Fog-computing, also known as mobile edge computing (MEC), is a novel and promising technology for future networks . In contrast to conventional base station (BS) and access point (AP), Fog-AP is capable working as a wireless AP, and provides caching and computing capabilities . The network composed by Fog-APs is generally referred to as the fog radio access network (Fog-RAN), which can improve the content delivery efficiency and support computation offloading . Therefore, the problem of caching scheme design in Fog-RAN attracted much attention from both industry and academia.
By pushing contents to the edge, the users can access the interested information within one hop, which significantly reduces the latency. In , an information-theoretic formulation of the caching problem is introduced. The basic structure and global caching gain is analyzed for a novel coded caching approach. In , the authors focused on content sharing among smart devices in the social IoT with D2D-based cooperative coded caching. In , the un-coded and coded caching schemes are studied for femtocells. The fountain code is considered, where the objective is to solve the cache assignment problem with a given maximum number of helpers a user can be connected to. In , the MDS code with random caching strategy is considered in D2D networks, where the fundamental limits of caching is obtained. In , the authors investigated the content delivery network and information centric network solution of 5G mobile communication network. In , both centralized algorithm and distributed algorithm are proposed for optimal caching placement problem in Fog-RAN. In , a caching based socially aware D2D communication framework is considered, where a hypergraph framework is summarized.
Previous works illusatrate great potential of edge caching for reducing the delay by pushing the user required content at network edge nodes. In this paper, we focus on the transmission scheme design and coding parameter optimization for coded caching in Fog-RAN . As the Fog-AP may encounter device failure, it is required to recover the stored content from other Fog-APs to a new Fog-AP. By using the coded caching scheme, the data can be recovered by transmitting data from other Fog-APs through backhaul link. Thus, the user can still access the interested content in one hop. On the other hand, the wireless channel may be in deep fading, some of the Fog-APs will be in outage in this case. By using coded caching, the user can recover the interested content by accessing any given number of Fog-APs. In this context, the reliability can be greatly improved if the coded caching scheme is applied. In this work, the objective is to choose the optimal subset of Fog-APs for each user, so that the required content can be deliveried to each user with the highest successful probability, i.e., the content outage probability is minimized. Therefore, we need to choose a coding scheme to minimize the size of data needed to be transmitted from Fog-APs to each user, i.e., the coding scheme with minimum storage size. On the other hand, the Fog-APs are assumed to be connected by reliable (wired or wireless) back-haul links with high bandwidth. The bandwidth requirement for regenerating the storing data in a new Fog-AP can be easily guaranteed. In this context, we choose the minimum storage regenerating (MSR) code with the optimal coding parameters. The transmission scheme design and coding parameter optimization problem is a complicated multi-objective probabilistic combinatorial optimization . Moreover, the requirement on solving this problem in a distributed manner further increases the difficulties.
It is well known there is (generally) no global optimum to multi-objective optimization problem . Therefore, we focus on the Pareto optimal solution (i.e., none of the objectives can be improved without degrading other objectives) with fairness constraint for coded caching problem in Fog-RAN. The key is the novel -matching approach, which can solve this problem with fixed coding parameters. Here is a function defined on vertices, which determines the maximum number of edges associated with a vertex in the matching . Specifically, the Fog-AP selection problem is first formulated as a random bipartite graph (RBG) based fairness maximum -matching problem. To explore not only the caching capability but also computing capability of Fog-APs, a fast message passing algorithm is proposed to select Fog-APs for each user in a distributed way. The memory usage of the proposed algorithm is only , where is the number of users and is the number of Fog-APs. Although, it is very difficult to derive the closed-form formulas for the optimal solution, a tight upper bound for the solution is still obtained by digging the properties of fairness maximum
-matching on RBG and conditional content outage probability. To obtain the optimal coding parameters, we define and derive the outage exponent region (OER), the region that contains all feasible outage exponent vectors of all users, and diversity-multiplexing region (DMR), the OER as the signal to noise ratio (SNR) tends to infinity, from content outage probability.111The diversity-multiplexing tradeoff (DMT), which is first proposed in , is used as a key performance metric in high SNR regime. Specifically, the DMT is the slope of the outage probability curve in the high SNR regime. According to the definition in , the DMT is the tradeoff between reliability (diversity gain) and efficiency (multiplexing gain) when SNR tends to infinity. The DMR is the DMT in multiple user scenario. The OER is a generalization of DMR in finite SNRs with multiple users. The obtained theoretical results illustrate that -matching approach is optimal in high SNR regime. Specifically, each user not only fairly shares the multiplexing gain but also achieves the full diversity gain, i.e.,the largest area of DMR. Simulation results illustrate the accuracy of the theoretical derivations, and verify the outage performance of the proposed scheme. In general, the contribution our work can be summarized as follows:
The optimal content transmission scheme from Fog-APs and users is proposed, which is formulated as a multi-objective probabilistic combinatorial optimization problem. The novel -matching approach is proposed for solving this problem. The decentralized algorithm is also obtained to select the optimal subset of Fog-APs for each user with low complexity and small storage cost.
