# Energy Efficient Caching in Backhaul-Aware Cellular Networks with Dynamic Content Popularity

Authors: Jiequ Ji; Kun Zhu; Ran Wang; Bing Chen; Chen Dai
Journal: Wireless Communications and Mobile Computing
 (2018)

---

## Abstract

Caching popular contents at base stations (BSs) has been regarded as an effective approach to alleviate the backhaul load and to improve the quality of service. To meet the explosive data traffic demand and to save energy consumption, energy efficiency (EE) has become an extremely important performance index for the 5th generation (5G) cellular networks. In general, there are two ways for improving the EE for caching, that is, improving the cache-hit rate and optimizing the cache size. In this work, we investigate the energy efficient caching problem in backhaul-aware cellular networks jointly considering these two approaches. Note that most existing works are based on the assumption that the content catalog and popularity are static. However, in practice, content popularity is dynamic. To timely estimate the dynamic content popularity, we propose a method based on shot noise model (SNM). Then we propose a distributed caching policy to improve the cache-hit rate in such a dynamic environment. Furthermore, we analyze the tradeoff between energy efficiency and cache capacity for which an optimization is formulated. We prove its convexity and derive a closed-form optimal cache capacity for maximizing the EE. Simulation results validate the proposed scheme and show that EE can be improved with appropriate choice of cache capacity.


---

## Body

## 1. Introduction

With the proliferation of new wireless devices and smart phones, as well as bandwidth-intensive applications, user demand for wireless data traffic has been growing tremendously, and the corresponding network load increases exponentially [1, 2]. For more effectively handling the explosive growth of mobile data services, caching at the edge of wireless network has been proposed and has become a hot research issue [3, 4]. Recent studies have shown that caching popular contents at the edge of wireless networks can not only reduce the backhaul cost, access latency, and energy consumption but also boost the throughput [5, 6].Meanwhile, energy efficiency (EE) has become a major performance metric for future 5G cellular networks [7]. However, only few works in the literature consider the energy efficiency analysis for cache-enabled cellular networks. In general, the EE of a cellular network with caching can be improved from two aspects: improving the cache-hit rate and optimizing the cache size. The work in [8] takes content popularity and request arrival density into account for deriving the EE. By analyzing the EE for the caching-enabled network, [9] elaborates whether and how caching at BSs can improve the EE. The authors in [10] show that introducing cache can improve the EE if the content catalog size is not infinitely large and placing cached contents at small cell base stations (SBSs) is more energy efficient. Furthermore, a user-centric cooperative model is proposed in [11], and the relationship between EE and the cache-hit rate is analyzed. In [12], the authors analyzed the impact of various factors on EE such as interference level, backhaul and cache capacity, user and BS density, power consumption parameters, BS work state, and content popularity. Also, the condition when the EE can benefit from caching is derived. This work also points out that the backhaul power consumption and caching power consumption are not negligible for analyzing EE. It is worth noting that there are tradeoffs between backhaul power consumption and caching power consumption. Specifically, caching more contents could reduce the backhaul load and correspondingly reduce the backhaul power consumption, at the cost of increasing the caching power consumption. However, it is still not clear what is the optimal amount of cached popular contents at the BS which could maximize the EE of the downlink system.Furthermore, all the above works are based on an ideal assumption that the content popularity is known and fixed. Popularity refers to the cumulative number of requests for each content, or the requested probability, or the requested arrival rate over a specific area and period of time [13]. In reality, content popularity is dynamic which needs to track an ever changing popularity profile of contents. Substantial number of experimental results have shown that requests for contents frequently appear during a certain period of time at a specific location, while the number of requests outside this time period is small [14]. Apparently, considering dynamic content popularity will pose some challenges, such as how to predict content popularity and how to analyze cache performance in such a dynamic environment [15].In this work, we investigate the energy efficient caching problem in backhaul-aware cellular networks by jointly improving cache-hit rate and deriving optimal cache size. And dynamic content popularity is considered. To timely estimate the dynamic content popularity, we propose a method based on shot noise model (SNM), which can predict the request hit rate of contents. Then we propose a distributed caching policy to improve the cache-hit rate in such a dynamic environment. Also, the content redundancy can be reduced by the proposed policy. Finally, we analyze the tradeoff between energy efficiency and cache capacity for which an optimization is formulated. Analytically closed-form solutions are obtained. And the simulation results show effectiveness of the proposed scheme under limited backhaul and cache capacity.The remainder of this paper is organized as follows: in Section2, we introduce the system model. The dynamic caching policy and distributed cache policy are proposed in Section 3. The EE optimization problem of a cache-enabled cellular network is formulated and analyzed in Section 4. Numerical results and analysis are provided in Section 5, and the conclusions are summarized in Section 6.


## 2. System Model

In this section, we describe our network, transmission, and BS sleep pattern models. For ease of reading, we summarize the notation used throughout in Notations.


### 2.1. Network Model

We consider a backhaul-aware cellular network consisting ofNb base stations. These BSs are randomly distributed on a two-dimensional Euclidean plane R2 by a homogeneous spatial Poisson point process (HPPP) Φb=di with density λb [16]. Each BS is equipped with Nt transmit antennas and serves multiple users with single antenna. Each BS has a cache device which is able to cache Nc contents and is connected to the core network through a backhaul link with limited backhaul capacity Cbh. An illustration of the network model is depicted in Figure 1.Figure 1
An illustration of the network model. A BS is deployed in a circle area with radius D to serve multiple users uniformly distributed in the area.In the network model, the users are modeled by an independent homogeneous Poisson Point Process with densityλu, where λu>λb. In the paper, we assume that user associating with BS is based on both content and signal-to-interference-plus-noise ratio (SINR). Specifically, if there are multiple BSs which have cached the requested content, a user will select the nearest one. Otherwise, a user will be associated with an BS with maximal receiving SINR. For mathematical tractability, we use a circular area with a radius of D to approximate the hexagonal shaped area, where a number of single antenna users are evenly distributed. The average user density of each cell is λu; then the probability density of u users in each cell can be expressed as (1)Pxu=λu/NbuΓuexp⁡-λuNb.DenotingCi=1,2…,Nc as the set of contents cached at the ith BS, i=1,2…,Nb. Each user requests different content, such as video, music, and news. We build the contents of the user request into a content catalog, that is, Ff=1,…,Ff. We assume that each content has equal size F bits in the content catalog, and the cache capacity of each BS is F·Nc bits. If the content requested by the user is cached in the local BS, the base station will directly deliver the content to the user from the caching device. And this user is called the cache-hit user, where the transmission rate is determined by the SINR of the downlink channel [17]. On the contrary, when the user requested content is not cached in the local BS, the BS firstly obtains the content from the core network through the backhaul link and then transmits it to the user through the radio link. Similarly, this user is called the cache-miss user.


### 2.2. Transmission Model

Spatial multiplexing is taken into consideration. At the same time, the number of simultaneously served users can not exceed the number of antennasNt. And a round-robin scheduling policy is applied [18]. To capture the essence of the problem and simplify the analysis, we assume that each BS serves users with zero-forcing beam-forming (ZFBF) [19], which is a spatial signal processing technique to eliminate multiuser interference. And the downlink transmission power for all users is the same. We introduce a parameter β to reflect the percentage of intercell interference (ICI) that can be eliminated by interference management methods (i.e., coordinated beam-forming and serial interference suppression) [20].The signal transmitted from a given BS is affected by two propagation losses before reaching the user:(1)
Path-loss: a distance dependent path-lossprub=rub-α, where rub is the distance from the BS to user and α is the path-loss exponent.(2)
Rayleigh fading:hub∈CNt×1 is the small-scale Rayleigh fading channel vector from the BS to user.Particularly, we describe the channel gain between theuth user and the bth BS as Gub, which can be expressed as (2)Gub=rub-αhub2,where hub2 denotes the integrated channel power gain between the uth user and the bth BS due to Rayleigh fading. With the above expressions, we can get the SINR of a typical user u as (3)SINRrub=Prub-αhub2∑j=1,j≠bNbβPrub,i-αhub,i2+σ2,where P is the transmit power of each base station and the random variable hub,i denotes the small-scale Rayleigh fading of the wireless channel and follows the exponential distribution. IBS→U=∑j=1,j≠bNbβPrub,i-αhub,i2 denotes the cumulative interference experienced from all BSs except the ith BS, and rub,i-α is the distance between the user and all base stations except the ith BS. σ2 is the variance of the white Gaussian noise.


