# Analysis of User Content Retrieval Delay Based on the Matern Hard-Core Point Process of Type II

Authors: Shuyuan Zhao; Jihong Zhao; Hua Qu; Gongye Ren
Journal: Wireless Communications and Mobile Computing
 (2018)

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## Abstract

The content retrieval delay is an important performance metric for enhancing user experience in wireless networks. In this paper, by modeling the locations of the base stations (BSs) as the Matern hard-core point process of type II (MHP), we analyze the content retrieval delay for a typical cache-enabled device in wireless networks under the most popular content policy. Since it is intractable to get the size distribution of a Voronoi cell in the MHP model, we propose an approximate formula based on the empirical result in the Poisson point process and derive the cellular load which denotes the number of the user devices connected to a randomly chosen BS. Since the probability generating functional for MHP does not exist, we also propose approximate methods for the coverage probability of the MHP model. At last, we derive the cumulative distribution function of the content retrieval delay. Simulation results validate the accuracy of our analytical conclusions for user content retrieval delay.


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## Body

## 1. Introduction

Over the past several years, the global mobile data traffic grows rapidly with the proliferation of smart terminals and is expected to increase to 30.6 exabytes per month by 2020 [1]. With the increasing demand for higher capacity and lower latency in wireless networks, the fifth generation (5G) wireless network has been proposed to improve the capacity of current networks by a factor of 1000 [2]. In recent years, studies show that a large portion of mobile data traffic is generated by repeated request of numerous popular contents [3]. Besides, the storage capacity of memory devices grows rapidly at a relatively low cost. Under these conditions, caching some popular contents at the network edge has been considered as a promising technique to alleviate the network traffic load and improve the quality of user service [4].Key notions, challenges, and research topics of wireless caching were provided in [5]. Two methods of caching at the network edge have been studied in the literature: caching at helper nodes [6] and caching at the mobile devices (MDs) [7–9]. In this paper, we focus on the device caching solely and assume the contents are cached in the devices during off-peak time (e.g., at night). Effective proactive caching policies can largely alleviate the burden of the base stations (BSs) during the peak hours, yielding high offloading gain [9–11].The content retrieval delay is an important performance metric for enhancing user experience, and in order to give a detailed performance analysis, we consider a popular-based caching strategy called the most-popular-content (MPC) policy [12, 13] in this paper. The spatial distribution of BSs and mobile devices strongly affects the performance analysis of the wireless networks. In recent years, stochastic geometry has been utilized to model the wireless networks, and the Poisson point process (PPP) has been shown to be a tractable system model for its useful mathematical properties [14]. Most works [15–17] focus on the caching problems in a scenario where the BSs or the user devices are distributed according to a homogeneous PPP. However, the BSs in the PPP model are assumed to be distributed independently, which deviates from the fact that the locations of the BSs have strong correlation. Therefore, in this paper, we propose to use the Matern hard-core point process of type II (MHP) [18–21] to model the distribution of BSs, which is a more general and realistic model. Since the probability generating functional (PGFL) for MHP does not exist, we focus on the approximation for the cellular load and coverage probability in carefully chosen cases.The contributions of this paper are summarized as follows:(i)
We analyze the content retrieval delay for a typical cache-enabled device with the MPC policy in the scenario where the BSs are distributed as the MHP model.(ii)
We propose a reasonable approximation for the probability density function of the size of a typical Voronoi cell in the MHP model and confirm its effectiveness through the simulation of the cellular load.(iii)
We evaluate the performance of two proposed methods for approximating the coverage probability in special cases and select the better one to calculate the delay.The rest of this paper is organized as follows. In Section2, we present our network and caching model and derive the cumulative distribution function of the content retrieval delay and then propose the approximation for the cellular load and coverage probability in the MHP model. In Section 3, we compare analytical results with the Monte Carlo simulation results. Finally, Section 4 provides our conclusions.


