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9162 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 9, SEPTEMBER 2019
Probabilistic Caching for Small-Cell Networks WithTerrestrial and Aerial Users
Fei Song , Jun Li , Senior Member, IEEE, Ming Ding , Senior Member, IEEE, Long Shi , Member, IEEE,Feng Shu , Member, IEEE, Meixia Tao , Fellow, IEEE, Wen Chen , Senior Member, IEEE,
and H. Vincent Poor , Fellow, IEEE
Abstract—The support for aerial users has become the focus ofrecent Third-Generation Partnership Project standardizations of5G, due to their high maneuverability and flexibility for on-demanddeployment. In this paper, probabilistic caching is studied for ultra-dense small-cell networks with terrestrial and aerial users, where adynamic on–off architecture is adopted under a sophisticated pathloss model incorporating both line-of-sight and non-line-of-sighttransmissions. Generally, this paper focuses on the successful down-load probability (SDP) of user equipments (UEs) from small-cellbase stations (SBSs) that cache the requested files under variouscaching strategies. To be more specific, the SDP is first analyzedusing stochastic geometry theory, by considering the distributionof such two-tier UEs and SBSs as homogeneous poisson pointprocesses. Second, an optimized caching strategy (OCS) is proposedto maximize the average SDP. Third, the performance limits of theaverage SDP are developed for the popular caching strategy (PCS)and the uniform caching strategy (UCS). Finally, the impacts ofthe key parameters, such as the SBS density, the cache size, theexponent of Zipf distribution, and the height of aerial user, areinvestigated on the average SDP. The analytical results indicatethat the UCS outperforms the PCS if the SBSs are sufficientlydense, while the PCS is better than the UCS if the exponent ofZipf distribution is large enough. Furthermore, the proposed OCSis superior to both the UCS and PCS.
Index Terms—Small-cell caching, successful download probabil-ity, UAV, optimization, stochastic geometry.
Manuscript received January 23, 2019; revised May 7, 2019 and June 21,2019; accepted July 16, 2019. Date of publication July 19, 2019; date of currentversion September 17, 2019. This work was supported in part by the National KeyR&D Program under Grant 2018YFB1004800, in part by the National NaturalScience Foundation of China under Grants 61872184, 61727802, 61571299,and 61671294, in part by the STCSM Key Fundamental Project under Grants16JC1402900 and 17510740700, in part by the National Science and TechnologyMajor Project under Grant 2018ZX03001009-002, and in part by the U.S.National Science Foundation under Grants CCF-0939370 and CCF-1513915.The review of this paper was coordinated by Dr. X. Ge. (Corresponding authors:Long Shi and Jun Li.)
F. Song, J. Li, and F. Shu are with the School of Electronic and OpticalEngineering, Nanjing University of Science Technology, Nanjing 210094, China(e-mail: [email protected]; [email protected]; [email protected]).
M. Ding is with the Data61, CSIRO, Sydney, NSW 2015, Australia (e-mail:[email protected]).
L. Shi is with the Science and Math Cluster, Singapore University of Tech-nology and Design, Singapore 487372 (e-mail: [email protected]).
M. Tao and W. Chen are with the Shanghai Institute of Advanced Communi-cations and Data Sciences, Department of Electronic Engineering, ShanghaiJiao Tong University, Shanghai 200240, China (e-mail: [email protected];[email protected]).
H. V. Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544 USA (e-mail: [email protected]).
Digital Object Identifier 10.1109/TVT.2019.2929839
I. INTRODUCTION
W ITH the dramatic proliferation of smart mobile devicesand various mobile applications, the global mobile data
traffic has been increasing rapidly in recent years. Cisco fore-casts that the traffic will increase sevenfold from 2016 to 2021,of which about 78 percent will be video streams by 2021 [1].This deluge of data has driven vendors and operators to seekevery possible tool at hand to improve network capacity [2]. Veryrecently, providing wireless connectivity for unmanned aerialvehicles (UAVs) has become an emerging research area [3]. TheUAV becomes increasingly popular for various commercial, in-dustrial, and public-safety applications [4]. Due to their high ma-neuverability and flexibility for on-demand deployment, UAVsequipped with advanced transceivers and batteries are gainingincreasing popularity in information technology applications[5], and have been widely used in delivery, communicationsand surveillance. As the UAV applications proliferate, securityissues in the UAV deployment have captured much attention inrecent years. Hence, the academia and industry have expandedpublic safety communications from the ground [6] to the air [7],[8]. Driven by the rising interest in aerial communications, theThird Generation Partnership Project (3GPP) has taken UAVssupported by Long Term Evolution (LTE) as a primary researchfocus [9].