The closed-form approximation formulas of the content outage probability is derived in this work. The DMR can be obtained to verify that the solution is optimal in high SNRs. These formulas provide a systematic way to design and evaluate the caching scheme for Fog-RANs in an analytic way.
The rest of this paper is organized as follows. Section II presents the system model and problem formulation. The RBG based -matching approach is presented in Section III, where the fast message passing based distributed Fog-AP selection algorithm is discussed in detail. Section IV defines and derives the closed-form formulas for content outage probabilities, OER, and DMR to obtain the optimal coding parameters. Section V presents the simulation results. Finally, Section VI concludes this paper.
Ii System Model and Problem Formulation
Ii-a Channel Model
Consider a dense Fog-RAN, as shown in Fig. 1, where users communicate with Fog-APs through orthogonal subchannels. The user set is denoted by , where is the -th user.222The script symbol denotes a set, whose cardinality is denoted by . The abbreviation is used to denote . The Fog-AP set is denoted by , where is the -th Fog-AP. One Fog-AP has a dedicated subchannel, each of which contains resource blocks (RBs) in the duration of the coherence time. For convenience, both the bandwidth and the duration of one RB are normalized as one. In this context, one Fog-AP can be accessed by users at most, each of which occupies one RB of the subchannel.
Suppose that the signal on each subchannel undergoes the independent Rayleigh slow fading. Then, the channel gain between user and Fog-AP , denoted by , is independent and follows the identical distribution of .333 denotes a circularly symmetric Gaussian distribution with mean and variance
denotes a circularly symmetric Gaussian distribution with mean
and variance. The received signal from Fog-AP to user , denoted by , is given by
is the additive Gaussain white noise (AWGN). As we assume one bit channel state information (CSI) is known at the transmitter, the mutual information between userand Fog-AP during one RB is given by
where is the average SNR at the receiver. Throughout the paper, the unit of information is “”. As we normalize both the bandwidth and the duration of one RB as one, the transmission rate is equal to per RB use.
Ii-B Coded Caching in Fog-RANs
To fully exploit the caching capability in Fog-APs, the coded caching scheme is adopted. Suppose the user requires the -th content , which composes the content set . The size of content is . Thus, there are files cached in Fog-RAN. As proposed in , the distributed storage code with parameters is used for content . Thus, will be cached in Fog-APs, each of which stores . Hence, the storage size of each Fog-AP is at least . If Fog-APs transmit each to user , it is capable of recovering the original content . For a new Fog-AP without caching any part of , it requires Fog-APs to transmit in total to regenerate in this new Fog-AP. It can be seen that the performance of the distributed storage code is characterized by . In this context, we have for and .
In the considered Fog-RAN, the contents should be first cached in Fog-APs based on some caching strategies [9, 4]. Then, the method in this work will be applied to optimize the transmission performance for the cached contents. Specifically, Fog-APs are connected by reliable (wired or wireless) back-haul links with high bandwidth . Thus, compared to the wireless channel between users and Fog-APs, the total regenerating bandwidth can be naturally assumed to be fulfilled.444If we take back-haul links into consideration, there will be a tradeoff between and . The proposed framework in this paper can be easily extended to this scenario. In contrast, due to channel fading, the mutual information between user and Fog-AP during one RB may be smaller than . In other words, the Fog-AP is in outage for user , referred to as Fog-AP outage. Thus, we need to minimize , i.e., apply MSR codes . Accordingly, we have the following optimal for MSR codes:
Ii-C Fog-AP Selection Problem
According to the property of MSR codes, the user must successfully access Fog-APs in order to recover the required content . Due to channel fading, however, the Fog-AP will be in outage for user , if . In this context, the user may still fail to recover , even the MSR code with is applied, i.e., the user may encounter content outage. Intuitively, with the fixed transmission rate from each Fog-AP to user in one RB, the content outage probability decreases if we decrease . However, as is equal to , the Fog-AP outage probability increases if we decrease . Therefore, an optimal exists for achieving the minimum content outage probability. On the other hand, because of the flexible architecture of the Fog-RAN, several Fog-APs are able to form an edge cloud to support mobile edge computing. Thus, the joint channel coding scheme, such as rotated -lattice code  or permutation code , can be applied in Fog-APs so as to further improve the outage performance. According to [16, 17, 18], the outage probability of user for content is given by
