Optimization and Analysis of Probabilistic Caching in N-tier Heterogeneous Networks

# Optimization and Analysis of Probabilistic Caching in N-tier Heterogeneous Networks

Kuikui Li, Chenchen Yang, Zhiyong Chen,  and Meixia Tao,  Manuscript received December 6, 2016; revised April 27, 2017 and September 15, 2017; accepted November 15, 2017. This work is supported by the National Natural Science Foundation of China under Grants 61571299, 61671291, 61329101, 61528103, and 61521062. This work was presented in part at the IEEE SPAWC 2017 [1]. The associate editor coordinating the review of this paper and approving it for publication was Prof. Chan-Byoung Chae. (Corresponding author: Zhiyong Chen.)The authors are with the Department of Electronic Engineering, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China. Z. Chen is also with the Cooperative Medianet Innovation Center, Shanghai, China. M. Tao is also with Shanghai Institute for Advanced Communication and Data Science (email: kuikuili@sjtu.edu.cn; zhanchifeixiang@sjtu.edu.cn; zhiyongchen@sjtu.edu.cn; mxtao@sjtu.edu.cn.).
###### Abstract

In this paper, we study the probabilistic caching for an -tier wireless heterogeneous network (HetNet) using stochastic geometry. A general and tractable expression of the successful delivery probability (SDP) is first derived. We then optimize the caching probabilities for maximizing the SDP in the high signal-to-noise ratio (SNR) regime. The problem is proved to be convex and solved efficiently. We next establish an interesting connection between -tier HetNets and single-tier networks. Unlike the single-tier network where the optimal performance only depends on the cache size, the optimal performance of -tier HetNets depends also on the BS densities. The performance upper bound is, however, determined by an equivalent single-tier network. We further show that with uniform caching probabilities regardless of content popularities, to achieve a target SDP, the BS density of a tier can be reduced by increasing the cache size of the tier when the cache size is larger than a threshold; otherwise the BS density and BS cache size can be increased simultaneously. It is also found analytically that the BS density of a tier is inverse to the BS cache size of the same tier and is linear to BS cache sizes of other tiers.

## I Introduction

The global mobile data traffic is estimated to increase to 30.6 exabytes per month by 2020, an eightfold growth over 2015, and the contribution by video is foreseen to increase from in 2015 to in 2020 [2]. To address this mobile data tsunami and hence meet the capacity requirement for the future 5G network[3], an effective and promising candidate solution is to deploy a dense network with heterogeneous base stations (BSs), such as macro BSs, relays, femto BSs and pico BSs [4]. The heterogeneous network (HetNet) can provide higher throughput and spectral efficiency. In the meantime, it also faces two challenges. One is the tremendous burden on the backhual link due to the explosive demand for video contents during the peak time and the other is high CAPEX and OPEX due to the denser BSs.

Recently, caching popular contents at BSs has been introduced as a promising technique to offload mobile data traffic in cellular networks [5, 6]. Unlike the communication resources, the storage resources are abundant, economical, and sustainable. By exploiting the abundance of the storage resources in wireless networks, significant gains in network capacity through caching can be expected [7], which enables caching to be an essential functionality of emerging wireless networks [8].

The aim of this work is to study how caching can address the aforementioned challenges in a multi-tier HetNet. In specific, we first would like to find out what is the optimal cache placement strategy in order to alleviate the traffic burden in backhaul links to the minimum. Second, we would like to find out if the deployment cost of dense BSs can be traded by BS cache storage, and if so, what are the tradeoffs and what conditions must be met in order for it to happen.

