Caching can alleviate peak-hour network congestion, provide traffic offloading, and improve users’ quality of experience by prefetching popular contents during off-peak times at the edge of wireless networks, such as base stations and user devices [1, 2, 3, 4]. Caching, at the same time, introduces new challenges for system design since the cache placement needs to be jointly considered with the content delivery by taking into account the wireless aspects of the network.
The caching design in multi-cell networks faces a well-known dilemma, i.e., to cache same contents or to cache different contents in the multiple edge nodes (ENs). To cache the same contents, one can achieve the physical-layer transmission cooperation (or diversity) gain and hence improve spectral efficiency and transmission reliability. To cache different contents, one can achieve the content diversity gain and hence increase the cache hits, thereby reducing backhaul traffic. Recently, much attention has been drawn to balancing these two gains [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. In 
, three heuristic caching schemes together with content-centric sparse multicast beamforming are proposed to balance the system total energy consumption and backhaul consumption in a cache-enabled cloud radio access network. It is shown with simulation that caching the most popular contents in each EN generally outperforms probabilistic caching for large user density. In, a small cell cooperation with threshold-based caching method is proposed to combine the advantages of distributed caching and physical layer cooperative transmission. In , the tradeoff between transmission cooperation gain and content diversity gain is investigated based on a caching scheme where each file is either cached at all the ENs entirely or equally split into subfiles and cached in each EN without overlapping. In 
, the authors studied the coded caching with maximum distance separable codes or random linear network codes in small-cell networks. It is shown that coded caching outperforms the most popular caching (MPC) strategy in terms of the average fractional offloaded traffic and average ergodic rate when the content popularity skewness parameter is small. Yet, this coded caching cannot exploit the transmission cooperation (or diversity) gain since each base station caches independent coded packets and each user adopts successive interference cancellation-based receiver. In, the authors studied the transmission cooperation gain in a interference channel with cache equipped at transmitters. In [12, 13, 14, 15], the tradeoff between storage and latency in interference networks with caches equipped at both the transmitter side and the receiver side are investigated. However, it is assumed in [12, 13, 14, 15] that the accumulated cache size at all the nodes (including transmitters and receivers) are large enough to collectively store the entire content library without the need for backhaul.
This work aims to exploit the tradeoff between transmission diversity and content diversity via partition-based caching in a multi-cell network with edge caching as shown in Fig. 1, where for simplicity, only one user is considered. Our proposed caching strategy allows that each file can be split into multiple subfiles and each subfile can be cached in multiple ENs. This enables partial ENs to transmit the subfiles cooperatively, which is called as partial transmission cooperation gain. It is seen that there is a tradeoff between the cache hit probability and transmission cooperation gain. We study this tradeoff by minimizing the average outage probability of the system. Our study shows that: (i) In the low SNR region, the ENs are encouraged to cache more fractions of the most popular files so as to better exploit the transmission diversity for the most popular content; (ii) In the high SNR region, the ENs are encouraged to cache more files with less fractions of each so as to better exploit the content diversity. Our numerical results show that our considered partial transmission cooperation scheme has a better performance than the full transmission cooperation scheme in .
Ii System Model
Ii-a Channel Model
Consider a multi-cell caching network as shown in Fig. 1, where there are cache-enabled ENs and one user. Each EN is connected to the server via a dedicated backhaul link. Each node is assumed to have single antenna. The file library consists of a set of files, denoted by . Each file has equal length of bits, where is assumed to be large enough. Each EN is assumed to have a local cache of size bits ( files) with
. The user requests a file from the file library according to a known probability distribution, i.e., is requested with probability . In this paper, the file popularity is modeled as the Zipf distribution with skewness parameter following the convention in [16, 17, 2], i.e.,
The communication takes place in two phases, a cache placement phase and a content delivery phase. During the cache placement phase, each EN fills up its cache memory via a backhaul link with some contents from the file library without having the knowledge of user request and channel state information. In the content delivery phase, the user requests a file from , where with probability . The channel gain, denoted by , from the -th EN to the user follows the distribution , and are independent of each other. The average transmission power of each EN is . Each EN-user link experiences an additive white Gaussian noise with distribution .