where , and is the set of Fog-APs accessable by user , which satisfies
To reduce the complexity in both signaling and computation, each user is only allowed to feedback CSI for each subchannel to Fog-APs at the beginning of each coherence time, i.e., we only knows whether the Fog-AP is in outage for user or not. Let denote the quantized CSI matrix, the entry at the -th row and -th column is given by
Therefore, the Fog-AP outage probability is given by
For convenience, we define
In this context, a Fog-AP selection scheme can be seen as a mapping from to , that is
According to the notation in  and footnote 1 of this paper, the optimal Fog-AP selection problem can be formulated as follows:
The problem P1 in Eq. (9) summarizes the optimal Fog-AP selection problem in an abstraction form. It will determine the optimal mapping and so as to minimize the content outage probability in Eq. (3). As a multi-objective probabilistic combinatorial optimization problem, P1 has no global optimum in general according to [12, 13]. Therefore, the Pareto optimal is proposed as a solution for multi-objective optimization problem. Specifically, the strong Pareto boundary of a multi-objective optimization problem consists of the attainable operating points that none of the objectives can be improved without degrading other objectives. Every point on the strong Pareto boundary is a Pareto optimal solution. In this work, we focus on the Pareto optimal solution with fairness consideration, denoted by and .555According to , the scalarization of this problem can be obtained by specifying a certain subjective tradeoff between the objectives. In this work, the Pareto optimal with fairness constraint is used as the scalarization method. As an another scalarization of the problem, the min-max formulation will be investigated in our future work. In the following, with fixed P1 is first reformulated and solved as P2 by the proposed -mathcing approach in Section III. The -matching approach guarantees that all of the users fairly share the total channel resource. Specifically, each user will be allocated the channel resource which is proportional to its rate requirement. In Section IV, the content outage probability, as a function of , is obtained in closed-form formulas, so that can be optimized. The theoretical results show that the -matching approach also achieves the optimal diversity order, i.e., the total diversity gain provided by the channel. Therefore, the Pareto optimal solution with fairness constraint is obtained by the proposed -matching approach.
Iii RBG based -Matching Approach
In this section, P1 with fixed is first reformulated as a combinatorial optimization problem P2 based on a given sample of . Then, the RBG based -matching method is proposed for solving P2, which is exactly the Pareto optimal mapping with fairness consideration for P1. To fully exploit the computation capability of Fog-APs, a fast message passing based distributed Fog-AP selection algorithm is proposed.
Iii-a Problem Re-formulation
At the beginning epoch of each coherence time,
can be obtained by channel estimation. The optimal Fog-AP selection problem will then be solved based on the known. Define the decision variable as
The decision matrix with being the entry at the -th row and -th column is denoted by . With the fixed and known , the mapping can be represented by the decision matrix . To achieve the Pareto optimal solution, i.e., minimum content outage probability, the objective is to maximize the total number of non-outage Fog-APs which are accessed by all of the users. The fairness constraint means that will access non-outage Fog-APs has the same probability , where and . In this context, the fair Pareto optimal mapping is equivalent to solve the following combinatorial optimization problem:
It can be seen that the content outage probability in P1 is minimized when combining the optimal solution of P2 and joint coding scheme for every sample of . The first constraint implies that each Fog-AP can only support users, each of which occupies one RB. The second constraint indicates that each user will access Fog-APs to fulfill the requirement of MSR codes. Due to channel fading, however, the accessible Fog-APs for user may not always be non-outage for every sample of . Therefore, the third constraint is introduced to guarantee fairness.
Iii-B -Matching based Fog-AP Selection
P2 is a complicated combinatorial optimization problem. The RBG based -matching method is proposed in the following to solve P2, which is also the Pareto optimal mapping with fairness consideration for P1. The preliminaries on RBG and -matching can be found in Appendix A.
The RBG model for Fog-RAN can be constructed as follows. All of the vertices in one partition class represent all of the users in , and all of the vertices in the other partition class represent all of the Fog-APs in . If , we join the vertex and the vertex with an edge . Otherwise, there will be no edge between and . Therefore, the probability space of the RBG model can be denoted by with , where is given by Eq. (6). A sample of is shown in Fig. 2.