A. Background of Wireless Caching
Recent research has unveiled that some popular files arerepeatedly requested by the user equipments (UEs), which takesa huge portion of the data traffic [10]. To reduce duplicated trans-missions, wireless caching has been proposed to pre-downloadthe popular files in cache devices at the wireless edges [11], [12].Unlike the traditional communication resources, the storageresources are abundant, economical, and sustainable, making thecaching technology even more promising in the modern com-munications [13]. For example, a scalable platform for imple-menting the caching technology is well-known as the small-cellcaching in the ultra-dense (UD) small-cell networks (SCNs),which has attracted significant attention as one of the enticingapproaches to meet the ever-increasing data traffic demands forthe fifth generation (5G) communication systems [2], [14]. In thesmall-cell caching, popular files are pre-downloaded into localcaches of small-cell base stations (SBSs) in the off-peak hours,and ready to be fetched by the UEs in the peak hours, alleviating
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SONG et al.: PROBABILISTIC CACHING FOR SMALL-CELL NETWORKS WITH TERRESTRIAL AND AERIAL USERS 9163
the backhaul congestion in wireless networks. In addition, small-cell caching makes the data traffic much closer to the mobileusers. Thus, the transmission latency can be reduced, and thequality of experience for users will be enhanced. However, eachdevice has a finite amount of storage, popular content should beseeded into the network in a way that maximizes the successfuldownload probability (SDP).
B. Related Work
Existing works have shown that the file placement of small-cell caching largely follows two approaches: deterministic place-ment and probabilistic placement. For deterministic placement,files are placed and optimized for specific networks by a de-terministic pattern [15]–[17]. In practice, the wireless channelsand the geographic distribution of UEs are time-variant. Thistriggers the optimal file placement strategy to be frequentlyupdated, which makes the file placement highly complicated. Tocope with this problem, probabilistic file placement considersthat each SBS randomly caches a subset of popular files witha certain caching probability in the stochastic networks. As aseminal work, [18] modelled the node locations as Homoge-neous Poisson Point Processes (HPPPs) and analyzed the generalperformance of the small-cell caching. Compared with cachingthe same copy of certain files in all SBSs, probabilistic fileplacement in the small-cell caching is more flexible and robust.
However, the cache-aided ground SBSs may not be able tosupport the users in high rise building scenarios [19] and inemergency situations where the ground infrastructures fail orthere is a sudden and temporary surge of traffic demand [20].The UAVs with high maneuverability and flexibility can be usedas flying relays [21] to dynamically cache the popular contentfiles from the cache-aided ground BSs and then effectivelydisseminate them to the users [22]. Zhao et al. [23] studiedthe cache-enabled UAVs that serve as flying BSs and refreshthe cached content files from macro BSs (MBSs). However, itis time and energy consuming for UAVs with limited batterycapacity to fly back to MBSs to update the cached content files.In view of this problem, the UD SCNs point out a promisingUAV-aided wireless caching scenario where UAVs are connectedto the nearby SBSs that are much denser than MBSs. In thisscenario, the role of UAV can be either terminal UE served bystatic BSs or flying relays that forward files to other UEs, aimingat alleviating the peak backhaul traffic and assisting the SBSs.Recent works [20], [21] and [24] have extended conventionalterrestrial cellular services to aerial users in the 5G networks.In addition, the 3GPP launched an investigation on enhancedLTE support and proposed a channel model for aerial UEs [25].Therefore, in this paper, we focus on the small-cell cellularnetworks with terrestrial users (TUs) and aerial users (AUs),i.e., UAVs.
