Probabilistic Small-Cell Caching: Performance Analysis and ... · D2D caching is indeed also a hot...

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016 1

Probabilistic Small-Cell Caching: PerformanceAnalysis and Optimization

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Youjia Chen, Ming Ding, Member, IEEE, Jun Li, Senior Member, IEEE, Zihuai Lin, Senior Member, IEEE,Guoqiang Mao, Senior Member, IEEE, and Lajos Hanzo, Fellow, IEEE

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Abstract—Small-cell caching utilizes the embedded storage of5small-cell base stations (SBSs) to store popular contents for the6sake of reducing duplicated content transmissions in networks and7for offloading the data traffic from macrocell base stations to SBSs.8In this paper, we study a probabilistic small-cell caching strategy,9where each SBS caches a subset of contents with a specific caching10probability. We consider two kinds of network architectures: 1)11The SBSs are always active, which is referred to as the always-on12architecture; and 2) the SBSs are activated on demand by mobile13users (MUs), which is referred to as the dynamic on–off archi-14tecture. We focus our attention on the probability that MUs can15successfully download content from the storage of SBSs. First, we16derive theoretical results of this successful download probability17(SDP) using stochastic geometry theory. Then, we investigate the18impact of the SBS parameters, such as the transmission power and19deployment intensity on the SDP. Furthermore, we optimize the20caching probabilities by maximizing the SDP based on our stochas-21tic geometry analysis. The intrinsic amalgamation of optimization22theory and stochastic geometry based analysis leads to our optimal23caching strategy, characterized by the resultant closed-form ex-24pressions. Our results show that in the always-on architecture, the25optimal caching probabilities solely depend on the content request26probabilities, while in the dynamic on–off architecture, they also27relate to the MU-to-SBS intensity ratio. Interestingly, in both ar-28chitectures, the optimal caching probabilities are linear functions29of the square root of the content request probabilities. Monte-30Carlo simulations validate our theoretical analysis and show that31the proposed schemes relying on the optimal caching probabilities32

Manuscript received February 11, 2016; revised June 25, 2016; acceptedAugust 24, 2016. Date of publication; date of current version. This workwas supported in part by the Fujian Provincial Natural Science Foundation(2016J01290); in part by the National Natural Science Foundation of Chinaunder Grant 61571128 and Grant 61501238; in part by the Jiangsu ProvincialScience Foundation (BK20150786); in part by the China Scholarship Coun-cil, the Specially Appointed Professor Program in Jiangsu Province, 2015;and in part by the Fundamental Research Funds for the Central Universities(30916011205). The review of this paper was coordinated by Dr. B. Canberk.

Y. Chen is with the School of Electrical and Information Engineering, Uni-versity of Sydney, Sydney, NSW 2006, Australia, also with Fujian NormalUniversity, Fuzhou 350007, China, and also with the Data61, CSIRO, CanberraACT 2600, Australia (e-mail: [email protected]).

M. Ding is with the Data61, CSIRO Canberra ACT, 2600, Australia (e-mail:[email protected]).

J. Li is with the School of Electronic and Optical Engineering, Nan-jing University of Science and Technology, Beijing 100044, China (e-mail:[email protected]).

Z. Lin is with the School of Electrical and Information Engineering, Univer-sity of Sydney, Sydney, NSW 2006, Australia, and also with the Data61, CSIROCanberra ACT, 2600, Australia (e-mail: [email protected]).

G. Mao is with the School of Computing and Communications, University ofTechnology Sydney, Ultimo, NSW 2007, Australia, and also with the Data61,CSIRO Canberra ACT, 2600, Australia (e-mail: [email protected]).

L. Hanzo is with the Department of Electronics and Computer Science,University of Southampton, Southampton SO17 1BJ, U.K. (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2016.2606765

are capable of achieving substantial SDP improvement, compared Q133with the benchmark schemes. 34

Index Terms—. 35

I. INTRODUCTION 36

I T IS forecast that at least a 100x network capacity increase 37

will be required to meet the traffic demands in 2020 [1]. As 38

a result, vendors and operators are now looking at using every 39

tool at hand to improve network capacity [2]. 40

In addition, a substantial contribution to the traffic explosion 41

comes from the repeated download of a small portion of popu- 42

lar contents, such as popular movies and videos [3]. Therefore, 43

intelligent caching in wireless networks has been proposed for 44

effectively reducing such duplicated transmissions of popular 45

contents, as well as for offloading the traffic from the over- 46

whelmed macrocells to small cells [4], [5]. Caching in third- 47

generation (3G) and fourth-generation (4G) wireless networks 48

was shown to be able to reduce the traffic by one third to two 49

thirds [6]. 50

Several caching strategies have been proposed for wireless 51

networks. Woo et al. [7] analyzed the strategy of caching con- 52

tents in the evolved packet core of local thermal equilibrium 53

(LTE) networks.The strategy of caching contents in the radio 54

access network, with an aim to place contents closer to mo- 55

bile users (MUs) was studied in [8] and [9]. The concept of 56

small-cell caching, referred to as “Femtocaching” in [9] and 57

[10], utilized small-cell base stations (SBS) in heterogeneous 58

cellular networks as distributed caching devices. Caching strate- 59

gies conceived for device-to-device (D2D) networks were in- 60

vestigated in [11]–[13], where the mobile terminals serve as 61

caching devices. The coexistence of small-cell caching and 62

D2D caching is indeed also a hot research direction. In [14], 63

Yang et al. considered the joint caching in both the relays and 64

a subset of the mobile terminals, which relies on the coex- 65

istence of small-cell caching and D2D caching. Moreover, a 66

coded caching scheme was proposed in [15] to improve system 67

performance. 68

In this paper, we focus on the small-cell caching because 69

1) the large number of SBSs in 4G and fifth-generation (5G) 70

networks already provide a promising basis for caching [2]; and 71

2) compared with D2D caching, small-cell caching has several 72

advantages, such as the abundance of power supply, fewer grave 73

security issues, and more reliable data delivery. As illustrated in 74

Fig. 1, with small-cell caching, popular contents are transmitted 75

and cached in the storage of the SBSs during off-peak hours. 76

Then in peak hours, if an MU can find its requested content in 77

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2 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016

Fig. 1. Small-cell caching.

a nearby SBS, the MU can directly download the content from78

such SBS.79

There are generally two approaches to implement the small-80

cell caching, i.e., the deterministic content placement and the81

nondeterministic content placement. In [9], [16], and [17], the82

deterministic contents placement was analyzed. In these works,83

the placement of popular contents was optimized using the in-84

formation of the network node locations and the statistical or in-85

stantaneous channel states. However, in practice, the geographic86

distribution of MUs and the wireless channels are time variant.87

Thus, the optimal content placement strategy has to be fre-88

quently updated in the deterministic content placement, leading89

to a high complexity and fewer tractable results. On the other90

hand, the nondeterministic content placement permits simple91

implementation and has a good tractability. In [18] and [14],92

the distributions of SBSs and MUs were modeled as homo-93

geneous Poisson point processes (HPPPs) to obtain a general94

performance analysis for the small-cell caching. However, in95

these works, all the SBSs were assumed to cache the same copy96

of certain popular contents. In [11], probabilistic content place-97

ment was proposed and analyzed in the context of D2D caching,98

where each mobile terminal caches a specific subset of the con-99

tents with a given caching probability. The throughput versus100

outage tradeoff was analyzed and the optimal caching distribu-101

tion was derived for a grid network relying on a particular proto-102

col model. The idea of probabilistic content placement was also103

investigated in the coded multicasting system [19]. Compared104

with caching the same copy of certain popular contents in all the105

SBSs, probabilistic content placement in small-cell caching can106

provide more flexibility. Therefore, in this paper, we focus on107

small-cell caching relying on probabilistic content placement,108

shortened as probabilistic small-cell caching (PSC) for brevity.109

In small-cell networks, there are two network architectures,110

namely, the always-on architecture and the dynamic on–off ar-111

chitecture. The always-on architecture is a common practice in112

the current cellular networks, where all the SBSs are always ac-113

tive. By contrast, in the dynamic on–off architecture, the SBSs114

are only active, when they are required to provide services to115

nearby MUs [20]. Aiming for saving energy consumption and116

mitigating unnecessary intercell interference, the dynamic on–117

off architecture has been proposed and it is currently under118

investigation in 3GPP as an important candidate of 5G tech-119

nologies in future dense and ultradense small-cell networks [2],120

[21], [22]. Energy consumption is of critical interest in future 5G121

systems [23], [24], especially in ultradense networks. Compared122

with the power-thirsty always-on architecture, where the energy123

consumption grows with the network’s densification, the energy124

consumption of the ultradense network relying on the dynamic 125

on–off architecture mainly depends on the density of MUs in 126

the network [2]. The in-depth investigation of the associated 127

energy consumption issues of wireless caching will constitute 128

our future work. 129

Against this background, we study the PSC under the above- 130

mentioned pair of network architectures. First, we use a stochas- 131

tic geometry to develop theoretical results of the probability 132

Pr(D) that MUs can successfully download contents from the 133

storage of SBSs. Second, we investigate the impact of the SBSs’ 134

parameters on Pr(D), namely, that of the transmission power 135

P and of the deployment intensity λs . In the always-on archi- 136

tecture, although Pr(D) monotonically increases with either P 137

or λs , it approaches a constant when P or λs is sufficiently 138

high. In the dynamic on–off architecture, Pr(D) reaches a con- 139

stant when P is high enough, while it keeps on increasing as 140

λs grows. Most importantly, we optimize the caching probabil- 141

ities for maximizing Pr(D) in the pair of network architectures 142

considered. We emphasize that it is quite a challenge to ap- 143

ply optimization theory to an objective function obtained from 144

stochastic geometry analysis, especially to derive a closed-form 145

expression for the optimal solution. Our results will demonstrate 146

that in the always-on architecture, the optimal subset of contents 147

to be cached depends on the content request probabilities, while 148

in the dynamic on–off architecture, it also depends on the MU- 149

to-SBS intensity ratio. Most interestingly, in both architectures, 150

the optimal caching probabilities can be expressed as linear 151

functions of the square root of the content request probabilities. 152

The rest of the paper is structured as follows. In Section II we 153

describe the system model, while in Section III we present the 154

definition of PSC and formulate the probability that MUs can 155

successfully download contents from the storage of SBSs. The 156

main analytical results characterizing this successful download 157

probability (SDP) are presented in Section IV. In Section V, 158

we optimize the caching probabilities in both of the network 159

architectures for maximizing the derived SDP. The accuracy of 160

the analytical results and the performance gains of optimization 161

are characterized by simulations in Section VI. Finally, our 162

conclusions are offered in Section VII. 163

II. SYSTEM MODEL 164

We consider a cellular network supporting multiple MUs by 165

the SBSs operating within the same frequency spectrum. We 166

model the distribution of the SBSs and that of the MUs as two 167

independent HPPPs, with the intensities of λs and λu , respec- 168

tively. The transmission power of the SBSs is denoted by P . 169

The path loss of the channel spanning from an SBS to an MU 170

is modeled as d−α , where d denotes the distance between them, 171

and α denotes the path-loss exponent. The multipath fading is 172

modeled as Rayleigh fading with a unit power, and hence the 173

channel’s power gain is denoted by h ∼ exp(1). All the channels 174

are assumed to be independently and identically distributed. 175

A. Network Architectures 176

We consider two network architectures. 177

1) Always-On Architecture: In this architecture, all the 178

SBSs are assumed to be active, i.e., all the SBSs are 179

CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 3

continuously transmitting signals. This architecture is com-180

monly employed in the operational cellular networks [25]. The181

rationale for this architecture is that the number of SBSs is182

usually much lower than that of MUs, and thus each and every183

SBS has to be turned ON to serve the MUs in its coverage.184

2) Dynamic On–Off Architecture: In this architecture, an185

SBS will be active only when it has to provide services to its as-186

sociated MUs. In future 5G networks, the intensity of deployed187

SBSs is expected to be comparable to or even potentially higher188

than the intensity of MUs [2]. In such ultradense networks, hav-189

ing an adequate received signal coverage is always guaranteed,190

since the distance between an MU and its serving SBS is short,191

but the interference becomes the dominant issue. With the goal192

of mitigating the potentially avoidable intercell interference and193

saving energy, the dynamic on–off architecture has been identi-194

fied as one of the key technologies in 5G networks [20]. With the195

dynamic on–off architecture, an SBS will switch to its idle mode,196

i.e., turn OFF its radio transmission, if there is no MU associated197

with it, otherwise, it will switch back to the active mode.198

B. File Request Model199

We consider a contents library consisting of M different files.200

Note that M does not represent the number of files available on201

the Internet, but the number of popular files that the MUs tend202

to access. We denote by qm the probability that the mth file Fm203

will be requested. By stacking qm into {qm : m = 1, · · · ,M},204

we can get the probability mass function (PMF) of requesting205

the M files. According to [26], the request- PMF of the files can206

be modeled as a Zipf distribution. More specifically, for Fm , its207

request probability qm is written as208

qm =1

mβ

∑Mi=1

1iβ

(1)

where β is the exponent of the Zipf distribution and a large β209

implies having an uneven popularity among those files. From210

(1), qm tends to zero, as M → ∞ when β < 1, while it con-211

verges to a constant value when β > 1. Note that (1) implies that212

the indices of the files are not randomly generated, but follow a213

descending order of their request probabilities.214

Due to the limited storage of SBSs, an SBS is typically unable215

to cache the entire file library. Therefore, we assume that the216

library is partitioned into N nonoverlapping subsets of files,217

referred to as file groups (FGs), and each SBS can cache only218

one of the N FGs. Note that the same FG can be redundantly219

stored in multiple SBSs. The scenario of FGs with overlapping220

subsets of files will be considered later, which will be compared221

with the nonoverlapping scenario. We denote the nth FG, n ∈222

{1, · · · , N} by Gn . The probability Qn that an MU requests a223

file in FG Gn , is thus given by224

Qn =∑

m, forFm ∈Gn

qm. (2)

III. PROBABILISTIC SMALL-CELL CACHING STRATEGY225

In this section, we introduce the PSC strategy, and formulate226

the probability that MUs can successfully download contents227

from the storage of the SBSs, which is an important performance 228

metric of small-cell caching. 229

Generally, caching consists of two phases: a contents place- 230

ment phase and a contents delivery phase [27]. In the contents 231

placement phase, popular contents are transmitted and cached 232

in the storage units of network devices that are close to MUs. In 233

the contents delivery phase, the popular cached contents can be 234

promptly retrieved for serving the MUs. 235

A. Contents Placement Phase 236

In the content placement phase of PSC, each SBS indepen- 237

dently caches FGGn with a specific caching probability, denoted 238

by Sn . Hence, from the perspective of the entire network, the 239

fraction of the SBSs that caches Gn equals to Sn . Since the dis- 240

tribution of SBSs in the network is modeled as an HPPP with the 241

intensity of λs , according to the thinning theorem of HPPP [28], 242

we can view the distribution of SBSs that cache Gn as a thinned 243

HPPP with the intensity of Snλs . 244

We assume that at a particular time instant, an MU can only 245

request one file, and hence, the distribution of MUs who request 246

the files in Gn can also be modeled as a thinned HPPP with 247

the intensity Qnλu . We treat the SBSs that cache Gn together 248

with the MUs that request the files in Gn as the nth tier of the 249

network, shortened as Tier-n. 250

B. Contents Delivery Phase 251

During the contents delivery phase, an MU that requests a 252

file in Gn will associate with the nearest SBS that caches Gn , 253

and then attempts to download the file from it. We assume 254

that only when the received signal-to-interference-and-noise- 255

ratio (SINR) at the MU is above a prescribed threshold, can the 256

requested file be successfully downloaded. 257

If the MU cannot download the requested file from the cached 258

SBS, the requested file would be transmitted to the MU from 259

a remote content provider, which means the data should flow 260

across the Internet, the cellular core network, and the backhaul 261

network, as illustrated in Fig. 1. 262

C. Probability of Successful Download 263

Recent surveys show that 96% of the operators consider 264

backhaul as one of the most important challenges to small- 265

cell deployments, and this issue is exacerbated in ultradense 266

networks [29], [30]. If an MU can successfully download a re- 267

quested file from storages of SBSs, the usage of the backhaul 268

network will be greatly reduced and the transmission latency 269

of a requested file will be significantly shortened. Therefore, 270

we assume that a successful download of a requested file from 271

storages of SBSs is always beneficial to the network perfor- 272

mance. Accordingly, we focus on our attention on this SDP as 273

the performance metric for small-cell caching in the following. 274

According to Slyvnyak’s theorem for HPPP [28], an existing 275

point in the process does not change the statistical distribution of 276

other points of the HPPP. Therefore, the probability that an MU 277

in Tier-n can successfully download the contents from SBSs 278

can be obtained by analyzing the probability that a typical MU 279


in Tier-n, say located at the origin, can successfully download280

the contents from its associated SBS in Tier-n.281

When the MU considered requests a file in Gn , its received282

SINR from its nearest SBS in Tier-n can be formulated as283

γn (z) =Phx0z

−α

∑xj ∈Φ\{x0} Phxj

‖xj‖−α + σ2(3)

where σ2 denotes the Gaussian noise power, z is the distance284

between the typical MU and its nearest SBS in Tier-n, xj repre-285

sents the locations of the interfering SBSs, Φ denotes the set of286

simultaneously active SBSs, and x0 is the location of the serving287

BS at a distance of z. Additionally, ‖xj‖ denotes the distance288

between xj and the typical MU, while hx0 and hxjdenote the289

corresponding channel gains.290

Since the intercell interference is the dominant factor deter-291

mining the signal quality in the operational cellular networks,292

especially when unity frequency reuse has been adopted for im-293

proving the spectrum efficiency, the minimum received SINR is294

used as the metric of successful reception. Let δ be the minimum295

SINR required for successful transmissions and Dn be the event296

that the typical Tier-n MU successfully receives the requested297

file from the associated Tier-n SBS. Then, the probability of Dn298

can be formulated as299

Pr(Dn ) = Pr[γn (z) ≥ δ]. (4)

