Enhancing Performance of Random Caching in Large-Scale Heterogeneous Wireless Networks with Random Discontinuous Transmission

Enhancing Performance of Random Caching in Large-Scale Heterogeneous Wireless Networks with Random Discontinuous Transmission

Wanli Wen, Ying Cui, Fu-Chun Zheng, Shi Jin and Yanxiang Jiang Y. Cui is with the Department of Electronic Engineering, Shanghai Jiao Tong University, China. W. Wen, F.-C. Zheng, S. Jin and Y. Jiang are with the National Mobile Communications Research Laboratory, Southeast University, Nanjing, China. This paper will be presented in part at IEEE WCNC 2018.
Abstract

To make better use of file diversity provided by random caching and improve the successful transmission probability (STP) of a file, we consider retransmissions with random discontinuous transmission (DTX) in a large-scale cache-enabled heterogeneous wireless network (HetNet) employing random caching. We analyze and optimize the STP in two mobility scenarios, i.e., the high mobility scenario and the static scenario. First, in each scenario, by using tools from stochastic geometry, we obtain a closed-form expression for the STP in the general signal-to-interference ratio (SIR) threshold regime. The analysis shows that a larger caching probability corresponds to a higher STP in both scenarios; random DTX can improve the STP in the static scenario and its benefit gradually diminishes when mobility increases. In each scenario, we also derive a closed-form expression for the asymptotic outage probability in the low SIR threshold regime. The asymptotic analysis shows that the diversity gain is jointly affected by random caching and random DTX in both scenarios. Then, in each scenario, we consider the maximization of the STP with respect to the caching probability and the BS activity probability, which is a challenging non-convex optimization problem. In particular, in the high mobility scenario, we obtain a globally optimal solution using interior point method. In the static scenario, we develop a low-complexity iterative algorithm to obtain a stationary point using alternating optimization. Finally, numerical results show that the proposed solutions achieve significant gains over existing baseline schemes and can well adapt to the changes of the system parameters to wisely utilize storage resources and transmission opportunities.

Random caching, retransmission, random discontinuous transmission (DTX), heterogeneous wireless networks, optimization, stochastic geometry.

I Introduction

With the proliferation of smart mobile devices and multimedia services, the global mobile data traffic is expected to increase exponentially in the coming years. However, the majority of such traffic is asynchronously but repeatedly requested by many users at different times and thus a tremendous amount of mobile data traffic have actually been redundantly generated over networks [1]. Motivated by this, caching at base stations (BSs) has been proposed as a promising approach for reducing delay and backhaul load [2]. When the coverage regions of different BSs overlap, a user can fetch the desired file from multiple adjacent BSs, and hence the performance can be increased by caching different files among BSs, i.e., providing file diversity. In [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], the authors analyze the performance of various caching designs in large-scale cache-enabled wireless networks. In particular, in [3] and [4], the authors study the most popular caching design, where each BS only stores the most popular files. As the most popular caching design can not provide any spatial file diversity, it may not yield the optimal network performance. To provide more spatial file diversity, the authors in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16] consider random caching. Specifically, in [5], the authors consider uniform caching where each BS randomly stores a file according to the uniform distribution. In [6], the authors consider i.i.d. caching where each BS stores a file in an i.i.d. manner. In [7, 8, 9, 10, 11, 12, 13, 14, 15, 16], besides analysis, the authors also consider the optimization of random caching to either maximize the cache hit probability [7, 8], the successful offloading probability [9] and the successful transmission probability (STP) [10, 11, 12, 13, 14, 16] or minimize the average caching failure probability [15]. Note that, in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], a user may associate with a relatively farther BS when nearer BSs do not cache the requested file [16]. In this case, the signal is usually weak compared with the interference, and the user may not successfully receive the requested file and benefit from the file diversity offered by random caching.