### 2.3. BS Sleep Pattern of the Network

The energy consumption of a BS is determined by its working state [21]. When a BS is in active state, the components (e.g., power amplifier, signal processing unit, and antennas) will generate energy consumption. On the other hand, if the BS switches to sleep state, some units will be closed, and parts of the energy consumption can be saved. Once a BS has no active user, it will be switched into the sleeping state. According to the system model, the probability of a BS being in sleep state can be expressed as (4)Ps=eλb/λu+λb.


## 2.1. Network Model

We consider a backhaul-aware cellular network consisting ofNb base stations. These BSs are randomly distributed on a two-dimensional Euclidean plane R2 by a homogeneous spatial Poisson point process (HPPP) Φb=di with density λb [16]. Each BS is equipped with Nt transmit antennas and serves multiple users with single antenna. Each BS has a cache device which is able to cache Nc contents and is connected to the core network through a backhaul link with limited backhaul capacity Cbh. An illustration of the network model is depicted in Figure 1.Figure 1
An illustration of the network model. A BS is deployed in a circle area with radius D to serve multiple users uniformly distributed in the area.In the network model, the users are modeled by an independent homogeneous Poisson Point Process with densityλu, where λu>λb. In the paper, we assume that user associating with BS is based on both content and signal-to-interference-plus-noise ratio (SINR). Specifically, if there are multiple BSs which have cached the requested content, a user will select the nearest one. Otherwise, a user will be associated with an BS with maximal receiving SINR. For mathematical tractability, we use a circular area with a radius of D to approximate the hexagonal shaped area, where a number of single antenna users are evenly distributed. The average user density of each cell is λu; then the probability density of u users in each cell can be expressed as (1)Pxu=λu/NbuΓuexp⁡-λuNb.DenotingCi=1,2…,Nc as the set of contents cached at the ith BS, i=1,2…,Nb. Each user requests different content, such as video, music, and news. We build the contents of the user request into a content catalog, that is, Ff=1,…,Ff. We assume that each content has equal size F bits in the content catalog, and the cache capacity of each BS is F·Nc bits. If the content requested by the user is cached in the local BS, the base station will directly deliver the content to the user from the caching device. And this user is called the cache-hit user, where the transmission rate is determined by the SINR of the downlink channel [17]. On the contrary, when the user requested content is not cached in the local BS, the BS firstly obtains the content from the core network through the backhaul link and then transmits it to the user through the radio link. Similarly, this user is called the cache-miss user.


## 2.2. Transmission Model

Spatial multiplexing is taken into consideration. At the same time, the number of simultaneously served users can not exceed the number of antennasNt. And a round-robin scheduling policy is applied [18]. To capture the essence of the problem and simplify the analysis, we assume that each BS serves users with zero-forcing beam-forming (ZFBF) [19], which is a spatial signal processing technique to eliminate multiuser interference. And the downlink transmission power for all users is the same. We introduce a parameter β to reflect the percentage of intercell interference (ICI) that can be eliminated by interference management methods (i.e., coordinated beam-forming and serial interference suppression) [20].The signal transmitted from a given BS is affected by two propagation losses before reaching the user:(1)
Path-loss: a distance dependent path-lossprub=rub-α, where rub is the distance from the BS to user and α is the path-loss exponent.(2)
Rayleigh fading:hub∈CNt×1 is the small-scale Rayleigh fading channel vector from the BS to user.Particularly, we describe the channel gain between theuth user and the bth BS as Gub, which can be expressed as (2)Gub=rub-αhub2,where hub2 denotes the integrated channel power gain between the uth user and the bth BS due to Rayleigh fading. With the above expressions, we can get the SINR of a typical user u as (3)SINRrub=Prub-αhub2∑j=1,j≠bNbβPrub,i-αhub,i2+σ2,where P is the transmit power of each base station and the random variable hub,i denotes the small-scale Rayleigh fading of the wireless channel and follows the exponential distribution. IBS→U=∑j=1,j≠bNbβPrub,i-αhub,i2 denotes the cumulative interference experienced from all BSs except the ith BS, and rub,i-α is the distance between the user and all base stations except the ith BS. σ2 is the variance of the white Gaussian noise.


## 2.3. BS Sleep Pattern of the Network

The energy consumption of a BS is determined by its working state [21]. When a BS is in active state, the components (e.g., power amplifier, signal processing unit, and antennas) will generate energy consumption. On the other hand, if the BS switches to sleep state, some units will be closed, and parts of the energy consumption can be saved. Once a BS has no active user, it will be switched into the sleeping state. According to the system model, the probability of a BS being in sleep state can be expressed as (4)Ps=eλb/λu+λb.


## 3. Distributed Caching with Dynamic Content Popularity

Appropriate modeling has important impacts on the performance analysis of caching policies. However, most existing works do not take the dynamic changes of popular content distribution into consideration. In this work, we consider dynamic content popularity and propose a method to handle the time-varying popularity based on Poisson Shot Noise Model (SNM) [22].Shot Noise Model is a “content-level” business model, which can reveal the dynamic changing of content catalogs and the arrival rates of different contents in the same area, and can be used to evaluate the performance of a cached network. SNM describes the process of each content generation (i.e., content from active to inactive) and each content request process [23]. In [24], the author assumes that the process of generating contents is a Poisson process, and each content is taken out after being active for certain period. The request process of each content is an independent nonhomogeneous Poisson process. In this paper, we assume that the process of generating contents and the request process follows the process of [24]. Figure 2 shows the Poisson Shot Noise Model, which includes two different request arrival rates and time during the lifetime. We consider the active lifetime and shape of all contents to be fixed in this model.Figure 2
An illustration of the Poisson Shot Noise Model. This figure shows the two different request arrival rates and times during the lifetime and sets the active lifetime and shape of all content fixed, whereμ1 and μ2 are the request arrival rate of a content 1 at time t1.The lifetime of each contentf is associated with a time-varying request, which can be described by following characteristics:(1)
The start time of request contenttf: the request start time of contents follows a Poisson process with average arrival rate λf (i.e., for a unit of time there is an average of λf contents start to be active). For rectangular pulses, the average total number of contents generated in time T0 is N¯f=λfT0 and the average number of active content is λfT, where T is the file life-cycle.(2)
Shape∧ft: the shape describes the regularity of the request arrival rate of file m changing with time (i.e., rectangular pulse). It is reported that the this feature has a smaller impact on the hit probability under the Least Recently Used (LRU) cache management policy.(3)
The total number of requestsNf: this feature represents the total number of times that content f is requested by a user during the active time.(4)
The life-cycle of contentTf: this feature describes the length of the request for the content f’s arrival time, which can be defined by rectangular pulses, so the file life-cycle Tf can be expressed as (5)Tf=1∫0∞∧f2tdt.We assume that all contents have the same life-cycle. Denote withtm the arrival time of each request content f. At time t the lifetime content catalog is expressed as (6)Fft=f:tf<t<tm+T.We can describe the alive contentf∈Ff by the active “age” of each content and the total number of requests Nf. The “age” of active content f at time t can be expressed by τft=t-tf, where the “age” refers to the active time of content f at time t since the content appears in the system. According to the above characteristics, for the shot noise model with file life-cycle T and rectangular pulse ∧f, the request arrival rate of a content f at time t can be expressed as μf(t)=∧fNft-tf.