## 2. Analysis of User Content Retrieval Delay

### 2.1. Network Model

Consider a downlink cellular network consisting of BSs and MDs, and all BSs operate in full loaded state. We use MHP to model the actual spatial distribution of BSs. The MHPΦm of BSs is generated from a PPP Φp with density λp as follows: associate each point with a random mark m~U(0,1), and then delete a point if there exist other points within the distance d with a smaller mark. The density of Φm is given by [19](1)λm=1-exp⁡-λpπd2πd2.Besides, we assume the MDs in service are scattered randomly based on a stationary PPP ΦD with density λD, and each MD is connected to the nearest BS. So the cell area of each BS forms a Voronoi tessellation. Also, we consider the standard path loss propagation model with path loss exponent α and assume that the small scale fading follows Rayleigh fading.


### 2.2. Caching Model

We assume that each MD requests only one file from a library ofL files of equal size S bits at a time. The files are indexed according to their popularity, ranking from the most popular (the 1st file) to the least popular (the Lth file). The request probability of the ith popular file follows the Zipf distribution [22]:(2)Pi=i-σ∑j=1Lj-σ,where σ characterizes the skewness of content popularity distribution.According to the MPC policy, for simplicity, we assume that each MD has the same cache capacity and cachesK (K≪L) most popular files, so the content hit probability denoted by Pmpc is(3)Pmpc=∑i=1Ki-σ∑j=1Lj-σ.So the user retrieval delay, denoted by Td, happens when a content request is retrieved from the connected BS (Td=0 means cache hit). Hence, the cumulative distribution function (CDF) of Td can be calculated as(4)PTd<t=PSRd<t=1-PmpcPRd>St+Pmpc.Considering the effect of the TDMA scheduling, the data rate Rd is given by [23](5)Rd=BNlog⁡1+SINRd,where N denotes the cellular load which means the total number of MDs served by the associated BS at the same time. With the assumption that all BSs operate in full loaded state, the complementary cumulative distribution function (CCDF) of Rd can be derived by the distribution of N and SINRd as follows:(6)PRd>St=PSINRd>2SN/tB-1=∑n=1∞PN=n∣N≥1PSINRd>2SN/tB-1∣N=n,where P(N=n) denotes the probability mass function of the cellular load N and PSINRd>2SN/tB-1 denotes the coverage probability of the MHP model. They will be derived in the following sections.