Increasing the number of transmissions of a file can increase the probability that eventually the file is successfully transmitted, at the cost of delay increase. Enabling retransmissions at BSs is an effectively way to improve the STP for some applications without strict delay requirements, e.g., elastic services. The interferences experienced by a user at different slots are usually correlated as they come from the same set of BSs [17]. In [17, 18, 19, 20, 21], the authors study retransmissions and show that the interference correlation significantly degrades the performance, e.g., the diversity gain [18] or the transmission delay [19, 20, 21]. In [17, 18, 19, 20, 21], the authors adopt random discontinuous transmission (DTX) together with retransmissions to effectively manage such interference correlation, by creating randomness for interferers, and analyze the performance in large-scale wireless networks. Note that, [17, 18, 19, 20, 21] do not consider caching at BSs.

Recently, [22] studies the effect of retransmissions on the performance of random caching, and analyzes and optimizes the STP in a large-scale single tier network. Note that [22] does not consider DTX, and hence the gain of retransmissions is limited due to the strong interference correlation across multiple retransmissions. Therefore, it is still not clear how retransmissions with random DTX can maximally improve the performance of random caching. Heterogeneous wireless networks (HetNets) can further improve the network capacity by deploying small BSs together with traditional macro BSs, to provide better time or frequency reuse. Caching at small BSs can effectively alleviate the backhaul capacity requirement in HetNets. For cache-enabled HetNets, it is also not known how to jointly design random caching and random DTX across different tiers.

In this paper, we would like to address the above issues. We consider a large-scale cache-enabled HetNet. We adopt random caching and a simple retransmission protocol with random DTX to improve the STP, which is defined as the probability that a file can be successfully transmitted to a user. Our focus is on the analysis and optimization of joint random caching and random DTX in two scenarios of user mobility, i.e., the high mobility scenario and the static scenario. The main contributions of the paper are summarized below.

• First, we analyze the STP in both scenarios. The random caching and retransmission with random DTX make the analysis very challenging. In each scenario, by carefully considering the joint impacts of random caching and random DTX on the distribution of the signal-to-interference ratio (SIR) in each slot, we derive closed-form expression for the STP in the general SIR threshold regime, utilizing tools from stochastic geometry. The analysis shows that a larger caching probability corresponds to a higher STP in both scenarios, which reveals the advantage of caching. In addition, the analysis reveals that random DTX can improve the STP in the static scenario and its benefit gradually diminishes when mobility increases. We also derive a closed-form expression for the asymptotic outage probability in the low SIR threshold regime, utilizing series expansion of some special functions. The asymptotic analysis shows that the diversity gain is jointly affected by random caching and random DTX.

• Next, we consider the maximization of the STP with respect to the caching probability and the BS activity probability in both scenarios, which is a challenging non-convex optimization problem. In the high mobility scenario, we obtain a globally optimal solution using interior point method. In the static scenario, we develop a low-complexity iterative algorithm to obtain a stationary point using alternating optimization.

• Finally, numerical results show that the proposed solutions achieve significant gains over existing baseline schemes and can well adapt to the changes of the system parameters to wisely utilize storage resources and transmission opportunities. As the maximum number of transmissions increases, more files are stored and more BSs are silenced for improving the STP.

Ii System Model

Ii-a Network Model

We consider a large-scale HetNet consisting of independent network tiers. We denote the set of tiers by . All network tiers are co-channel deployed. The BS locations in tier are modeled by an independent homogeneous Poisson point process (PPP) with density . Let be the superposition of , , i.e., , which denotes the locations of all tiers of BSs in the network. Each BS in tier has one transmit antenna with transmission power . For the propagation model, we consider a general power-law path-loss model in which a transmitted signal from a BS with distance , is attenuated by a factor , where denotes the path-loss exponent. For the small-scale fading model, we assume Rayleigh fading. Since a HetNet is primarily interference-limited, we ignore the thermal noise for simplicity.