### I-a Related Work

Caching has the potential to alleviate the heavy burden on the capacity-limited backhaul link and also improves user-perceived experience [9]. Utilizing the tool of stochastic geometry [10], the work [11] formulates the caching problem in a scenario where small BSs are distributed according to a homogeneous Poisson Point Process (HPPP). The authors in [12] consider a two-tier HetNet and derive a closed-form expression of the outage probability by jointly considering spectrum allocation and storage constraints. In [13], the authors consider a 3-tier HPPP-based HetNet with caching and theoretically elaborate the average ergodic rate, outage probability and delay. Considering an HPPP-based cache-enabled small cell network, a closed form expression of the outage probability and the optimal BS density to achieve a target hit probability are derived in [14]. The work [15] proposes a cluster-centric small cell network and designs cooperative transmission scheme to balance transmit diversity and content diversity. It is worth noting that these works mainly focus on the performance analysis of cache-enabled wireless networks for given caching strategies, such as caching the most popular contents.

Caching strategy is an important issue for cache-enabled wireless networks. Previous works on the optimal caching strategy design can be classified into two trends based on whether channel fading and interference are considered. The early trend focuses on the connection topology only while ignoring channel fading and interference. The authors in [16] formulate a cache placement problem in distributed helper stations to minimize the average download delay with both uncoded and coded caching. It is shown in [16] that the optimal caching problem in a wireless network with fixed connection topology is an NP-hard problem (without coding). In [17], a joint routing and caching design problem is studied to maximize the content requests served by small BSs. By reducing the NP-hard optimization problem to a variant of the facility location problem, algorithms with approximation guarantees are established. The second trend takes into account channel fading and interference for caching optimization by mostly utilizing the tool of stochastic geometry. The work [18] proposes an optimal randomized caching policy to maximize the total hit probability and overviews different coverage models to evaluate the performance. The works [19, 20, 21] optimize the probabilistic caching strategy to maximize the successful download probability in small cell networks. Further, a closed-form expression for the optimal caching probabilities is obtained in the noise-limited scenario in [20]. In [22], a greedy algorithm is proposed to find the optimal caching strategy to minimize the average bit error rate. The work [23] studies the problem of joint caching, routing, and channel assignment for video delivery over coordinated multicell systems of the future Internet.

Recently, caching strategy optimization is extended to wireless heterogeneous networks. The combination of the optimal caching and the network heterogeneity brings more gains in network capacity. Utilizing the tool of stochastic geometry, the works [24, 25] investigate the optimal probabilistic caching at helper stations while assuming deterministic caching at macro stations to maximize the successful transmission probability in a two-tier HetNet. Based on [18], the work [26] considers different types of BSs with different cache capacities. The cache optimization problem for the first type of BSs is solved by assuming that the placement strategy for other types of BSs is given. The joint probabilistic caching optimization problem for all types of BSs is yet not considered, and little analytical insight on the cache design and system performance is available. In general, the joint optimization for probabilistic cache placements in different tiers of a HetNet is very challenging due to the different tier association probabilities brought by the content diversity as well as the complicated interference distribution by the nature of network heterogeneity.

Furthermore, a tradeoff between the small BS density and total storage size is firstly presented in [11], where each small BS caches the most popular contents. Using the optimal caching scheme, [25] shows that the helper density can be traded by the cache size to achieve a target area spectral efficiency. Note that the tradeoff studies in [11, 25] are conducted numerically only without theoretical analysis. Deriving and analyzing the tradeoff theoretically has not been solved. In [27], the authors address the question that how much caching is needed to achieve the linear capacity scaling in the dense wireless network based on scaling law method.

### I-B Contributions

In this work, we first investigate the optimal probabilistic caching to maximize the successful delivery probability (SDP) in a general -tier () wireless cache-enabled HetNet. We next establish an interesting connection between -tier HetNets and single-tier networks. We then address the tradeoffs between the BS caching capability and the BS density analytically based on the uniform caching strategy. The main contributions are summarized as follows:

• Analyzing and optimizing the SDP for the -tier HetNet: We derive the tier association probability and the SDP by modeling the BS locations in the HetNet as -tier independent HPPPs. The optimal probabilistic caching problem for maximizing the SDP is then formulated. We prove that this problem is concave in the high signal-to-noise ratio (SNR) regime. The sufficient and necessary conditions for the optimal solution are derived.