### 2.3. Probability of the Cellular Load

The probability mass function of the cellular loadN in a typical Voronoi cell can be given by the law of total probability as follows:(7)PN=n=∫0∞PN=n∣X=x·fXxdx,where X denotes the size of a typical Voronoi cell and fX(x) denotes the probability density function of it. However, we cannot determine fX(x) in the MHP model directly, a problem that has not been studied in the literature before. So firstly we consider the situation that the BSs are distributed as a homogeneous PPP Φp with density λp. The normalized size distribution function of Poisson Voronoi cells is accurately approximated in [24] by the Monte Carlo method:(8)fpx=3.53.5Γ3.5x2.5e-3.5x,where x denotes the random variable of a cell size normalized by the value 1/λp.Consider an MHPΦm of density λm with a minimum distance d generated from the PPP Φp with density λp as mentioned above. We can get the minimum cell size in the situation that the distance between the target BS and all neighbor BSs is d, as shown in Figure 1: Smin=3d2/2. When d is small enough, there are very few cases that the distance between two BSs is less than d. So ignoring the invalid situations x<Smin, fm(x) is close to fp(x) when d is small enough. Compared with fp(x), we can give an approximation for the normalized size distribution of a Voronoi cell of the MHP model, denoted by fm(x):(9)fmx≈0,x∈0,Sminfpx1-∫0Sminλmfpxdx,x∈Smin,+∞.Notice that x is normalized by the value 1/λm. When x∈[Smin,+∞), fm(x) can be simplified as follows:(10)fmx≈fpx1-∫0Sminλmfpxdx=3.53.5x2.5e-3.5x/Γ3.51-∫0Sminλm3.53.5x2.5e-3.5x/Γ3.5dx=a3.53.5·x2.5e-3.5xΓ3.5-γ3.5,7Sminλm/2,where (a) follows: ∫0uxv-1e-μxdx=μ-vγ(v,μu) [25], and γ(v,t)=∫0txv-1e-xdx denotes the lower incomplete gamma function. So the probability mass function of the cellular load in the MHP model can be derived in the following theorem, according to (7) and (10).Figure 1
The minimum size of a typical Voronoi cell in the MHP model with a minimum distanced.Theorem 1.
Whend is small enough, the probability mass function of the cellular load in the MHP model can be approximated by(11)PN=n=3.53.5λD/λmnλD/λm+3.5-n+3.5Γn+3.5,SminλD/λm+3.5λmΓ3.5-γ3.5,7Sminλm/2n!,and(12)PN=n∣N≥1=PN=n1-PN=0.Proof.
See AppendixA.Considering general cases, whend is a random variable, compared fm(x) with fp(x) which have the same density λ, we can find several useful characteristics of fm(x) as follows obviously:(1)
Whenx→Smin,  fm(x)→0.(2)
Whend→0,  fm(x)→fp(x).(3)
For a specific value ofx,f p ( x ) = x n o   B S   c l o s e r   t h a n   d + x { a t   l e a s t   o n e   B S   c l o s e r   t h a n   d } / a l l   s i t u a t i o n s and fm(x)=xnoBScloserthand/allsituationsnoBS  closerthand, where xno BS closer thand denotes the situations that a cell has a size of x and there is no BS located closer than d, and xat least one BS closer thand denotes the situations that a cell has a size of x and there is at least one BS located closer than d. For all x, the situations x{atleastoneBScloserthand} can be considered as a relatively fixed value, so we can get that fm(x)/fp(x) increases with the increase of fp(x).Based on the three characteristics mentioned above, we consider a conjecture offm(x) as follows:(13)fmx=C×3.53.5Γ3.5x2.5e-3.5x1+γ×x-Sx,where C denotes the bias and γ denotes a predefined constant depending on the value of λm and d. Since ∫Smin∞fm(x)dx=1, we can get(14)C=Γ3.51+γ3.53.51+γ×3.5+3.5γ-3.5+2.5γΓ3.5+2.5γ,3.5Smin1+γ-Smin3.5+3.5γ-2.5+2.5γΓ2.5+2.5γ,3.5Smin1+γ.Similar to (11), the probability mass function of the cellular load in the MHP model can be derived in the following theorem.Theorem 2.
Whend is a random variable, the probability mass function of the cellular load in the MHP model is(15)PN=n=C3.53.51+γλD/λmnΓ3.51+γn!×λDλm+3.51+γ-n+3.5+2.5γΓn+3.5+2.5γ,λDλm+3.51+γSmin-SminλDλm+3.51+γ-n+2.5+2.5γΓn+2.5+2.5γ,λDλm+3.51+γSmin,where γ is a predefined constant depending on the value of λm and d.Proof.
See AppendixB.In Section3, we give detailed numerical simulation about the conjecture and can get that if γ is appropriately predefined, the analysis (15) suits the simulation result well.