From the stochastic cellular network model, the BS locationsare supposed to follow an HPPP distribution [26], [27]. Ref. [28]proposed an optimal geographic placement in wireless cellularnetworks modelled by HPPP. Furthermore, a trade-off betweenthe SBS density and the storage size was presented in [18], whereeach SBS caches the most popular files. In [29], the librarywas divided into N file groups and the probabilistic caching
probability of each file group was optimized to maximize theSDP in SCNs. Utilizing stochastic geometry, [30] optimizedprobabilistic caching at helper stations in a two-tier heteroge-neous network, where one tier of multi-antenna MBSs coexistswith the other tier of helpers with caches. However, to our bestknowledge, most existing works on small-cell caching consid-ered the path loss models without differentiating line-of-sight(LoS) from non-line-of-sight (NLoS) transmissions. It is wellknown that LoS transmission may occur when the distancebetween a transmitter and a receiver is small and no shelter,and NLoS transmission is common in indoor environments andin central business districts. In our previous works [31], [32], weconsidered both multi-slope piece-wise path loss function andprobabilistic LoS or NLoS transmission in cellular networks.Furthermore, for ease of exposition, we ignored the antennaheight difference between SBSs and UEs in the performanceanalysis due to the dominance of the horizontal distance. How-ever, the antenna height difference becomes non-negligible asthe distance between an UE and its serving SBS decreases. Toverify this, [33] clarified that the height difference between UEsand BSs imposes a significant impact on coverage probabilityand area spectral efficiency.
Regarding the SBS activity, there are two network archi-tectures in the SCNs, namely, the always-on architecture andthe dynamic on-off architecture. The always-on architectureis commonly used in the current cellular networks, where allthe SBSs are always active. By contrast, in the dynamic on-offarchitecture, the SBSs are only active when they are required toprovide services to UEs [34]. To mitigate inter-cell interference,the dynamic on-off architecture will thrive as an important 5Gtechnology in the UD SCNs, which is also investigated in 3GPP[2]. Therefore, in this paper, we focus on the dynamic on-offarchitecture.
C. Contributions
In this work, we analyze the average SDP that UEs can suc-cessfully download files from the storage of SBSs and optimizethe caching probability of each file. We consider an UD SCNwith UEs including TUs and AUs under a general path lossmodel that incorporates both LoS and NLoS paths. Furthermore,we consider the dynamic on-off architecture. Our goal is tomaximize the average SDP. To be concise, the contributions ofthis article are summarized as follows:� We investigate the average SDP by considering the 3GPP
path loss models for TUs and UAVs respectively (seeSection IV).
� We propose the optimized caching strategy (OCS) to max-imize the average SDP of UD SCNs by optimizing cachingprobability of each content (see Section V).
� We analyze the performance limits of the SDP with theuniform caching strategy (UCS) and the popular cachingstrategy (PCS) under a single-slope path loss model, re-spectively (see Section VI). First, we show that the OCSis superior to both the UCS and PCS. Second, we revealthat the UCS outperforms the PCS if the SBS density islarge enough, while the PCS is better than the UCS if theexponent of Zipf distribution grows sufficiently large.
9164 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 9, SEPTEMBER 2019
TABLE IMAIN NOTATIONS AND THEIR DEFINITIONS
Fig. 1. System model of a small-cell network consisting of the SBS andtwo-tier UEs.
The rest of the paper is organized as follows. We describethe system model in Section II and study the probabilisticcaching strategy in Section III. Section IV presents our analyticalresults of SDP. Section V proposes the OCS. Section VI showsthe impacts of network parameters under the UCS and PCS.Section VII provides simulations and numerical results. Finally,Section VIII concludes this paper. Table I lists main notationsand symbols used in this paper.
II. SYSTEM MODEL
As illustrated in Fig. 1, we consider a small-cell cellularnetwork where the SBSs serve two-tier UEs including TUsas ground users and UAVs as AUs over the same frequencyspectrum. We assume that the SBSs, the TUs and the UAVsare deployed according to three independent HPPPs with theheight of hBS, hTU and hAU, respectively. With reference to [5]and [24], consider that all the UAVs are positioned at the sameheight.1 Let λs be the density of SBSs, λTU be the density of
1This paper considers that the locations of UAVs follow a 2-D HPPP withthe same height. As Fig. 7 will show, the change of the UAV height affects itsaverage SDP but imposes no performance impact on TUs. We remark that theoptimization of the caching probabilities (to be discussed in Section V) whenthe UAVs are deployed as a 3-D HPPP is quite challenging and will be left asour future work.
TUs, and λAU be the density of UAVs. Second, each UE2 isassociated with an SBS with the smallest path loss. The transmitpower of each SBS is denoted by P .