• Highlighting the connection between -tier HetNets and single-tier networks: We further study the optimal caching problem in special cases, and find that the maximum SDP of single-tier networks only depends on the cache size while that of -tier HetNets is also determined by the BS densities and transmit powers. Moreover, in the high SNR regime, we prove that there exists a single-tier network such that the maximum SDP of the -tier HetNet is upper bounded by that of the single-tier network. When all tiers of BSs have the same cache size, the -tier HetNet performs the same as the single-tier network, regardless of the network heterogeneity.

• Presenting insights on the impacts of the key network parameters: We first show that the optimal performance of single-tier networks is independent of the BS density and transmit power. Then, under uniform caching strategy, we analytically present the impacts of the BS cache size, density and transmit power of each tier on the system performance. Numerical results also verify our analytical results.

• Revealing the tradeoffs between the BS density and the BS cache size: With uniform caching strategy, our analysis reveals that, to maintain a target SDP, the network parameters are related as follows: increasing the BS caching capability can reduce the BS density when its cache size is larger than a threshold ; the BS density is inversely proportional to the cache size in the same tier, i.e., . Here, denotes the index of the tier and is independent of and . For the different tiers, we prove that the BS density is a linear function of the cache size , i.e., , for , where , and are independent of and . Likewise, we reveal the similar tradeoffs between the BS transmit power and the BS cache size.

The rest of this paper is organized as follows. Section II presents the system model. The performance metric is analyzed in Section III. In Section IV, we formulate and solve the optimal caching problem. Then, the impacts and tradeoffs of the network parameters are shown in section V. The numerical and simulation results are presented in Section VI, and the conclusions are drawn in Section VII.

## Ii System Model

### Ii-a Network and Caching Model

We consider a general wireless cache-enabled HetNet consisting of tiers of BSs, where the BSs in different tiers are distinguished by their transmit powers, spatial densities, biasing factors, and cache sizes111These model assumptions indicate that BSs in different tiers have different traffic load to handle, and also reflect the demand heterogeneity in different BSs.. The locations of BSs in each tier are spatially distributed according to an independent HPPP, denoted as with density for . A three-tier HetNet including macro BSs, relays and pico BSs, is illustrated in Fig. 1. Consider the downlink transmission. Time is divided into discrete slots with equal duration and we study one slot of the system. For the wireless channel, both large-scale fading and small-scale fading are considered. The large-scale fading is modeled by a standard distance-dependent path loss attenuation with path loss exponent . The Rayleigh fading channel is considered as the small-scale fading, i.e., . Each user receiver experiences an additive noise that obeys zero-mean complex Gaussian distribution with variance .

Consider a database consisting of contents denoted by , and all the contents are assumed to have equal length222Note that the extension to the general case where contents are of different lengths is quite straightforward since the contents can be divided into chunks with equal size.. Each user only requests one single content at each time slot. The content popularity distribution is identical among all users, represented by , where each user requests the -th content with probability and . The content popularity is assumed to be known a prior for cache placement. Without loss of generality, we assume . Each BS is equipped with a cache storage. The cache capacities of -tier BSs are denoted as , where each BS in the -th tier can store at most () contents.

When a user submits a content request, the content will be delivered directly from the local cache of a BS that has cached it. If the content is not cached in any BS, it will be downloaded from the core network through backhaul links. Since the main purpose of this paper is to optimize the caching strategy to offload the backhaul traffic, we only consider the transmission of the cached contents at BSs, same as [19].