Considering the request probabilities of Gn and based on the300

result of Pr(Dn ), we obtain the average probability that the301

MUs can successfully download contents from the storage of302

the SBSs, denoted by Pr(D), as303

Pr(D) =N∑

n=1

Qn · Pr(Dn ). (5)

In essence, Pr(D) quantifies the weighted sum of the SDP,304

where the weights are the request probabilities reflecting the305

importance of the files.306

IV. PERFORMANCE ANALYSIS OF SMALL-CELL CACHING307

In this section, we derive the SDP Pr(D) for the pair of308

network architectures. Some special cases are also considered309

with an aim to obtain more insights into the design of PSC.310

A. Always-On Architecture311

Our main result on the probability Pr(D) for the always-on312

architecture is summarized in Theorem 1.313

Theorem 1: In the always-on architecture, the probability314

Pr(D) is given by315

Pr(D) =N∑

n=1

QnPr(Dn )

=N∑

n=1

Qn

∫ ∞

0πSnλs exp

(

−zαδσ2

P

)

exp(−πλsz

2 ((1 − Sn )C(δ, α) + SnA(δ, α) + Sn ))dz2

(6)

where A(δ, α) � δ 2α−2 2F1(1, 1 − 2

α ; 2 − 2α ;−δ), and C(δ, α) 316

� 2α δ

2α B( 2

α , 1 − 2α ). Furthermore, 2F1(·) denotes the hyperge- 317

ometric function, and B(·) represents the beta function [31]. 318

Proof: See Appendix A. � 319

From (6), we conclude that the probability Pr(D) increases 320

as the transmission power P grows, because exp(− zα δσ 2

P ) in- 321

creases with P . Since it remains a challenge to obtain deeper 322

insights from (6), which is not a closed-form expression, two 323

special cases are examined in the sequel to gain deeper insight 324

on the performance behavior of Pr(D). 325

1) Path-Loss Exponent α = 4: According to 3GPP measure- 326

ment [32], the typical value of the path-loss exponent for SBSs 327

in practical environments is around 4. Substituting this typical 328

value of α = 4 into (6), we have 329

Pr(D) |α=4 =N∑

n=1

QnπSn

√π

4δ

Pλ2s

σ2erfcx

(π

2·

√Pλ2

s

δσ2

(Sn +

π

2

√δ(1 − Sn ) + Sn

√δ arctan

√δ))

(7)

where erfcx(x) � exp(x2)erfc(x) is the scaled complementary 330

error function [33]. 331

Regarding the relationship between Pr(D) and λs , we pro- 332

pose Corollary 1. 333

Corollary 1: In the always-on architecture, for the special 334

case of α = 4, Pr(D) monotonically increases with the increase 335

of λs . 336

Proof: See Appendix B. � 337

From the results obtained in (6) that Pr(D) increases as P 338

grows, and based on Corollary 1, we conclude that when α = 4, 339

the SDP Pr(D) can be improved by either increasing the SBSs’ 340

transmission power P or the SBSs’ deployment intensity λs . 341

Furthermore, since (7) can be viewed as a function of the variable 342

Pλ2s , the effect of increasing P to kP on Pr(D) is equivalent to 343

increasing λs to√

kλs , where k is a positive constant. 344

Moreover, according to the property of the function erfcx(x), 345

i.e., limx→∞ erfcx(x) = 1√πx

, we have 346

limP →∞

Pr(D) |α=4 = limλs →∞

Pr(D) |α=4

=N∑

n=1

QnSn

π2

√δ + (

√δ arctan

√δ + 1 − π

2

√δ)Sn

. (8)

From (8), we have Remark 1. 347

Remark 1: In the always-on architecture, given σ2 and δ, the 348

value of Pr(D) monotonically grows with the increase of P and 349

λs , and it converges to a constant, when P or λs is sufficiently 350

large. 351

2) Neglecting Noise, i.e., σ2 = 0: In an interference-limited 352

network, where the noise level is much lower than the interfer- 353

ence, the impact of the noise can be neglected. In such cases, 354

we assume that σ2 = 0, and it follows that Pr(D) in (6) can be 355

rewritten as 356

Pr(D) |σ 2→0 =N∑

n=1

QnSn

SnA(δ, α) + (1 − Sn )C(δ, α) + Sn. (9)


From (9), we have Remark 2.357

Remark 2: In the always-on architecture operating in an358

interference-limited network, the probability of successful359

download depends only on the request probabilities and caching360

probabilities of the FGs, i.e., Qn and Sn .361

Note that in the scenario, where the different FGs may have362

an overlapping subset of files, the probability Pr(D) still has363

the same formulation as (6). However, all the subscripts n in364

(6) should be changed to m, because we should consider both365

the request probability and the caching probability of each file366

Fm , i.e., Sm and Qm , instead of each FG Gn . Therefore, in this367

scenario, the specific SBSs that cache Fm and the MUs that368

request Fm are viewed as Tier-m. Since all the derivations are369

the same, our main results summarized in Theorem 1 as well370

as the aforementioned corollary and remarks, are still valid in371

conjunction with the subscript m. Hence we omit the analysis372

for this scenario with overlapping subsets of files for brevity.373

B. Dynamic On–Off Architecture374

As mentioned, in the dynamic on–off architecture an SBS is375

only active, when it has to provide services for the associated376

MUs. Specifically, an SBS in Tier-n is only active, when there377

is at least one MU in Tier-n located in its Voronoi cell. Hence,378

the probability that an SBS in Tier-n is active, which is de-379

noted by Pr (An ), should be considered for the dynamic on–off380

architecture.381

Our main result on the probability Pr(D) for the dynamic382

on–off architecture is summarized in Theorem 2.383

Theorem 2: In the dynamic on–off architecture, the proba-384

bility Pr(D) is given by385

Pr(D) =N∑

n=1

QnPr(Dn )

=N∑

n=1

Qn

∫ ∞

0πSnλs exp

(

−zαδσ2

P

)

exp

(

− πλsz2

( N∑

i=1,i =n

Pr(Ai)SiC(δ, α) + Pr(An )SnA(δ, α) + Sn

))

dz2 (10)

where Pr(An ) denotes the probability that an SBS in Tier-n is386

in the active mode, and387

Pr(An ) ≈ 1 −(

1 +Qnλu

3.5Snλs

)−3.5

. (11)

Proof: See Appendix C. �388

Compared to Pr(D) in the always-on architecture, Pr(D) in389

the dynamic on–off architecture also depends on the intensity390

of the MUs λu . The reason behind this is that the number of391

active SBSs in the network depends on the number of MUs in392

the network.393


Remark 3: In the dynamic on–off architecture, given σ2 and395

δ, the value of Pr(D) monotonically increases with the increase396

of the transmission power P .397


network, substituting σ2 = 0 into (10), we have 399

Pr(D) |σ 2→0 =

N∑

n=1

QnSn

Pr(An )SnA(δ, α) +∑N

i=1,i =n Pr(Ai)SiC(δ, α) + Sn

.

(12)


Remark 4: In the dynamic on–off architecture operating in 401

an interference-limited network, the probability of successful 402

download Pr(D) is independent of P , and depends only on Qn , 403

Sn as well as on the MU-to-SBS intensity ratio λu/λs . 404

When considering the scenario of FGs with overlapping sub- 405

sets of files, the average probability Pr(D) cannot be formulated 406

as the sum of Pr(Dn ) as in (5). Furthermore, we cannot formu- 407

late Pr(D) as Pr(D) =∑M

m=1 Pr(Dm ), which we propose for 408

the overlapping scenario in the always-on architecture. This is 409

because in the dynamic on–off architecture the active probability 410

of an SBS depends on the specific FG that it caches. Therefore, 411

the analysis of Pr(D) in the dynamic on–off architecture con- 412

sidering the scenario with overlapping subsets of files requires 413

further investigations as part of our future research. 414

V. OPTIMIZATION OF THE CACHING PROBABILITY 415

A larger Pr(D) always benefits the network because of 1) the 416

backhaul saving and 2) the low-latency transmission of local 417

contents from SBSs [2]. Based on such facts, in this section, we 418

concentrate on maximizing Pr(D) by optimally designing the 419

caching probabilities of the contents in the system, denoted by 420

{SOptn : n = 1, . . . , N}. 421

Note that there is a paucity of literature on applying opti- 422

mization theory relying on an objective function obtained from 423

stochastic geometry analysis, especially, when aiming for deriv- 424

ing a closed-form expression of the optimal solution. In order to 425

facilitate this optimization procedure, we ensure the mathemat- 426

ical tractability of the objective function by using a simple user 427

association strategy and neglect the deleterious effects of noise. 428

A. Always-On Architecture 429

From (9), we can formulate the optimization problem of max- 430

imizing Pr(D) as 431

max{Sn }

Pr(D) = max{Sn }

N∑

n=1

QnSn

(1 − Sn )C(δ, α) + SnA(δ, α) + Sn

s.t.N∑

n=1

Sn = 1

Sn ≥ 0, n = 1, . . . , N.

(13)

The solution of Problem (13) is presented in Theorem 3. 432

Theorem 3: In the always-on architecture, the optimal 433

caching scheme, which is denoted by the file caching PMF 434

{Soptn }, that maximizes the average probability of successful 435


download, is given by436

Soptn =

⎡

⎢⎢⎢

√Qn

ξ − C(δ, α)

A(δ, α) − C(δ, α) + 1

⎤

⎥⎥⎥

+

, n = 1, . . . , N (14)

where√

ξ =∑N ∗

n = 1

√Qn

(N ∗−1)C (δ,α)+A(δ,α)+1 , �Ω + � max{Ω, 0}, and437

N ∗, 1 ≤ N ∗ ≤ N satisfies the constraint that Sn ≥ 0 ∀n.438

Proof: It can be shown that the optimization Prob-439

lem (13) is concave and can be solved by invoking the440

Karush−Kuhn−Tucker conditions [34]. The conclusion then441

follows. �442

From (14), when the request probability obeys Qn >443

ξC2(δ, α), Gn is cached with a caching probability of Soptn , oth-444

erwise, it is not cached. This optimal strategy implies that ideally445

the SBSs should cache the specific files with high request prob-446

abilities, while those files with low request probabilities should447

not be cached at all due to the limited storage of SBSs in the net-448

work. Moreover, we can see that from (14) the optimal caching449

probability of an FG is a linear function of the square root of its450

request probability.451

Regarding the scenario of FGs associated with overlapping452

subsets of files, as we mentioned before, Pr(D) in this scenario453

has the same formulation as that in the nonoverlapping scenario.454

Therefore, the optimal caching probability ofFm in the scenario455

of FGs having overlapping subsets of files can be formulated as456

SOptm = min

⎧⎪⎨

⎪⎩

⎡

⎢⎢⎢

√Qm

ξ − C(δ, α)

A(δ, α) − C(δ, α) + 1

⎤

⎥⎥⎥

+

, 1

⎫⎪⎬

⎪⎭(15)

where√

ξ =∑M ∗

m = 1

√Qm

(M ∗−V )C (δ,α)+V (A(δ,α)+1) , and M ∗(1 ≤ M ∗ ≤457

M), satisfies the constraint that 0 ≤ Sm ≤ 1 ∀m, and V de-458

notes the number of files in each FG.459

Compared with the nonoverlapping scenario, the presence of460

overlapping subsets among the FGs provides a higher grade of461

diversity in the system. However, based on our simulations to462

be discussed in the sequel, we find that the gain of maximum463

Pr(D) obtained as a benefit of this diversity is limited, while the464

algorithm associated with the optimal caching strategy of (15)465

is more complex than that of (14).466


In this architecture, as shown in (11), the probability Pr(An )468

that an SBS in Tier-n is in the active mode, is a function of the469

ratio Qnλu/Snλs . Since the intensity of SBSs is much higher470

than the intensity of the MUs in this architecture, i.e., we have471

λs � λu , the SBS activity probability Pr(An ) in (11) can be472

approximated as473

Pr(An ) ≈ Qnλu

Snλs. (16)

Substituting (16) into (12) and (5), we can formulate the op-474

timization problem of maximizing the successful downloading475

probability as 476

max{Sn ,εn }

Pr(D) =

max{Sn ,εn }

N∑

n=1

QnSn

Qnλu

λsA(δ, α) · εn +

∑i:i =n Qi

λu

λsC(δ, α) · εi +Sn

s.t.N∑

n=1

Sn = 1

Sn ≥ 0, n = 1, . . . , N

εn =

{1, if Sn > 0

0, if Sn = 0.(17)

Different from the optimization problem in (13), the variable 477

εn is introduced to indicate whether Gn is cached. Due to the 478

existence of εn , which implies 2N hypotheses of file caching 479

states, Problem (17) is difficult to solve. Nevertheless, we man- 480

age to find the solution and summarize it in Theorem 4. 481

Theorem 4: The optimal caching scheme, i.e., the optimal 482

file caching PMF {SOptn }, that maximizes the average probabil- 483

ity of successful download, is given by 484

SOptn

=

⎧⎪⎨

⎪⎩

ζK

√QnξK C(δ, α) − Q2

n (C(δ, α) − A(δ, α))−(ξK

λu

λsC(δ, α) − Qn

λu

λs(C(δ, α) − A(δ, α))

), n ≤ K

0, K < n ≤ N.

(18)

where 485

ξK �K∑

i=1

Qi

ζK �1 + KξK

λu

λsC(δ, α) − ξK

λu

λs(C(δ, α) − A(δ, α))

∑Ki=1

√QiξK C(δ, α) − Q2

i (C(δ, α) − A(δ, α)). (19)

Regarding K, we have 486

K = arg maxk

{Dk : k = 1, 2, . . . , N̂

}(20)

where 487

Dk � ξk

−λu

λs

(∑kn=1

√QnξkC(δ, α) − Q2

n (C(δ, α) − A(δ, α)))2

1 + kξkλu

λsC(δ, α) − ξk

λu

λs(C(δ, α) − A(δ, α))

(21)

and 488

N̂ =

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

N, ifλu

λs< aN

N − 1, if aN ≤ λu

λs< aN −1

· · ·1, if a2 ≤ λu

λs.

(22)


Algorithm 1: Optimal Caching Probabilities in theDynamic On–Off Architecture.

1: Set j = N .2: Compute ξj =

∑ji=1 Qi , and ϑj and aj in (24) and (23).

3: Compare λu

λswith aj . If λu

λs< aj , go to Step 4;

otherwise, set j = j − 1 and go to Step 2.4: Set N̂ = j.5: Compute ξk =

∑ki=1 Qi and Dk in (21), k = 1, · · · , N̂ .

6: Set K = arg maxk

{Dk}.

7: Compute ξK and ζK in (19), then compute SOptn in (18).