The horizontal distance between an SBS and an UE is denotedby r. Moreover, the absolute antenna height difference betweenan UE and an SBS is denoted by h, and the distance between anUE and an SBS is denoted by l. Let lTU be the distance betweena TU and an SBS, and lAU be the distance between an AU and anSBS. As such, the height differences are h1 = hBS − hTU forTUs and h2 = hAU − hBS for UAVs respectively. Hence, thedistance l can be expressed as
l =√
r2 + h2. (1)
Regarding the UAV acting as the aerial UE, recent works suchas [3], [20], [21] and [24] have studied a variety of communi-cation scenarios where conventional terrestrial cellular servicesare extended to aerial UEs.
Considering the downlink transmission, the small-scale effectin the network is assumed to be Rayleigh fading, and the pathloss model embraces both LoS and NLoS paths as large-scalefading. The link from any UE to the typical SBS has a LoSpath with probability PrL(r, h) or a NLoS path with probability1 − PrL(r, h), respectively. According to [25] and [35], thepiece-wise LoS probability function is given by
PrL(r, h) =
⎧⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
PrL1 (r, h), 0 < r ≤ d1(h)
PrL2 (r, h), d1(h) < r ≤ d2(h)
......
PrLK(r, h), r > dK−1(h)
(2)
where PrLk (r, h), k ∈ {1, 2, ...,K} is the k-th piece probabilityfunction that an UE and an SBS separated by a horizontal dis-tance dk−1(h) < r ≤ dk(h) has a LoS path. In addition, dk(h)varies as the height difference h changes. The details of dk(h)for TUs and UAVs are provided in Sections IV-C and IV-D,respectively.
Furthermore, with reference to [25] and [35], we adopt ageneral and practical path loss model in [31], where the pathloss with respect to the distance l is modeled as (3), as shown atthe bottom of the next page. In (3), AL
k and ANLk denote the path
losses of the LoS path and NLoS path at a reference distanceof l = 1 respectively. Moreover, αL
k and αNLk denote the path
loss exponents of the LoS path and NLoS path respectively. Itis worthwhile noting that the values of AL
k , ANLk , αL
k and αNLk
for the TUs are different from those for the UAVs. For TUs,AL
k , ANLk , αL
k and αNLk are constants from field tests in [35]. For
UAVs, ALk and ANL
k are constants, while αLk and αNL
k vary withdifferent height ranges [25].
In the dynamic on-off architecture, an SBS is only active whenit is required to serve the associated UEs. At any time, a typicalUE only associates with an intended SBS, while other activeSBSs are regarded as interferers. Since both TU and UAV canbe regarded as the typical UE in this model, the two-tier UEs needto be projected onto the same plane for this architecture. As such,
2By the UE, we mean either the TU or the UAV.
SONG et al.: PROBABILISTIC CACHING FOR SMALL-CELL NETWORKS WITH TERRESTRIAL AND AERIAL USERS 9165
we can get the total density of two-tier UEs λu = λTU + λAU.Hence, the probability that an SBS is active is given by [36]
Pron ≈ 1 −(
1 + λu
qλs
)−q
, (4)
where q = 3.5 is a tight lower bound of q, especially for denseSCNs.
III. PROBABILISTIC UD SCNS CACHING STRATEGY
Suppose that a library consists of N popular files each withequal length. Note that N represents the number of popularfiles that the UEs tend to access rather than the number of filesavailable on the Internet. Furthermore, each file is requestedaccording to its popularity, known as a priori information.
LetQn represents the probability that then-th file is requestedby the UEs. Q = [Q1, Q2, . . . , QN ] collects the request proba-bility mass functions (PMFs) of all N files. Similar to existingworks [13], [29] and [37], we model the PMF of each file requestas Zipf distribution, and the request probability is given by
Qn =1nβ
∑Ni=1
1iβ
, (5)
where β is the exponent of the Zipf distribution. A larger βimplies a more uneven popularity among those files.