We adopt the probabilistic caching strategy and assume all the BSs in a same tier use the same caching probabilities. Each BS caches contents with the given probabilities independently of other BSs. Define as the caching probability matrix where denotes the probability that the BSs in the -th tier caches the -th content. It must satisfy

 0≤pij≤1,  ∀i∈N,∀j∈M (1) M∑j=1pij≤Qi,  ∀i∈N. (2)

Note that the conditions (1) and (2) are sufficient and necessary for the existence of a random content placement policy requiring no more than slots of storage at each BS in tier for [18]. Also note that if a BS realizes the caching strategy by caching each file at random with the given probability but independently of other files, the actual cache memory in the BS can be exceeded or wasted. To strictly meet the instantaneous cache size constraint (2) at each BS, a novel content placement approach is proposed in Section II-C of [18]333Note that the file ordering in this content placement approach can be varied arbitrarily at each particular realization if different file combinations are desired.. This approach brings dependency among the caching events of different files. But such cache dependency is irrelevant to the analysis in this work.

### Ii-B Probability of Tier Association

Without loss of generality, we carry out our analysis for a typical user, denoted as , located at the origin as in [10]. In the cache-enable HetNet, the user association policy does not only depend on the received signal strength but also the requested and cached contents. Specifically, when requests content , it is associated with the strongest BS among those that have cached content from all the tiers based on the average received signal power. Denote the distance between and the nearest BS caching content in the -th tier by . According to our tier association policy, the index of the tier that is associated with for content is:

 i(j)=argmaxl∈N{BlSlr−βl|j}, (3)

where and are the association bias factor and the transmit power of BSs in the -th tier, respectively. For notation simplicity, we assume , , in the rest of the paper.

It is essential to determine each tier’s association probability when a user requests a content. Since each BS caches contents independently of other BSs, the locations of the BSs caching content in the -th tier can be modeled as a thinned HPPP with density [28]444Note that application of the thinning property of HPPP is based on the assumption that each BS caches contents independently of other BSs, hence it is not affected by the realization method of [18].. Then, we have the following lemma.

###### Lemma 1.

The probability of associated with the -th tier for content is given by

 Wi|j =λipijSi2β∑Nl=1λlpljSl2β. (4)
###### Proof.

Please refer to Appendix -A. ∎

This lemma states that the association probability is determined directly by the density and the transmit power of the thinned HPPP .

## Iii Performance Analysis

In this section, we analyze the SDP for a given probabilistic caching scheme . Consider that all the BSs operate in the fully loaded state and share the common bandwidth [10]. By using an orthogonal multiple access strategy within a cell, the intra-cell interference is thus not considered here and only the interference introduced by inter-tier cells and intra-tier other cells is incorporated into analysis. Given that sends a request for content and is associated with the -th tier, then the received instantaneous signal-to-interference-plus-noise ratio (SINR) of is given by

 SINRi|j=Si|hi,o|2R−βi|jσ2+∑Nl=1∑k∈Φl∖{nio}Sl|hl,k|2d−βl,k, (5)

where is the distance from to its serving BS in the -tier tier, denotes the distance between and the -th interfering BS in the -th tier, is the small-scale fading channel gain between and the serving BS (the -th interfering BS). The delivery of content from tier is successful when the received SINR of is larger than a threshold . Thus, the SDP of content from tier can be expressed as555Note that the SDP in a cache-enabled network depends on both the average received signal strength and the caching distribution, which is different from the traditional coverage probability where is always associated with the strongest BS.

 Ci|j≜ERi|j[P[% SINRi|j>τ|Ri|j]]. (6)

Recall that the locations of BSs caching content in the -tier can be modeled as a thinned HPPP, then the probability density function (PDF) of is given below.666 and can also be obtained by applying the thinned HPPP to Lemma and Lemma of [29], respectively.

###### Lemma 2.

The PDF of is

 fRi|j(r)=2πpijλiWi|je−π∑Nl=1pljλl(SlSi)2βr2r. (7)
###### Proof.