489

Furthermore, the segmentation parameter aj , j = 2, . . . , N490

is given by491

aj =ϑj

(ϑj ξj − Qj )(C(δ, α) − A(δ, α)) + (1 − jϑj )ξjC(δ, α)(23)

where492

ϑj �

√QjξjC(δ, α) − Q2

j (C(δ, α) − A(δ, α))∑j

i=1

√QiξjC(δ, α) − Q2

i (C(δ, α) − A(δ, α)). (24)

Proof: See Appendix D. �493

To get a better understanding of Theorem 4, we propose494

Algorithm 1 to implement Theorem 4.495

From Theorem 4, we have the following remarks.496

Remark 5: In the always-on architecture, the optimal number497

of FGs to be cached depends only on {Qn : n = 1, · · · , N}. By498

contrast, in the dynamic on–off architecture, the optimal number499

of FGs to be cached depends not only on {Qn} but on the MU-500

to-SBS intensity ratio λu/λs in the network as well.501

Remark 6: According to (22), given λu , more FGs tend to be502

cached in the SBSs, when λs becomes higher. Moreover, when503

the intensity of SBSs is not sufficiently high to cache all the504

FGs, the SBSs should cache the specific files with relatively high505

request probabilities, which is consistent with the conclusion for506

the always-on architecture.507

Remark 7: In (18), with a practical region of the SINR508

threshold and path-loss exponent from 3GPP, i.e., for δ ∈509

[0.5, 3] and α ∈ (2, 4], we have ξK C(δ, α) � Qn (C(δ, α) −510

A(δ, α)), and the optimal caching probability SOptn ≈511

ζK

√Qn

λu

λsξK C(δ, α) − ξK

λu

λsC(δ, α). From (14) and (18), it512

is interesting to observe that the optimal caching scheme in both513

the always-on architecture and in the dynamic on–off architec-514

ture follow a square root law, i.e., SOptn is a linear function of515 √

Qn .516

VI. NUMERICAL AND SIMULATION RESULTS517

In this section, we present both our numerical and Monte-518

Carlo simulation results of Pr(D) in various scenarios. In the519

Monte-Carlo simulations, the performance is averaged over520

1000 network deployments, where in each deployment SBSs521

and MUs are randomly distributed in an area of 5 × 5 km ac-522

cording to an HPPP distribution. The intensity of MUs in the523

network is 200/km2. The transmission power of the SBSs, the524

Fig. 2. Numerical and simulation results of Pr(D) of the O-PSC strategy inthe always-on architecture.

noise power, the path-loss exponent, and the SINR threshold are 525

set to 30 dBm, –104 dBm, 4 and 0.25(−6 dB), respectively [32]. 526

In the simulations of the always-on architecture, the deployment 527

intensity of SBSs is set to 80/km2, while in the simulations of 528

the dynamic on–off architecture, the intensity is set to 400/km2. 529

Furthermore, we consider a file library consisting of M = 100 530

files, and we partition the file library into N = 10 FGs with a 531

simple grouping strategy that the mth file belongs to Gn if 532

m ∈ [MN (n − 1) + 1, . . . , M

N n] ∀n ∈ {1, . . . , N}. Note that the 533

specific choice of the file grouping strategy is beyond the scope 534

of this paper and it does not affect our results, because it only 535

changes the specific values of the request-PMF {Qn}. 536

In addition, we consider the following two PSC strategies. 537

1) The request probability based PSC (RP-PSC) [12], where 538

the caching probability of one FG equals to its request 539

probability, i.e., Sn = Qn . Intuitively, a particular FG is 540

more popular than another, the RP-PSC strategy will des- 541

ignate more SBSs to cache it. This strategy is evaluated 542

as a benchmark in our simulations. 543

2) The proposed optimized PSC (O-PSC) based on (14) in 544

the always-on architecture and (18) in the dynamic on–off 545

architecture, where Sn = Soptn . 546


Fig. 2 compares the numerical and the simulation results con- 548

cerning Pr(D) of the O-PSC strategy. First, it can be seen that 549

the numerical results closely match the simulation results in all 550

scenarios. In the following, we will focus on the analytical re- 551

sults only, due to the accuracy of our analytical results. Second, 552

Pr(D) increases with the Zipf exponent β. With a larger β, the 553

request probabilities of files are more unevenly distributed. In 554

such cases, a few FGs dominate the requests and caching such 555

popular FGs gives a large Pr(D). Third, Pr(D) will be lower, if 556

the value of δ becomes higher. This is because when the SINR 557

threshold is increased, the probability that the received SINR 558

from the SBS storing the file exceeds this threshold is reduced. 559

Finally, we can see that Pr(D) increases as the number of FGs 560


Fig. 3. Pr(D) of the O-PSC strategy with different P and λs in the always-onarchitecture.

Fig. 4. Comparison of Pr(D) versus δ of the RP-PSC and O-PSC strategiesin the always-on architecture.

decreases. Since each SBS only caches one FG, decreasing the561

number of FGs implies that each SBS caches more files. Hence,562

this Pr(D) improvement comes from increasing the stored con-563

tents in each SBS.564

Fig. 3 shows the SDP Pr(D) for the O-PSC strategy when the565

transmission power P of SBSs varies within 20–40 dBm and the566

deployment intensity λs of SBSs varies within 10–400/km2. To567

highlight the asymptotic behavior of Pr(D) with the growth of568

P , we set the noise power to −50 dBm. We can see from the569

figure that Pr(D) increases monotonically with P or λs . The570

value of Pr(D) remains constant, when P or λs is sufficiently571

high. This result illustrates the limit of Pr(D) in the always-on572

architecture shown in (8).573

In Fig. 4, we plot Pr(D) versus the SINR threshold δ to com-574

pare the performances of the RP-PSC and O-PSC strategies. We575

can see that the proposed O-PSC strategy exhibits a significantly576

better performance than the RP-PSC strategy. With the number577

of FGs N = 10, the performance gain in terms of Pr(D) pro-578

vided by the O-PSC strategy ranges from 20% to 50%, when579

Fig. 5. Comparison of Pr(D) versus β of the RP-PSC and O-PSC strategiesin the always-on architecture.

δ varies from 0.1 to 1. When δ is high, the probability that 580

MUs can directly download the files from the storage of SBSs 581

becomes small. In such cases, the advantage of optimizing the 582

caching probabilities of the FGs is more obvious. 583

Even more significant Pr(D) improvement can be observed 584

for the case of N = 20 than that for N = 10. A larger number of 585

FGs means that less contents can be cached in each SBS, which 586

implies a very limited storage capacity. In such cases, the benefit 587

of optimizing the caching probabilities is more significant. 588

Fig. 5 compares Pr(D) in the context of RP-PSC and O-PSC 589

strategies versus the Zipf exponents β. First, we can see that 590

the proposed O-PSC strategy greatly outperforms the RP-PSC 591

strategy in terms of Pr(D). With the number of FGs N = 20, 592

the performance gain of Pr(D) ranges from 65% to 20% when 593

β varies from 0.5 to 1.5. In other words, the Pr(D) improve- 594

ment decreases, as β grows. The reason behind this trend is 595

that for a large β, a small fraction of FGs dominate the file 596

requests. Once the SBSs cache these very popular FGs, Pr(D) 597

will become sufficiently high. Thus, the additional gain given 598

by the optimization of caching probabilities becomes smaller. 599

Furthermore, compared with the case N = 10, the Pr(D) im- 600

provement when N = 20 is more significant. The reason for 601

this phenomenon has been explained above. 602

Fig. 6 compares Pr(D) in conjunction with the O-PSC strate- 603

gies in the overlapping and nonoverlapping scenarios. Since the 604

total number of files in our simulations is 100, in the figure, the 605

curves of “FGNo = 10” and “FGNo = 20” are compared against 606

the curves of “FilesPerGroup = 10” and “FilesPerGroup = 5,” 607

respectively. We can see that the performance of SDP in the sce- 608

nario of FGs having overlapping subsets of files is better than 609

that of the nonoverlapping subsets of files. The reason for this 610

observation is that allowing overlapping amongst the different 611

FGs provides a beneficial diversity of the FGs. Furthermore, we 612

can see that when the SINR threshold is increased, the advan- 613

tage of the overlapping scenario wanes. This is because when 614

the SINR threshold is high, the O-PSC strategy tends to cache 615

fewer popular files and the diversity of FGs becomes of limited 616

benefit here. 617


Fig. 6. Comparison of Pr(D) versus δ in the overlapping and nonoverlappingscenarios in the always-on architecture.

Fig. 7. Numerical and simulation results of Pr(D) of the O-PCP strategy inthe dynamic on–off architecture.


Fig. 7 shows our comparison between the numerical and sim-619

ulation results of Pr(D) for the O-PSC strategy. We can see620

from this figure that the numerical results closely match the621

simulation results in all scenarios. Similar phenomena can be622

observed as in the always-on architecture.623

1) Pr(D) decreases upon increasing the SINR threshold δ.624

2) Pr(D) increases with the Zipf exponent β.625

3) Pr(D) increases when the number of FGs decreases.626

The reasons behind these trends are the same as those dis-627

cussed for the always-on architecture. Moreover, compared to628

Fig. 2, the value of Pr(D) in the dynamic on–off architecture629

of Fig. 7 is shown to be higher. The reason is that the dynamic630

on–off technique efficiently mitigates the potential avoidable631

interference in the network.632

Fig. 8 shows the performance of Pr(D) for the O-PSC strategy633

in the dynamic on–off architecture, when the transmission power634

Fig. 8. Pr(D) with different P and λs in the dynamic on–off architecture.

Fig. 9. Comparison of Pr(D) of the RP-PSC and O-PSC strategies versus δin the dynamic on–off architecture.

P of SBSs varies from 20 to 40 dBm and the SBS intensity λs 635

varies from 200 to 2000/km2. We can see from this figure that 636

Pr(D) increases monotonically, when either P or λs increases. 637

Moreover, we can see that when P increases to a sufficiently high 638

value, any further increase of P will no longer improve Pr(D). 639

However, the increase of λs will always improve Pr(D), as seen 640

in (12). 641

Fig. 9 compares Pr(D) of the RP-PSC and O-PSC strategies, 642

when the SINR threshold δ varies. It can be seen from the fig- 643

ure that compared to the RP-PSC strategy, Pr(D) is obviously 644

improved by the optimal caching PMF {SOptn } in the O-PSC 645

strategy. With the Zipf exponent β = 1, the performance gain 646

of Pr(D) ranges from 7% to 30%, when δ varies from 0.1 to 1. 647

This observation is similar to that in the always-on architecture. 648

That is, the Pr(D) improvement achieved by the O-PSC strat- 649

egy is more pronounced, when the SINR threshold is higher. 650

Furthermore, the Pr(D) improvement is higher when the Zipf 651

exponent β is lower. The reason for this is explained above. 652


Fig. 10. Comparison of Pr(D) of the RP-PSC and O-PSC strategies versusλs in the dynamic on–off architecture.

Furthermore, in order to verify the optimality of the solution653

given by our algorithm, we plot the optimal solution obtained654

from the exhaustive search over all legitimate file caching states,655

denoted by “Exh. Search” in the figure. Observed from the656

figure that our solution exactly matches the optimal solution of657

“Exh. Search,” which confirms our statement that the proposed658

solution achieves global optimality.659

In Fig. 10, we portray Pr(D) of the RP-PSC and the O-660

PSC strategies versus the SBS intensity λs . First, it can be661

seen that compared with the RP-PSC strategy, the optimization662

of the caching probabilities in the O-PSC strategy improves663

Pr(D) in all scenarios. This Pr(D) improvement achieved by664

the O-PSC strategy wanes slightly when λs increases because665

when the SBS intensity is higher, each MU becomes capable of666

associating with multiple SBSs, and thus, the probability that667

MUs can successfully download contents from SBSs will be668

higher. In such a case, the Pr(D) improvement obtained by the669

optimization of the FG caching probabilities remains limited. In670

addition, we verify the optimality of our solution by comparing671

it to the optimal solution obtained from the exhaustive search.672

VII. CONCLUSION673

In this paper, based on stochastic geometry theory, we ana-674

lyzed the performance of the PSC in a pair of network architec-675

tures. Specifically, we analyzed the probability Pr(D) that MUs676

can successfully download contents from the storage of SBSs.677

We concluded that increasing the SBSs’ transmission power678

P or their deployment intensity λs is capable of increasing the679

SDP. However, in the always-on architecture, Pr(D) remains680

constant when P or λs is sufficiently high, while in the dynamic681

on–off architecture, Pr(D) always increases as λs grows.682

Furthermore, in order to maximize Pr(D), we optimized the683

caching probabilities of the FGs. Our results demonstrated that684

in the always-on architecture, the optimal subset of FGs depends685

on the contents request probabilities. In the dynamic on–off ar-686

chitecture, a piecewise defined function of MU-to-SBS intensity687

ratio λu/λs was introduced in order to find the optimal subset 688

of FGs to be cached. Interestingly, a similar optimal caching 689

probability law was found for both architectures, i.e., SOptn is a 690

linear function of√

Qn . Our simulation results showed that the 691

proposed optimal caching probabilities of the FGs achieve a 692

substantial gain in both architecture in terms of Pr(D) compared 693

to the benchmark Sn = Qn , because more caching resources 694

are devoted to the more popular files in the proposed scheme. 695

APPENDIX APROOF OF THEOREM 1 696

In Tier-n of the always-on architecture, where the inten- 697

sity of the SBSs is Snλs , the PDF of z, i.e., the distance be- 698

tween the typical MU and its nearest SBS, follows fZ (z) = 699

2πSnλsz exp(−πSnλsz2). From (3) and (4), we have 700

Pr(Dn ) = Pr (γn (z) ≥ δ)

=∫ ∞

0Pr

[Phx0z

−α


‖xj‖−α + σ2� δ

]

fZ (z) dz

(a)=

∫ ∞

0EI [exp (−zαδI)] exp

(

−zαδσ2

P

)

2πSnλsz exp(−πSnλsz2)dz (25)

where (a) is obtained by hx0 ∼ exp(1) and I �∑

xj ∈Φ\{x0} 701

hxj‖xj‖−α represents the interference. 702

The interference I consists of two independent parts: 1) I1: the 703

SBSs in other tiers, which are dispersed across the entire area of 704

the network, and 2) I2: the SBSs in the nth tier, whose distances 705

from the typical MU are larger than z. Due to the independence 706

of I1 and I2, we have EI [exp (−zαδI)] = EI1 [exp (−zαδI1)] · 707

EI2 [exp (−zαδI2)]. 708

Since the distribution of the SBSs in Tier-i is viewed as an 709

HPPP φi with Siλs and therefore, we have 710

EI1 [exp (−zαδI1)]

= Ehx j,xj

⎡

⎢⎣

∏

xj ∈∑N

i = 1, i = n φi

exp(−zαδhxj

‖xj‖−α)

⎤

⎥⎦

(b)= Exj

⎡

⎢⎣

∏

xj ∈∑N

i = 1, i = n φi

1

1 + zαδ ‖xj‖−α

⎤

⎥⎦

(c)= exp

⎛

⎝−N∑

i=1,i =n

Siλs

∫

R2

(

1 − 1

1 + δzα ‖xj‖−α

)

dxj

⎞

⎠

= exp

⎛

⎝−2π

N∑

i=1,i =n

Siλs1α

δ2α B

(2α

, 1 − 2α

)

z2

⎞

⎠ (26)

where (b) uses hxj∼ exp(1), and (c) uses E

[∏v∈Φ ξ (v)

]= 711

exp(−λΦ

∫(1 − ξ (v)) dv

). 712


As for I2, we have713


= exp(

−Snλs2π

∫ ∞

z

(

1 − 1


)

‖xj‖ d ‖xj‖)

(d)= exp

(

−Snλsπδ2α z2 2

α

∫ ∞

δ−1

l2α −1

1 + ldl

)

= exp(

−Snλsπz2 2δ

α − 2 2F1

(

1, 1 − 2α

; 2 − 2α

;−δ

))

(27)

where (d) uses l � δ−1z−α ‖xj‖α .714

Our proof is completed by plugging (26) and (27) into715

(25). �716

APPENDIX BPROOF OF COROLLARY 1717

Since we have Pr(D) =∑N

n=1 Qn Pr(Dn ), to prove that718

Pr(D) increases with the increase of λs , we only have to prove719

that Pr(Dn ) increases monotonically upon increasing λs ∀n.720

Thus, in the following, we focus our attention on the proof that721∂ Pr(Dn )

∂λs> 0.722

To simplify our discourse, we use C1 � πSn

2σ

√πPδ , and723

C2 � π2σ

√Pδ

(Sn + π

2

√δ(1 − Sn ) + Sn

√δ arctan

√δ)

.724

Obviously, we have C1 > 0 and C2 > 0. Then, Pr(Dn ) can be725

rewritten as726

Pr(Dn ) = C1λs exp(C22λ2

s)erfc(C2λs). (28)

Hence, we have727

∂ Pr(Dn )∂λs

= C1λs exp(C22λ2

s) (1 − erf(C2λs))

=(C1 exp(C2

2λ2s) + C1λs exp(C2

2λ2s)2C2

2λs

)erfc(C2λs)

− C1λs exp(C22λ2

s)2√π

C2 exp(−C22λ2

s)

= C1 exp(C22λ2

s)(1 + 2C22λ2

s)erfc(C2λs) − C1C2λs2√π

.