### 2.4. Approximation of the Coverage Probability

The proposed interference model is illustrated in Figure2. Since MHP is a stationary point process, we assume that the attached BS is located at the origin without loss of generality. r denotes the distance between MD and the attached BS, v denotes the distance between the attached BS and any interfering BS, and li denotes the distance between MD and any interfering BS i,  li=v2+r2-2rvcos⁡(θ-φ). We assume that the small scale fading follows Rayleigh fading, h~exp⁡(μ), and the noise is neglected, σ2=0. The MD is in coverage when its SINRd is larger than some threshold T. So we can get the following lemma.Figure 2
The interference model, where the attached BS is located at the origin.Lemma 3.
The coverage probability of a typical MD can be given by(16)PSINRd>T=∫02π∫0∞fr2πEΦm!o∏i∈Φm11+Trαli-αdrdφ,where f(r) is the probability density function of the distance between a typical MD and its associated BS, which can be approximated as follows [26]:(17)fr=2πλmre-πλmr2.Proof.
See AppendixC.Since the PGFL of MHP is unknown, it is difficult to get the exact value ofEΦm!o∏i∈Φm1/1+Trαli-α. So in this paper, we consider several reasonable approximations for the coverage probability of the MHP model in different situations. We first give the second-order product density ρm2(v) of the MHP [19] which will be used in the following approximations:(18)ρm2v=λm2,ifv≥2d2Vv1-exp⁡-λpπd2-2πd21-exp⁡-λpVvπd2VvVv-πd2,if2d>v≥d0,ifv<d,(19)Vv=2πd2-2d2cos-1⁡v2d+vd2-v24.Then we analyze the coverage probability as follows:(1) Considering general cases, whend is a random nonnegative variable, we consider two methods to approximate the coverage probability, as shown in the following two propositions.Proposition 4.
A rough approximation for the coverage probability of a typical MD using the second-order Taylor expansion can be given by(20)P1SINRd>T≈∫02π∫0∞λmre-πλmr2ωdrdφ,where(21)ω≈1-λm-1∫R2ρm2vg1r,v,θ,φdx+12λm-1∫R2ρm2vg12r,v,θ,φdx=1-λm-1∫02π∫max⁡d,2rcos⁡θ-φ∞ρm2vg1r,v,θ,φvdvdθ+12λm-1∫02π∫max⁡d,2rcos⁡θ-φ∞ρm2vg12r,v,θ,φvdvdθ,(22)g1r,v,θ,φ=ln⁡1+Tr2v2+r2-2rvcos⁡θ-φα/2.Proof.
See AppendixD.Proposition 5.
A lower bound for the coverage probability using Jensen’s inequality is given by(23)P1SINRd>T≥∫02π∫0∞λmre-πλmr2e-ηdrdφ,where(24)η=λm-1∫02π∫max⁡d,2rcos⁡θ-φ∞ρm2vg1r,v,θ,φvdvdθ.Proof.
See AppendixE.(2) Whend is small enough, according to Proposition 1 in [26], a lower bound is given by(25)P2SINRd>T≥∫02π∫0∞λmre-πλmr2e-ςdrdφ,where(26)ς=λm-1∫02π∫max⁡d,2rcos⁡θ-φ∞ρm2vg2r,v,θ,φvdvdθ,(27)g2r,v,θ,φ=1+T-1r2v2+r2-2rvcos⁡θ-φ-α/2-1.According to the analysis proposed above, we will evaluate the performance of the approximations for the cellular load and coverage probability with numerical results in different situations, mainly including Theorems 1, 2, Propositions 4, 5, and lower bound (25). Also, the content retrieval delay will be simulated according to (4) and (6).


## 2.1. Network Model

Consider a downlink cellular network consisting of BSs and MDs, and all BSs operate in full loaded state. We use MHP to model the actual spatial distribution of BSs. The MHPΦm of BSs is generated from a PPP Φp with density λp as follows: associate each point with a random mark m~U(0,1), and then delete a point if there exist other points within the distance d with a smaller mark. The density of Φm is given by [19](1)λm=1-exp⁡-λpπd2πd2.Besides, we assume the MDs in service are scattered randomly based on a stationary PPP ΦD with density λD, and each MD is connected to the nearest BS. So the cell area of each BS forms a Voronoi tessellation. Also, we consider the standard path loss propagation model with path loss exponent α and assume that the small scale fading follows Rayleigh fading.


## 2.2. Caching Model

We assume that each MD requests only one file from a library ofL files of equal size S bits at a time. The files are indexed according to their popularity, ranking from the most popular (the 1st file) to the least popular (the Lth file). The request probability of the ith popular file follows the Zipf distribution [22]:(2)Pi=i-σ∑j=1Lj-σ,where σ characterizes the skewness of content popularity distribution.According to the MPC policy, for simplicity, we assume that each MD has the same cache capacity and cachesK (K≪L) most popular files, so the content hit probability denoted by Pmpc is(3)Pmpc=∑i=1Ki-σ∑j=1Lj-σ.So the user retrieval delay, denoted by Td, happens when a content request is retrieved from the connected BS (Td=0 means cache hit). Hence, the cumulative distribution function (CDF) of Td can be calculated as(4)PTd<t=PSRd<t=1-PmpcPRd>St+Pmpc.Considering the effect of the TDMA scheduling, the data rate Rd is given by [23](5)Rd=BNlog⁡1+SINRd,where N denotes the cellular load which means the total number of MDs served by the associated BS at the same time. With the assumption that all BSs operate in full loaded state, the complementary cumulative distribution function (CCDF) of Rd can be derived by the distribution of N and SINRd as follows:(6)PRd>St=PSINRd>2SN/tB-1=∑n=1∞PN=n∣N≥1PSINRd>2SN/tB-1∣N=n,where P(N=n) denotes the probability mass function of the cellular load N and PSINRd>2SN/tB-1 denotes the coverage probability of the MHP model. They will be derived in the following sections.