Each user has one receive antenna. We consider a discrete-time system with time being slotted. Let denote the slot index. Two scenarios of user mobility are considered: the high mobility scenario and the static scenario [19]. In the high mobility scenario, in each slot, the user locations follow an independent homogeneous PPP, i.e., a new realization of the PPP for the user locations is drawn in each slot, and the user locations are independent over time. Mathematically, the high mobility scenario is equivalent to the case where the user locations follow an independent homogeneous PPP and stay fixed over time, and in each slot, new realizations of the independent PPPs for the tiers of BSs are drawn [19]. In the static scenario, the user locations stay fixed over time and follow an independent PPP. Note that, the mobility scenario in a practical network is between the two scenarios, and hence the results in this paper provide some theoretical performance bounds for a practical network. In both scenarios, without loss of generality (w.l.o.g.), we can study the performance of a typical user , which is located at the origin , according to Slivnyak’s Theorem [23].

Let denote the set of files in the HetNet. For ease of analysis, as in [10, 13], we assume that all files have the same size, and file popularity distribution is identical among all users.111Note that, the results in this paper can be easily extended to the case of different file sizes. To be specific, we can consider file combinations of the same total size, but formed by files of possibly different sizes. The probability that file is requested by each user is , where . Thus, the file popularity distribution is given by , which is assumed to be known a priori.222Note that, the file popularity evolves at a slower timescale and various learning methodologies can be employed to estimate the file popularity over time [24]. In addition, w.l.o.g., we assume that , i.e., the popularity rank of file is . Assume that at the beginning of slot 1, each user randomly requests a file according to the file popularity distribution . We shall consider the delivery of each requested file over consecutive slots. The network consists of cache-enabled BSs. In particular, each BS in tier is equipped with a cache of size to store different files out of .

Ii-B Random Caching and Retransmissions with Random DTX

To provide high spatial file diversity, we consider a random caching design similar to the one in [12], where file is stored at each BS in tier according to a certain probability , called the caching probability of file in tier . Denote , where , as the caching distribution of the files in the -tier HetNet. Note that, the random caching design is parameterized by . We have [13, 12]:333To implement the random caching design, we randomly place a file combination of different files at each BS in tier according to a corresponding caching probability for file combinations. The detailed relationship between and the caching probability for file combinations can be found in [12].

 0≤Tn,k≤1,n∈N,k∈K, (1) ∑n∈NTn,k=Ck,k∈K. (2)

Let denote the point process of the BSs in tier which store file . Note that, , . Under the random caching design, , are independent PPPs with densities , .

Consider a user requesting file at the beginning of slot 1. If file is not stored in any tier, the user will not be served. Otherwise, the user is associated with the BS which not only stores file but also provides the maximum average received signal strength (RSS) [8] among all BSs in the -tier HetNet, referred to as its serving BS. Note that, in the high mobility scenario, the user association changes from slot to slot, while in the static scenario, the user association does not change over slots. Under this content-based user association, in each slot, a user may not be associated with the BS which provides the maximum average RSS if it has not stored file . As a result, a user may suffer from more severe inter-cell interference under this content-based user association than under the traditional connection-based user association.

The transmission of a file in one slot is more likely to fail, if is not associated with the BS providing the maximum average RSS. Increasing the number of transmissions of a file can increase the probability that eventually the file is successfully transmitted, at the cost of delay increase. In addition, there are some applications without strict delay requirements, e.g., elastic services. Therefore, for those applications, we consider a simple retransmission protocol in which a file is repeatedly transmitted until it is successfully received or the number of transmissions exceeds .444Note that, we consider the case that a user will not request any new file until the current file request is served or expires (i.e., the number of transmissions exceeds ).

However, in a practical HetNet, the interference suffered by a user is temporally correlated since it comes from the same set of interferers in different time slots [20]. Such correlation makes the SIRs temporally correlated and thus dramatically decreases the performance gain of retransmission. In order to manage such correlation, we consider random DTX at the BSs [21, 20], where each BS has two possible transmission states in each slot, i.e., the active state and the inactive state. Specifically, in each slot, a BS is active with probability , called the activity probability, and is inactive with probability , independent of the BS location and slot.555Note that, a user cannot be served in a slot if its serving BS is inactive in this slot. Note that, the random DTX design is parameterized by . The density of active BSs in tier is .

