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
4 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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)
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 5
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
From (10), we have Remark 3.394
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
1) Neglecting Noise, i.e., σ2 = 0: In an interference-limited 398
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)
From (12), we have Remark 4. 400
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
6 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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
B. Dynamic On–Off Architecture467
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)
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 7
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
A. Always-On Architecture 547
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
8 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 9
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.
B. Dynamic On–Off Architecture618
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
10 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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 ∈Φ\{x0} Phxj
‖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
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 11
As for I2, we have713
EI2 [exp (−zαδI2)]
= exp(
−Snλs2π
∫ ∞
z
(
1 − 1
1 + zαδ ‖xj‖−α
)
‖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 ) =∫ ∞
0EI [exp (−zαδI)] exp
(
−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
EI1 [exp (−zαδI1)]
= 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
12 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 13
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
14 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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
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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.
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
4 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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)
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 5
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
From (10), we have Remark 3.394
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
1) Neglecting Noise, i.e., σ2 = 0: In an interference-limited 398
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)
From (12), we have Remark 4. 400
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
6 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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
B. Dynamic On–Off Architecture467
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)
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 7
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
A. Always-On Architecture 547
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
8 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 9
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.
B. Dynamic On–Off Architecture618
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
10 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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 ∈Φ\{x0} Phxj
‖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
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 11
As for I2, we have713
EI2 [exp (−zαδI2)]
= exp(
−Snλs2π
∫ ∞
z
(
1 − 1
1 + zαδ ‖xj‖−α
)
‖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 ) =∫ ∞
0EI [exp (−zαδI)] exp
(
−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
EI1 [exp (−zαδI1)]
= 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
12 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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
CHEN et al.: PROBABILISTIC SMALL-CELL CACHING: PERFORMANCE ANALYSIS AND OPTIMIZATION 13
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
14 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 00, NO. 00, 2016
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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