Please refer to Appendix -B. ∎

Note that different from the conventional network without caching where each user is associated with the strongest BS and there only exists one type of interfering BSs, in the multi-tier cache-enabled HetNet considered in this work, the interferences to when it is associated with the -th tier for content can be divided into two groups. The first group of interferences come from all the BSs (except the serving BS ) in each tier that have stored content , the locations of which can be modeled as a thinned HPPP, denoted as with density for . The second group comes from all the BSs in each tier that do not cache content , the locations of which can also be modeled as a thinned HPPP, denoted as with density for . For the first interference group, the distance from to each interfering BS in is at least times of the distance from to its serving BS according to (30) caused by our association policy. For the second group, the interfering BSs could be very close to . By carefully handling these two groups of interferences in -tier HetNets, we derive an analytical expression for in the following proposition.

###### Proposition 1.

The SDP is

 Ci|j=2πpijλiWi|j∫∞0rexp(−τrβσ2Si) exp⎛⎝−πN∑l=1λl(SlSi)2β(pljH(τ,β)+(1−plj)D(τ,β)+plj)r2⎞⎠dr, (8)

where , and . Furthermore, denotes the Gauss hypergeometric function, and is the Beta function defined as .

###### Proof.

Please refer to Appendix -C. ∎

By the law of total probability, the average SDP for is given by

 C≜M∑j=1N∑i=1tjWi|jCi|j. (9)

Substituting (8) and (4) into (9), we obtain a tractable expression of as follows

 C=M∑j=1N∑i=12πpijtjλi∫∞0rexp(−τrβσ2Si) exp⎛⎝−πN∑l=1λl(SlSi)2β(pljH(τ,β)+(1−plj)D(τ,β)+plj)r2⎞⎠dr. (10)

In the interference-limited scenario, where the noise power is very small compared with the interference power and hence can be neglected, the expression (10) can be simplified.

###### Corollary 1.

In the interference-limited scenario, i.e., , the SDP can be simplified as

 C′=M∑j=1∑Ni=1λiS2βipijtj∑Nl=1λlSl2β[T(τ,β)plj+D(τ,β)], (11)

where .

###### Proof.

Substituting to (10), we then obtain (11). ∎

Equation (10) and (11) show a tractable expression and a closed-form expression for the SDP in the general regime and high SNR regime, respectively. The performance metric depends on four main factors: the number of tiers , the caching probabilities , the BS densities and transmit powers . In the rest of this paper, we shall focus on the interference-limited regime with high SNR.

## Iv Caching Optimization and Analysis

In this section, we formulate and solve the optimal caching problem for maximizing the SDP in the high SNR regime. Further, by considering the optimal caching problem in special cases, we establish an interesting connection between -tier HetNets and single-tier networks.

### Iv-a Caching Optimization for General Case

The optimal caching problem of maximizing the SDP is formulated as

 P1:maxP C′(P) \mathnormals.t. (???),(???)
###### Proposition 2.

Problem is a concave optimization problem.

###### Proof.

Please refer to Appendix -D. ∎

By Proposition 2, we can use the standard interior point method to solve [30]. Let denote the optimal solution of . By the Karush-Kuhn-Tucker (KKT) conditions, the sufficient and necessary conditions for can be stated in the following lemma.

###### Lemma 3.

The optimal solution of Problem satisfies the following sufficient and necessary conditions:

 p∗ij=min⎧⎨⎩⎡⎣1Gi⎛⎝√VijEηi−N∑k=1,≠iGkp∗kj−E⎞⎠⎤⎦+,1⎫⎬⎭, (12)

for , where , , , , and is the Lagrangian multiplier that satisfies 777From (39), it can be shown by contradiction that the maximum SDP is achieved only when constraint (2) holds with equality. The similar proof is given by Lemma 2 of [18]. In the rest of this paper, we use (2) with equality as the constraint. for .

###### Proof.

Please refer to Appendix -E. ∎

Furthermore, according to (39), we have the following remark to state the impact of the BS cache size of each tier.

###### Remark 1.

For , the maximum SDP increases with the cache size ().