Ii-C Performance Metric

Suppose that the typical user requests file at the beginning of slot 1. Let denote the index of the tier with which is associated and denote the index of the serving BS of . We denote and as the distance and the fading power coefficient between BS and in slot , respectively. Assume , , are i.i.d., according to the exponential distribution with unit mean. Let be the set of active BSs in tier in slot . When requests file and file is transmitted by BS , the signal-to-interference ratio (SIR) of in slot is given by

 SIRn,0(t)=Pk0hk0,l0,0(t)Xk0,l0,0(t)−α(t)\mathbbm1(l0∈Bak0(t))∑l∈Φk0∖{l0}Pk0hk0,l,0(t)X−αk0,l,0(t)\mathbbm1(l∈Bak0(t))+∑j∈K∖{k0}∑l∈ΦjPjhj,l,0(t)X−αj,l,0(t)\mathbbm1(l∈Baj(t)), (3)

where denotes the indicator function.

We say that file is successfully transmitted to in slot if is greater than or equal to a given threshold , i.e., . Let denote the event that file is successfully transmitted to in slot and denote the complementary event of , i.e., the event that file is not successfully transmitted to in slot . The probability that file is successfully transmitted to in consecutive slots, referred to as the successful transmission probability (STP) of file , under the adopted simple retransmission protocol in the high mobility and static scenarios, is given by

 qn,i(Tn,β)=1−Pr(Scn(1),Scn(2),⋯,Scn(M)),i∈{hm,st}. (4)

In the high mobility scenario, since the events (or ), , are i.i.d., the STP of file in (4) can be expressed as

 qn,hm(Tn,β) = 1−M∏t=1Pr(Scn(t))=1−(1−Pr(Sn))M, (5)

where is the STP of file in one slot. Here, we have dropped the index in , as , , are i.i.d.

In the static scenario, as the locations of BSs and do not change, the events , , are correlated. Let denote the event that file is successfully transmitted to in slot , conditioned on . Similarly, denotes the complementary event of . Note that, the events (or ), , are i.i.d. due to the fact that the fading power coefficients are i.i.d. with respect to (w.r.t.) . Thus, in the static scenario, the STP of file in (4) can be expressed as

 qn,st(Tn,β) = \mathbbmEΦ(1−Pr(Scn|Φ(1),Scn|Φ(2),⋯,Scn|Φ(M))) (6) =

where is the expectation operation and . Note that, . Here, we have dropped the index in , as , , are i.i.d..

Note that, , , is a linear function when , and a concave function when . Thus, by Jensen’s inequality, we have , where the equality holds when , implying that mobility has a positive effect on the STP. The intuitions are given as follows. In the static scenario, the locations of BSs during consecutive slots stay fixed, leading to temporal SIR correlation. That is, if the transmission in one slot fails, there is a higher chance that the transmission in another slot also fails. In contrast, in the high mobility scenario, the locations of BSs during consecutive slots are independent, and hence there is no correlation among SIRs during consecutive slots. Consequently, a user has a higher chance to experience a favorable transmission channel with high SIR within transmissions. Therefore, mobility increases temporal diversity, leading to the STP increase.

Users are mostly concerned about whether their requested files can be successfully received. Therefore, in this paper, we adopt the probability that a randomly requested file by the typical user is successfully transmitted in consecutive slots, referred to as the STP, as the network performance metric. By total probability theorem, the STP in the high mobility and static scenarios is given by

 qi(T,β) = ∑n∈Nanqn,i(Tn,β),i∈{hm,st}, (7)

where and are the design parameters of random caching and random DTX, respectively.

Iii High Mobility Scenario

In this section, we consider the high mobility scenario. We first analyze the STP and then maximize the STP by optimizing the design parameters of random caching and random DTX.

Iii-a Performance Analysis

In this part, we analyze the STP in the general SIR threshold regime and the low SIR threshold regime, respectively. To be specific, we only need to analyze the STP of file , i.e., , in the two regimes. Then, by (7), we can directly obtain the STP, i.e., .