Due to the limited storage, each SBS cannot cache the entirefile library. In this context, we consider that the files are indepen-dently placed in different SBSs. Suppose that a cache memoryof size M is available on each SBS. In the file placement phase,each SBS store the n-th file in its local cache with a cachingprobability Sn, yielding
N∑
n=1
Sn ≤ M, (6)
0 ≤ Sn ≤ 1, ∀n. (7)
We emphasize that then-th tier of SBS is formed by a group ofSBSs that cache then-th file. Given that each SBS independentlycaches the files, the distribution of SBSs that cache the n-th fileis viewed as a thinned HPPP with density of Snλs. In addition,given that each UE only requests a single content at each timeslot, the distribution of UEs who request the n-th file can alsobe modeled as a thinned HPPP with density of Qnλu. In thefollowing, we consider three types of caching strategies:
1) UCS: Each SBS caches each file randomly with equalprobability [13].
2) PCS: Each SBS only caches the most popular files [13].3) OCS: Each SBS caches each file with optimized probabil-
ity for SDP maximization (see Section V).
IV. PERFORMANCE ANALYSIS OF SMALL-CELL CACHING
In this section, we derive the SDP for the dynamic on-off ar-chitecture. Some cases adopted by the 3GPP are also considered.
A. Received SINR
The received signal power of a typical UE from its associatedSBS can be written as
Prs = Pζ(r, h)g, (8)
where the channel gain of the Rayleigh fading g follows anindependent and identically distributed (i.i.d.) exponential dis-tribution with unit mean.
Consequently, the signal-to-interference-and-noise-ratio(SINR) at the typical UE can be expressed as
SINR =Prs
σ2 + IZ, (9)
where σ2 is noise power, and Z is the set of interfering SBSswith the total interference being
IZ =∑
z∈ZPζ (rz, h)gz, (10)
where gz denotes the channel gain between the typical user andthe z-th interfering SBS, also following an i.i.d. exponentialdistribution with unit mean.
B. Successful Download Probability
Let Dn be the event that the typical UE can successfullyreceive the requested n-th file from the associated n-th tier ofSBS. In this paper, we consider that Dn occurs if the SINR ofthe UE is no less than a targeted value δ. As such, the SDP ofDn can be formulated as
Pr (Dn) = Pr (SINR > δ) . (11)
Recall that the SBSs in the n-th tier and the UEs that requestthen-th file form two independent thinned HPPPs with densitiesSnλs and Qnλu respectively. Let An be the event that the SBSin the n-th tier is active, we rewrite the probability that an SBSin the n-th tier is active as
Pr(An) ≈ 1 −(
1 + Qnλu
qSnλs
)−q
, (12)
ζ(r, h) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
ζ1(r, h) =
{AL
1 l−αL
1 , LoS with PrL1 (r, h)
ANL1 l−αNL
1 , NLoS with(1 − PrL1 (r, h)
) , 0 < r ≤ d1(h)
ζ2(r, h) =
{AL
2 l−αL
2 , LoS with PrL2 (r, h)
ANL2 l−αNL
2 , NLoS with(1 − PrL2 (r, h)
) , d1(h) < r ≤ d2(h)
......
ζK(r, h) =
{AL
K l−αLK , LoS with PrLK(r, h)
ANLK l−αNL
K , NLoS with(1 − PrLK(r, h)
) , r > dK−1(h)
(3)
9166 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 9, SEPTEMBER 2019
where we replace λs and λu in (4) with Snλs and Qnλu
respectively.Theorem 1: Given a particular value of Snλs, the SDP of Dn
is given by
Pr (Dn) =
K∑
k=1
(TLk + TNL
k
), (13)
where
TLk =
∫ dk(h)
dk−1(h)
Pr
[PζLk (r, h)g
σ2 + IZ> δ
]fLk (r, h)dr, (14)
TNLk =
∫ dk(h)
dk−1(h)
Pr
[PζNL
k (r, h)g
σ2 + IZ> δ
]fNLk (r, h)dr, (15)
and fLk (r, h) and fNL
k (r, h) are the probability density functions(PDFs) of LoS path and NLoS path, respectively. Let d0 = 0 anddK = ∞. Moreover, we have
fLk(r, h) = exp
(−∫ r1
02πSnλs
(1−PrLk (u, h)
)udu
)
× exp
(−∫ r
02πSnλsPr
Lk (u, h)udu
)
× PrLk (r, h)2πrSnλs, dk−1(h)<r≤dk(h),(16)
fNLk (r, h) = exp
(−∫ r2
02πSnλsPr
Lk (u, h)udu
)
× exp
(−∫ r
02πSnλs
(1 − PrLk (u, h)
)udu
)
× (1−PrLk (r, h)
)2πrSnλs, dk−1(h)<r≤dk(h),
(17)
where r1 = argr1
{ζNL(r1, h) = ζLk (r, h)
}and r2 = argr2{
ζL(r2, h) = ζNLk (r, h)
}.