### 3.1. Request Hit Rate with Estimated Popularity

In this section, we assume that we do not know the popularity of each content; then we estimate the dynamic content popularity by observations of the past user’s requests to the file in the caching controller and try to maximize the request hit rate in lifetimeT. However, considering that there is no traffic power for transmitting a content from the core network to the BSs, we can achieve the time horizon request hit rate optimization via solving the individual problem of maximizing the request hit rate in each time instance.In this model, we choose the pointwise traversal method to study the request hit rate in each time instance. For ease of analysis, we choose to study the case wheret=0 and observe the random events in the time interval -T,0. We use binary caching vector Xf(t) to determine whether the content is cached at time t, where xf(t)=1 represents the fact that content f is cached in the BS at time t, and xf(t)=0 otherwise. Due to the limited cache capacity, the constraint ∑f=1f=Ffxft≤Cca must be satisfied in each time interval.In the following, we consider using binary caching vectorXf(t) to maximize the instantaneous request hit probability at the origin. Taking the fact into account that contents popularity μf is unknown, the instantaneous request hit probability at the origin can be expressed as Rx=∑f=1f=Ffxfμf. Next we would like to optimize the expected request hit rate which can be expressed as follows: (7)ERx∣Nf,τf=∑f=1f=FfxfEμf∣Nf,τf.With SNM, predictive popularity can be expressed as (8)Eμf∣Nf,τf=∫0μfμfμfτfNfe-μfτf/Nffμfdμf∫0μfμfτfNfe-μfτf/Nffμfdμf,where fμf is the power-law density and τf is the “age” of active content f at time t. Accordingly, the request hit rate maximization problem based on predicted popularity can be formulated as (9)R∗Nf,τf=arg⁡max∀f,xf∈0,1∑f=0f=FfxfEμf∣Nf,τfsubject  to∑0Fxf≤Cca.It can be seen from the above expression that we can calculate the instantaneous request hit rate by the value of Nf and τf.


### 3.2. Age-Based Threshold Caching Policy

To solve the optimal request hit probability at every time instant, we design a caching policy with age-based threshold, and this policy does not need to calculate the estimationμf^=Eμf∣Nf,τf. Intuitively, the “age” τf affects the estimation μf^. Due to the uncertainty in the estimation μf^ is low; then if we want to maximize the request hit rate we must cache more contents with larger ages. Accordingly, we propose an age-based threshold A~τ which depends on the content age τ. Under this policy, if the content satisfies Nf≥A~τf, it will be cached. When the content is fixed and the age is known, the cache decision depends only on the total number of requests Nf, which makes caching decision more simpler.Note that an important progress of the age-based threshold estimation is to computeA~τ. We can compute the age threshold function A~τ by iteratively filling cache capacity Cca in the algorithms, which make the marginal request hit rate improvements Eμf∣Nf=A~τ,τf=τ for each request age τ approximately equal. The details can be found in Algorithm 1.Algorithm 1:Age-based threshold policy.
Step  1: Selecting Parameter. (i) Cca is the cache capacity, (ii) λ is the average request rate, (iii) Tλ is the average total number of request content at any time instant, (iv)γca=Cca/λT is the fraction of caching content catalog, (v)Wμ^f is the function of distribution. (vi) Define θγca is the γcath tipper-percentile of Wμ^f, where θ γ c a = W μ ^ f - 1 1 - γ c a. Step  2: Setting the Age-Based Threshold. Denote N~f(τ) is the age-based threshold. N ~ f τ = min ⁡ f ∈ N : E μ f ∣ N f = f , τ f = τ ≥ θ γ c a Step  3: Caching Content Vector. Using binary caching vector Xf(t) to determine whether the content is cached at timet. X f t = 1 , if  N f ≥ N ~ τ f , 0 , if otherwise, Step  4: Caching Capacity. We define a constraint condition for the capacity of the cache, if ∑0Fxf>Cca, we will select∑0Fxf-Cca,∀f∈F with xf=1, otherwise set xf=0 for content.


### 3.3. The Correlation of Content Request Popularity

Due to differences in social and cultural backgrounds among users, restrictions on specific office areas, or the impact of certain group activities, content popularity among BSs may vary. However, there are still possible correlations. We assume that the contents exhibit geographical correlations and hence there exist groups of contents which are very popular in subset of local caches [25–27]. For local caches with the same popularity, we can more effectively constrain the aggregation to a subset of caches that witness similar traffic patterns by aggregating all cached request contents.Request Model for Correlated Popularity. We design a model for the correlation of content request popularity μfi. The aggregate total popularity is expressed as μfNb=Nbμfiτ,∀f,i∈1⋯Nb, and the global average request hit rate is EμfZ=μ¯. We obtain the correlation between the intercell request arrival rate μmi through community detection [28] and inhomogeneous random graphs [29] as follows:(i)
Each contentf is associated with a property vector Mf, where Mf is the property of the content f (e.g., the probability of the content as comedy). In order to simplify the model we consider that Mf is an independent uniformly distributed random variable taking values in 0,1.(ii)
Each BSi is also associated with a property vector Li, where Li is the property of the BSs i (e.g., the probability of the user request comedy in intercell i). The vector Li is selected uniformly and randomly in 0,1.(iii)
We assume thatMf and Li are uniformly distributed in 0,1. We define Kx,y=gx-y, where g(·) is a symmetric and continuous function with period of 1, strictly decreasing on [0,1/2], and ∫01gx-ydy=1 for all x∈0,1.When the content popularity of each intercell is different, according to [30], the request arrival rate of content f in BS can be expressed as (10)μfi=μfZKMf,Li∑0i∈NbKMf,Li,∀f,i,where Z are the contents in the core network. As the number of caching contents Z increases in the core network and the kernel function g· satisfies some base conditions, the normalization constant almost surely becomes deterministic: (11)1Z∑i=0i=ZKMf,Li′⟶limZ→∝⁡∫01gMf-ydy=1μfi⟶limZ→∝⁡μfZKMf,LiZ.In the section, the request hit rate of the future contents is predicted according to the request information of the historical contents of each BS, and then each BS caches the most popularNc different contents.


### 3.4. BS Cache Policy of the Network

Research shows that multiple users could access the same content, like some videos, news, music, and so on. In this case, the network will be flooded with requests of same contents, which can largely increase the latency, eventually leading to network congestion. Therefore, under the limited cache capacity, assuming that the user’s choice distribution matches with the dynamic content popularity, then each BS can cache most popular content to reduce service latency and backhaul congestion in such a dynamic environment.A content catalog is defined asFf=1,2,…,Ff, and the size of each content f in the catalog has a length of L(f) Mbytes. According to the description of Section 3, we can predict the content popularity, and the instantaneous popularity follows the Zipf distribution [31]. Each user requests a content from the content catalog Ff, and the probability of requesting the fth content is (12)PFf=∑i=1Ff1/iλu/λb-1fλu/λb,where λu/λb characterizes the steepness of the distribution. The value of λu/λb depends on the average number of users per BS, and higher values of λu/λb correspond to steeper distributions which means that the higher the number of users attached to a BS is, the more accurately the trend is sampled.As mentioned above, each BS caches most popularNc content according to the dynamic popularity distribution. Hence, the cache-hit rate of each BS can be obtained by the Zipf distribution: (13)H=∑f=1Ncfλu/λb∑i=1Ffiλu/λb.It can be proved from (13) that H converges to 1 when the content catalog Ff in the BS goes to infinity. Consider the relation between maximal EE of the network and the cache size. To this end, we provide the cache-hit rate H for large values of Nc and Ff. In order to show the impact of the cache capacity and content library Ff on cache-hit rate, we analyze a normalized cache size η≜Nc/Ff,0<Nc≤Ff,η∈0,1. Thus (13) is normalized as (14)H=ln⁡Nc+ΘNc-1ln⁡Ff+ΘFf-1≈ln⁡Ncln⁡Ff≈1+ln⁡ηln⁡Ff,where the approximation in (14) is accurate when Nc≫1 and Ff≫1.Cache-hit rate affects the performance of wireless networks, like throughput, energy efficiency, user quality of service (QoS), and so on. In order to increase cache-hit rate, we propose a distributed content caching strategy in backhaul-aware cellular networks, where four adjacent BSs cache different popular contents. According to the system model, we can find that each user is associated with a BS based on both content and SINR. Meanwhile, if there are several BSs caching the requested content, the user selects one whose distance to him is the shortest and delivers the content via radio link directly. Otherwise, user associates with a BS whose SINR is maximal from its neighbouring BSs and then gets the requested content through the backhaul link from the core network. As illustrated in Figure3, each hexagon represents a BS with cached popular content. This distributed content caching strategy can reduce content redundancy by caching different contents in different BS. When each BS caches Nc contents with the distributed cache strategy, each user can requests 4Nc popular contents.Figure 3
An illustration of distributed caching policy. The BS marked with “a,” “b,” “c,” and “d” caches different popular contents and each user associates with the nearest BS that caches the user’s requested contents.