(29)

According to [35], the continued fraction expansion of the728

complementary error function is729

erfc(z) =z√π

exp(−z2)1

z2 + a1

1+ a 2z 2+

a 31+ ···

, am =m

2. (30)

From (30), we have erfc(z) > z√π

exp(−z2) 1z 2+ 1

2. Substituting730

C2λs for z, we have731

exp(C22λ2

s)erfc(C2λs) >C2λs√

π

1

C22λ2

s + 12

. (31)

Substituting (31) into (29), we can prove that ∂ Pr(Dn )∂λs

> 0,732

which implies that Pr(D) increases monotonically upon in-733

creasing λs . �734

APPENDIX CPROOF OF THEOREM 2 735

Similar to the derivation in Appendix A, in the dynamic on– 736

off architecture, the intensity of SBSs in Tier-n is also Snλs . 737

Thus, in Tier-n the distance z between the typical MU and 738

its nearest SBS follows the same PDF fZ (z) in the always-on 739

architecture. It follows that we have a similar formulation for 740

Pr(Dn ) in the dynamic on–off architecture, yielding 741

Pr(Dn ) =∫ ∞


(

−zαδσ2

P

)

2πSnλsz exp(−πSnλsz2)dz. (32)

In the dynamic on–off architecture, the interference I only 742

arrives from the SBSs in the active mode. According to [36], 743

the activity probability Pr(An ) of the SBSs in Tier-n, can be 744

formulated as 745

Pr(An ) ≈ 1 −(

1 +Qnλu

3.5Snλs

)−3.5

.

As in Appendix A, we divide the interference into two parts: 746

I = I1 + I2. The first part of interference I1 is inflicted by the 747

active SBSs in any Tier-i, i = n, which can be viewed as a 748

homogeneous PPP with the intensity of Pr(Ai)Siλs . Hence, 749

we update (26) as follows: 750


= exp

⎛

⎝−2π

N∑

i=1:i =n

Pr(Ai)Siλs1α

δ2α B

(2α

, 1 − 2α

)

z2

⎞

⎠ .

(33)

The second part of the interference I2 comes from the active 751

SBSs in Tier-n located in the area outside the circle with radius 752

z. We update (27) as follows: 753

EI2 [exp (−zαδI2)] = exp (−Pr(An )

Snλsπz2 2δ

α − 2 2F1

(

1, 1 − 2α

; 2 − 2α

;−δ

))

. (34)

Integrating (33) and (34) into (32) completes the proof. � 754

APPENDIX DPROOF OF THEOREM 4 755

Note that in the following proof, we simplify the notation by 756

introducing a � λu

λs, C � C(δ, α), and A � A(δ, α). 757

First, we investigate the optimization Problem (17) for a given 758

indicator vector ε. Let us denote by N ∗ the number of ones in 759

ε, and by {nj} the subscript of the ones in N ∗. Then, we have 760


a new optimization problem represented as761

max{Sn j

}

N ∗∑

j=1

QnjSnj

QnjaA +

∑i:i =j Qni

aC + Snj

s.t.N ∗∑

j=1

Snj= 1

Snj> 0 ∀j = 1, . . . N ∗.

(35)

If we neglect the constraint Snj> 0, the solution to Prob-762

lem (35) is presented in Lemma 1.763

Lemma 1: Neglecting the constraint Snj> 0, the optimal764

solution for Problem (35) is given by765

SOptnj

= ζ√

QnjCξ − Q2

nj(C − A) −

[ξaC − Qnj

a(C − A)]

(36)where we have ζ � 1+N ∗ξaC−ξa(C−A)

∑N ∗i = 1

√Qn i

ξC−Q 2n i

(C−A)and ξ �766

∑N ∗

j=1 Qnj.767

Proof: See Appendix E.768

From (36), we propose Lemma 2.769

Lemma 2: Given the request probabilities of two FGs770

cached, where Qni> Qnj

, according to (36), we have SOptni

>771

SOptnj

.772

Proof: See Appendix F.773

Based on Lemma 2, we have SOptnj ∗ = min {SOpt

nj } where774

nj ∗ = arg minnj{Qnj

}. Hence, the constraint Snj> 0, ∀j =775

1, . . . N ∗, is equivalent to Snj ∗ > 0. In order to ensure that776

SOptnj ∗ > 0, based on (36), we have777

a < anj ∗ , anj ∗ �ϑnj ∗

(ϑnj ∗ ξ − Qnj ∗ )(C−A) + (1 − N ∗ϑnj ∗ )ξC(37)

where778

ϑnj ∗ �

√Qnj ∗ ξC + Q2

nj ∗(A − C)

∑N ∗

i=1

√Qni

ξC + Q2ni

(A − C). (38)

Hence, (36) only becomes the optimal solution of Problem (35),779

when a meets the requirement (37).780

Substituting the optimal solution in (36) into (35), we obtain781

the maximum value of Pr(D) for the given indicator vector ε,782

yielding783

DN ∗ = ξ −a(∑N ∗

j=1

√Qnj

Cξ + Q2nj

(A − C))2

1 + N ∗ξaC + ξa(A − C). (39)

Second, we extend the Problem (35) to Problem (17). Based784

on the analysis above, given the indicator vector ε1, when a <785

aε1 in (37), we can obtain the maximum Pr(D) denoted by Dε1786

in (39). For ε2, if we have aε2 > aε1 , then provided a < aε1787

holds, we have a < aε2 . Thus, ε1 and ε2 are both reasonable788

for this optimization problem. Through the comparison of Dε1789

and Dε2 , we can find the right choice between ε1 and ε2. Then790

obtain the optimal solution of {Sn} in form of (36).791

Using {Qn}, we can obtain the segmentation parameters for a792

in (37). The smallest segmentation parameter is obtained when ε793

contains N ones, which is denoted by aN . When a < aN , i.e., λs 794

is high enough, all FGs can be cached in SBSs. Then, with the in- 795

crease of a, i.e., the decrease of λs , some FGs cannot be cached, 796

where a reduced number of ones appear in ε. Since we have 797

Q1 > Q2 > · · · > QN , the unpopular FGs will be discarded one 798

by one. Accordingly, we can obtain both εi as well as the seg- 799

mentation parameter ai . As a result, a piecewise defined function 800

regarding a is obtained like the number of ones in ε is shown 801

in (20). � 802

APPENDIX EPROOF OF LEMMA1 803

Neglecting the constraint Snj> 0, it becomes plausible that 804

Problem (35) is a concave maximization problem. Adopting the 805

Lagrange multiplier Λ, we have 806

Λ(S, λ)

=N ∗∑

j=1

QnjSnj

QnjaA+

∑N ∗

i=1:i =j QniaC + Snj

+λ

⎛

⎝N ∗∑

j=1

Snj− 1

⎞

⎠.

(40)

Using ξ �∑N ∗

j=1 Qnjand ∂Λ

∂Sn j= 0, we have 807

QnjaCξ + Q2

nja(A − C)

(aCξ + a(A − C)Qnj

+ Snj

)2 + λ = 0 ∀nj . (41)

Since∑N ∗

j=1 Snj= 1, we have 808

SOptnj

= ζ√

QnjaCξ + Q2

nja(A − C) −

[ξaC + Qnj

a(A − C)]

(42)

where 809

ζ � 1 + N ∗ξaC + ξa(A − C)∑N ∗

i=1

√Qni

ξaC + Q2ni

a(A − C). (43)

� 810

APPENDIX FPROOF OF LEMMA 2 811

First, based on the optimal solution given in (36), we have 812

∂SOptnj

∂Qnj

= ζ

√a

2

Cξ + 2Qnj(A − C)

√Qnj

Cξ + Q2nj

(A − C)+ a(C − A). (44)

Since C(α, δ) > A(α, δ) > 0, we have∂S Opt

n j

∂Qn j≥ 0 when Qnj

≤ 813

ξ2

CC−A , which means SOpt

nj increases with the growth of Qnj, 814

when Qnjis no bigger than ξ

2C

C−A . 815

1) Since Qnj≤ ξ, if C

C−A ≥ 2, for all Qnj,

∂S Optn j

∂Qn j> 0, and 816

the proof is completed. 817

2) For CC−A < 2, we consider the following case. Since 818

CC−A > 1, we have ξ

2C

C−A > ξ2 . Because

∑j Qnj

= ξ, among 819


the N ∗ FGs cached, there is only one FG associated with820

Qnj> ξ

2C

C−A . We denote the request probability of this popular821

file by Q1 and its caching probability by SOpt1 . Since the request822

probabilities of other cached FGs must be less than ξ2

CC−A , and823

∂S Optn j

∂Qn j> 0 when Qnj

in this region, the highest caching prob-824

ability among these less popular FGs occurs when only two825

FGs are cached. That is, the other FG with request probability826

Q2 = ξ − Q1. Denoted by SOpt2 its caching probability. We have827

SOpt1 − SOpt

2 = ζ√

a

(√Q1Cξ + Q2

1(A − C)

−√

Q2Cξ + Q22(A − C)

)

+ (Q1 − Q2)a(C − A). (45)

Since Q1Cξ + Q21(A − C) − Q2Cξ − Q2

2(A − C) = (Q1 −828

Q2)ξaA > 0, we have SOpt1 − SOpt

2 > 0. Thus, for the dominate829

FG, its caching probability also dominates.830

Combining the two parts above, we complete the proof. �831

ACKNOWLEDGMENT832

The authors would like to thank Dr. H. (Henry) Chen from833

the University of Sydney, Australia, for his helpful discussions834

and suggestions.835

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Youjia Chen received the B.S. and M.S degrees in 938communication engineering from Nanjing Univer- 939sity, Nanjing, China, in 2005 and 2008, respectively. 940She is currently working toward the Ph.D. degree in 941wireless engineering with the University of Sydney, 942Sydney, Australia. 943

From 2008 to 2009, she was with Alcatel Lu- 944cent Shanghai Bell. From August 2009 until now, 945she has been with the College of Photonic and Elec- 946trical Engineering, Fujian Normal University, China. 947Her research interests include resource management, 948

load balancing, and caching strategy in heterogeneous cellular networks. 949950


Ming Ding (M’12) received the B.S. and M.S. de-951grees (with first class Hons.) in electronics engineer-952ing and Ph.D. degree in signal and information pro-953cessing from Shanghai Jiao Tong University (SJTU),954Shanghai, China, in 2004, 2007, and 2011, respec-955tively.956

From September 2007 to September 2011, while957at the same time working as a Researcher/Senior958Researcher Sharp Laboratories of China (SLC), af-959ter achieving the Ph.D. degree, he continued work-960ing with SLC as a Senior Researcher/Principal Re-961

searcher until September 2014, when he joined National Information and Com-962munications Technology Australia (NICTA). In July 2016, Commonwealth Sci-963entific and Industrial Research Organization (CSIRO) and NICTA joined forces964to create Data61, where he continued as a Senior Research Scientist in this new965R&D center in Sydney, NSW, Australia. He has authored more than 30 papers966in IEEE journals and conferences, all in recognized venues, and about 20 3GPP967standardization contributions, as well as a Springer book entitled Multi-point Co-968operative Communication Systems: Theory and Applications (Springer-Verlag,9692013). In addition, as the first inventor, he holds 15 CN, seven JP, three US, two970KR patents, and has co-authored another 100+ patent applications on 4G/5G971technologies.972

Dr. Ding has been the Guest Editor/Cochair/TPC member of several IEEE973top-tier journals/conferences, e.g., the IEEE JOURNAL ON SELECTED AREAS IN974COMMUNICATIONS, the IEEE COMMUNICATIONS MAGAZINE, the IEEE Globe-975com Workshops, etc. He received the Presidents Award from the SLC in 2012976for his inventions and publications and served as one of the key members in977the 4G/5G standardization team when it was awarded in 2014 as the Sharp978Company Best Team: LTE 2014 Standardization Patent Portfolio.979

980

Jun Li (M’09–SM’16) received the Ph.D. degree in981electronic engineering from Shanghai Jiao Tong Uni-982versity, Shanghai, China in 2009.983

From January 2009 to June 2009, he was a Re-984search Scientist with the Department of Research and985Innovation, Alcatel Lucent Shanghai Bell. From June9862009 to April 2012, he was a Postdoctoral Fellow with987the School of Electrical Engineering and Telecommu-988nications, University of New South Wales, Australia.989From April 2012 to June 2015, he was a Research990Fellow with the School of Electrical Engineering,991

University of Sydney, Australia. From June 2015 to the present, he has been992a Professor with the School of Electronic and Optical Engineering, Nanjing993University of Science and Technology, Nanjing, China. His research interests994include network information theory, channel coding theory, wireless network995coding, and cooperative communications.996

997

Zihuai Lin (S’98–M’06–SM’10) received the Ph.D.998degree in electrical engineering from Chalmers Uni-999versity of Technology, Gothenburg, Sweden, in 2006.1000

Prior to his Ph.D. degree, he has held positions at1001Ericsson Research, Stockholm, Sweden. Following1002having received the Ph.D. degree, he was a Research1003Associate Professor with Aalborg University, Aal-1004borg, Denmark, and is currently with the School of1005Electrical and Information Engineering, University of1006Sydney, Sydney, Australia. His research interests in-1007clude source/channel/network coding, coded modu-1008

lation, MIMO, OFDMA, SC-FDMA, radio resource management, cooperative1009communications, small-cell networks, 5G cellular systems, etc.1010

1011

Guoqiang Mao (S’98–M’02–SM’08) received the 1012Ph.D. degree in telecommunications engineering 1013from Edith Cowan University, Perth, Australia, in 10142002. 1015

Between 2002 and 2014, he was with the School 1016of Electrical and Information Engineering, University 1017of Sydney, Sydney, Australia. He joined the Univer- 1018sity of Technology Sydney in February 2014 as a 1019Professor of wireless networking and the Director of 1020Center for real-time information networks. The Cen- 1021ter is among the largest university research centers in 1022

Australia in the field of wireless communications and networking. He has pub- 1023lished about 200 papers in international conferences and journals, which have 1024been cited more than 4000 times. His research interest includes intelligent trans- 1025port systems, applied graph theory and its applications in telecommunications, 1026Internet of Things, wireless sensor networks, wireless localization techniques, 1027and network performance analysis. 1028

Dr. Mao is an Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICA- 1029TIONs (since 2014) and the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 1030(since 2010). He received Top Editor award for outstanding contributions to the 1031IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY in 2011, 2014, and 2015. 1032He is a Cochair of the IEEE Intelligent Transport Systems Society Technical 1033Committee on Communication Networks. He has served as a Chair, Cochair, 1034and Technical Program Committee Member in a large number of international 1035conferences. 1036

1037

Lajos Hanzo (F’08) received the M.S. degree in elec- 1038tronics and the Ph.D. degree from the Technical Uni- 1039versity of Budapest, Budapest, Hungary, in 1976 and 10401983, respectively. He received the prestigious Doc- 1041tor of Sciences research degree in wireless commu- 1042nications from the University of Southampton, U.K., 1043in 2004. 1044

In 2016, he was admitted to the Hungarian 1045Academy of Science, Budapest, Hungary. During his 104640-year career in telecommunications, he has held 1047various research and academic posts in Hungary, Ger- 1048

many, and the U.K. Since 1986, he has been with the School of Electronics and 1049Computer Science, University of Southampton, U.K., where he holds the Chair 1050in telecommunications. He has successfully supervised 111 Ph.D. students, co- 1051authored 20 John Wiley/IEEE Press books on mobile radio communications, 1052totalling in excess of 10 000 pages, published 1600+ research contributions on 1053IEEE Xplore, acted both as Technical Program Committee member and General 1054Chair of IEEE conferences, presented keynote lectures, and received a number 1055of distinctions. Currently he is directing a 60-strong academic research team, 1056working on a range of research projects in the field of wireless multimedia 1057communications sponsored by industry; the Engineering and Physical Sciences 1058Research Council (EPSRC), U.K.; and the European Research Council.s Ad- 1059vanced Fellow Grant. He is an enthusiastic supporter of industrial and academic 1060liaison, and he offers a range of industrial courses. He has 25 000+ citations and 1061an H-index of 60. For further information on research in progress and associated 1062publications, see http://www-mobile.ecs.soton.ac.uk. Dr. Hanzo is also a Gov- 1063ernor of the IEEE Vehicular Technology Society. During 2008–2012, he was 1064the Editor-in-Chief of the IEEE Press and a Chaired Professor with Tsinghua 1065University, Beijing, China. In 2009, he received an honorary doctorate award 1066by the Technical University of Budapest and in 2015, from the University of 1067Edinburgh, Edinburgh, U.K., as well as the Royal Society.s Wolfson Research 1068Merit Award. He is a Fellow of the Royal Academy of Engineering, The Insti- 1069tution of Engineering and Technology, and EURASIP. 1070

1071

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http://www.ieee.org/documents/taxonomy_v101.pdf. 1074