## 2.3. Probability of the Cellular Load

The probability mass function of the cellular loadN in a typical Voronoi cell can be given by the law of total probability as follows:(7)PN=n=∫0∞PN=n∣X=x·fXxdx,where X denotes the size of a typical Voronoi cell and fX(x) denotes the probability density function of it. However, we cannot determine fX(x) in the MHP model directly, a problem that has not been studied in the literature before. So firstly we consider the situation that the BSs are distributed as a homogeneous PPP Φp with density λp. The normalized size distribution function of Poisson Voronoi cells is accurately approximated in [24] by the Monte Carlo method:(8)fpx=3.53.5Γ3.5x2.5e-3.5x,where x denotes the random variable of a cell size normalized by the value 1/λp.Consider an MHPΦm of density λm with a minimum distance d generated from the PPP Φp with density λp as mentioned above. We can get the minimum cell size in the situation that the distance between the target BS and all neighbor BSs is d, as shown in Figure 1: Smin=3d2/2. When d is small enough, there are very few cases that the distance between two BSs is less than d. So ignoring the invalid situations x<Smin, fm(x) is close to fp(x) when d is small enough. Compared with fp(x), we can give an approximation for the normalized size distribution of a Voronoi cell of the MHP model, denoted by fm(x):(9)fmx≈0,x∈0,Sminfpx1-∫0Sminλmfpxdx,x∈Smin,+∞.Notice that x is normalized by the value 1/λm. When x∈[Smin,+∞), fm(x) can be simplified as follows:(10)fmx≈fpx1-∫0Sminλmfpxdx=3.53.5x2.5e-3.5x/Γ3.51-∫0Sminλm3.53.5x2.5e-3.5x/Γ3.5dx=a3.53.5·x2.5e-3.5xΓ3.5-γ3.5,7Sminλm/2,where (a) follows: ∫0uxv-1e-μxdx=μ-vγ(v,μu) [25], and γ(v,t)=∫0txv-1e-xdx denotes the lower incomplete gamma function. So the probability mass function of the cellular load in the MHP model can be derived in the following theorem, according to (7) and (10).Figure 1
The minimum size of a typical Voronoi cell in the MHP model with a minimum distanced.Theorem 1.
Whend is small enough, the probability mass function of the cellular load in the MHP model can be approximated by(11)PN=n=3.53.5λD/λmnλD/λm+3.5-n+3.5Γn+3.5,SminλD/λm+3.5λmΓ3.5-γ3.5,7Sminλm/2n!,and(12)PN=n∣N≥1=PN=n1-PN=0.Proof.
See AppendixA.Considering general cases, whend is a random variable, compared fm(x) with fp(x) which have the same density λ, we can find several useful characteristics of fm(x) as follows obviously:(1)
Whenx→Smin,  fm(x)→0.(2)
Whend→0,  fm(x)→fp(x).(3)
For a specific value ofx,f p ( x ) = x n o   B S   c l o s e r   t h a n   d + x { a t   l e a s t   o n e   B S   c l o s e r   t h a n   d } / a l l   s i t u a t i o n s and fm(x)=xnoBScloserthand/allsituationsnoBS  closerthand, where xno BS closer thand denotes the situations that a cell has a size of x and there is no BS located closer than d, and xat least one BS closer thand denotes the situations that a cell has a size of x and there is at least one BS located closer than d. For all x, the situations x{atleastoneBScloserthand} can be considered as a relatively fixed value, so we can get that fm(x)/fp(x) increases with the increase of fp(x).Based on the three characteristics mentioned above, we consider a conjecture offm(x) as follows:(13)fmx=C×3.53.5Γ3.5x2.5e-3.5x1+γ×x-Sx,where C denotes the bias and γ denotes a predefined constant depending on the value of λm and d. Since ∫Smin∞fm(x)dx=1, we can get(14)C=Γ3.51+γ3.53.51+γ×3.5+3.5γ-3.5+2.5γΓ3.5+2.5γ,3.5Smin1+γ-Smin3.5+3.5γ-2.5+2.5γΓ2.5+2.5γ,3.5Smin1+γ.Similar to (11), the probability mass function of the cellular load in the MHP model can be derived in the following theorem.Theorem 2.
Whend is a random variable, the probability mass function of the cellular load in the MHP model is(15)PN=n=C3.53.51+γλD/λmnΓ3.51+γn!×λDλm+3.51+γ-n+3.5+2.5γΓn+3.5+2.5γ,λDλm+3.51+γSmin-SminλDλm+3.51+γ-n+2.5+2.5γΓn+2.5+2.5γ,λDλm+3.51+γSmin,where γ is a predefined constant depending on the value of λm and d.Proof.
See AppendixB.In Section3, we give detailed numerical simulation about the conjecture and can get that if γ is appropriately predefined, the analysis (15) suits the simulation result well.