## 3.1. Request Hit Rate with Estimated Popularity

In this section, we assume that we do not know the popularity of each content; then we estimate the dynamic content popularity by observations of the past user’s requests to the file in the caching controller and try to maximize the request hit rate in lifetimeT. However, considering that there is no traffic power for transmitting a content from the core network to the BSs, we can achieve the time horizon request hit rate optimization via solving the individual problem of maximizing the request hit rate in each time instance.In this model, we choose the pointwise traversal method to study the request hit rate in each time instance. For ease of analysis, we choose to study the case wheret=0 and observe the random events in the time interval -T,0. We use binary caching vector Xf(t) to determine whether the content is cached at time t, where xf(t)=1 represents the fact that content f is cached in the BS at time t, and xf(t)=0 otherwise. Due to the limited cache capacity, the constraint ∑f=1f=Ffxft≤Cca must be satisfied in each time interval.In the following, we consider using binary caching vectorXf(t) to maximize the instantaneous request hit probability at the origin. Taking the fact into account that contents popularity μf is unknown, the instantaneous request hit probability at the origin can be expressed as Rx=∑f=1f=Ffxfμf. Next we would like to optimize the expected request hit rate which can be expressed as follows: (7)ERx∣Nf,τf=∑f=1f=FfxfEμf∣Nf,τf.With SNM, predictive popularity can be expressed as (8)Eμf∣Nf,τf=∫0μfμfμfτfNfe-μfτf/Nffμfdμf∫0μfμfτfNfe-μfτf/Nffμfdμf,where fμf is the power-law density and τf is the “age” of active content f at time t. Accordingly, the request hit rate maximization problem based on predicted popularity can be formulated as (9)R∗Nf,τf=arg⁡max∀f,xf∈0,1∑f=0f=FfxfEμf∣Nf,τfsubject  to∑0Fxf≤Cca.It can be seen from the above expression that we can calculate the instantaneous request hit rate by the value of Nf and τf.


## 3.2. Age-Based Threshold Caching Policy

To solve the optimal request hit probability at every time instant, we design a caching policy with age-based threshold, and this policy does not need to calculate the estimationμf^=Eμf∣Nf,τf. Intuitively, the “age” τf affects the estimation μf^. Due to the uncertainty in the estimation μf^ is low; then if we want to maximize the request hit rate we must cache more contents with larger ages. Accordingly, we propose an age-based threshold A~τ which depends on the content age τ. Under this policy, if the content satisfies Nf≥A~τf, it will be cached. When the content is fixed and the age is known, the cache decision depends only on the total number of requests Nf, which makes caching decision more simpler.Note that an important progress of the age-based threshold estimation is to computeA~τ. We can compute the age threshold function A~τ by iteratively filling cache capacity Cca in the algorithms, which make the marginal request hit rate improvements Eμf∣Nf=A~τ,τf=τ for each request age τ approximately equal. The details can be found in Algorithm 1.Algorithm 1:Age-based threshold policy.
Step  1: Selecting Parameter. (i) Cca is the cache capacity, (ii) λ is the average request rate, (iii) Tλ is the average total number of request content at any time instant, (iv)γca=Cca/λT is the fraction of caching content catalog, (v)Wμ^f is the function of distribution. (vi) Define θγca is the γcath tipper-percentile of Wμ^f, where θ γ c a = W μ ^ f - 1 1 - γ c a. Step  2: Setting the Age-Based Threshold. Denote N~f(τ) is the age-based threshold. N ~ f τ = min ⁡ f ∈ N : E μ f ∣ N f = f , τ f = τ ≥ θ γ c a Step  3: Caching Content Vector. Using binary caching vector Xf(t) to determine whether the content is cached at timet. X f t = 1 , if  N f ≥ N ~ τ f , 0 , if otherwise, Step  4: Caching Capacity. We define a constraint condition for the capacity of the cache, if ∑0Fxf>Cca, we will select∑0Fxf-Cca,∀f∈F with xf=1, otherwise set xf=0 for content.


## 3.3. The Correlation of Content Request Popularity

Due to differences in social and cultural backgrounds among users, restrictions on specific office areas, or the impact of certain group activities, content popularity among BSs may vary. However, there are still possible correlations. We assume that the contents exhibit geographical correlations and hence there exist groups of contents which are very popular in subset of local caches [25–27]. For local caches with the same popularity, we can more effectively constrain the aggregation to a subset of caches that witness similar traffic patterns by aggregating all cached request contents.Request Model for Correlated Popularity. We design a model for the correlation of content request popularity μfi. The aggregate total popularity is expressed as μfNb=Nbμfiτ,∀f,i∈1⋯Nb, and the global average request hit rate is EμfZ=μ¯. We obtain the correlation between the intercell request arrival rate μmi through community detection [28] and inhomogeneous random graphs [29] as follows:(i)
Each contentf is associated with a property vector Mf, where Mf is the property of the content f (e.g., the probability of the content as comedy). In order to simplify the model we consider that Mf is an independent uniformly distributed random variable taking values in 0,1.(ii)
Each BSi is also associated with a property vector Li, where Li is the property of the BSs i (e.g., the probability of the user request comedy in intercell i). The vector Li is selected uniformly and randomly in 0,1.(iii)
We assume thatMf and Li are uniformly distributed in 0,1. We define Kx,y=gx-y, where g(·) is a symmetric and continuous function with period of 1, strictly decreasing on [0,1/2], and ∫01gx-ydy=1 for all x∈0,1.When the content popularity of each intercell is different, according to [30], the request arrival rate of content f in BS can be expressed as (10)μfi=μfZKMf,Li∑0i∈NbKMf,Li,∀f,i,where Z are the contents in the core network. As the number of caching contents Z increases in the core network and the kernel function g· satisfies some base conditions, the normalization constant almost surely becomes deterministic: (11)1Z∑i=0i=ZKMf,Li′⟶limZ→∝⁡∫01gMf-ydy=1μfi⟶limZ→∝⁡μfZKMf,LiZ.In the section, the request hit rate of the future contents is predicted according to the request information of the historical contents of each BS, and then each BS caches the most popularNc different contents.


## 3.4. BS Cache Policy of the Network

Research shows that multiple users could access the same content, like some videos, news, music, and so on. In this case, the network will be flooded with requests of same contents, which can largely increase the latency, eventually leading to network congestion. Therefore, under the limited cache capacity, assuming that the user’s choice distribution matches with the dynamic content popularity, then each BS can cache most popular content to reduce service latency and backhaul congestion in such a dynamic environment.A content catalog is defined asFf=1,2,…,Ff, and the size of each content f in the catalog has a length of L(f) Mbytes. According to the description of Section 3, we can predict the content popularity, and the instantaneous popularity follows the Zipf distribution [31]. Each user requests a content from the content catalog Ff, and the probability of requesting the fth content is (12)PFf=∑i=1Ff1/iλu/λb-1fλu/λb,where λu/λb characterizes the steepness of the distribution. The value of λu/λb depends on the average number of users per BS, and higher values of λu/λb correspond to steeper distributions which means that the higher the number of users attached to a BS is, the more accurately the trend is sampled.As mentioned above, each BS caches most popularNc content according to the dynamic popularity distribution. Hence, the cache-hit rate of each BS can be obtained by the Zipf distribution: (13)H=∑f=1Ncfλu/λb∑i=1Ffiλu/λb.It can be proved from (13) that H converges to 1 when the content catalog Ff in the BS goes to infinity. Consider the relation between maximal EE of the network and the cache size. To this end, we provide the cache-hit rate H for large values of Nc and Ff. In order to show the impact of the cache capacity and content library Ff on cache-hit rate, we analyze a normalized cache size η≜Nc/Ff,0<Nc≤Ff,η∈0,1. Thus (13) is normalized as (14)H=ln⁡Nc+ΘNc-1ln⁡Ff+ΘFf-1≈ln⁡Ncln⁡Ff≈1+ln⁡ηln⁡Ff,where the approximation in (14) is accurate when Nc≫1 and Ff≫1.Cache-hit rate affects the performance of wireless networks, like throughput, energy efficiency, user quality of service (QoS), and so on. In order to increase cache-hit rate, we propose a distributed content caching strategy in backhaul-aware cellular networks, where four adjacent BSs cache different popular contents. According to the system model, we can find that each user is associated with a BS based on both content and SINR. Meanwhile, if there are several BSs caching the requested content, the user selects one whose distance to him is the shortest and delivers the content via radio link directly. Otherwise, user associates with a BS whose SINR is maximal from its neighbouring BSs and then gets the requested content through the backhaul link from the core network. As illustrated in Figure3, each hexagon represents a BS with cached popular content. This distributed content caching strategy can reduce content redundancy by caching different contents in different BS. When each BS caches Nc contents with the distributed cache strategy, each user can requests 4Nc popular contents.Figure 3
An illustration of distributed caching policy. The BS marked with “a,” “b,” “c,” and “d” caches different popular contents and each user associates with the nearest BS that caches the user’s requested contents.