It can be seen that each sample of , denoted by a bipartite graph , corresponds to a snapshot of Fog-RAN and vice versa. For any given , associate to as
As the maximum -matching may not be unique on some samples, to guarantee the third constraint in problem P2, we need to generate on with for every , where is the number of non-outage Fog-APs allocated to . The maximum -matching satisfying the fairness constraint is referred to as the fairness maximum -matching and denoted by . In this context, the Fog-AP set selected for user can be constructed as
where contains all of the vertices that are incident with an edge in . contains Fog-APs, each of which satisfies
where is the indicator function defined as
Clearly, if , is an empty set for any . Otherwise, can be generated by randomly selecting non-saturated outage Fog-APs to user . The discussion above implies that the optimal solution of P2 is given by:
In the case of , the content outage probability for user in P1 also depends on the applied joint coding scheme. Therefore, after generating the optimal solution of P2 as Eq. (16), the optimal joint coding scheme will then be applied on for user . An example of is shown in Fig. 2. It can be seen that , and are -saturated, and thus are not in outage. The outage state of , however, is determined by the sum capacity achieved by the applied coding scheme.
Iii-C Distributed Fog-AP Selection Algorithm
Based on the fast message passing method in , a distributed Fog-AP selection algorithm is proposed to solve problem P2 in this subsection. As discussed in Section III-B, the optimal Fog-AP selection can be obtained by finding the fairness maximum -matching . By representing the maximum -matching problem in a factor graph, it can be mapped into a marginal probability computation problem on the probabilistic graphical model. Therefore, the maximum -matching problem can then be solved by passing messages between the adjacent vertices on the probabilistic graphical model. The fairness is guaranteed by adding randomness in the message passing process. The basic message passing algorithm for solving the fairness maximum -matching problem is similar to the algorithm in , whose total computation cost scales as .
To reduce the memory usage to , an improved message passing algorithm proposed in  is applied to solve the fairness maximum -matching problem. A key step of this algorithm is the selection operation, which finds the -th largest element in a set for a given . The selection operation over can be written as
For convenience, we define
where is the quantized CSI matrix, and
where is the decision matrix in problem P2. The message passing algorithm maintains a belief value for each edge in , which can be denoted by a matrix . Similarly, we define
The belief value for the edge at iteration is denoted by the entry , where for and for . is updated with the following rule
Define as the negation of the -th selection and for the -th selection. and are updated with the following rules
Let denote the entry at the -th row and -th column of at the -th iteration. The resulting belief searching rule can then be written as
The estimation of after each iteration will be updated as
To further reduce the running time, a variation of Algorithm 1 is introduced, where Steps 7-8 are improved. Since does not change during the message passing, the algorithm can compute the index cache with cache size initially, where is the index of the -th largest weight connected to node .
As for and entry , the values are similarly sorted and stored in the index vector after each iteration. After updating process of each step, we maintain a set of the greatest beliefs obtained so far. Thus, the current estimation for and at each stage are
At each iteration, the greatest possible unseen belief is bounded as the sum of the least weight seen so far from the sorted weight cache and the least value so far from the cache. The sufficient selection procedure will exit when becomes less than or equals to the sum, since further comparisons are unnecessary. The details of this algorithm is shown in Algorithm 2.
Algorithm 1 is the key result of the proposed -matching approach, which will be implemented in each Fog-AP. The procedure in each coherence time is listed as follows:
At the beginning of each coherence time, the user will feedback one bit CSI according to the channel estimation result to show if the Fog-AP is in outage for a user.
Algorithm 1 will be executed between Fog-APs and users to select the Fog-APs for each user. This algorithm will converge very quickly because of its low complexity.
After Algorithm 1, the data will be transmitted from the selected Fog-APs to each user.
Iv Content Outage Performance Analysis and DMR Optimal MSR Code
In this section, the outage probability will be analyzed for problem P2. With the closed-form formula for content outage probability, the optimal can be obtained, so that the optimization problem P1 is solved.
Iv-a Conditional Content Outage Probability
As discussed before, may not always exist, so that may be smaller than for user . Thus, we need first calculate the tight upper bound of the content outage probability under the condition that non-outage Fog-APs can be accessed by user . With a generated , the conditional content outage probability of user is written by
We then have the following theorem.
See Appendix B.
Iv-B Content Outage Probability
The content outage probability of the proposed RBG based -matching approach is analyzed in this subsection. The exact performance analysis is intractable, so that we focus on the first order approximations in both high and low SNR regimes.
To analyze the content outage probability, we need to study the properties of the maximum -matching on RBG first. Due to space limitation, we introduce the following lemma, where the detailed proof can be found in ,.
Let be the ordered sequence of , where . Denote the ordered sequence of as , where has the same order as . Define
The following lemma can then be established from Lemma 2.
Let be a bipartite graph with and . is defined as Eq. (12). If exists such that:
the only unsaturated vertex is in , the cardinality of is given by
has only one unsaturated vertex, the cardinality of is given by
See Appendix C.
If , the content outage probability of in high SNRs is given by
If , the content outage probability of in high SNRs is given by
In low SNRs, the content outage probability of is then given by
See Appendix D.