## 2.4. Approximation of the Coverage Probability

The proposed interference model is illustrated in Figure2. Since MHP is a stationary point process, we assume that the attached BS is located at the origin without loss of generality. r denotes the distance between MD and the attached BS, v denotes the distance between the attached BS and any interfering BS, and li denotes the distance between MD and any interfering BS i,  li=v2+r2-2rvcos⁡(θ-φ). We assume that the small scale fading follows Rayleigh fading, h~exp⁡(μ), and the noise is neglected, σ2=0. The MD is in coverage when its SINRd is larger than some threshold T. So we can get the following lemma.Figure 2
The interference model, where the attached BS is located at the origin.Lemma 3.
The coverage probability of a typical MD can be given by(16)PSINRd>T=∫02π∫0∞fr2πEΦm!o∏i∈Φm11+Trαli-αdrdφ,where f(r) is the probability density function of the distance between a typical MD and its associated BS, which can be approximated as follows [26]:(17)fr=2πλmre-πλmr2.Proof.
See AppendixC.Since the PGFL of MHP is unknown, it is difficult to get the exact value ofEΦm!o∏i∈Φm1/1+Trαli-α. So in this paper, we consider several reasonable approximations for the coverage probability of the MHP model in different situations. We first give the second-order product density ρm2(v) of the MHP [19] which will be used in the following approximations:(18)ρm2v=λm2,ifv≥2d2Vv1-exp⁡-λpπd2-2πd21-exp⁡-λpVvπd2VvVv-πd2,if2d>v≥d0,ifv<d,(19)Vv=2πd2-2d2cos-1⁡v2d+vd2-v24.Then we analyze the coverage probability as follows:(1) Considering general cases, whend is a random nonnegative variable, we consider two methods to approximate the coverage probability, as shown in the following two propositions.Proposition 4.
A rough approximation for the coverage probability of a typical MD using the second-order Taylor expansion can be given by(20)P1SINRd>T≈∫02π∫0∞λmre-πλmr2ωdrdφ,where(21)ω≈1-λm-1∫R2ρm2vg1r,v,θ,φdx+12λm-1∫R2ρm2vg12r,v,θ,φdx=1-λm-1∫02π∫max⁡d,2rcos⁡θ-φ∞ρm2vg1r,v,θ,φvdvdθ+12λm-1∫02π∫max⁡d,2rcos⁡θ-φ∞ρm2vg12r,v,θ,φvdvdθ,(22)g1r,v,θ,φ=ln⁡1+Tr2v2+r2-2rvcos⁡θ-φα/2.Proof.
See AppendixD.Proposition 5.
A lower bound for the coverage probability using Jensen’s inequality is given by(23)P1SINRd>T≥∫02π∫0∞λmre-πλmr2e-ηdrdφ,where(24)η=λm-1∫02π∫max⁡d,2rcos⁡θ-φ∞ρm2vg1r,v,θ,φvdvdθ.Proof.
See AppendixE.(2) Whend is small enough, according to Proposition 1 in [26], a lower bound is given by(25)P2SINRd>T≥∫02π∫0∞λmre-πλmr2e-ςdrdφ,where(26)ς=λm-1∫02π∫max⁡d,2rcos⁡θ-φ∞ρm2vg2r,v,θ,φvdvdθ,(27)g2r,v,θ,φ=1+T-1r2v2+r2-2rvcos⁡θ-φ-α/2-1.According to the analysis proposed above, we will evaluate the performance of the approximations for the cellular load and coverage probability with numerical results in different situations, mainly including Theorems 1, 2, Propositions 4, 5, and lower bound (25). Also, the content retrieval delay will be simulated according to (4) and (6).