Iii-A1 Performance Analysis in General SIR Threshold Regime

In this part, we analyze in the general SIR threshold regime, using tools from stochastic geometry. To calculate , based on (5), we first need to analyze the distribution of the SIR, . Under random caching, there are three types of interferers for : i) all the other BSs in the same tier as the serving BS of which have stored the desired file of (apart from the serving BS of ), ii) all the BSs in the same tier as the serving BS of which have not stored the desired file of , and iii) all the BSs in other tiers. In addition, under random DTX, the serving BS of is active with probability , and the number of interferers of is times that for the case where the BSs are always active. By jointly considering the impacts of random caching and random DTX on , we can derive the distribution of and then , as summarized in the following theorem.

Theorem 1 (STP in High Mobility Scenario)

The STP of file in the high mobility scenario is given by

 (8)

where , , , and and denote the Gauss hypergeometric function and Gamma function, respectively.

Proof: See Appendix A.

Theorem 1 provides a closed-form expression for in the general SIR threshold regime. From Theorem 1, we can see that the system parameters , , , , , , , , and jointly affect in a complex manner.

Based on Theorem 1, we characterize how changes with and , as summarized blow.

Lemma 1 (Effects of Random Caching and Random DTX)
• is an increasing and concave function of , for all .

• is an increasing and concave function of .

The first result in Lemma 1 shows that a larger corresponds to a larger , which reveals the advantage of caching. This is because the average distance between a user requesting file and its serving BS decreases with . The second result in Lemma 1 shows that a larger corresponds to a larger , which reveals that it is not beneficial to apply random DTX in the high mobility scenario. To understand this result, we first study the STP of file in one slot, i.e., . By setting in (8), we have

 Pr(Sn)=β∑k∈KzkTn,kW(β)∑k∈KzkTn,k+V(β)(1−∑k∈KzkTn,k),n∈N. (9)

It is easy to verify that is an increasing function of , implying that the penalty of random DTX in signal reduction overtakes its advantage in interference reduction in one slot. In the high mobility scenario, , are i.i.d., implying that random DTX has no further benefit of reducing interference correlation. Therefore, random DTX cannot improve the STP in the high mobility scenario. Fig. 3 plots versus and , respectively, verifying Theorem 1 and Lemma 1.

Iii-A2 Performance Analysis in Low SIR Threshold Regime

To further obtain insights, in this part, we analyze the outage probability of file which is defined as , in the low SIR threshold regime, i.e., , where the (normalized) target bit rate .666Different types of files may have different target bit rates. For instance, some video files such as MPEG 1, MPEG 4 and H.323 [26] and audio files such as CD and MP3 [27] require relatively low target bit rates. Let denote the -dimensional all-one vector. Here, denotes the transpose operation. Denote . For ease of illustration, in the following, we consider four cases.

• Case i): File is stored at each BS and random DTX is not applied, i.e., and .

• Case ii): File is not stored at any BS and random DTX is not applied, i.e., and .

• Case iii): File is stored at each BS and random DTX is applied, i.e., and .

• Case iv): File is not stored at any BS and random DTX is applied, i.e., and .

By analyzing the four cases, we have the following result.777Note that, when means

Lemma 2 (Outage Probability in High Mobility Scenario When θ→0)

In the high mobility scenario, when , we have

 ¯qn,hm(Tn,β)∼(1−β)M+chm(Tn,β),n∈N, (10)

where

 chm(Tn,β)≜⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩θM(2α−2)M,case i),θ2Mα((1∑k∈KzkTn,k−1)Γ(1+2α)Γ(1−2α))M,case ii),θ(1−β)M−1Mβ22α−2,case iii),θ2α(1−β)M−1Mβ2(1∑k∈KzkTn,k−1)Γ(1+2α)Γ(1−2α),case iv). (11)

Proof: See Appendix B.