## 4. Energy Efficiency Optimization

In recent years, the research on green communications has made great progress. It shows that the energy efficiency of the cellular network has become an essential factor in the study of its performance, which is the key to reduce operating costs and is aiming to provide wireless access in the way of sustainable development and rational utilization of resources [32]. We define the EE as the ratio of the average throughput of the network to the average total power consumption at the BS, which can be expressed as (15)EE=R¯P¯t,where R¯ is the average throughput and P¯t is the average power consumption.


### 4.1. Average Throughput of the Network

If useru1,u2,…,uc request contents are cached at local BS, the user can immediately obtain the requested content from the local cache via the radio links, where the data rate is limited only by the capacity of the wireless channel. On the contrary, if the contents requested by users uc+1,…,ub are not cached at local BSs, the BSs need to obtain the contents from the core network via the backhaul links and then transmit the contents to users through radio links, so the data rate is subject to backhaul link capacity and radio channel capacity. Considering the above two cases, the downlink average throughput of the bth cell can be expressed as (16)R¯=1-PsR¯2+R¯1-R¯21+ln⁡ηln⁡Ff,R¯1=EB∑u=1uclog21+SINRub,R¯2=Emin⁡B∑U=uc+1Ublog21+SINRub,Cbh,where R¯1 is the average sum rate of the cache-hit users and R¯2 is the average sum rate of the cache-miss users.


#### 4.1.1. The Average Sum Rate of Cache-Hit Users

Considering that the SINR for every user shown in (3) are identically distributed, the average sum rate of the cache-hit users R¯1 can be approximated as (17)R¯1=ucBElog21+rub-αhub2∑j=1,j≠bNbβubrub,i-αhub,i2+σ2/P≈ucBlog2Nt-ub+1ub+∫0Dlog2rub-α2rubD2drub-ucBlog2eλbu/λu+λbuβ2ϕD-α+σ2P≈ucBα2ln⁡2+log2Nt-ub+1ubeuλb/λu+uλbβ2ϕ+σ2/PDα,where ϕ is a constant only depending on the path-loss exponent α when Nb→∞, rub is the distance from bth BS to the uth user.


#### 4.1.2. The Average Sum Rate of Cache-Miss Users

Considering the transmit power of base stations to be the same, the cumulative interference from all BS approaches to its expectationEIBS→U when Nb→∞ according to the law of the ultradense BSs deployment. This suggests that the comparison of Cbh and B∑U=uc+1Ublog2(1+SINRub) depends on the distance from each user to the local base station when the number of BSs is very large. Therefore, the average sum of the cache-miss users can be approximated as (18)R¯2=Emin⁡B∑U=uc+1Ublog21+SINRub,Cbh≈Emin⁡ub-ucR3+αB2ln⁡2∑uc+1ub2ln⁡Drub,Cbh,R3≈log2Nt-ub+1euλb/λu+uλbβ2ϕ+σ2/PDα.Given that the user’s distributions are independent and random, we can apply the probability density function of 2rub/D2 to proving that 2ln⁡D/rub,u∈1⋯ub,b∈1⋯Nb are random variables which are independently and exponentially distributed. Thus, we can get that x=∑uc+1ub2ln⁡D/rub is a random variable following Gamma distribution Gub-uc,1. In this case, when the probability density function ∑uc+1ub2ln⁡D/rub>0, we can derive fx=xub-uc-1e-x/ub-uc-1!. Therefore, when Cbh≤ub-ucR3, (18) becomes R¯2=Cbh. When Cbh>ub-ucR3, (18) can be approximated as (19)R¯2≈∫0∞min⁡AR3+αB2ln2x,Cbhfxdx≈∫0ZAR3+αB2ln2xfxdx+∫Z∞Cbhfxdx≈AαB2ln⁡2γB+R¯3γC+CbhΓC,where A=ub-uc, B=ub-uc+1,Z, C=ub-uc,Z, Z=2ln⁡2/αBCbh-AR3, Γa,b=e-b∑i=0b-1bi/i!, and γa,b=1-Γa,b.In summary, the average sum rate of the cache-miss users can be approximated as(20)R¯2≈AαB2ln⁡2γB+R¯3γC+CbhΓC,ifCbh>AR3,Cbh,ifCbh≤AR3.(i)
Whenλb is large and Nt=ub, by simplifying (17), the average number of bits totally transmitted can be approximated as follows: (21)R¯1≈Bα2ln⁡2+log2Nteuλb/λu+uλbβ2ϕ+σ2/PDα.(ii)
Whenub-uc=1 and Nt=ub, we can get z=2ln⁡2/αB(Cbh-R¯e), γB=1-e-z1+z, γ(C)=1-e-z and ΓC=e-z, R3≈R4≈log2Nt/euλb/λu+uλbβ2ϕ+σ2/PDα, R5=α/2ln⁡22-(R4-Cbh)/α. So the formula R¯bh can be approximated as (22)R¯2≈Bα2ln⁡2+R3-R4,ifCbh>R4,Cbh,ifCbh≤AR4.


### 4.2. Average Total Power Consumption of the Network

In this work, we investigate the performance of EE in cellular networks with caching at BSs. To this end, we need to calculate the overall energy consumption caused by the transmission and storage of contents.We have known that when the BS is in sleeping state, some units are turned off, such that the BS reduces part of the power consumption. As a result, we can divide the circuit power consumption and transmit power consumption into two states according to the working state of BS, respectively.Finally, the overall power consumption of the network can be divided into four parts:(i)
The average total transmit power of the base station in the entire downlink can be expressed as(23)Pt=NbρP1-Ps,ifζBS=1,0,ifζBS=0.(ii)
The average total power consumption of the power supply circuit can be expressed as(24)Pc=NbP11-Ps,ifζBS=1,NbPsP2,ifζBS=0,whereζBS is the active state of the base station, ρ reflects the impact of power amplifier, P is the transmit power of the BS in active mode, P1 is the supply power of the BS in active mode, and P2 is the circuit power of the BS in sleeping mode.(iii)
The average caching power consumption: in this paper, we consider DRAM as cache hardware. According to the power model in [33], if the power coefficient of the cache hardware is fixed, the average caching power consumption is related to the size and the number of content cached in the local caching device. The average caching power consumption of BS can be expressed as (25)P3=NcFω1=ηFfFω1,whereω1 is the power coefficient of caching hardware and η is the normalized cache capacity.(iv)
The average backhaul power consumption: we use fiber as the backhaul link (with capacity 1 Gbps). When we download contents, the backhaul link will generate power consumption, and the average backhaul power consumption of BS can be expressed as(26)P4=1-Ps1-Hω2R¯2=-1-Psln⁡ηlnFfω2R¯2,whereω2=P1/Cbh1 is the power coefficient of backhaul equipment and P1 denotes the power consumed by the backhaul for supporting the maximum data rate Cbh1. From formula (26), we can find that the energy consumption for backhauling will reduce sharply with a little decrease of the cache size, but decreasing more and more slowly with the growth of cache size. Therefore, we need to find the best cache capacity to minimize power consumption.Then, the average total power consumption at all BSs can be expressed as(27)P¯t=Pt+Pc+P3+P4.