Q2. Author: Please update Ref. [2]. 1075

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Q4. Author: Please provide complete bibliographic details in Refs. [29] and [30]. 1077

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016 1

Probabilistic Small-Cell Caching: PerformanceAnalysis and Optimization

1

2

Youjia Chen, Ming Ding, Member, IEEE, Jun Li, Senior Member, IEEE, Zihuai Lin, Senior Member, IEEE,Guoqiang Mao, Senior Member, IEEE, and Lajos Hanzo, Fellow, IEEE

3

4

Abstract—Small-cell caching utilizes the embedded storage of5small-cell base stations (SBSs) to store popular contents for the6sake of reducing duplicated content transmissions in networks and7for offloading the data traffic from macrocell base stations to SBSs.8In this paper, we study a probabilistic small-cell caching strategy,9where each SBS caches a subset of contents with a specific caching10probability. We consider two kinds of network architectures: 1)11The SBSs are always active, which is referred to as the always-on12architecture; and 2) the SBSs are activated on demand by mobile13users (MUs), which is referred to as the dynamic on–off archi-14tecture. We focus our attention on the probability that MUs can15successfully download content from the storage of SBSs. First, we16derive theoretical results of this successful download probability17(SDP) using stochastic geometry theory. Then, we investigate the18impact of the SBS parameters, such as the transmission power and19deployment intensity on the SDP. Furthermore, we optimize the20caching probabilities by maximizing the SDP based on our stochas-21tic geometry analysis. The intrinsic amalgamation of optimization22theory and stochastic geometry based analysis leads to our optimal23caching strategy, characterized by the resultant closed-form ex-24pressions. Our results show that in the always-on architecture, the25optimal caching probabilities solely depend on the content request26probabilities, while in the dynamic on–off architecture, they also27relate to the MU-to-SBS intensity ratio. Interestingly, in both ar-28chitectures, the optimal caching probabilities are linear functions29of the square root of the content request probabilities. Monte-30Carlo simulations validate our theoretical analysis and show that31the proposed schemes relying on the optimal caching probabilities32

Manuscript received February 11, 2016; revised June 25, 2016; acceptedAugust 24, 2016. Date of publication; date of current version. This workwas supported in part by the Fujian Provincial Natural Science Foundation(2016J01290); in part by the National Natural Science Foundation of Chinaunder Grant 61571128 and Grant 61501238; in part by the Jiangsu ProvincialScience Foundation (BK20150786); in part by the China Scholarship Coun-cil, the Specially Appointed Professor Program in Jiangsu Province, 2015;and in part by the Fundamental Research Funds for the Central Universities(30916011205). The review of this paper was coordinated by Dr. B. Canberk.

Y. Chen is with the School of Electrical and Information Engineering, Uni-versity of Sydney, Sydney, NSW 2006, Australia, also with Fujian NormalUniversity, Fuzhou 350007, China, and also with the Data61, CSIRO, CanberraACT 2600, Australia (e-mail: [email protected]).

M. Ding is with the Data61, CSIRO Canberra ACT, 2600, Australia (e-mail:[email protected]).

J. Li is with the School of Electronic and Optical Engineering, Nan-jing University of Science and Technology, Beijing 100044, China (e-mail:[email protected]).

Z. Lin is with the School of Electrical and Information Engineering, Univer-sity of Sydney, Sydney, NSW 2006, Australia, and also with the Data61, CSIROCanberra ACT, 2600, Australia (e-mail: [email protected]).

G. Mao is with the School of Computing and Communications, University ofTechnology Sydney, Ultimo, NSW 2007, Australia, and also with the Data61,CSIRO Canberra ACT, 2600, Australia (e-mail: [email protected]).

L. Hanzo is with the Department of Electronics and Computer Science,University of Southampton, Southampton SO17 1BJ, U.K. (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2016.2606765

are capable of achieving substantial SDP improvement, compared Q133with the benchmark schemes. 34

Index Terms—. 35

I. INTRODUCTION 36

I T IS forecast that at least a 100x network capacity increase 37

will be required to meet the traffic demands in 2020 [1]. As 38

a result, vendors and operators are now looking at using every 39

tool at hand to improve network capacity [2]. 40

In addition, a substantial contribution to the traffic explosion 41

comes from the repeated download of a small portion of popu- 42

lar contents, such as popular movies and videos [3]. Therefore, 43

intelligent caching in wireless networks has been proposed for 44

effectively reducing such duplicated transmissions of popular 45

contents, as well as for offloading the traffic from the over- 46

whelmed macrocells to small cells [4], [5]. Caching in third- 47

generation (3G) and fourth-generation (4G) wireless networks 48

was shown to be able to reduce the traffic by one third to two 49

thirds [6]. 50

Several caching strategies have been proposed for wireless 51

networks. Woo et al. [7] analyzed the strategy of caching con- 52

tents in the evolved packet core of local thermal equilibrium 53

(LTE) networks.The strategy of caching contents in the radio 54

access network, with an aim to place contents closer to mo- 55

bile users (MUs) was studied in [8] and [9]. The concept of 56

small-cell caching, referred to as “Femtocaching” in [9] and 57

[10], utilized small-cell base stations (SBS) in heterogeneous 58

cellular networks as distributed caching devices. Caching strate- 59

gies conceived for device-to-device (D2D) networks were in- 60

vestigated in [11]–[13], where the mobile terminals serve as 61

caching devices. The coexistence of small-cell caching and 62

D2D caching is indeed also a hot research direction. In [14], 63

Yang et al. considered the joint caching in both the relays and 64

a subset of the mobile terminals, which relies on the coex- 65

istence of small-cell caching and D2D caching. Moreover, a 66

coded caching scheme was proposed in [15] to improve system 67

performance. 68

In this paper, we focus on the small-cell caching because 69

1) the large number of SBSs in 4G and fifth-generation (5G) 70

networks already provide a promising basis for caching [2]; and 71

2) compared with D2D caching, small-cell caching has several 72

advantages, such as the abundance of power supply, fewer grave 73

security issues, and more reliable data delivery. As illustrated in 74

Fig. 1, with small-cell caching, popular contents are transmitted 75

and cached in the storage of the SBSs during off-peak hours. 76

Then in peak hours, if an MU can find its requested content in 77

0018-9545 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.


Fig. 1. Small-cell caching.

a nearby SBS, the MU can directly download the content from78

such SBS.79

There are generally two approaches to implement the small-80

cell caching, i.e., the deterministic content placement and the81

nondeterministic content placement. In [9], [16], and [17], the82

deterministic contents placement was analyzed. In these works,83

the placement of popular contents was optimized using the in-84

formation of the network node locations and the statistical or in-85

stantaneous channel states. However, in practice, the geographic86

distribution of MUs and the wireless channels are time variant.87

Thus, the optimal content placement strategy has to be fre-88

quently updated in the deterministic content placement, leading89

to a high complexity and fewer tractable results. On the other90

hand, the nondeterministic content placement permits simple91

implementation and has a good tractability. In [18] and [14],92

the distributions of SBSs and MUs were modeled as homo-93

geneous Poisson point processes (HPPPs) to obtain a general94

performance analysis for the small-cell caching. However, in95

these works, all the SBSs were assumed to cache the same copy96

of certain popular contents. In [11], probabilistic content place-97

ment was proposed and analyzed in the context of D2D caching,98

where each mobile terminal caches a specific subset of the con-99

tents with a given caching probability. The throughput versus100

outage tradeoff was analyzed and the optimal caching distribu-101

tion was derived for a grid network relying on a particular proto-102

col model. The idea of probabilistic content placement was also103

investigated in the coded multicasting system [19]. Compared104

with caching the same copy of certain popular contents in all the105

SBSs, probabilistic content placement in small-cell caching can106

provide more flexibility. Therefore, in this paper, we focus on107

small-cell caching relying on probabilistic content placement,108

shortened as probabilistic small-cell caching (PSC) for brevity.109

In small-cell networks, there are two network architectures,110

namely, the always-on architecture and the dynamic on–off ar-111

chitecture. The always-on architecture is a common practice in112

the current cellular networks, where all the SBSs are always ac-113

tive. By contrast, in the dynamic on–off architecture, the SBSs114

are only active, when they are required to provide services to115

nearby MUs [20]. Aiming for saving energy consumption and116

mitigating unnecessary intercell interference, the dynamic on–117

off architecture has been proposed and it is currently under118

investigation in 3GPP as an important candidate of 5G tech-119

nologies in future dense and ultradense small-cell networks [2],120

[21], [22]. Energy consumption is of critical interest in future 5G121

systems [23], [24], especially in ultradense networks. Compared122

with the power-thirsty always-on architecture, where the energy123

consumption grows with the network’s densification, the energy124

consumption of the ultradense network relying on the dynamic 125

on–off architecture mainly depends on the density of MUs in 126

the network [2]. The in-depth investigation of the associated 127

energy consumption issues of wireless caching will constitute 128

our future work. 129

Against this background, we study the PSC under the above- 130

mentioned pair of network architectures. First, we use a stochas- 131

tic geometry to develop theoretical results of the probability 132

Pr(D) that MUs can successfully download contents from the 133

storage of SBSs. Second, we investigate the impact of the SBSs’ 134

parameters on Pr(D), namely, that of the transmission power 135

P and of the deployment intensity λs . In the always-on archi- 136

tecture, although Pr(D) monotonically increases with either P 137

or λs , it approaches a constant when P or λs is sufficiently 138

high. In the dynamic on–off architecture, Pr(D) reaches a con- 139

stant when P is high enough, while it keeps on increasing as 140

λs grows. Most importantly, we optimize the caching probabil- 141

ities for maximizing Pr(D) in the pair of network architectures 142

considered. We emphasize that it is quite a challenge to ap- 143

ply optimization theory to an objective function obtained from 144

stochastic geometry analysis, especially to derive a closed-form 145

expression for the optimal solution. Our results will demonstrate 146

that in the always-on architecture, the optimal subset of contents 147

to be cached depends on the content request probabilities, while 148

in the dynamic on–off architecture, it also depends on the MU- 149

to-SBS intensity ratio. Most interestingly, in both architectures, 150

the optimal caching probabilities can be expressed as linear 151

functions of the square root of the content request probabilities. 152

The rest of the paper is structured as follows. In Section II we 153

describe the system model, while in Section III we present the 154

definition of PSC and formulate the probability that MUs can 155

successfully download contents from the storage of SBSs. The 156

main analytical results characterizing this successful download 157

probability (SDP) are presented in Section IV. In Section V, 158

we optimize the caching probabilities in both of the network 159

architectures for maximizing the derived SDP. The accuracy of 160

the analytical results and the performance gains of optimization 161

are characterized by simulations in Section VI. Finally, our 162

conclusions are offered in Section VII. 163

II. SYSTEM MODEL 164

We consider a cellular network supporting multiple MUs by 165

the SBSs operating within the same frequency spectrum. We 166

model the distribution of the SBSs and that of the MUs as two 167

independent HPPPs, with the intensities of λs and λu , respec- 168

tively. The transmission power of the SBSs is denoted by P . 169

The path loss of the channel spanning from an SBS to an MU 170

is modeled as d−α , where d denotes the distance between them, 171

and α denotes the path-loss exponent. The multipath fading is 172

modeled as Rayleigh fading with a unit power, and hence the 173

channel’s power gain is denoted by h ∼ exp(1). All the channels 174

are assumed to be independently and identically distributed. 175

A. Network Architectures 176

We consider two network architectures. 177

1) Always-On Architecture: In this architecture, all the 178

SBSs are assumed to be active, i.e., all the SBSs are 179


continuously transmitting signals. This architecture is com-180

monly employed in the operational cellular networks [25]. The181

rationale for this architecture is that the number of SBSs is182

usually much lower than that of MUs, and thus each and every183

SBS has to be turned ON to serve the MUs in its coverage.184

2) Dynamic On–Off Architecture: In this architecture, an185

SBS will be active only when it has to provide services to its as-186

sociated MUs. In future 5G networks, the intensity of deployed187

SBSs is expected to be comparable to or even potentially higher188

than the intensity of MUs [2]. In such ultradense networks, hav-189

ing an adequate received signal coverage is always guaranteed,190

since the distance between an MU and its serving SBS is short,191

but the interference becomes the dominant issue. With the goal192

of mitigating the potentially avoidable intercell interference and193

saving energy, the dynamic on–off architecture has been identi-194

fied as one of the key technologies in 5G networks [20]. With the195

dynamic on–off architecture, an SBS will switch to its idle mode,196

i.e., turn OFF its radio transmission, if there is no MU associated197

with it, otherwise, it will switch back to the active mode.198

B. File Request Model199

We consider a contents library consisting of M different files.200

Note that M does not represent the number of files available on201

the Internet, but the number of popular files that the MUs tend202

to access. We denote by qm the probability that the mth file Fm203

will be requested. By stacking qm into {qm : m = 1, · · · ,M},204

we can get the probability mass function (PMF) of requesting205

the M files. According to [26], the request- PMF of the files can206

be modeled as a Zipf distribution. More specifically, for Fm , its207

request probability qm is written as208

qm =1

mβ

∑Mi=1

1iβ

(1)

where β is the exponent of the Zipf distribution and a large β209

implies having an uneven popularity among those files. From210

(1), qm tends to zero, as M → ∞ when β < 1, while it con-211

verges to a constant value when β > 1. Note that (1) implies that212

the indices of the files are not randomly generated, but follow a213

descending order of their request probabilities.214

Due to the limited storage of SBSs, an SBS is typically unable215

to cache the entire file library. Therefore, we assume that the216

library is partitioned into N nonoverlapping subsets of files,217

referred to as file groups (FGs), and each SBS can cache only218

one of the N FGs. Note that the same FG can be redundantly219

stored in multiple SBSs. The scenario of FGs with overlapping220

subsets of files will be considered later, which will be compared221

with the nonoverlapping scenario. We denote the nth FG, n ∈222

{1, · · · , N} by Gn . The probability Qn that an MU requests a223

file in FG Gn , is thus given by224

Qn =∑

m, forFm ∈Gn

qm. (2)

III. PROBABILISTIC SMALL-CELL CACHING STRATEGY225

In this section, we introduce the PSC strategy, and formulate226

the probability that MUs can successfully download contents227

from the storage of the SBSs, which is an important performance 228

metric of small-cell caching. 229

Generally, caching consists of two phases: a contents place- 230

ment phase and a contents delivery phase [27]. In the contents 231

placement phase, popular contents are transmitted and cached 232

in the storage units of network devices that are close to MUs. In 233

the contents delivery phase, the popular cached contents can be 234

promptly retrieved for serving the MUs. 235

A. Contents Placement Phase 236

In the content placement phase of PSC, each SBS indepen- 237

dently caches FGGn with a specific caching probability, denoted 238

by Sn . Hence, from the perspective of the entire network, the 239

fraction of the SBSs that caches Gn equals to Sn . Since the dis- 240

tribution of SBSs in the network is modeled as an HPPP with the 241

intensity of λs , according to the thinning theorem of HPPP [28], 242

we can view the distribution of SBSs that cache Gn as a thinned 243

HPPP with the intensity of Snλs . 244

We assume that at a particular time instant, an MU can only 245

request one file, and hence, the distribution of MUs who request 246

the files in Gn can also be modeled as a thinned HPPP with 247

the intensity Qnλu . We treat the SBSs that cache Gn together 248

with the MUs that request the files in Gn as the nth tier of the 249

network, shortened as Tier-n. 250

B. Contents Delivery Phase 251

During the contents delivery phase, an MU that requests a 252

file in Gn will associate with the nearest SBS that caches Gn , 253

and then attempts to download the file from it. We assume 254

that only when the received signal-to-interference-and-noise- 255

ratio (SINR) at the MU is above a prescribed threshold, can the 256

requested file be successfully downloaded. 257

If the MU cannot download the requested file from the cached 258

SBS, the requested file would be transmitted to the MU from 259

a remote content provider, which means the data should flow 260

across the Internet, the cellular core network, and the backhaul 261

network, as illustrated in Fig. 1. 262

C. Probability of Successful Download 263

Recent surveys show that 96% of the operators consider 264

backhaul as one of the most important challenges to small- 265

cell deployments, and this issue is exacerbated in ultradense 266

networks [29], [30]. If an MU can successfully download a re- 267

quested file from storages of SBSs, the usage of the backhaul 268

network will be greatly reduced and the transmission latency 269

of a requested file will be significantly shortened. Therefore, 270

we assume that a successful download of a requested file from 271

storages of SBSs is always beneficial to the network perfor- 272

mance. Accordingly, we focus on our attention on this SDP as 273

the performance metric for small-cell caching in the following. 274

According to Slyvnyak’s theorem for HPPP [28], an existing 275

point in the process does not change the statistical distribution of 276

other points of the HPPP. Therefore, the probability that an MU 277

in Tier-n can successfully download the contents from SBSs 278

can be obtained by analyzing the probability that a typical MU 279


in Tier-n, say located at the origin, can successfully download280

the contents from its associated SBS in Tier-n.281

When the MU considered requests a file in Gn , its received282

SINR from its nearest SBS in Tier-n can be formulated as283

γn (z) =Phx0z

−α


‖xj‖−α + σ2(3)

where σ2 denotes the Gaussian noise power, z is the distance284

between the typical MU and its nearest SBS in Tier-n, xj repre-285

sents the locations of the interfering SBSs, Φ denotes the set of286

simultaneously active SBSs, and x0 is the location of the serving287

BS at a distance of z. Additionally, ‖xj‖ denotes the distance288

between xj and the typical MU, while hx0 and hxjdenote the289

corresponding channel gains.290

Since the intercell interference is the dominant factor deter-291

mining the signal quality in the operational cellular networks,292

especially when unity frequency reuse has been adopted for im-293

proving the spectrum efficiency, the minimum received SINR is294

used as the metric of successful reception. Let δ be the minimum295

SINR required for successful transmissions and Dn be the event296

that the typical Tier-n MU successfully receives the requested297

file from the associated Tier-n SBS. Then, the probability of Dn298

can be formulated as299

Pr(Dn ) = Pr[γn (z) ≥ δ]. (4)