### Iv-B Caching Optimization for Special Cases

#### Iv-B1 Optimization for N=1

When , the network degrades to the single-tier network. Denote as the caching strategy and as the cache size for the single-tier network, then the optimal caching probabilities in (12) become:

 p∗j=min⎧⎨⎩⎡⎣1T(τ,β)√tjD(τ,β)η∗−D(τ,β)T(τ,β)⎤⎦+,1⎫⎬⎭, ∀j∈M (13)

where satisfies and can be found by bisection method. 888Note that the optimal caching probability in this special case is consistent with the prior works on single-tier networks in [19, 21]. Our work extends the probabilistic caching strategy optimization for a single-tier network to that for a general -tier HetNet and contributes to presenting the impacts and essential tradeoffs of the heterogeneous network parameters.

Based on (13), we have the following result.

###### Corollary 2.

The optimal caching probability decreases with the index , i.e, increases with the content popularity . Besides, increases with the cache size .

###### Proof.

Please refer to Appendix -F. ∎

By (11) for , the maximum SDP for the single-tier network, denoted by , is

 C′∗1=M∑j=1p∗jtjT(τ,β)p∗j+D(τ,β), (14)

thus we have the following remark.

###### Remark 2.

In the interference-limited regime, the maximum SDP of single-tier networks is independent of the BS density, transmit power, and only depends on the cache size. This is because the serving BS and interfering BSs have the same caching resource, and the increase in signal power is counter-balanced by the increase in interference power. Similar performance independency on the BS density and transmit power also exists for traditional networks without cache [10].

#### Iv-B2 Connection between 1-tier and N-tier HetNets

We observe from (11) that the SDP in the high SNR regime depends on the caching probabilities through . Thus, by defining , then we can formulate a new problem below,

 P2:maxx M∑j=1xjtjT(τ,β)xj+D(τ,β) (15) \mathnormals.t. 0≤xj≤1,  ∀j∈M (16) M∑j=1xj=∑Ni=1λiSi2βQi∑Ni=1λiSi2β. (17)

From (14) and (15), it is seen that is identical with the caching optimization problem in a single-tier network with being the caching probability vector and being the equivalent cache size. By further comparing and , we obtain the following proposition which states a general relationship between the optimal performance of an -tier HetNet and that of a single-tier network.

###### Proposition 3.

Let be the optimal objective of for the considered -tier HetNet. For the single-tier network with cache size , we have

 C′∗≤C′∗1

with equality if for .

###### Proof.

Please refer to Appendix -G. ∎

Proposition 3 states that in the interference-limited regime, the optimal performance of an -tier HetNet with BS cache sizes , BS densities , and BS transmit powers is upper bounded by that of a single-tier network with BS cache size , arbitrary BS density, and arbitrary BS transmit power. Further, their performances are the same when the cache sizes are the same for all tiers in the -tier HetNet.

#### Iv-B3 Optimization for Qi=Qj, ∀i, j∈N

In this case, Proposition 3 shows that is equivalent to . Based on (13), the optimal solution of can be further given below.

###### Corollary 3.

When for , there exists an optimal solution of satisfying for , where the optimal solution of follows (13) with and .

###### Proof.

Problem is the new constructed single-tier caching optimization problem. Thus, its optimal solution is the same as (13) where let and . In the second part of the proof of Proposition 3, we show that when for , is equivalent to , and with is also an optimal solution of . Thus, Corollary 3 is proved. ∎

Based on Remark 2 and Proposition 3, we have the following remark.

###### Remark 3.

In the interference-limited regime, when all the BSs in the -tier HetNet have the same cache size, the maximum SDP of the HetNet is independent of the network heterogeneity in the BS density and transmit power.