From Lemma 2, we can see that both and significantly affect the asymptotic behaviours of when , but in different manners. Fig. 4 plots versus and indicates that Lemma 2 provides a good approximation for when is small. In addition, from Fig. 4, we observe that the rates of decay to zero of when in the four cases are different. In the following, we further characterize such rate in each case, referred to as the diversity gain [28], i.e.,

 dhm=limθ→0log(¯qn,hm(Tn,β))logθ. (12)

Note that the definition in (12) is similar to the usual definition of diversity gain as the rate of decay to zero of the error probability in the high SNR regime [29]. A larger diversity gain implies a faster decay to zero of with decreasing . From Lemma 2, we have the following result.

Lemma 3 (Diversity Gain in High Mobility Scenario)

The diversity gain in the high mobility scenario is given by

 dhm=⎧⎪⎨⎪⎩M,case i),2Mα,case ii),0,cases iii) and iv). (13)

Lemma 3 tells us that as long as random DTX is applied, there is no diversity gain. Without random DTX, caching a file at every BS can achieve the full diversity gain , and caching a file only at some BSs, irrespective of the caching probability, achieves the same smaller diversity gain (as and ). Fig. 4 verifies Lemma 3.

Iii-B Performance Optimization

By substituting (8) into (7), the STP in the high mobility scenario is calculated as

 qhm(T,β)=∑n∈Nan⎛⎝1−(1−β∑k∈KzkTn,kW(β)∑k∈KzkTn,k+V(β)(1−∑k∈KzkTn,k))M⎞⎠. (14)

The caching distribution and BS activity probability significantly affect the STP in the high mobility scenario. We would like to maximize in (14) by jointly optimizing and . Specifically, we have the following optimization problem.

Problem 1 (Optimization of Random Caching and Random DTX in High Mobility Scenario)
 q∗hm≜maxT,β qhm(T,β) s.t. (???),(???),β∈(0,1],

where denotes an optimal value and denotes an optimal solution.

Problem 1 maximizes a non-concave function over a convex set, and hence is non-convex. In general, it is difficult to obtain a globally optimal solution of a non-convex problem. By exploring properties of the objective function , in the following, we can obtain a globally optimal solution of Problem 1.

Recall that Lemma 1 shows that increases with for all . Thus, we know that . It remains to obtain by maximizing w.r.t. , i.e., solving the following problem.

Problem 2 (Optimization of Random Caching in High Mobility Scenario)
 q∗hm=maxT qhm(T,1) s.t. (???),(???).

Recall that Lemma 1 shows that is a concave function of , implying that in (14) is a concave function of . Thus, Problem 2 is a convex optimization problem and can be efficiently solved by the interior point method. Consider a special case of for all , i.e., equal cache size across all tiers. Using KKT conditions, the optimal solution of Problem 2 can be characterized as follows.

Lemma 4 (Optimal Solution of Problem 2 When Ck=C, k∈K)

When for all , an optimal solution of Problem 2 is given by888Note that, for all , can be obtained by solving using the bisection method, and can be obtained by solving using the bisection method.

 T∗n,k=⎧⎪⎨⎪⎩0,if η∗≥g(0),1,if η∗≤g(1),g−1(η∗),otherwise,n∈N,k∈K,

where denotes the inverse function of function , given by

 g(x)≜anM(1−xxW(1)+(1−x)V(1))M−1V(1)(xW(1)+(1−x)V(1))2,

and satisfies .

Based on and an optimal solution of Problem 2, we can obtain a globally optimal solution of Problem 1.

Iv Static Scenario

In this section, we consider the static scenario. We first analyze the STP and then maximize the STP by optimizing the design parameters of random caching and random DTX.

Iv-a Performance Analysis

In this part, we analyze the STP in the general SIR threshold regime and the low SIR threshold regime, respectively. To be specific, we only need to analyze the STP of file , i.e., , in the two regimes. Then, by (7), we can directly obtain the STP, i.e., .