Considering the request probabilities of Gn and based on the300

result of Pr(Dn ), we obtain the average probability that the301

MUs can successfully download contents from the storage of302

the SBSs, denoted by Pr(D), as303

Pr(D) =N∑

n=1

Qn · Pr(Dn ). (5)

In essence, Pr(D) quantifies the weighted sum of the SDP,304

where the weights are the request probabilities reflecting the305

importance of the files.306

IV. PERFORMANCE ANALYSIS OF SMALL-CELL CACHING307

In this section, we derive the SDP Pr(D) for the pair of308

network architectures. Some special cases are also considered309

with an aim to obtain more insights into the design of PSC.310

A. Always-On Architecture311

Our main result on the probability Pr(D) for the always-on312

architecture is summarized in Theorem 1.313

Theorem 1: In the always-on architecture, the probability314

Pr(D) is given by315

Pr(D) =N∑

n=1

QnPr(Dn )

=N∑

n=1

Qn

∫ ∞

0πSnλs exp

(

−zαδσ2

P

)

exp(−πλsz

2 ((1 − Sn )C(δ, α) + SnA(δ, α) + Sn ))dz2

(6)

where A(δ, α) � δ 2α−2 2F1(1, 1 − 2

α ; 2 − 2α ;−δ), and C(δ, α) 316

� 2α δ

2α B( 2

α , 1 − 2α ). Furthermore, 2F1(·) denotes the hyperge- 317

ometric function, and B(·) represents the beta function [31]. 318

Proof: See Appendix A. � 319

From (6), we conclude that the probability Pr(D) increases 320

as the transmission power P grows, because exp(− zα δσ 2

P ) in- 321

creases with P . Since it remains a challenge to obtain deeper 322

insights from (6), which is not a closed-form expression, two 323

special cases are examined in the sequel to gain deeper insight 324

on the performance behavior of Pr(D). 325

1) Path-Loss Exponent α = 4: According to 3GPP measure- 326

ment [32], the typical value of the path-loss exponent for SBSs 327

in practical environments is around 4. Substituting this typical 328

value of α = 4 into (6), we have 329

Pr(D) |α=4 =N∑

n=1

QnπSn

√π

4δ

Pλ2s

σ2erfcx

(π

2·

√Pλ2

s

δσ2

(Sn +

π

2

√δ(1 − Sn ) + Sn

√δ arctan

√δ))

(7)

where erfcx(x) � exp(x2)erfc(x) is the scaled complementary 330

error function [33]. 331

Regarding the relationship between Pr(D) and λs , we pro- 332

pose Corollary 1. 333

Corollary 1: In the always-on architecture, for the special 334

case of α = 4, Pr(D) monotonically increases with the increase 335

of λs . 336

Proof: See Appendix B. � 337

From the results obtained in (6) that Pr(D) increases as P 338

grows, and based on Corollary 1, we conclude that when α = 4, 339

the SDP Pr(D) can be improved by either increasing the SBSs’ 340

transmission power P or the SBSs’ deployment intensity λs . 341

Furthermore, since (7) can be viewed as a function of the variable 342

Pλ2s , the effect of increasing P to kP on Pr(D) is equivalent to 343

increasing λs to√

kλs , where k is a positive constant. 344

Moreover, according to the property of the function erfcx(x), 345

i.e., limx→∞ erfcx(x) = 1√πx

, we have 346

limP →∞

Pr(D) |α=4 = limλs →∞

Pr(D) |α=4

=N∑

n=1

QnSn

π2

√δ + (

√δ arctan

√δ + 1 − π

2

√δ)Sn

. (8)


Remark 1: In the always-on architecture, given σ2 and δ, the 348

value of Pr(D) monotonically grows with the increase of P and 349

λs , and it converges to a constant, when P or λs is sufficiently 350

large. 351


network, where the noise level is much lower than the interfer- 353

ence, the impact of the noise can be neglected. In such cases, 354

we assume that σ2 = 0, and it follows that Pr(D) in (6) can be 355

rewritten as 356

Pr(D) |σ 2→0 =N∑

n=1

QnSn

SnA(δ, α) + (1 − Sn )C(δ, α) + Sn. (9)



Remark 2: In the always-on architecture operating in an358

interference-limited network, the probability of successful359

download depends only on the request probabilities and caching360

probabilities of the FGs, i.e., Qn and Sn .361

Note that in the scenario, where the different FGs may have362

an overlapping subset of files, the probability Pr(D) still has363

the same formulation as (6). However, all the subscripts n in364

(6) should be changed to m, because we should consider both365

the request probability and the caching probability of each file366

Fm , i.e., Sm and Qm , instead of each FG Gn . Therefore, in this367

scenario, the specific SBSs that cache Fm and the MUs that368

request Fm are viewed as Tier-m. Since all the derivations are369

the same, our main results summarized in Theorem 1 as well370

as the aforementioned corollary and remarks, are still valid in371

conjunction with the subscript m. Hence we omit the analysis372

for this scenario with overlapping subsets of files for brevity.373


As mentioned, in the dynamic on–off architecture an SBS is375

only active, when it has to provide services for the associated376

MUs. Specifically, an SBS in Tier-n is only active, when there377

is at least one MU in Tier-n located in its Voronoi cell. Hence,378

the probability that an SBS in Tier-n is active, which is de-379

noted by Pr (An ), should be considered for the dynamic on–off380

architecture.381

Our main result on the probability Pr(D) for the dynamic382

on–off architecture is summarized in Theorem 2.383

Theorem 2: In the dynamic on–off architecture, the proba-384

bility Pr(D) is given by385

Pr(D) =N∑

n=1

QnPr(Dn )

=N∑

n=1

Qn

∫ ∞

0πSnλs exp

(

−zαδσ2

P

)

exp

(

− πλsz2

( N∑

i=1,i =n

Pr(Ai)SiC(δ, α) + Pr(An )SnA(δ, α) + Sn

))

dz2 (10)

where Pr(An ) denotes the probability that an SBS in Tier-n is386

in the active mode, and387

Pr(An ) ≈ 1 −(

1 +Qnλu

3.5Snλs

)−3.5

. (11)

Proof: See Appendix C. �388

Compared to Pr(D) in the always-on architecture, Pr(D) in389

the dynamic on–off architecture also depends on the intensity390

of the MUs λu . The reason behind this is that the number of391

active SBSs in the network depends on the number of MUs in392

the network.393


Remark 3: In the dynamic on–off architecture, given σ2 and395

δ, the value of Pr(D) monotonically increases with the increase396

of the transmission power P .397


network, substituting σ2 = 0 into (10), we have 399

Pr(D) |σ 2→0 =

N∑

n=1

QnSn

Pr(An )SnA(δ, α) +∑N

i=1,i =n Pr(Ai)SiC(δ, α) + Sn

.

(12)


Remark 4: In the dynamic on–off architecture operating in 401

an interference-limited network, the probability of successful 402

download Pr(D) is independent of P , and depends only on Qn , 403

Sn as well as on the MU-to-SBS intensity ratio λu/λs . 404

When considering the scenario of FGs with overlapping sub- 405

sets of files, the average probability Pr(D) cannot be formulated 406

as the sum of Pr(Dn ) as in (5). Furthermore, we cannot formu- 407

late Pr(D) as Pr(D) =∑M

m=1 Pr(Dm ), which we propose for 408

the overlapping scenario in the always-on architecture. This is 409

because in the dynamic on–off architecture the active probability 410

of an SBS depends on the specific FG that it caches. Therefore, 411

the analysis of Pr(D) in the dynamic on–off architecture con- 412

sidering the scenario with overlapping subsets of files requires 413

further investigations as part of our future research. 414

V. OPTIMIZATION OF THE CACHING PROBABILITY 415

A larger Pr(D) always benefits the network because of 1) the 416

backhaul saving and 2) the low-latency transmission of local 417

contents from SBSs [2]. Based on such facts, in this section, we 418

concentrate on maximizing Pr(D) by optimally designing the 419

caching probabilities of the contents in the system, denoted by 420

{SOptn : n = 1, . . . , N}. 421

Note that there is a paucity of literature on applying opti- 422

mization theory relying on an objective function obtained from 423

stochastic geometry analysis, especially, when aiming for deriv- 424

ing a closed-form expression of the optimal solution. In order to 425

facilitate this optimization procedure, we ensure the mathemat- 426

ical tractability of the objective function by using a simple user 427

association strategy and neglect the deleterious effects of noise. 428


From (9), we can formulate the optimization problem of max- 430

imizing Pr(D) as 431

max{Sn }

Pr(D) = max{Sn }

N∑

n=1

QnSn

(1 − Sn )C(δ, α) + SnA(δ, α) + Sn

s.t.N∑

n=1

Sn = 1

Sn ≥ 0, n = 1, . . . , N.

(13)

The solution of Problem (13) is presented in Theorem 3. 432

Theorem 3: In the always-on architecture, the optimal 433

caching scheme, which is denoted by the file caching PMF 434

{Soptn }, that maximizes the average probability of successful 435


download, is given by436

Soptn =

⎡

⎢⎢⎢

√Qn

ξ − C(δ, α)

A(δ, α) − C(δ, α) + 1

⎤

⎥⎥⎥

+

, n = 1, . . . , N (14)

where√

ξ =∑N ∗

n = 1

√Qn

(N ∗−1)C (δ,α)+A(δ,α)+1 , �Ω + � max{Ω, 0}, and437

N ∗, 1 ≤ N ∗ ≤ N satisfies the constraint that Sn ≥ 0 ∀n.438

Proof: It can be shown that the optimization Prob-439

lem (13) is concave and can be solved by invoking the440

Karush−Kuhn−Tucker conditions [34]. The conclusion then441

follows. �442

From (14), when the request probability obeys Qn >443

ξC2(δ, α), Gn is cached with a caching probability of Soptn , oth-444

erwise, it is not cached. This optimal strategy implies that ideally445

the SBSs should cache the specific files with high request prob-446

abilities, while those files with low request probabilities should447

not be cached at all due to the limited storage of SBSs in the net-448

work. Moreover, we can see that from (14) the optimal caching449

probability of an FG is a linear function of the square root of its450

request probability.451

Regarding the scenario of FGs associated with overlapping452

subsets of files, as we mentioned before, Pr(D) in this scenario453

has the same formulation as that in the nonoverlapping scenario.454

Therefore, the optimal caching probability ofFm in the scenario455

of FGs having overlapping subsets of files can be formulated as456

SOptm = min

⎧⎪⎨

⎪⎩

⎡

⎢⎢⎢

√Qm

ξ − C(δ, α)

A(δ, α) − C(δ, α) + 1

⎤

⎥⎥⎥

+

, 1

⎫⎪⎬

⎪⎭(15)

where√

ξ =∑M ∗

m = 1

√Qm

(M ∗−V )C (δ,α)+V (A(δ,α)+1) , and M ∗(1 ≤ M ∗ ≤457

M), satisfies the constraint that 0 ≤ Sm ≤ 1 ∀m, and V de-458

notes the number of files in each FG.459

Compared with the nonoverlapping scenario, the presence of460

overlapping subsets among the FGs provides a higher grade of461

diversity in the system. However, based on our simulations to462

be discussed in the sequel, we find that the gain of maximum463

Pr(D) obtained as a benefit of this diversity is limited, while the464

algorithm associated with the optimal caching strategy of (15)465

is more complex than that of (14).466


In this architecture, as shown in (11), the probability Pr(An )468

that an SBS in Tier-n is in the active mode, is a function of the469

ratio Qnλu/Snλs . Since the intensity of SBSs is much higher470

than the intensity of the MUs in this architecture, i.e., we have471

λs � λu , the SBS activity probability Pr(An ) in (11) can be472

approximated as473

Pr(An ) ≈ Qnλu

Snλs. (16)

Substituting (16) into (12) and (5), we can formulate the op-474

timization problem of maximizing the successful downloading475

probability as 476

max{Sn ,εn }

Pr(D) =

max{Sn ,εn }

N∑

n=1

QnSn

Qnλu

λsA(δ, α) · εn +

∑i:i =n Qi

λu

λsC(δ, α) · εi +Sn

s.t.N∑

n=1

Sn = 1

Sn ≥ 0, n = 1, . . . , N

εn =

{1, if Sn > 0

0, if Sn = 0.(17)

Different from the optimization problem in (13), the variable 477

εn is introduced to indicate whether Gn is cached. Due to the 478

existence of εn , which implies 2N hypotheses of file caching 479

states, Problem (17) is difficult to solve. Nevertheless, we man- 480

age to find the solution and summarize it in Theorem 4. 481

Theorem 4: The optimal caching scheme, i.e., the optimal 482

file caching PMF {SOptn }, that maximizes the average probabil- 483

ity of successful download, is given by 484

SOptn

=

⎧⎪⎨

⎪⎩

ζK

√QnξK C(δ, α) − Q2

n (C(δ, α) − A(δ, α))−(ξK

λu

λsC(δ, α) − Qn

λu

λs(C(δ, α) − A(δ, α))

), n ≤ K

0, K < n ≤ N.

(18)

where 485

ξK �K∑

i=1

Qi

ζK �1 + KξK

λu

λsC(δ, α) − ξK

λu

λs(C(δ, α) − A(δ, α))

∑Ki=1

√QiξK C(δ, α) − Q2

i (C(δ, α) − A(δ, α)). (19)

Regarding K, we have 486

K = arg maxk

{Dk : k = 1, 2, . . . , N̂

}(20)

where 487

Dk � ξk

−λu

λs

(∑kn=1

√QnξkC(δ, α) − Q2

n (C(δ, α) − A(δ, α)))2

1 + kξkλu

λsC(δ, α) − ξk

λu

λs(C(δ, α) − A(δ, α))

(21)

and 488

N̂ =

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

N, ifλu

λs< aN

N − 1, if aN ≤ λu

λs< aN −1

· · ·1, if a2 ≤ λu

λs.

(22)


Algorithm 1: Optimal Caching Probabilities in theDynamic On–Off Architecture.

1: Set j = N .2: Compute ξj =

∑ji=1 Qi , and ϑj and aj in (24) and (23).

3: Compare λu

λswith aj . If λu

λs< aj , go to Step 4;

otherwise, set j = j − 1 and go to Step 2.4: Set N̂ = j.5: Compute ξk =

∑ki=1 Qi and Dk in (21), k = 1, · · · , N̂ .

6: Set K = arg maxk

{Dk}.

7: Compute ξK and ζK in (19), then compute SOptn in (18).