Ii-B Caching Model
We proposed a partition-based caching scheme. The cache memory at each EN is split into parts, with each part of size bits and used to cache bits of file , for . Here, and are all design variables satisfying the following constraint:
In this paper, we only consider that are all integers. Define
Here, can be viewed as the total cache size (normalized by file size ) in the network edge allocated to file . For each file , , it is partitioned into disjoint subfiles of equal size, denoted by
Each subfile is cached in all the ENs in set . Thus, every bit of each file is cached simultaneously at distinct ENs. Based on this partitioning, each EN , for , stores a set of subfiles in the library denoted by
Remark 1: The caching scheme in  only considered the special case with or . In this paper, we consider the more general case where (or equivalently ) can be optimized within .
Ii-C Delivery Model and Performance Metric
In the content delivery phase, assume that the file is requested. The delivery scheme is elaborated as follows.
Ii-C1 (cache hit)
For this case, the file is cached by the ENs. Each subfile is transmitted by its associated set of caching ENs cooperatively and different subfiles are transmitted sequentially over the time independent to each other. The channel thus becomes a multiple-intput single-output (MISO) channel. The capacity of this MISO channel is given by
The transmission of file is said to be in outage if the transmission rate of any subfile falls below a target rate, denoted by . By this definition, the outage probability of transmitting file , , is given by
The transmission diversity of file , , is given by
Ii-C2 (cache miss)
For this case, the ENs will fetch the content from the backhaul link and then transmit to the user. To simplify our analysis, we do not consider any specific delivery scheme for the cache miss case. We only treat this as a transmission outage (from the local cache of ENs) for file . That is, the transmission of is in outage if it is not cached. The transmission diversity of file , , is 0.
Ii-C3 System performance metrics
After defining the outage probability for each file request, the average outage probability of the system is given by
The transmission diversity of the system is defined as
The transmission diversity for cache hit is defined as
The content diversity is defined as .
It is seen from (11) that the performance metric can be viewed as the sum of the cache hit transmission outage probability and the cache miss probability . With the increase of the content diversity , the transmission diversity decreases because decreases. Thus, the second term in (11) becomes smaller while the first term in (11) becomes larger. The average outage probability of the system is able to characterize the tradeoff between transmission diversity and content diversity.
Iii Outage Probability Analysis ()
In this section, we present the outage probability of the file request .
Theorem 1: When file () is requested, the outage probability is given by (14) on top of this page with
, and the transmission diversity is .
The diversity result in this theorem is intuitive since every bit of file is cached in distinct ENs. However, the proof of the outage probability is rather challenging due to the correlation of each rate expression in (9) when .
Iii-a Proof of Outage probability
By definition in (9), we have
For simplicity of presentation, we define . Since are independent and identically distributed with , are
statistically independent variables with standard exponential distribution. Letdenote the ordered sample . Then, (15c) can be rewritten as
Note that are no longer independent of each other , making the distribution of difficult to obtain. Thus, we introduce the following lemma first.
Lemma 1: Let denote the ordered variables in a sample of from the standard exponential distribution, then can be expressed as
where are statistically independent variables with standard exponential distribution.
See Appendix A.
With Lemma 1, we can express as
Thus far, it is seen that the calculation of (16) is equivalent to showing the distribution of a linear combination of standard exponential variables without order. To proceed the analysis, we present a useful lemma in  as below:
Lemma 2 ([18, Theorem 3]): Let
be independent and identically distributed standard exponential random variables. Then, the CDF ofis given by
where , and represents the -th divided difference of a function with arguments 111For the calculation of divided difference, we refer interested readers to Appendix B for details..