Iv-A1 Performance Analysis in General SIR Threshold Regime

In this part, we analyze in the general SIR threshold regime, using tools from stochastic geometry. It is challenging to calculate , as , are correlated. To address this challenge, by using the binomial expansion theorem, we first rewrite in (6) as . Note that, conditioned on , , are i.i.d.. Thus, we can first analyze the distribution of , conditioned on , and then derive by deconditioning on . Thus, we have the following theorem.

Theorem 2 (STP in Static Scenario)

The STP of file in the static scenario is given by,

 qn,st(Tn,β)=M∑m=1(Mm)(−1)m+1βm∑k∈KzkTn,kFm(β)∑k∈KzkTn,k+Gm(β)(1−∑k∈KzkTn,k),n∈N, (15)

where is given by Theorem 1, and are given, respectively, by

 Fm(β) = m∑i=0(mi)βi(1−β)m−i2F1(−2α,i;1−2α;−θ), (16) Gm(β) = m∑i=0(mi)βi(1−β)m−iΓ(i+2α)Γ(i)Γ(1−2α)θ2α. (17)

Proof: See Appendix C.

Similarly to Theorem 1, Theorem 2 provides a closed-form expression for in the general SIR threshold regime. However, the expression in Theorem 2 is more complex than that in Theorem 1, due to the correlations across the consecutive slots. Fig. 7 plots versus and , respectively, in the static scenario, verifying Theorem 2. It is worth noting that unlike the high mobility scenario, it is hard to analytically characterize how changes with and . However, from Fig. 7, we can observe some properties in the static scenario for the considered setup. Specifically, Fig. 7(a) shows that is an increasing function of , for all , which reveals the advantage of caching. Fig. 7(b) shows that in the static scenario, for some , there exists an optimal BS activity probability that maximizes , in contrast to the case in the high mobility scenario, where for any given . Although the penalty of random DTX in signal reduction may overtake its advantage in interference reduction in one particular slot (see (9)), random DTX is also able to reduce interference correlation across different slots in the static scenario [20]. When , the advantages of random DTX outweigh its penalty and when , its penalty outweighs its advantages.

Iv-A2 Performance Analysis in Low SIR Threshold Regime

To further obtain insights, in this part, we analyze the outage probability of file which is defined as , in the low SIR threshold regime, i.e., . By considering the same four cases as in Lemma 2, we have the following result.

Lemma 5 (Outage Probability in Static Scenario When θ→0)

In the static scenario, when , we have

 ¯qn,st(Tn,β)∼(1−β)M+cst(Tn,β),n∈N, (18)

where

 cst(Tn,β)≜⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩θM∂M∂xM(1F1(−2α;1−2α;x))−1|x=0,case i),θ2α(1∑k∈KzkTn,k−1)Γ(1−2α)∑Mm=1(Mm)×(−1)m+1βm∑mi=0(mi)βi(1−β)m−iΓ(i+2α)Γ(i),cases ii and iv% ),θ(1−β)M−1Mβ22α−2,case iii). (19)

Here, is the confluent hypergeometric function of the first kind.

Proof: See Appendix D.

From Lemma 5, we can see that both the caching probability and the BS activity probability significantly affect the asymptotic behaviours of when , but in different manners. Fig. 8 plots versus and indicates that Lemma 5 provides a good approximation for when is small.

Similarly, we further characterise the diversity gain in the static scenario, i.e.,

 dst=limθ→0log(¯qn,st(Tn,β))logθ. (20)

From Lemma 5, we have the following result.

Lemma 6 (Diversity Gain in Static Scenario)

The diversity gain in the static scenario is given by

 dst=⎧⎪⎨⎪⎩M,case i),2α,case ii),0,cases iii) and iv). (21)

Lemma 6 can be interpreted in the same way as Lemma 3. Comparing the diversity gains for case ii) in Lemma 6 and Lemma 3, we know that for case ii), the diversity gain in the high mobility scenario is times of that in the static scenario due to the fact that there is no interference correlation in the high mobility scenario.