Proof: See Appendix A. �To specify (14) and (15), we further have
Pr
[PζLk (r, h) g
σ2 + IZ>δ
]=exp
(− δσ2
PζLk (r, h)
)LIZ
(δ
PζLk (r, h)
),
(18)
Pr
[PζNL
k (r, h)g
σ2 + IZ>δ
]=exp
(− δσ2
PζNLk (r, h)
)LIZ
(δ
PζNLk (r, h)
),
(19)
where LIZ (·) is the Laplace transform of IZ . We note that IZat the typical UE that requests the n-th file comes from twoindependently portions: 1) IZ1, caused by the SBSs in other tierswhich locate in the entire area of the network, and 2) IZ2, causedby the SBSs in the n-th tier whose distances with the typical UEare larger than l. Based on the observation, IZ = IZ1 + IZ2.
Going forward, Lemma 1 below computes LIZ
(δ
PζLk (r,h)
)in
(18) and LIZ
(δ
PζNLk (r,h)
)in (19).
Lemma 1:
LIZ
(δ
Pζk(r, h)
)=EIZ
[exp
(− δIZPζk(r, h)
)]
=EIZ1
[exp
(− δIZ1
Pζk(r, h)
)]EIZ2
[exp
(− δIZ2
Pζk(r, h)
)].
(20)
For LoS,
EIZ1
[exp
(− δIZ1
PζLk (r,h)
)]= exp
⎧⎨
⎩−2π
N∑
i=1,i�=n
Pr(Ai)Siλs
×[∫ ∞
0
PrL(u, h)u
1 + ζLk (r, h)(δζL(u, h))−1 du
+
∫ ∞
0
[1 − PrL(u, h)
]u
1 + ζLk (r, h)(δζNL(u, h))−1 du
]⎫⎬
⎭, (21)
EIZ2
[exp
(− δIZ2
PζLk (r, h)
)]
=exp
{
−2πPr (An)Snλs
[∫ ∞
r
PrL(u, h)u
1 + ζLk (r, h)(δζL(u, h))−1 du
+
∫ ∞
r1
[1 − PrL(u, h)
]u
1 + ζLk (r, h)(δζNL(u, h))−1 du
]}
. (22)
For NLoS,
EIZ1
[exp
(− δIZ1
PζNLk (r, h)
)]=exp
⎧⎨
⎩−2π
N∑
i=1,i�=n
Pr (Ai)Siλs
×[∫ ∞
0
PrL(u, h)u
1 + ζNLk (r, h)(δζL(u, h))−1 du
+
∫ ∞
0
[1 − PrL(u, h)
]u
1 + ζNLk (r, h)(δζNL(u, h))−1 du
⎤
⎦
⎫⎬
⎭, (23)
EIZ2
[exp
(− δIZ2
PζNLk (r, h)
)]
=exp
{
−2πPr(An)Snλs
[∫ ∞
r2
PrL(u, h)u
1 + ζNLk (r, h)(δζL(u, h))−1 du
+
∫ ∞
r
[1 − PrL(u, h)
]u
1 + ζNLk (r, h)(δζNL(u, h))−1 du
]}
. (24)
Proof: See Appendix B. �LetPr (An)Snλs be the density of active SBSs in then-th tier.
Eqn. (12) implies that the density of the active SBSs increasesas the UE density goes up. From Theorem 1 and Lemma 1,the increase of the UE density degrades its SDP.