### 4.3. Energy Efficiency of the Network

Substituting (16) and (27) into (15), we can approximate the network EE as (28)EE≈1-PsR¯1lnFf+R¯1-R¯2ln⁡ηPt+Pcln⁡Ff+ω1FηFfln⁡Ff-1-Psω2R¯2ln⁡η.The goal of this paper is to maximize the energy efficiency of the downlink wireless network by finding out the optimal cache content in the local base station. Mathematically, we have the following optimization problem:(29)maximizeη∈0,1EEηsubject  to∑f=1ηFff-λu/λb∑j=1Ffj-λu/λb<1,∀Nc∈Ff,Cbh≤Cbh′,∀Nb∈BS,Psω2R¯2ω1F>∑i=1Ffi-1,∀Nc∈Ff,where ∑f=1ηFff(-λu/λb)/∑j=1Ffj(-λu/λb)<1 represents the probability of the cache-hit rate, Cbh≤Cbh′ is the limit of backhaul capacity, and Psω2R¯2/ω1F>∑i=1Ffi-1 is the condition that the cache can increase energy efficiency. The following result can be obtained for the maximization of EE and the analysis is carried out later.Proposition 1.
Consider that the BS has limited cache capacity and letη0 denote the optimal cache size that maximizes EE. Then (30)η0=QFfWQeM,where WQeM is a Lambert-W function satisfying fWx=Wxexp⁡Wx.Proof.
(i)
We first derive the derivative ofη in (28); we obtain (31)dEEdηη=η0=0⟹Qη0Ff+ln⁡1η0Ff-M=0,Q=R¯1R¯2/R¯1-R¯2ω21-Ps+Pt+Pcω1F,M=R¯2R¯1-R¯2ln⁡Ff-1, whereM and Q are considered as constants. (ii)
Letf(η)=Q/ηFf+ln⁡1/ηFf-M, and the derivative of η in the equation is (32)f′η=-M+ηFfFfη2. (iii)
Sof′η<0 when η>0; then fη decreases with the increase of the value η. When η=η0, the equation fη=0; then we can get (33)fη<0,ifη>η0,fη>0,ifη<η0. (iv)
Sincef(η) decreases with η, we can obtain that EE monotonically increases when η<η0 and EE monotonically decreases when η>η0. Through the above descriptions, we find that the EE can be maximized when η=η0.


## 4.1. Average Throughput of the Network

If useru1,u2,…,uc request contents are cached at local BS, the user can immediately obtain the requested content from the local cache via the radio links, where the data rate is limited only by the capacity of the wireless channel. On the contrary, if the contents requested by users uc+1,…,ub are not cached at local BSs, the BSs need to obtain the contents from the core network via the backhaul links and then transmit the contents to users through radio links, so the data rate is subject to backhaul link capacity and radio channel capacity. Considering the above two cases, the downlink average throughput of the bth cell can be expressed as (16)R¯=1-PsR¯2+R¯1-R¯21+ln⁡ηln⁡Ff,R¯1=EB∑u=1uclog21+SINRub,R¯2=Emin⁡B∑U=uc+1Ublog21+SINRub,Cbh,where R¯1 is the average sum rate of the cache-hit users and R¯2 is the average sum rate of the cache-miss users.


### 4.1.1. The Average Sum Rate of Cache-Hit Users

Considering that the SINR for every user shown in (3) are identically distributed, the average sum rate of the cache-hit users R¯1 can be approximated as (17)R¯1=ucBElog21+rub-αhub2∑j=1,j≠bNbβubrub,i-αhub,i2+σ2/P≈ucBlog2Nt-ub+1ub+∫0Dlog2rub-α2rubD2drub-ucBlog2eλbu/λu+λbuβ2ϕD-α+σ2P≈ucBα2ln⁡2+log2Nt-ub+1ubeuλb/λu+uλbβ2ϕ+σ2/PDα,where ϕ is a constant only depending on the path-loss exponent α when Nb→∞, rub is the distance from bth BS to the uth user.


### 4.1.2. The Average Sum Rate of Cache-Miss Users

Considering the transmit power of base stations to be the same, the cumulative interference from all BS approaches to its expectationEIBS→U when Nb→∞ according to the law of the ultradense BSs deployment. This suggests that the comparison of Cbh and B∑U=uc+1Ublog2(1+SINRub) depends on the distance from each user to the local base station when the number of BSs is very large. Therefore, the average sum of the cache-miss users can be approximated as (18)R¯2=Emin⁡B∑U=uc+1Ublog21+SINRub,Cbh≈Emin⁡ub-ucR3+αB2ln⁡2∑uc+1ub2ln⁡Drub,Cbh,R3≈log2Nt-ub+1euλb/λu+uλbβ2ϕ+σ2/PDα.Given that the user’s distributions are independent and random, we can apply the probability density function of 2rub/D2 to proving that 2ln⁡D/rub,u∈1⋯ub,b∈1⋯Nb are random variables which are independently and exponentially distributed. Thus, we can get that x=∑uc+1ub2ln⁡D/rub is a random variable following Gamma distribution Gub-uc,1. In this case, when the probability density function ∑uc+1ub2ln⁡D/rub>0, we can derive fx=xub-uc-1e-x/ub-uc-1!. Therefore, when Cbh≤ub-ucR3, (18) becomes R¯2=Cbh. When Cbh>ub-ucR3, (18) can be approximated as (19)R¯2≈∫0∞min⁡AR3+αB2ln2x,Cbhfxdx≈∫0ZAR3+αB2ln2xfxdx+∫Z∞Cbhfxdx≈AαB2ln⁡2γB+R¯3γC+CbhΓC,where A=ub-uc, B=ub-uc+1,Z, C=ub-uc,Z, Z=2ln⁡2/αBCbh-AR3, Γa,b=e-b∑i=0b-1bi/i!, and γa,b=1-Γa,b.In summary, the average sum rate of the cache-miss users can be approximated as(20)R¯2≈AαB2ln⁡2γB+R¯3γC+CbhΓC,ifCbh>AR3,Cbh,ifCbh≤AR3.(i)
Whenλb is large and Nt=ub, by simplifying (17), the average number of bits totally transmitted can be approximated as follows: (21)R¯1≈Bα2ln⁡2+log2Nteuλb/λu+uλbβ2ϕ+σ2/PDα.(ii)
Whenub-uc=1 and Nt=ub, we can get z=2ln⁡2/αB(Cbh-R¯e), γB=1-e-z1+z, γ(C)=1-e-z and ΓC=e-z, R3≈R4≈log2Nt/euλb/λu+uλbβ2ϕ+σ2/PDα, R5=α/2ln⁡22-(R4-Cbh)/α. So the formula R¯bh can be approximated as (22)R¯2≈Bα2ln⁡2+R3-R4,ifCbh>R4,Cbh,ifCbh≤AR4.


## 4.1.1. The Average Sum Rate of Cache-Hit Users

Considering that the SINR for every user shown in (3) are identically distributed, the average sum rate of the cache-hit users R¯1 can be approximated as (17)R¯1=ucBElog21+rub-αhub2∑j=1,j≠bNbβubrub,i-αhub,i2+σ2/P≈ucBlog2Nt-ub+1ub+∫0Dlog2rub-α2rubD2drub-ucBlog2eλbu/λu+λbuβ2ϕD-α+σ2P≈ucBα2ln⁡2+log2Nt-ub+1ubeuλb/λu+uλbβ2ϕ+σ2/PDα,where ϕ is a constant only depending on the path-loss exponent α when Nb→∞, rub is the distance from bth BS to the uth user.