489

Furthermore, the segmentation parameter aj , j = 2, . . . , N490

is given by491

aj =ϑj

(ϑj ξj − Qj )(C(δ, α) − A(δ, α)) + (1 − jϑj )ξjC(δ, α)(23)

where492

ϑj �

√QjξjC(δ, α) − Q2

j (C(δ, α) − A(δ, α))∑j

i=1

√QiξjC(δ, α) − Q2

i (C(δ, α) − A(δ, α)). (24)

Proof: See Appendix D. �493

To get a better understanding of Theorem 4, we propose494

Algorithm 1 to implement Theorem 4.495

From Theorem 4, we have the following remarks.496

Remark 5: In the always-on architecture, the optimal number497

of FGs to be cached depends only on {Qn : n = 1, · · · , N}. By498

contrast, in the dynamic on–off architecture, the optimal number499

of FGs to be cached depends not only on {Qn} but on the MU-500

to-SBS intensity ratio λu/λs in the network as well.501

Remark 6: According to (22), given λu , more FGs tend to be502

cached in the SBSs, when λs becomes higher. Moreover, when503

the intensity of SBSs is not sufficiently high to cache all the504

FGs, the SBSs should cache the specific files with relatively high505

request probabilities, which is consistent with the conclusion for506

the always-on architecture.507

Remark 7: In (18), with a practical region of the SINR508

threshold and path-loss exponent from 3GPP, i.e., for δ ∈509

[0.5, 3] and α ∈ (2, 4], we have ξK C(δ, α) � Qn (C(δ, α) −510

A(δ, α)), and the optimal caching probability SOptn ≈511

ζK

√Qn

λu

λsξK C(δ, α) − ξK

λu

λsC(δ, α). From (14) and (18), it512

is interesting to observe that the optimal caching scheme in both513

the always-on architecture and in the dynamic on–off architec-514

ture follow a square root law, i.e., SOptn is a linear function of515 √

Qn .516

VI. NUMERICAL AND SIMULATION RESULTS517

In this section, we present both our numerical and Monte-518

Carlo simulation results of Pr(D) in various scenarios. In the519

Monte-Carlo simulations, the performance is averaged over520

1000 network deployments, where in each deployment SBSs521

and MUs are randomly distributed in an area of 5 × 5 km ac-522

cording to an HPPP distribution. The intensity of MUs in the523

network is 200/km2. The transmission power of the SBSs, the524

Fig. 2. Numerical and simulation results of Pr(D) of the O-PSC strategy inthe always-on architecture.

noise power, the path-loss exponent, and the SINR threshold are 525

set to 30 dBm, –104 dBm, 4 and 0.25(−6 dB), respectively [32]. 526

In the simulations of the always-on architecture, the deployment 527

intensity of SBSs is set to 80/km2, while in the simulations of 528

the dynamic on–off architecture, the intensity is set to 400/km2. 529

Furthermore, we consider a file library consisting of M = 100 530

files, and we partition the file library into N = 10 FGs with a 531

simple grouping strategy that the mth file belongs to Gn if 532

m ∈ [MN (n − 1) + 1, . . . , M

N n] ∀n ∈ {1, . . . , N}. Note that the 533

specific choice of the file grouping strategy is beyond the scope 534

of this paper and it does not affect our results, because it only 535

changes the specific values of the request-PMF {Qn}. 536

In addition, we consider the following two PSC strategies. 537

1) The request probability based PSC (RP-PSC) [12], where 538

the caching probability of one FG equals to its request 539

probability, i.e., Sn = Qn . Intuitively, a particular FG is 540

more popular than another, the RP-PSC strategy will des- 541

ignate more SBSs to cache it. This strategy is evaluated 542

as a benchmark in our simulations. 543

2) The proposed optimized PSC (O-PSC) based on (14) in 544

the always-on architecture and (18) in the dynamic on–off 545

architecture, where Sn = Soptn . 546


Fig. 2 compares the numerical and the simulation results con- 548

cerning Pr(D) of the O-PSC strategy. First, it can be seen that 549

the numerical results closely match the simulation results in all 550

scenarios. In the following, we will focus on the analytical re- 551

sults only, due to the accuracy of our analytical results. Second, 552

Pr(D) increases with the Zipf exponent β. With a larger β, the 553

request probabilities of files are more unevenly distributed. In 554

such cases, a few FGs dominate the requests and caching such 555

popular FGs gives a large Pr(D). Third, Pr(D) will be lower, if 556

the value of δ becomes higher. This is because when the SINR 557

threshold is increased, the probability that the received SINR 558

from the SBS storing the file exceeds this threshold is reduced. 559

Finally, we can see that Pr(D) increases as the number of FGs 560


Fig. 3. Pr(D) of the O-PSC strategy with different P and λs in the always-onarchitecture.

Fig. 4. Comparison of Pr(D) versus δ of the RP-PSC and O-PSC strategiesin the always-on architecture.

decreases. Since each SBS only caches one FG, decreasing the561

number of FGs implies that each SBS caches more files. Hence,562

this Pr(D) improvement comes from increasing the stored con-563

tents in each SBS.564

Fig. 3 shows the SDP Pr(D) for the O-PSC strategy when the565

transmission power P of SBSs varies within 20–40 dBm and the566

deployment intensity λs of SBSs varies within 10–400/km2. To567

highlight the asymptotic behavior of Pr(D) with the growth of568

P , we set the noise power to −50 dBm. We can see from the569

figure that Pr(D) increases monotonically with P or λs . The570

value of Pr(D) remains constant, when P or λs is sufficiently571

high. This result illustrates the limit of Pr(D) in the always-on572

architecture shown in (8).573

In Fig. 4, we plot Pr(D) versus the SINR threshold δ to com-574

pare the performances of the RP-PSC and O-PSC strategies. We575

can see that the proposed O-PSC strategy exhibits a significantly576

better performance than the RP-PSC strategy. With the number577

of FGs N = 10, the performance gain in terms of Pr(D) pro-578

vided by the O-PSC strategy ranges from 20% to 50%, when579

Fig. 5. Comparison of Pr(D) versus β of the RP-PSC and O-PSC strategiesin the always-on architecture.

δ varies from 0.1 to 1. When δ is high, the probability that 580

MUs can directly download the files from the storage of SBSs 581

becomes small. In such cases, the advantage of optimizing the 582

caching probabilities of the FGs is more obvious. 583

Even more significant Pr(D) improvement can be observed 584

for the case of N = 20 than that for N = 10. A larger number of 585

FGs means that less contents can be cached in each SBS, which 586

implies a very limited storage capacity. In such cases, the benefit 587

of optimizing the caching probabilities is more significant. 588

Fig. 5 compares Pr(D) in the context of RP-PSC and O-PSC 589

strategies versus the Zipf exponents β. First, we can see that 590

the proposed O-PSC strategy greatly outperforms the RP-PSC 591

strategy in terms of Pr(D). With the number of FGs N = 20, 592

the performance gain of Pr(D) ranges from 65% to 20% when 593

β varies from 0.5 to 1.5. In other words, the Pr(D) improve- 594

ment decreases, as β grows. The reason behind this trend is 595

that for a large β, a small fraction of FGs dominate the file 596

requests. Once the SBSs cache these very popular FGs, Pr(D) 597

will become sufficiently high. Thus, the additional gain given 598

by the optimization of caching probabilities becomes smaller. 599

Furthermore, compared with the case N = 10, the Pr(D) im- 600

provement when N = 20 is more significant. The reason for 601

this phenomenon has been explained above. 602

Fig. 6 compares Pr(D) in conjunction with the O-PSC strate- 603

gies in the overlapping and nonoverlapping scenarios. Since the 604

total number of files in our simulations is 100, in the figure, the 605

curves of “FGNo = 10” and “FGNo = 20” are compared against 606

the curves of “FilesPerGroup = 10” and “FilesPerGroup = 5,” 607

respectively. We can see that the performance of SDP in the sce- 608

nario of FGs having overlapping subsets of files is better than 609

that of the nonoverlapping subsets of files. The reason for this 610

observation is that allowing overlapping amongst the different 611

FGs provides a beneficial diversity of the FGs. Furthermore, we 612

can see that when the SINR threshold is increased, the advan- 613

tage of the overlapping scenario wanes. This is because when 614

the SINR threshold is high, the O-PSC strategy tends to cache 615

fewer popular files and the diversity of FGs becomes of limited 616

benefit here. 617


Fig. 6. Comparison of Pr(D) versus δ in the overlapping and nonoverlappingscenarios in the always-on architecture.

Fig. 7. Numerical and simulation results of Pr(D) of the O-PCP strategy inthe dynamic on–off architecture.


Fig. 7 shows our comparison between the numerical and sim-619

ulation results of Pr(D) for the O-PSC strategy. We can see620

from this figure that the numerical results closely match the621

simulation results in all scenarios. Similar phenomena can be622

observed as in the always-on architecture.623

1) Pr(D) decreases upon increasing the SINR threshold δ.624

2) Pr(D) increases with the Zipf exponent β.625

3) Pr(D) increases when the number of FGs decreases.626

The reasons behind these trends are the same as those dis-627

cussed for the always-on architecture. Moreover, compared to628

Fig. 2, the value of Pr(D) in the dynamic on–off architecture629

of Fig. 7 is shown to be higher. The reason is that the dynamic630

on–off technique efficiently mitigates the potential avoidable631

interference in the network.632

Fig. 8 shows the performance of Pr(D) for the O-PSC strategy633

in the dynamic on–off architecture, when the transmission power634

Fig. 8. Pr(D) with different P and λs in the dynamic on–off architecture.

Fig. 9. Comparison of Pr(D) of the RP-PSC and O-PSC strategies versus δin the dynamic on–off architecture.

P of SBSs varies from 20 to 40 dBm and the SBS intensity λs 635

varies from 200 to 2000/km2. We can see from this figure that 636

Pr(D) increases monotonically, when either P or λs increases. 637

Moreover, we can see that when P increases to a sufficiently high 638

value, any further increase of P will no longer improve Pr(D). 639

However, the increase of λs will always improve Pr(D), as seen 640

in (12). 641

Fig. 9 compares Pr(D) of the RP-PSC and O-PSC strategies, 642

when the SINR threshold δ varies. It can be seen from the fig- 643

ure that compared to the RP-PSC strategy, Pr(D) is obviously 644

improved by the optimal caching PMF {SOptn } in the O-PSC 645

strategy. With the Zipf exponent β = 1, the performance gain 646

of Pr(D) ranges from 7% to 30%, when δ varies from 0.1 to 1. 647

This observation is similar to that in the always-on architecture. 648

That is, the Pr(D) improvement achieved by the O-PSC strat- 649

egy is more pronounced, when the SINR threshold is higher. 650

Furthermore, the Pr(D) improvement is higher when the Zipf 651

exponent β is lower. The reason for this is explained above. 652


Fig. 10. Comparison of Pr(D) of the RP-PSC and O-PSC strategies versusλs in the dynamic on–off architecture.

Furthermore, in order to verify the optimality of the solution653

given by our algorithm, we plot the optimal solution obtained654

from the exhaustive search over all legitimate file caching states,655

denoted by “Exh. Search” in the figure. Observed from the656

figure that our solution exactly matches the optimal solution of657

“Exh. Search,” which confirms our statement that the proposed658

solution achieves global optimality.659

In Fig. 10, we portray Pr(D) of the RP-PSC and the O-660

PSC strategies versus the SBS intensity λs . First, it can be661

seen that compared with the RP-PSC strategy, the optimization662

of the caching probabilities in the O-PSC strategy improves663

Pr(D) in all scenarios. This Pr(D) improvement achieved by664

the O-PSC strategy wanes slightly when λs increases because665

when the SBS intensity is higher, each MU becomes capable of666

associating with multiple SBSs, and thus, the probability that667

MUs can successfully download contents from SBSs will be668

higher. In such a case, the Pr(D) improvement obtained by the669

optimization of the FG caching probabilities remains limited. In670

addition, we verify the optimality of our solution by comparing671

it to the optimal solution obtained from the exhaustive search.672

VII. CONCLUSION673

In this paper, based on stochastic geometry theory, we ana-674

lyzed the performance of the PSC in a pair of network architec-675

tures. Specifically, we analyzed the probability Pr(D) that MUs676

can successfully download contents from the storage of SBSs.677

We concluded that increasing the SBSs’ transmission power678

P or their deployment intensity λs is capable of increasing the679

SDP. However, in the always-on architecture, Pr(D) remains680

constant when P or λs is sufficiently high, while in the dynamic681

on–off architecture, Pr(D) always increases as λs grows.682

Furthermore, in order to maximize Pr(D), we optimized the683

caching probabilities of the FGs. Our results demonstrated that684

in the always-on architecture, the optimal subset of FGs depends685

on the contents request probabilities. In the dynamic on–off ar-686

chitecture, a piecewise defined function of MU-to-SBS intensity687

ratio λu/λs was introduced in order to find the optimal subset 688

of FGs to be cached. Interestingly, a similar optimal caching 689

probability law was found for both architectures, i.e., SOptn is a 690

linear function of√

Qn . Our simulation results showed that the 691

proposed optimal caching probabilities of the FGs achieve a 692

substantial gain in both architecture in terms of Pr(D) compared 693

to the benchmark Sn = Qn , because more caching resources 694

are devoted to the more popular files in the proposed scheme. 695

APPENDIX APROOF OF THEOREM 1 696

In Tier-n of the always-on architecture, where the inten- 697

sity of the SBSs is Snλs , the PDF of z, i.e., the distance be- 698

tween the typical MU and its nearest SBS, follows fZ (z) = 699

2πSnλsz exp(−πSnλsz2). From (3) and (4), we have 700

Pr(Dn ) = Pr (γn (z) ≥ δ)

=∫ ∞

0Pr

[Phx0z

−α


‖xj‖−α + σ2� δ

]

fZ (z) dz

(a)=

∫ ∞


(

−zαδσ2

P

)

2πSnλsz exp(−πSnλsz2)dz (25)

where (a) is obtained by hx0 ∼ exp(1) and I �∑

xj ∈Φ\{x0} 701

hxj‖xj‖−α represents the interference. 702

The interference I consists of two independent parts: 1) I1: the 703

SBSs in other tiers, which are dispersed across the entire area of 704

the network, and 2) I2: the SBSs in the nth tier, whose distances 705

from the typical MU are larger than z. Due to the independence 706

of I1 and I2, we have EI [exp (−zαδI)] = EI1 [exp (−zαδI1)] · 707

EI2 [exp (−zαδI2)]. 708

Since the distribution of the SBSs in Tier-i is viewed as an 709

HPPP φi with Siλs and therefore, we have 710


= Ehx j,xj

⎡

⎢⎣

∏

xj ∈∑N

i = 1, i = n φi

exp(−zαδhxj

‖xj‖−α)

⎤

⎥⎦

(b)= Exj

⎡

⎢⎣

∏

xj ∈∑N

i = 1, i = n φi

1


⎤

⎥⎦

(c)= exp

⎛

⎝−N∑

i=1,i =n

Siλs

∫

R2

(

1 − 1

1 + δzα ‖xj‖−α

)

dxj

⎞

⎠

= exp

⎛

⎝−2π

N∑

i=1,i =n

Siλs1α

δ2α B

(2α

, 1 − 2α

)

z2

⎞

⎠ (26)

where (b) uses hxj∼ exp(1), and (c) uses E

[∏v∈Φ ξ (v)

]= 711

exp(−λΦ

∫(1 − ξ (v)) dv

). 712


As for I2, we have713


= exp(

−Snλs2π

∫ ∞

z

(

1 − 1


)

‖xj‖ d ‖xj‖)

(d)= exp

(

−Snλsπδ2α z2 2

α

∫ ∞

δ−1

l2α −1

1 + ldl

)

= exp(

−Snλsπz2 2δ

α − 2 2F1

(

1, 1 − 2α

; 2 − 2α

;−δ

))

(27)

where (d) uses l � δ−1z−α ‖xj‖α .714

Our proof is completed by plugging (26) and (27) into715

(25). �716

APPENDIX BPROOF OF COROLLARY 1717

Since we have Pr(D) =∑N

n=1 Qn Pr(Dn ), to prove that718

Pr(D) increases with the increase of λs , we only have to prove719

that Pr(Dn ) increases monotonically upon increasing λs ∀n.720

Thus, in the following, we focus our attention on the proof that721∂ Pr(Dn )

∂λs> 0.722

To simplify our discourse, we use C1 � πSn

2σ

√πPδ , and723

C2 � π2σ

√Pδ

(Sn + π

2

√δ(1 − Sn ) + Sn

√δ arctan

√δ)

.724

Obviously, we have C1 > 0 and C2 > 0. Then, Pr(Dn ) can be725

rewritten as726

Pr(Dn ) = C1λs exp(C22λ2

s)erfc(C2λs). (28)

Hence, we have727

∂ Pr(Dn )∂λs

= C1λs exp(C22λ2

s) (1 − erf(C2λs))

=(C1 exp(C2

2λ2s) + C1λs exp(C2

2λ2s)2C2

2λs

)erfc(C2λs)

− C1λs exp(C22λ2

s)2√π

C2 exp(−C22λ2

s)

= C1 exp(C22λ2

s)(1 + 2C22λ2

s)erfc(C2λs) − C1C2λs2√π

.