## 3. Numerical Results

In this section, we evaluate the performance of the proposed approximations for the cellular load and the coverage probability both in general cases and in special cases (d is small enough) and validate the accuracy of the numerical results for user content retrieval delay with the MPC policy.


### 3.1. Simulation Setup

The simulation is conducted in a square area with side length 3km. The MHP model of BSs is generated from a PPP model with the densityλp=5×10-6/m2. The density of MDs is λD=5×10-5/m2. The bandwidth of all BSs is 20 MHz, and the path loss parameter is 4. The size of the file library is 1000, and the size of each file is 100 Mbits. In the MPC policy, each MD can cache 50 files.


### 3.2. The Cellular Load Simulation

According to (10) and (11), we get the analytical results of the cellular load in the MHP model as shown in Figure 3. It is impressive that the numerical results closely match the simulation results when d is small enough as shown in Figures 3(a) and 3(b), and it also means that the approximate size distribution of the MHP model is effective. But in general cases, when d is 150, 200m or larger, it seems that the approximation of (10) does not work well as shown in Figures 3(c) and 3(d).Figure 3
Numerical results of the cellular load with (11), compared with the simulation results. (a) d=50m, (b) d=100m, (c) d=150m, and (d) d=200m.
(a)
d = 50 m, cellular load (b)
d = 100 m, cellular load (c)
d = 150 m, cellular load (d)
d = 200 m, cellular loadIn Figure4, we conduct the simulation of the conjecture for the size distribution and cellular load in general cases according to (13) and (15). We can easily find that γ has notable positive correlation with d and when d=200,300,400m, the numerical results agree well with the simulation results with the parameter γ=0.4,0.8,1.2, respectively. This is because with the increase of d, the situations x{atleastoneBScloserthand} become greater, which leads to the greater change of fm(x)/fp(x), and the value of γ should be predefined larger then. By comparing the numerical with simulation results, the conjecture (13) proposed above will work well when γ is carefully predefined according to d.Figure 4
Numerical results of the cellular load with conjecture (15), γ = 0.4, 0.8, 1.2, compared with (11) and the simulation results. (a) d=100m, (b) d=200m, (c) d=300m, and (d) d=400m.
(a)
d = 100 m, cellular load (b)
d = 200 m, cellular load (c)
d = 300 m, cellular load (d)
d = 400 m, cellular load


### 3.3. The Coverage Probability Simulation

In Figure5, we evaluate the performance of Propositions 4 and 5 and lower bound (25) with the MHP model of d=50m and d=300m, respectively. Although the lower bound (25) is not very close to the simulation results when d=300m, it still performs much better than (20) and (23). Therefore, we will always use the lower bound (25) for the coverage probability to conduct the simulation of the user content retrieval delay.Figure 5
Coverage probability based on Propositions4 and 5 and lower bound (25), compared with the simulation results, d=50m and d=300m.
(a)
d = 50 m, SINR threshold (dB) (b)
d = 300 m, SINR threshold (dB)


### 3.4. User Content Retrieval Delay Simulation

In Figure6, we use Theorem 1 and (25) together to analyze the content retrieval delay when d=50m, while Theorem 2 and (25) are used when d=300m. Figure 6 shows that the analytical results are close to the simulation results, and the CDF of the content retrieval delay increases with the growth of the popularity exponent σ, since MDs can get the requested content from themselves more probably.Figure 6
CDF of the user content retrieval delay,d=50m and d=300m.
(a)
d = 50 m, delay(s) (b)
d = 300 m, delay(s)According to the numerical results above, we can get the following conclusions:(1)
Whend is small enough, Theorem 1 for the cellular load and lower bound (25) for the coverage probability performs well compared with the simulation results.(2)
Whend is a random nonnegative variable, especially when d is large, Theorem 2 for the cellular load with carefully predefined γ performs well. Besides, although lower bound (25) is derived when d is small enough, it still performs better than Propositions 4 and 5.