## 4.1.2. The Average Sum Rate of Cache-Miss Users

Considering the transmit power of base stations to be the same, the cumulative interference from all BS approaches to its expectationEIBS→U when Nb→∞ according to the law of the ultradense BSs deployment. This suggests that the comparison of Cbh and B∑U=uc+1Ublog2(1+SINRub) depends on the distance from each user to the local base station when the number of BSs is very large. Therefore, the average sum of the cache-miss users can be approximated as (18)R¯2=Emin⁡B∑U=uc+1Ublog21+SINRub,Cbh≈Emin⁡ub-ucR3+αB2ln⁡2∑uc+1ub2ln⁡Drub,Cbh,R3≈log2Nt-ub+1euλb/λu+uλbβ2ϕ+σ2/PDα.Given that the user’s distributions are independent and random, we can apply the probability density function of 2rub/D2 to proving that 2ln⁡D/rub,u∈1⋯ub,b∈1⋯Nb are random variables which are independently and exponentially distributed. Thus, we can get that x=∑uc+1ub2ln⁡D/rub is a random variable following Gamma distribution Gub-uc,1. In this case, when the probability density function ∑uc+1ub2ln⁡D/rub>0, we can derive fx=xub-uc-1e-x/ub-uc-1!. Therefore, when Cbh≤ub-ucR3, (18) becomes R¯2=Cbh. When Cbh>ub-ucR3, (18) can be approximated as (19)R¯2≈∫0∞min⁡AR3+αB2ln2x,Cbhfxdx≈∫0ZAR3+αB2ln2xfxdx+∫Z∞Cbhfxdx≈AαB2ln⁡2γB+R¯3γC+CbhΓC,where A=ub-uc, B=ub-uc+1,Z, C=ub-uc,Z, Z=2ln⁡2/αBCbh-AR3, Γa,b=e-b∑i=0b-1bi/i!, and γa,b=1-Γa,b.In summary, the average sum rate of the cache-miss users can be approximated as(20)R¯2≈AαB2ln⁡2γB+R¯3γC+CbhΓC,ifCbh>AR3,Cbh,ifCbh≤AR3.(i)
Whenλb is large and Nt=ub, by simplifying (17), the average number of bits totally transmitted can be approximated as follows: (21)R¯1≈Bα2ln⁡2+log2Nteuλb/λu+uλbβ2ϕ+σ2/PDα.(ii)
Whenub-uc=1 and Nt=ub, we can get z=2ln⁡2/αB(Cbh-R¯e), γB=1-e-z1+z, γ(C)=1-e-z and ΓC=e-z, R3≈R4≈log2Nt/euλb/λu+uλbβ2ϕ+σ2/PDα, R5=α/2ln⁡22-(R4-Cbh)/α. So the formula R¯bh can be approximated as (22)R¯2≈Bα2ln⁡2+R3-R4,ifCbh>R4,Cbh,ifCbh≤AR4.


## 4.2. Average Total Power Consumption of the Network

In this work, we investigate the performance of EE in cellular networks with caching at BSs. To this end, we need to calculate the overall energy consumption caused by the transmission and storage of contents.We have known that when the BS is in sleeping state, some units are turned off, such that the BS reduces part of the power consumption. As a result, we can divide the circuit power consumption and transmit power consumption into two states according to the working state of BS, respectively.Finally, the overall power consumption of the network can be divided into four parts:(i)
The average total transmit power of the base station in the entire downlink can be expressed as(23)Pt=NbρP1-Ps,ifζBS=1,0,ifζBS=0.(ii)
The average total power consumption of the power supply circuit can be expressed as(24)Pc=NbP11-Ps,ifζBS=1,NbPsP2,ifζBS=0,whereζBS is the active state of the base station, ρ reflects the impact of power amplifier, P is the transmit power of the BS in active mode, P1 is the supply power of the BS in active mode, and P2 is the circuit power of the BS in sleeping mode.(iii)
The average caching power consumption: in this paper, we consider DRAM as cache hardware. According to the power model in [33], if the power coefficient of the cache hardware is fixed, the average caching power consumption is related to the size and the number of content cached in the local caching device. The average caching power consumption of BS can be expressed as (25)P3=NcFω1=ηFfFω1,whereω1 is the power coefficient of caching hardware and η is the normalized cache capacity.(iv)
The average backhaul power consumption: we use fiber as the backhaul link (with capacity 1 Gbps). When we download contents, the backhaul link will generate power consumption, and the average backhaul power consumption of BS can be expressed as(26)P4=1-Ps1-Hω2R¯2=-1-Psln⁡ηlnFfω2R¯2,whereω2=P1/Cbh1 is the power coefficient of backhaul equipment and P1 denotes the power consumed by the backhaul for supporting the maximum data rate Cbh1. From formula (26), we can find that the energy consumption for backhauling will reduce sharply with a little decrease of the cache size, but decreasing more and more slowly with the growth of cache size. Therefore, we need to find the best cache capacity to minimize power consumption.Then, the average total power consumption at all BSs can be expressed as(27)P¯t=Pt+Pc+P3+P4.


## 4.3. Energy Efficiency of the Network

Substituting (16) and (27) into (15), we can approximate the network EE as (28)EE≈1-PsR¯1lnFf+R¯1-R¯2ln⁡ηPt+Pcln⁡Ff+ω1FηFfln⁡Ff-1-Psω2R¯2ln⁡η.The goal of this paper is to maximize the energy efficiency of the downlink wireless network by finding out the optimal cache content in the local base station. Mathematically, we have the following optimization problem:(29)maximizeη∈0,1EEηsubject  to∑f=1ηFff-λu/λb∑j=1Ffj-λu/λb<1,∀Nc∈Ff,Cbh≤Cbh′,∀Nb∈BS,Psω2R¯2ω1F>∑i=1Ffi-1,∀Nc∈Ff,where ∑f=1ηFff(-λu/λb)/∑j=1Ffj(-λu/λb)<1 represents the probability of the cache-hit rate, Cbh≤Cbh′ is the limit of backhaul capacity, and Psω2R¯2/ω1F>∑i=1Ffi-1 is the condition that the cache can increase energy efficiency. The following result can be obtained for the maximization of EE and the analysis is carried out later.Proposition 1.
Consider that the BS has limited cache capacity and letη0 denote the optimal cache size that maximizes EE. Then (30)η0=QFfWQeM,where WQeM is a Lambert-W function satisfying fWx=Wxexp⁡Wx.Proof.
(i)
We first derive the derivative ofη in (28); we obtain (31)dEEdηη=η0=0⟹Qη0Ff+ln⁡1η0Ff-M=0,Q=R¯1R¯2/R¯1-R¯2ω21-Ps+Pt+Pcω1F,M=R¯2R¯1-R¯2ln⁡Ff-1, whereM and Q are considered as constants. (ii)
Letf(η)=Q/ηFf+ln⁡1/ηFf-M, and the derivative of η in the equation is (32)f′η=-M+ηFfFfη2. (iii)
Sof′η<0 when η>0; then fη decreases with the increase of the value η. When η=η0, the equation fη=0; then we can get (33)fη<0,ifη>η0,fη>0,ifη<η0. (iv)
Sincef(η) decreases with η, we can obtain that EE monotonically increases when η<η0 and EE monotonically decreases when η>η0. Through the above descriptions, we find that the EE can be maximized when η=η0.


## 5. Numerical and Simulation Results

In this section, we evaluate the energy efficiency of downlink networks with caching at BS. Meanwhile, we investigate the impacts of some system parameters on throughput and energy efficiency, for example, interference levelβ, the ratio of user density to BS density λu/Nb, and backhaul capacity Cbh.In the work, the distribution of BS and user depends on HPPP in the whole downlink network, and the user density is set to beλu=35. In order to facilitate calculation, we record the parameters in Table 1 that will be used throughout the numerical analysis.Table 1
Numerical calculation parameter settings.
Parameters Value Number of BSs:Nb 36. Round radius of cells:D 40. Number of antennas:Nt 4. The transmission bandwidth:B 21 MHz. The noise power:σ2 −90. The path-loss model:α 1. The size of each content:F 30 MB The size of catalog:Ff 1 0 5 ρ,P,P1,P2 15.2,22,3.95,10.3 The power coefficient of caching:ω1 6.25 × 1 0 - 7 The power coefficient of backhaul:ω2 5.35 × 1 0 - 5


### 5.1. Caching and Backhaul Power Consumption

In this section, we validate whether both the energy consumption for content update and the power consumption for transmit content influence the total power consumption, where the transmission power involves the power consumed by the core network transmitting the content to the BS through backhaul link.(i)
Caching power consumption: in intervalT, we assume that x percent of the cached dynamic contents are updated in BSs. According to (25), the percentage of caching energy consumption with respect to the average total energy consumption at all BSs during T can be expressed as (34)Eca=xFηFfω1TP¯t,wherexFηFfω1 is the caching energy consumed for updating contents. In Section 3, we consider that many contents are dynamic changes at lifetime T, and the popularity of many contents changes slowly. We set up several sets of data in Table 2 to validate the assumption that the cache energy consumption for content updates can not be ignored when the content popularity changes slowly. According to the result of Eca in the table, we find that Eca>3%. Then, the caching energy consumption for content update can not be ignored when content popularity is dynamic changes in lifetime T.(ii)
Backhaul power consumption: we assume thatx percent of the requested contents are transmitted over the backhaul link; the percentage of backhaul power consumption to the total power consumption in interval T is (35)Ebh=-x1-Psln⁡η/ln⁡Ffω2R¯2TP¯t.According to the above parameters, we can obtainEbh>10%; then we must consider the backhaul power consumption for the total power consumption in all BS.Table 2
Cache and backhaul power consumption.
x T η E c a Ebh 10% 12 0.01 3.5% 10.5% 15% 12 0.05 4.2% 10.9% 20% 12 0.05 4.4% 12% 20% 12 0.1 4.9% 11.6%Based on the above analysis, we have a knowledge that the power consumption of cache and backhaul can not be ignored in the total power consumption when analyzing EE.