Traditionally, without caching ability at BSs, the outage probability is independent of the number of tiers, the BS densities and transmit powers in the interference-limited -tier HetNets [29]. By introducing caching resource into the system, (11) states that the maximum SDP generally depends on the number of tiers , the BS cache size , the BS density and transmit power . The intuition behind this observation is that the caching resource changes the decision of a user to access a BS. In the multi-tier cache-enabled HetNet, the decision not only depends on the received SINR, but also depends on the contents cached at BSs. In order to further understand the impact of the cache size on , we theoretically illustrate the relationship between the cache size and the BS densitytransmit power in the next section.

## V Analysis on Network Parameters under Uniform Cache

In this section, with the uniform caching strategy where each content is cached with equal probabilities regardless of content popularities, the equivalence between the SDP of an -tier HetNet and that of a single-tier network is obtained. Based on this property, we further investigate the impacts of the key network parameters, i.e, the BS cache size, density and transmit power on the system performance. Finally, the tradeoffs of the BS density , transmit power and cache size are found.

### V-a The Equivalence under Uniform Cache

Consider the uniform caching strategy where and for single-tier and -tier HetNets, respectively. By substituting and into (11) and (15), respectively, the equivalence of the system performance of -tier HetNets and 1-tier Networks can be established.

###### Proposition 4.

In the interference-limited regime, the SDP of a -tier HetNet with the caching strategy equals that of the single-tier network with the caching strategy , and is given by

 C′(Pu)=C′1(P1,u)=M∑j=1QetjT(τ,β)Qe+D(τ,β)M, (18)

where the cache size of the single-tier network .

###### Proof.

Substituting and into (11) and (15), respectively, we then have (18). ∎

Based on (13) and , we have the following corollary.

###### Corollary 4.

In the interference-limited regime, consider the scenario that the video content popularity distribution is an uniform distribution, i.e., , , then the uniform caching strategies and are the optimal probabilistic caching strategies for single-tier networks and -tier HetNets, respectively.

###### Proof.

When , , we have from (13). Since , we thus have for , and the maximum . Then from Proposition 3 and Proposition 4, we have , and hence is also the optimal solution of . ∎

### V-B Impacts of λi, Si, Qi, ∀i∈N

Proposition 4 further states that, with uniform caching regardless of content popularity, the SDP performance of -tier HetNets depends on all the system parameters through the equivalent cache size only. Then, the impacts of the BS density and cache size of tier can be obtained in the following two lemmas.

###### Lemma 4.

The SDP increases with and when . Otherwise, decreases with or .

###### Proof.

From (18), we can see that increases with . Then we have

 ∂Qe∂λi =S2βi(Qi∑Nj=1λjSj2β−∑Nj=1λjSj2βQj)(∑Nj=1λjSj2β)2 =S2βi(Qi∑Nj=1,≠iλjSj2β−∑Nj=1,≠iλjSj2βQj)(∑Nj=1λjSj2β)2, ∂Qe∂Si =2βλiS2β−1i(Qi∑Nj=1,≠iλjSj2β−∑Nj=1,≠iλjSj2βQj)(∑Nj=1λjSj2β)2.

Obviously, when , we have that and , which also means that increases with and 999This is not contradict with Remark 2 in Section IV, because the BS densities , transmit powers and cache sizes affect the cache size in here based on .. Due to =, we thus have Lemma 4. ∎

###### Lemma 5.

The SDP increases with () and the increasing speed is monotonic to , .

###### Proof.

Due to , increases with for . Since increases with and , increases with . We thus have this lemma. ∎

###### Remark 4.

From Lemma 4 and Lemma 5, we can observe two important features in the -tier cache-enabled HetNet:

• Increasing the density or transmit power of the BSs from the tier with small cache size decreases the system performance. This somewhat surprising result is actually intuitive since such BSs (e.g., pico or femto) with small cache only provide little service but bring strong interferences to other BSs (e.g., macro or relay).

• If the BS transmit power and density of the -th tier are both the largest among all the tiers, it is most effective to increase the performance of the network by increasing the BS cache size of the -th tier.