When , are all equal to 1 from (18b). The calculation of the divided difference is complicated. Thus, we obtain the distribution of this special case directly by finding the distribution of the sum of
standard exponential variables. From the property of the Gamma distribution,follows Gamma distribution with shape and scale . The first part of the theorem is proved.
Iii-B Proof of transmission diversity
We first consider the case . For the simplicity of presentation, we define
By using the series expansion
the outage probability in (14) can be rewritten as
To proceed the analysis, we introduce the following lemma.
Lemma 3: For given and , we have
See Appendix C.
With Lemma 3, (22) can be rewritten as
The transmission diversity can be calculated as
Iv Minimum Outage Probability of the System
In the previous sections, we have obtained the outage probability for each file request for given design parameters and . The average outage probability of the system can be minimized by optimizing and . This is formulated as follows.
Note that the above problem is an integer programming problem. A brute-force approach can be used to find the global optimal solution. In particular, there are at most possible choices of . The search space can be reduced by exploiting the optimality condition that . Nevertheless, it is worth mentioning that the cache placement occurs in the off-peak time and hence the optimization problem can be solved off-line.
Next, we present the tradeoff between the transmission diversity for cache hit and the content diversity for each file under uniform file demand, i.e., for all . For this case, it is seen from (29) that the optimal , , are the same due to the symmetry of each file. Thus, can be expressed as
When , this means the optimal file splitting is to cache all the first files at all the ENs. When , it is seen that , which indicates is inversely proportional to .
V Numerical Results
In this section, we provide numerical results to show the performance of the system. The target data rate for each message is set to be 1 bit/s/Hz. The file popularity is modeled as the Zipf distribution with parameter . We consider the scenario where there are ENs and files in the library. The outage probability performance compared with  is illustrated in Fig. 2. When and , i.e., the cache size of each EN is small, it is seen from Fig. 2 that the transmission diversity of the system are the same with the two strategies, i.e., when and when . However, when becomes larger, the transmission diversity of the system of our proposed caching strategy increases ( when , when , and when ) while that of the strategy proposed in  remains to be 1. This is because the authors in  only considered the full transmission cooperation of all the ENs, which occupies too much storage at the EN for each file. Our proposed strategy consider the partial transmission cooperation. The storage occupied at the EN for each file is less than that of the full transmission cooperation scheme, which can better exploit both transmission diversity and content diversity. Furthermore, it is seen that the outage probability of the scheme proposed in  remains the same in the low SNR region while the outage probability of our proposed scheme decreases when SNR increases in the low SNR region.
The optimal selection of of our proposed scheme is given in TABLE I. In the low SNR region, it is seen from TABLE I that is small such that we can select a large value for each nonzero to better exploit the transmission diversity for the most popular content. In the high SNR region, it is seen from TABLE I that is large such that more content can be cached in the EN to increase the cache hit rate such that the content diversity of the system increases.
In this paper, we studied the tradeoff between transmission diversity and content diversity in a multi-cell network with edge caching. We proposed a partition-based caching scheme and a partial transmission cooperation delivery scheme which can exploit both transmission diversity and content diversity. We obtained two main results for this tradeoff. In the low SNR region, the ENs are encouraged to cache more fractions of the most popular files so as to better exploit the transmission diversity for the most popular content. In the high SNR region, the ENs are encouraged to cache more files with less fractions of each so as to better exploit the content diversity. In the future work, we are interested in studying the tradeoff between transmission diversity and content diversity in more complicated cache-aided wireless networks.
Appendix A: Proof of Lemma 1
Appendix B: Calculation of divided difference
Appendix C: Proof of Lemma 3
From (35), we can rewrite as
When , it is seen that and thus we have . When , it is seen that the last column of is the same as the last column of when , which indicates when . Thus, when . When , it is seen that is a full-rank matrix and and thus with probability 1.
Thus far, the lemma is proved.
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