### 5.2. Impact of Parameters on Throughput

We compare the simulation results of the average throughput per cell with the numerical results obtained from (16) versus Cbh in Figure 4. We can see that the throughput increases with increasing backhaul capacity. Moreover, the throughput increases with Cbh more sharply when β is small.Figure 4
Average throughput versus backhaul capacity.To show the impact of different cache size on the average throughput per cell, we provide the numerical results derived from (16) versus η in Figure 5. It can be seen when the interference level is the same and the throughput of the downlink increases with the cache capacity. Moreover, we can also find when β is small, the average throughput increases with η more sharply. This suggests that the throughput can be boosted more efficiently by caching at the BS if the ICI level can be reduced.Figure 5
Average throughput versus cache capacity.


### 5.3. Impact of Parameters on EE

According to the derivation of the optimal cache capacity for EE maximization in Section3, we derive numerical results of EE with both backhaul capacity and cache capacity. As shown in Figure 6, when only a small portion of contents are cached at each BS, the EE firstly increases rapidly with the increasing backhaul capacity (i.e., η=0.001); then the performance gain approaches a constant. This is due to the fact that though the throughput is increased and the backhaul power consumption is reduced by caching, ICI limits the final throughput.Figure 6
EE versus backhaul capacity.In Figure7, we show the numerical results of EE obtained from (28) versus the normalized cache capacity with different ratios of BS density to user density. It can be seen that the EE firstly increase with η quickly and then decreases gradually. This suggests that there is an optimal normalized cache capacity that maximizes EE. Moreover, with the same cache capacity, EE increases with λb/λu.Figure 7
EE versus cache capacityη.In Section3, we propose a distributed caching strategy to cache different contents at four adjacent BSs. Figure 8 shows the simulation results of EE for nondistributed caching strategy, distributed caching strategy which caches different contents at two adjacent base stations, and distributed caching strategy which caches different contents at four neighbouring BSs. We can see that distributed caching can achieve higher EE as shown in the figure. This is because when each BS uses a distributed cache strategy to cache Nc contents, each user can access 4Nc cache contents; that is, the cache-hit rate is increased. No matter what kind of cache policy is adopted, energy efficiency firstly increases and then decreases, which indicates the existence of an optimal cache capacity to maximize energy efficiency.Figure 8
Impact of user with distributed caching.


## 5.1. Caching and Backhaul Power Consumption

In this section, we validate whether both the energy consumption for content update and the power consumption for transmit content influence the total power consumption, where the transmission power involves the power consumed by the core network transmitting the content to the BS through backhaul link.(i)
Caching power consumption: in intervalT, we assume that x percent of the cached dynamic contents are updated in BSs. According to (25), the percentage of caching energy consumption with respect to the average total energy consumption at all BSs during T can be expressed as (34)Eca=xFηFfω1TP¯t,wherexFηFfω1 is the caching energy consumed for updating contents. In Section 3, we consider that many contents are dynamic changes at lifetime T, and the popularity of many contents changes slowly. We set up several sets of data in Table 2 to validate the assumption that the cache energy consumption for content updates can not be ignored when the content popularity changes slowly. According to the result of Eca in the table, we find that Eca>3%. Then, the caching energy consumption for content update can not be ignored when content popularity is dynamic changes in lifetime T.(ii)
Backhaul power consumption: we assume thatx percent of the requested contents are transmitted over the backhaul link; the percentage of backhaul power consumption to the total power consumption in interval T is (35)Ebh=-x1-Psln⁡η/ln⁡Ffω2R¯2TP¯t.According to the above parameters, we can obtainEbh>10%; then we must consider the backhaul power consumption for the total power consumption in all BS.Table 2
Cache and backhaul power consumption.
x T η E c a Ebh 10% 12 0.01 3.5% 10.5% 15% 12 0.05 4.2% 10.9% 20% 12 0.05 4.4% 12% 20% 12 0.1 4.9% 11.6%Based on the above analysis, we have a knowledge that the power consumption of cache and backhaul can not be ignored in the total power consumption when analyzing EE.


## 5.2. Impact of Parameters on Throughput

We compare the simulation results of the average throughput per cell with the numerical results obtained from (16) versus Cbh in Figure 4. We can see that the throughput increases with increasing backhaul capacity. Moreover, the throughput increases with Cbh more sharply when β is small.Figure 4
Average throughput versus backhaul capacity.To show the impact of different cache size on the average throughput per cell, we provide the numerical results derived from (16) versus η in Figure 5. It can be seen when the interference level is the same and the throughput of the downlink increases with the cache capacity. Moreover, we can also find when β is small, the average throughput increases with η more sharply. This suggests that the throughput can be boosted more efficiently by caching at the BS if the ICI level can be reduced.Figure 5
Average throughput versus cache capacity.


## 5.3. Impact of Parameters on EE

According to the derivation of the optimal cache capacity for EE maximization in Section3, we derive numerical results of EE with both backhaul capacity and cache capacity. As shown in Figure 6, when only a small portion of contents are cached at each BS, the EE firstly increases rapidly with the increasing backhaul capacity (i.e., η=0.001); then the performance gain approaches a constant. This is due to the fact that though the throughput is increased and the backhaul power consumption is reduced by caching, ICI limits the final throughput.Figure 6
EE versus backhaul capacity.In Figure7, we show the numerical results of EE obtained from (28) versus the normalized cache capacity with different ratios of BS density to user density. It can be seen that the EE firstly increase with η quickly and then decreases gradually. This suggests that there is an optimal normalized cache capacity that maximizes EE. Moreover, with the same cache capacity, EE increases with λb/λu.Figure 7
EE versus cache capacityη.In Section3, we propose a distributed caching strategy to cache different contents at four adjacent BSs. Figure 8 shows the simulation results of EE for nondistributed caching strategy, distributed caching strategy which caches different contents at two adjacent base stations, and distributed caching strategy which caches different contents at four neighbouring BSs. We can see that distributed caching can achieve higher EE as shown in the figure. This is because when each BS uses a distributed cache strategy to cache Nc contents, each user can access 4Nc cache contents; that is, the cache-hit rate is increased. No matter what kind of cache policy is adopted, energy efficiency firstly increases and then decreases, which indicates the existence of an optimal cache capacity to maximize energy efficiency.Figure 8
Impact of user with distributed caching.


## 6. Conclusion

In this paper, we have investigated the energy efficient caching problem in backhaul-aware cellular networks jointly considering the cache-hit rate and the optimal cache. Considering that the content popularity is dynamic, we have proposed a policy with “age” threshold based on the shot noise model to timely estimate content popularity. Then, we have further proposed a distributed caching policy to improve the hit rate. Finally, based on the analysis of the tradeoff between energy efficiency and cache capacity, we have formulated an optimization problem and have derived a closed-form solution of optimal cache capacity for maximizing the EE. The numerical results show that the EE can be improved with appropriate choice of cache capacity. Meanwhile, compared with the nondistributed cache scheme, the distributed cache scheme can further improve the energy efficiency by increasing the cache-hit rate.



---

*Source: 7532049-2018-04-08.xml*