higher payoff [28].
To evaluate the stability at the fixed point , which is obtained by solving
, the eigenvalues of the Jacobian matrix corresponding to the replicator dynamics need to be evaluated. The fixed point is stable if all eigenvalues have a negative real part
[30].For the mode selection game in this paper, the replicator dynamics can be expressed as follows:
(21) 
where follows that .
Hence, we could derive the evolution algorithm as described in Algorithm 1. Also, the stability of the equilibrium is clarified in Theorem 2.
2.
For a potential D2D mode user with a stable equilibrium point, the interior evolutionary equilibrium in game is asymptotically stable.
Proof.
See Appendix E. ∎
Iva Comparison with Max Rate Based Access Mode Selection Algorithm
For comparison purpose, we consider a max rate based access mode selection algorithm under the same system model. Max rate based access mode selection algorithm needs to sort both the D2D mode and FAP mode rate list based on QuickSort algorithms to have ordered FAP and MBS rate lists, and then control each user to access to its node with max rate. We derive the max rate based access mode selectio algorithm as described in Algorithm 2.
V Numerical Results
In this section, the accuracy of the above ergodic rate expressions and the performance of the proposed algorithm are evaluated by using Matlab with Monte Carlo simulation method. The simulation parameters are listed as follows in Table I, and the simulation network topology is shown in Fig. 3.
Parameters  Value  Description 

Number of video contents  
30m  D2D distance threshold  
Cache size of D2D transmit users  
Cache size of FAPs  
300MHz  Bandwidth of D2D users  
100MHz  Bandwidth of FAPs  
13dBm  Transmit power of D2D transmit user  
23dBm  Transmit power of FAP  
Intensity of content require users  
Intensity of D2D transmit users  
Intensity of FAPs  
Intensity of gateways  
4 [29]  Path loss exponent 
Fig. 4 shows the ergodic rate achieved by the D2D mode and FAP mode in FRAN with the varying SIR QoS thresholds . The analytical results closely match with the corresponding simulation results, which validates our analysis in Section III. It can be observed that the ergodic rate of both D2D and FAP user access mode decreases as the SIR threshold increases. This is because the larger suggests that the user is more strict in the quality of SIR, which leads to ergodic rate decreasing.
The proportion and the achieved average payoff per user accessing different mode during the evolution process are shown in Fig. 5 and Fig. 6. It can be seen that the proposed user access mode selection algorithm can reach the evolutionary equilibrium very fast, within less than 15 iterations. More specifically, as can be seen from Fig. 5, we could find that after several iterations all curves with different proportion at the initial point at last achieve a much better average payoff than only consider max rate based user access mode selection algorithm.
On the other hand, the trajectories of the proportion of users selecting each access mode are illustrated in Fig. 6. It can be observed that the proportions eventually converge to the equilibrium after changing in different directions. At the same time, those curves about payoff finally converge to the network average payoff.
Vi Conclusion
In this paper, an evolutionary game theory based algorithm to solve the problem of user access mode selection in downlink FRAN is proposed. Stochastic geometry is used to derive the payoff expressions by taking into account the different nodes locations, interferes as well as the delay cost. The evolutionary equilibrium has been considered to be the stable solution for which all users receive identical payoff from accessing different access modes. The simulation results show that the evolutionary game based access mode selection algorithm can reach a much higher payoff than the max rate based algorithm.
a Proof of
According to the definition of coverage probability, we have
(22) 
where (a) follows from the Laplace transform of and the independence of and [22][23]. (b) follows from letting in the Laplace transforms of and , which are given by
(23) 
(24) 
Using the independence of the fading random variables in
,we have(25) 
where follows , follows from probability generating functional (PGFL) of PPP [24] and, results from algebraic manipulation after converting from Cartesian to polar coordinates and follows from using some properties of Gamma function, and can be obtained in a similar way.
B Proof of
For a positive continuous random variable
, we can use the following formula for computing its expectation(26) 
Next, we focus on the second term of (26), after changing variables with , and the expression of this term can be given as
(27) 
where , and the proof is finished.
C Proof of
According to the definition of coverage probability, we have
(28) 
where (a) follows the setting of and the independence between intertier interference and intratier interference , equation (b) follows the definition of the Laplace transform for , , where and can be derived by using a similar way as (25), and the proof is finished.
D Proof of lemma 1
we first drive the coverage probability of FAP mode with and , which can be denoted as
(29) 
where (a) follows the property of the inverse trigonometric functions that if A is smaller than 1, i.e., .
E Proof of Theorem 2
For the system to have a stable equilibrium point, all eigenvalues of the Jacobian of the system of equations should have a negative real part [30]. As we have one system equation, an equivalence to this condition is that the Jacobian should be negative definite. To this end, we first denote by the right hand side of (21). Accordingly, we have
(30) 
According to the definition, at the equilibrium point. Hence, . Then we calculate and as (31) and (31) on the top of next page
(31) 
(32) 
It is obvious that for any , we have ,and . Hence this proves , and is strictly negative for any nonzero value of . Therefore, evaluated at any interior equilibrium point is negative. This completes the proof.
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