### V-C Tradeoffs of Qi, λi, and Si, ∀i∈N

In the preceding subsection, we describe the impacts of , , and on . Now we will present the tradeoffs of these network parameters at a target SDP.

#### V-C1 Tradeoffs of one tier parameters

Lemma 4 and 5 show that increasing the BS density, transmit power or cache size influence the SDP by changing . This suggests that, as long as does not change, one can interchange different types of system parameters to maintain the same system performance, and hence can obtain the tradeoffs of these network parameters at a target SDP. In the following, we elaborate the tradeoff between the BS density (transmit power ) and the cache size within each tier for .

Given a target SDP , the communication resource ( and ) and the caching resource () of all the tiers without the -th tier, i.e., and , we can obtain based on (18), thereby gaining the tradeoffs of the -th tier’s cache size , BS density and transmit power , as illustrated in the following theorem.

###### Theorem 1.

With the uniform caching strategy, given a target determined by , and the fixed values , , for and , the network parameters , , satisfy the following tradeoffs

 λi =K1Qi−Qe, for given Si, ∀i∈N, (19) Si =(K2Qi−Qe)β2, for given%  λi,∀i∈N, (20)

where

 K1 =N∑j=1,≠iλj(SjSi)2β(Qe−Qj), (21) K2 =N∑j=1,≠i(λjλi)Sj2β(Qe−Qj). (22)
###### Proof.

Please refer to Appendix -H. ∎

Interestingly, we can observe from Theorem 1 that is inversely proportional to , while is a power function of with a negative exponent (). Accordingly, it is natural to ask that if the BS density can be reduced by increasing the BS caching capability. If yes, what is the condition? Thus, we have the following corollary to answer this question.

###### Corollary 5.

To maintain the same SDP, we have the following results for the -th tier

• The BS density and transmit power decrease with the cache size , i.e., and ,, when .

• The BS density and transmit power increase with the cache size , i.e., and , when .

###### Proof.

Please refer to Appendix -I. ∎

It is worth mentioning that increasing the BS caching capability of one tier does not always reduce the BS density or transmit power of this tier to achieve the same performance in a multi-tier NetHet, in contrast to single-tier caching networks [11]. Taking for example, the tradeoff between and is shown in Fig. 2(a). The target SDP determined by is . Note that if a tier is deployed with larger cache size (), the more the BS cache size () increases, the less the BS density () becomes, in order to maintain the same . However, if the tier is deployed with small cache size (), increasing both the BS cache size and density can keep the same performance, as show in Fig. 2(a). This is because the increase in BS density of the tier with small cache size only improves its own tier association probability, and also causes the stronger interference to other tiers with larger cache sizes. Therefore, more users are associated with the tier with small cache size, and the SDP decreases because this tier can only serve fewer content requests. The tradeoff between and shown in Fig. 2(b) is similar to that of and .

#### V-C2 Tradeoffs of different tier parameters

Similarly, we can find the tradeoffs of the parameters in different tiers, as specified in Theorem 2.

###### Theorem 2.

When , , and () satisfy the following tradeoffs for any two different tiers, remains constant.

 λj =K3−K4QiK5, (23) Sj =(K3−K4QiK6)β2, (24)

for , where

 K3 =QeN∑k=1,≠jλkSk2β−N∑k=1,≠i,jλkSk2βQk, (25) K4 =λiS2βi, (26) K5 =S2βj(Qj−Qe), (27) K6 =λj(Qj−Qe). (28)
###### Proof.

When keeps constant, will be a fixed value. We consider the tradeoffs between the -th tier parameters , and the -th tier parameter assuming that other parameters are constants. Then according to , we have

 QeN∑k=1,≠jλkSk2β−N∑k=1,≠i,jλkSk2βQk=λiS2βiQi+λjSj2β(Qj−Qe), (29)

Substituting (25), (26), (27), (28) into (29), we thus have (23), (24). ∎