Iv-B Performance Optimization

By substituting (15) into (6), the STP in the static scenario is calculated as

 qst(T,β)=∑n∈NanM∑m=1(Mm)(−1)m+1βm∑k∈KzkTn,kFm(β)∑k∈KzkTn,k+Gm(β)(1−∑k∈KzkTn,k). (22)

The caching distribution and BS activity probability significantly affects the STP in the static scenario. We would like to maximize in (22) by jointly optimizing and . Specifically, we have the following optimization problem.

Problem 3 (Optimization of Random Caching and Random DTX in Static Scenario)
 q∗st≜maxT,β qst(T,β) s.t. (???),(???),β∈(0,1], (23)

where denotes the optimal value and denotes the optimal solution.

Problem 3 maximizes a non-concave function over a convex set, and hence is non-convex in general. Recall that in Section II-C, when , we have , implying that when , Problem 3 can be solved by using the same method as for Problem 1. Thus, in the following, we focus on solving Problem 3 when . Note that, as is differentiable, in general, we can obtain a stationary point of Problem 3 when , using the gradient projection method with a diminishing stepsize.999Note that a stationary point is a point that satisfies the necessary optimality conditions of a non-convex optimization problem, and it is the classic goal in the design of iterative algorithms for non-convex optimization problems. However, the rate of convergence of the gradient projection method is strongly dependent on the choices of stepsize. If it is chosen improperly, it may take a large number of iterations to meet some convergence criterion, especially when the number of variables in Problem 3 is large. To address this issue, we propose a more efficient algorithm to obtain a stationary point of Problem 3, based on alternating optimization. Specifically, we partition the variables in Problem 3 into two blocks, i.e., and , and separate the constraint sets of these two blocks. Then, we solve a random caching optimization problem and a random DTX optimization problem alternatively.

Iv-B1 Random Caching Optimization

First, we consider the optimization of the random caching probability while fixing .

Problem 4 (Optimization of Random Caching for Given β)
 maxT qst(T,β) s.t. (???),(???).

To solve Problem 4, we first analyze its structural properties. Let and denote the sets of all the odd and even numbers in the set , respectively. We rewrite in (22) as

 qst(T,β)=q1(T,β)−q2(T,β),

where is given by

 qi(T,β)=∑n∈Nan∑m∈Mi(Mm)βm∑k∈KzkTn,kFm(β)∑k∈KzkTn,k+Gm(β)(1−∑k∈KzkTn,k),i=1,2.

It can be easily verified that is a concave function of . Thus, Problem 4 is a difference-of-convex (DC) programming problem and can be solved based on the convex-concave procedure (CCP) [30]. The basic idea of the CCP is to linearize the convex terms of the objective function (i.e., in ) to obtain a concave objective for a maximization problem, and then solve a sequence of convex problems successively. Specifically, at iteration , we solve the following problem:

 T(j) ≜argmaxTq1(T,β)−~q2(T,β;T(j−1)) s.t.(???),(???),

where , and denotes the gradient of at .

Since and are concave and linear w.r.t. , respectively, the optimization in (IV-B1) is a convex problem which can be efficiently solved by the interior point method. The details of the proposed iterative algorithm are summarized in Algorithm 1. Note that, it has been shown in [30] that the sequence generated by Algorithm 1 converges to a stationary point of Problem 4.

Next, we consider a special case of for all , i.e., equal cache size across all tiers. In this case, the optimization in (IV-B1) is convex. Similarly, using KKT conditions, we can obtain an optimal solution of the problem in (IV-B1) as follows.

Lemma 7 (Optimal Solution of Problem in (Iv-B1) When Ck=C for all k∈K)

When for all , an optimal solution of problem in (IV-B1) is given by

 T(j)n,k=⎧⎪⎨⎪⎩0,if η∗≥f(0),1,if η∗≤f(1),f−1(η∗),otherwise,n∈N,k∈K,

where denotes the inverse function of function , given by

 f(x) ≜ an∑m∈M1(Mm)βmGm(β)(Fm(β)x+Gm(β)(1−x))2−an∑m∈M2(Mm)βmGm(β)(Fm(β)T(j−1)n,k+Gm(β)(1−T(j−1)n,k))