(29)

According to [35], the continued fraction expansion of the728

complementary error function is729

erfc(z) =z√π

exp(−z2)1

z2 + a1

1+ a 2z 2+

a 31+ ···

, am =m

2. (30)

From (30), we have erfc(z) > z√π

exp(−z2) 1z 2+ 1

2. Substituting730

C2λs for z, we have731

exp(C22λ2

s)erfc(C2λs) >C2λs√

π

1

C22λ2

s + 12

. (31)

Substituting (31) into (29), we can prove that ∂ Pr(Dn )∂λs

> 0,732

which implies that Pr(D) increases monotonically upon in-733

creasing λs . �734

APPENDIX CPROOF OF THEOREM 2 735

Similar to the derivation in Appendix A, in the dynamic on– 736

off architecture, the intensity of SBSs in Tier-n is also Snλs . 737

Thus, in Tier-n the distance z between the typical MU and 738

its nearest SBS follows the same PDF fZ (z) in the always-on 739

architecture. It follows that we have a similar formulation for 740

Pr(Dn ) in the dynamic on–off architecture, yielding 741

Pr(Dn ) =∫ ∞


(

−zαδσ2

P

)

2πSnλsz exp(−πSnλsz2)dz. (32)

In the dynamic on–off architecture, the interference I only 742

arrives from the SBSs in the active mode. According to [36], 743

the activity probability Pr(An ) of the SBSs in Tier-n, can be 744

formulated as 745

Pr(An ) ≈ 1 −(

1 +Qnλu

3.5Snλs

)−3.5

.

As in Appendix A, we divide the interference into two parts: 746

I = I1 + I2. The first part of interference I1 is inflicted by the 747

active SBSs in any Tier-i, i = n, which can be viewed as a 748

homogeneous PPP with the intensity of Pr(Ai)Siλs . Hence, 749

we update (26) as follows: 750


= exp

⎛

⎝−2π

N∑

i=1:i =n

Pr(Ai)Siλs1α

δ2α B

(2α

, 1 − 2α

)

z2

⎞

⎠ .

(33)

The second part of the interference I2 comes from the active 751

SBSs in Tier-n located in the area outside the circle with radius 752

z. We update (27) as follows: 753

EI2 [exp (−zαδI2)] = exp (−Pr(An )

Snλsπz2 2δ

α − 2 2F1

(

1, 1 − 2α

; 2 − 2α

;−δ

))

. (34)

Integrating (33) and (34) into (32) completes the proof. � 754

APPENDIX DPROOF OF THEOREM 4 755

Note that in the following proof, we simplify the notation by 756

introducing a � λu

λs, C � C(δ, α), and A � A(δ, α). 757

First, we investigate the optimization Problem (17) for a given 758

indicator vector ε. Let us denote by N ∗ the number of ones in 759

ε, and by {nj} the subscript of the ones in N ∗. Then, we have 760


a new optimization problem represented as761

max{Sn j

}

N ∗∑

j=1

QnjSnj

QnjaA +

∑i:i =j Qni

aC + Snj

s.t.N ∗∑

j=1

Snj= 1

Snj> 0 ∀j = 1, . . . N ∗.

(35)

If we neglect the constraint Snj> 0, the solution to Prob-762

lem (35) is presented in Lemma 1.763

Lemma 1: Neglecting the constraint Snj> 0, the optimal764

solution for Problem (35) is given by765

SOptnj

= ζ√

QnjCξ − Q2

nj(C − A) −

[ξaC − Qnj

a(C − A)]

(36)where we have ζ � 1+N ∗ξaC−ξa(C−A)

∑N ∗i = 1

√Qn i

ξC−Q 2n i

(C−A)and ξ �766

∑N ∗

j=1 Qnj.767

Proof: See Appendix E.768

From (36), we propose Lemma 2.769

Lemma 2: Given the request probabilities of two FGs770

cached, where Qni> Qnj

, according to (36), we have SOptni

>771

SOptnj

.772

Proof: See Appendix F.773

Based on Lemma 2, we have SOptnj ∗ = min {SOpt

nj } where774

nj ∗ = arg minnj{Qnj

}. Hence, the constraint Snj> 0, ∀j =775

1, . . . N ∗, is equivalent to Snj ∗ > 0. In order to ensure that776

SOptnj ∗ > 0, based on (36), we have777

a < anj ∗ , anj ∗ �ϑnj ∗

(ϑnj ∗ ξ − Qnj ∗ )(C−A) + (1 − N ∗ϑnj ∗ )ξC(37)

where778

ϑnj ∗ �

√Qnj ∗ ξC + Q2

nj ∗(A − C)

∑N ∗

i=1

√Qni

ξC + Q2ni

(A − C). (38)

Hence, (36) only becomes the optimal solution of Problem (35),779

when a meets the requirement (37).780

Substituting the optimal solution in (36) into (35), we obtain781

the maximum value of Pr(D) for the given indicator vector ε,782

yielding783

DN ∗ = ξ −a(∑N ∗

j=1

√Qnj

Cξ + Q2nj

(A − C))2

1 + N ∗ξaC + ξa(A − C). (39)

Second, we extend the Problem (35) to Problem (17). Based784

on the analysis above, given the indicator vector ε1, when a <785

aε1 in (37), we can obtain the maximum Pr(D) denoted by Dε1786

in (39). For ε2, if we have aε2 > aε1 , then provided a < aε1787

holds, we have a < aε2 . Thus, ε1 and ε2 are both reasonable788

for this optimization problem. Through the comparison of Dε1789

and Dε2 , we can find the right choice between ε1 and ε2. Then790

obtain the optimal solution of {Sn} in form of (36).791

Using {Qn}, we can obtain the segmentation parameters for a792

in (37). The smallest segmentation parameter is obtained when ε793

contains N ones, which is denoted by aN . When a < aN , i.e., λs 794

is high enough, all FGs can be cached in SBSs. Then, with the in- 795

crease of a, i.e., the decrease of λs , some FGs cannot be cached, 796

where a reduced number of ones appear in ε. Since we have 797

Q1 > Q2 > · · · > QN , the unpopular FGs will be discarded one 798

by one. Accordingly, we can obtain both εi as well as the seg- 799

mentation parameter ai . As a result, a piecewise defined function 800

regarding a is obtained like the number of ones in ε is shown 801

in (20). � 802

APPENDIX EPROOF OF LEMMA1 803

Neglecting the constraint Snj> 0, it becomes plausible that 804

Problem (35) is a concave maximization problem. Adopting the 805

Lagrange multiplier Λ, we have 806

Λ(S, λ)

=N ∗∑

j=1

QnjSnj

QnjaA+

∑N ∗

i=1:i =j QniaC + Snj

+λ

⎛

⎝N ∗∑

j=1

Snj− 1

⎞

⎠.

(40)

Using ξ �∑N ∗

j=1 Qnjand ∂Λ

∂Sn j= 0, we have 807

QnjaCξ + Q2

nja(A − C)

(aCξ + a(A − C)Qnj

+ Snj

)2 + λ = 0 ∀nj . (41)

Since∑N ∗

j=1 Snj= 1, we have 808

SOptnj

= ζ√

QnjaCξ + Q2

nja(A − C) −

[ξaC + Qnj

a(A − C)]

(42)

where 809

ζ � 1 + N ∗ξaC + ξa(A − C)∑N ∗

i=1

√Qni

ξaC + Q2ni

a(A − C). (43)

� 810

APPENDIX FPROOF OF LEMMA 2 811

First, based on the optimal solution given in (36), we have 812

∂SOptnj

∂Qnj

= ζ

√a

2

Cξ + 2Qnj(A − C)

√Qnj

Cξ + Q2nj

(A − C)+ a(C − A). (44)

Since C(α, δ) > A(α, δ) > 0, we have∂S Opt

n j

∂Qn j≥ 0 when Qnj

≤ 813

ξ2

CC−A , which means SOpt

nj increases with the growth of Qnj, 814

when Qnjis no bigger than ξ

2C

C−A . 815

1) Since Qnj≤ ξ, if C

C−A ≥ 2, for all Qnj,

∂S Optn j

∂Qn j> 0, and 816

the proof is completed. 817

2) For CC−A < 2, we consider the following case. Since 818

CC−A > 1, we have ξ

2C

C−A > ξ2 . Because

∑j Qnj

= ξ, among 819


the N ∗ FGs cached, there is only one FG associated with820

Qnj> ξ

2C

C−A . We denote the request probability of this popular821

file by Q1 and its caching probability by SOpt1 . Since the request822

probabilities of other cached FGs must be less than ξ2

CC−A , and823

∂S Optn j

∂Qn j> 0 when Qnj

in this region, the highest caching prob-824

ability among these less popular FGs occurs when only two825

FGs are cached. That is, the other FG with request probability826

Q2 = ξ − Q1. Denoted by SOpt2 its caching probability. We have827

SOpt1 − SOpt

2 = ζ√

a

(√Q1Cξ + Q2

1(A − C)

−√

Q2Cξ + Q22(A − C)

)

+ (Q1 − Q2)a(C − A). (45)

Since Q1Cξ + Q21(A − C) − Q2Cξ − Q2

2(A − C) = (Q1 −828

Q2)ξaA > 0, we have SOpt1 − SOpt

2 > 0. Thus, for the dominate829

FG, its caching probability also dominates.830

Combining the two parts above, we complete the proof. �831

ACKNOWLEDGMENT832

The authors would like to thank Dr. H. (Henry) Chen from833

the University of Sydney, Australia, for his helpful discussions834

and suggestions.835

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Youjia Chen received the B.S. and M.S degrees in 938communication engineering from Nanjing Univer- 939sity, Nanjing, China, in 2005 and 2008, respectively. 940She is currently working toward the Ph.D. degree in 941wireless engineering with the University of Sydney, 942Sydney, Australia. 943

From 2008 to 2009, she was with Alcatel Lu- 944cent Shanghai Bell. From August 2009 until now, 945she has been with the College of Photonic and Elec- 946trical Engineering, Fujian Normal University, China. 947Her research interests include resource management, 948

load balancing, and caching strategy in heterogeneous cellular networks. 949950


Ming Ding (M’12) received the B.S. and M.S. de-951grees (with first class Hons.) in electronics engineer-952ing and Ph.D. degree in signal and information pro-953cessing from Shanghai Jiao Tong University (SJTU),954Shanghai, China, in 2004, 2007, and 2011, respec-955tively.956

From September 2007 to September 2011, while957at the same time working as a Researcher/Senior958Researcher Sharp Laboratories of China (SLC), af-959ter achieving the Ph.D. degree, he continued work-960ing with SLC as a Senior Researcher/Principal Re-961

searcher until September 2014, when he joined National Information and Com-962munications Technology Australia (NICTA). In July 2016, Commonwealth Sci-963entific and Industrial Research Organization (CSIRO) and NICTA joined forces964to create Data61, where he continued as a Senior Research Scientist in this new965R&D center in Sydney, NSW, Australia. He has authored more than 30 papers966in IEEE journals and conferences, all in recognized venues, and about 20 3GPP967standardization contributions, as well as a Springer book entitled Multi-point Co-968operative Communication Systems: Theory and Applications (Springer-Verlag,9692013). In addition, as the first inventor, he holds 15 CN, seven JP, three US, two970KR patents, and has co-authored another 100+ patent applications on 4G/5G971technologies.972

Dr. Ding has been the Guest Editor/Cochair/TPC member of several IEEE973top-tier journals/conferences, e.g., the IEEE JOURNAL ON SELECTED AREAS IN974COMMUNICATIONS, the IEEE COMMUNICATIONS MAGAZINE, the IEEE Globe-975com Workshops, etc. He received the Presidents Award from the SLC in 2012976for his inventions and publications and served as one of the key members in977the 4G/5G standardization team when it was awarded in 2014 as the Sharp978Company Best Team: LTE 2014 Standardization Patent Portfolio.979

980

Jun Li (M’09–SM’16) received the Ph.D. degree in981electronic engineering from Shanghai Jiao Tong Uni-982versity, Shanghai, China in 2009.983

From January 2009 to June 2009, he was a Re-984search Scientist with the Department of Research and985Innovation, Alcatel Lucent Shanghai Bell. From June9862009 to April 2012, he was a Postdoctoral Fellow with987the School of Electrical Engineering and Telecommu-988nications, University of New South Wales, Australia.989From April 2012 to June 2015, he was a Research990Fellow with the School of Electrical Engineering,991

University of Sydney, Australia. From June 2015 to the present, he has been992a Professor with the School of Electronic and Optical Engineering, Nanjing993University of Science and Technology, Nanjing, China. His research interests994include network information theory, channel coding theory, wireless network995coding, and cooperative communications.996

997

Zihuai Lin (S’98–M’06–SM’10) received the Ph.D.998degree in electrical engineering from Chalmers Uni-999versity of Technology, Gothenburg, Sweden, in 2006.1000

Prior to his Ph.D. degree, he has held positions at1001Ericsson Research, Stockholm, Sweden. Following1002having received the Ph.D. degree, he was a Research1003Associate Professor with Aalborg University, Aal-1004borg, Denmark, and is currently with the School of1005Electrical and Information Engineering, University of1006Sydney, Sydney, Australia. His research interests in-1007clude source/channel/network coding, coded modu-1008

lation, MIMO, OFDMA, SC-FDMA, radio resource management, cooperative1009communications, small-cell networks, 5G cellular systems, etc.1010

1011

Guoqiang Mao (S’98–M’02–SM’08) received the 1012Ph.D. degree in telecommunications engineering 1013from Edith Cowan University, Perth, Australia, in 10142002. 1015

Between 2002 and 2014, he was with the School 1016of Electrical and Information Engineering, University 1017of Sydney, Sydney, Australia. He joined the Univer- 1018sity of Technology Sydney in February 2014 as a 1019Professor of wireless networking and the Director of 1020Center for real-time information networks. The Cen- 1021ter is among the largest university research centers in 1022

Australia in the field of wireless communications and networking. He has pub- 1023lished about 200 papers in international conferences and journals, which have 1024been cited more than 4000 times. His research interest includes intelligent trans- 1025port systems, applied graph theory and its applications in telecommunications, 1026Internet of Things, wireless sensor networks, wireless localization techniques, 1027and network performance analysis. 1028

Dr. Mao is an Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICA- 1029TIONs (since 2014) and the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 1030(since 2010). He received Top Editor award for outstanding contributions to the 1031IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY in 2011, 2014, and 2015. 1032He is a Cochair of the IEEE Intelligent Transport Systems Society Technical 1033Committee on Communication Networks. He has served as a Chair, Cochair, 1034and Technical Program Committee Member in a large number of international 1035conferences. 1036

1037

Lajos Hanzo (F’08) received the M.S. degree in elec- 1038tronics and the Ph.D. degree from the Technical Uni- 1039versity of Budapest, Budapest, Hungary, in 1976 and 10401983, respectively. He received the prestigious Doc- 1041tor of Sciences research degree in wireless commu- 1042nications from the University of Southampton, U.K., 1043in 2004. 1044

In 2016, he was admitted to the Hungarian 1045Academy of Science, Budapest, Hungary. During his 104640-year career in telecommunications, he has held 1047various research and academic posts in Hungary, Ger- 1048

many, and the U.K. Since 1986, he has been with the School of Electronics and 1049Computer Science, University of Southampton, U.K., where he holds the Chair 1050in telecommunications. He has successfully supervised 111 Ph.D. students, co- 1051authored 20 John Wiley/IEEE Press books on mobile radio communications, 1052totalling in excess of 10 000 pages, published 1600+ research contributions on 1053IEEE Xplore, acted both as Technical Program Committee member and General 1054Chair of IEEE conferences, presented keynote lectures, and received a number 1055of distinctions. Currently he is directing a 60-strong academic research team, 1056working on a range of research projects in the field of wireless multimedia 1057communications sponsored by industry; the Engineering and Physical Sciences 1058Research Council (EPSRC), U.K.; and the European Research Council.s Ad- 1059vanced Fellow Grant. He is an enthusiastic supporter of industrial and academic 1060liaison, and he offers a range of industrial courses. He has 25 000+ citations and 1061an H-index of 60. For further information on research in progress and associated 1062publications, see http://www-mobile.ecs.soton.ac.uk. Dr. Hanzo is also a Gov- 1063ernor of the IEEE Vehicular Technology Society. During 2008–2012, he was 1064the Editor-in-Chief of the IEEE Press and a Chaired Professor with Tsinghua 1065University, Beijing, China. In 2009, he received an honorary doctorate award 1066by the Technical University of Budapest and in 2015, from the University of 1067Edinburgh, Edinburgh, U.K., as well as the Royal Society.s Wolfson Research 1068Merit Award. He is a Fellow of the Royal Academy of Engineering, The Insti- 1069tution of Engineering and Technology, and EURASIP. 1070

1071

QUERIES 1072

Q1. Author: Please supply index terms/keywords for your paper. To download the IEEE Taxonomy go to 1073

http://www.ieee.org/documents/taxonomy_v101.pdf. 1074

Q2. Author: Please update Ref. [2]. 1075

Q3. Author: Please provide page range in Ref. [18]. 1076

Q4. Author: Please provide complete bibliographic details in Refs. [29] and [30]. 1077

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