## 3.1. Simulation Setup

The simulation is conducted in a square area with side length 3km. The MHP model of BSs is generated from a PPP model with the densityλp=5×10-6/m2. The density of MDs is λD=5×10-5/m2. The bandwidth of all BSs is 20 MHz, and the path loss parameter is 4. The size of the file library is 1000, and the size of each file is 100 Mbits. In the MPC policy, each MD can cache 50 files.


## 3.2. The Cellular Load Simulation

According to (10) and (11), we get the analytical results of the cellular load in the MHP model as shown in Figure 3. It is impressive that the numerical results closely match the simulation results when d is small enough as shown in Figures 3(a) and 3(b), and it also means that the approximate size distribution of the MHP model is effective. But in general cases, when d is 150, 200m or larger, it seems that the approximation of (10) does not work well as shown in Figures 3(c) and 3(d).Figure 3
Numerical results of the cellular load with (11), compared with the simulation results. (a) d=50m, (b) d=100m, (c) d=150m, and (d) d=200m.
(a)
d = 50 m, cellular load (b)
d = 100 m, cellular load (c)
d = 150 m, cellular load (d)
d = 200 m, cellular loadIn Figure4, we conduct the simulation of the conjecture for the size distribution and cellular load in general cases according to (13) and (15). We can easily find that γ has notable positive correlation with d and when d=200,300,400m, the numerical results agree well with the simulation results with the parameter γ=0.4,0.8,1.2, respectively. This is because with the increase of d, the situations x{atleastoneBScloserthand} become greater, which leads to the greater change of fm(x)/fp(x), and the value of γ should be predefined larger then. By comparing the numerical with simulation results, the conjecture (13) proposed above will work well when γ is carefully predefined according to d.Figure 4
Numerical results of the cellular load with conjecture (15), γ = 0.4, 0.8, 1.2, compared with (11) and the simulation results. (a) d=100m, (b) d=200m, (c) d=300m, and (d) d=400m.
(a)
d = 100 m, cellular load (b)
d = 200 m, cellular load (c)
d = 300 m, cellular load (d)
d = 400 m, cellular load


## 3.3. The Coverage Probability Simulation

In Figure5, we evaluate the performance of Propositions 4 and 5 and lower bound (25) with the MHP model of d=50m and d=300m, respectively. Although the lower bound (25) is not very close to the simulation results when d=300m, it still performs much better than (20) and (23). Therefore, we will always use the lower bound (25) for the coverage probability to conduct the simulation of the user content retrieval delay.Figure 5
Coverage probability based on Propositions4 and 5 and lower bound (25), compared with the simulation results, d=50m and d=300m.
(a)
d = 50 m, SINR threshold (dB) (b)
d = 300 m, SINR threshold (dB)


## 3.4. User Content Retrieval Delay Simulation

In Figure6, we use Theorem 1 and (25) together to analyze the content retrieval delay when d=50m, while Theorem 2 and (25) are used when d=300m. Figure 6 shows that the analytical results are close to the simulation results, and the CDF of the content retrieval delay increases with the growth of the popularity exponent σ, since MDs can get the requested content from themselves more probably.Figure 6
CDF of the user content retrieval delay,d=50m and d=300m.
(a)
d = 50 m, delay(s) (b)
d = 300 m, delay(s)According to the numerical results above, we can get the following conclusions:(1)
Whend is small enough, Theorem 1 for the cellular load and lower bound (25) for the coverage probability performs well compared with the simulation results.(2)
Whend is a random nonnegative variable, especially when d is large, Theorem 2 for the cellular load with carefully predefined γ performs well. Besides, although lower bound (25) is derived when d is small enough, it still performs better than Propositions 4 and 5.


## 4. Conclusions

We analyze the content retrieval delay of cache-enabled devices in wireless networks, by modeling the distribution of the BSs with the MHP. Due to the intractability of the MHP, we give some approximations of the cellular load and the coverage probability, both in general cases and in special cases for numerical evaluation. The simulations show that our approximations are reasonable and accurate. The approximations of the MHP are quite useful for the analysis in the cache-enabled networks, and we only consider the simplest scenario with the baseline caching strategy to verify the reliability in this paper. As a promising future direction, we will focus on the optimization of the probabilistic caching strategy for maximizing the cache hit probability or the successful delivery probability of the requested content.



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*Source: 2303289-2018-09-04.xml*