Finally, considering the request probabilities of allN files, weobtain the average SDP that the UEs can successfully download
SONG et al.: PROBABILISTIC CACHING FOR SMALL-CELL NETWORKS WITH TERRESTRIAL AND AERIAL USERS 9167
all possible files as
Pr =
N∑
n=1
Qn Pr (Dn). (25)
C. A 3GPP Path Loss Model for TUs
For the TUs, we show a path loss function adopted by 3GPP[35], i.e.,
ζt (r, h1) =
{AL
t l−αL
t , with PrLt (r, h1)
ANLt l−αNL
t , with(1 − PrLt (r, h1)
) , (26)
together with a linear LoS probability function also adopted by3GPP, i.e.,
PrLt (r, h1) =
{1 − l
l0, 0 < r ≤ dT
0, r > dT, (27)
where l0 is the cut-off distance of the LoS link, and dT �d1(h1) =
√l20 − h2
1.We remark that the path loss model in (26) and (27) is a
special case of the general path loss model in (3) with the follow-ing substitutions: N = 2, ζLt,1 (r, h1) = ζLt,2 (r, h1) = AL
t l−αL
t ,
ζNLt,1 (r, h1) = ζNL
t,2 (r, h1) = ANLt l−αNL
t , PrLt,1 (r, h1) = 1 − ll0
and PrLt,2 (r, h1) = 0.For the 3GPP path loss model, by Theorem 1, the probability
that the typical TU successfully receives the requested n-th filefrom the associated n-th tier SBS becomes
Prt (Dn)=2∑
k=1
(TLt,k + TNL
t,k
)
=
∫ dT
0exp
(
−δσ2lαLt
PALt
)
LIZ
(δlα
Lt
PALt
)
fLt,1 (r, h1) dr+0
+
∫ dT
0exp
(
−δσ2lαNLt
PANLt
)
LIZ
(δlα
NLt
PANLt
)
fNLt,1 (r, h1) dr
+
∫ ∞
dT
exp
(
−δσ2lαNLt
PANLt
)
LIZ
(δlα
NLt
PANLt
)
fNLt,2 (r, h1) dr
=TLt,1 + TNL
t,1 + TNLt,2 , (28)
whereTLt,2 = 0, becausePrLt (r, h1) = 0 when r > dT. For LoS,
fLt,1 (r, h1) and LIZ (
δlαLt
PALt) are calculated by (16), (21) and (22).
For NLoS, fNLt,1 (r, h1), fNL
t,2 (r, h1) and LIZ (δl
αNLt
PANLt
) are calcu-lated by (17), (23) and (24).
D. A 3GPP Path Loss Model for UAVs
For the UAVs, we consider a path loss function adopted by3GPP [25], i.e.,
ζa (r, h2) =
{AL
a l−αL
a , with PrLa (r, h2)
ANLa l−αNL
a , with(1 − PrLa (r, h2)
) , (29)
together with a LoS probability function also adopted by 3GPP[25], i.e.,
PrLa (r, h2)=
⎧⎨
⎩
1, 0<r≤dA
dA
r +exp(−rp1
)(1− dA
r
), r>dA
, (30)
where
p1 = 233.98log10 (hAU)− 0.95, (31)
dA�d1(h2)= max (294.05log10 (hAU)−432.94, 18) . (32)
From this 3GPP channel model, the applicability rangein terms of UAV height is 22.5 m ≤ hAU ≤ 300 m [25].Note that the path loss model in (29) and (30) is alsoa special case of (3) with the following substitutions:N = 2, ζLa,1 (r, h2) = ζLa,2 (r, h2) = AL
a l−αL
a , ζNLa,1 (r, h2) =
ζNLa,2 (r, h2) = ANL
a l−αNLa ,PrLa,1(r, h2) = 1 andPrLa,2(r, h2) =
dA
r + exp(
−rp1
) (1 − dA
r
).
From Theorem 1, the probability that the typical UAV suc-cessfully receives the requested n-th file can be given by
Pra (Dn) =
∫ dA
0exp(−δσ2lα
La
PALa
)LIZ
(δlα
La
PALa
)
fLa,1 (r, h2) dr
+
∫ ∞
dA
exp
(
−δσ2lαLa
PALa
)
LIZ
(δlα
La
PALa
)
fLa,2 (r, h2) dr+0
+
∫ ∞
dA
exp
(
−δσ2lαNLa
PANLa
)
LIZ
(δlα
NLa
PANLa
)
fNLa,2 (r, h2) dr
=TLa,1 + TL
a,2 + TNLa,2 , (33)
where TNLa,1 = 0, because PrNL
a (r, h2) = 0 when 0 < r ≤ dA.
For LoS, fLa,1 (r, h2), fL
a,2 (r, h2) and LIZ
(δlα
La
PALa
)are calculated
by (16), (21) and (22). For NLoS, fNLa,2 (r, h2) and LIZ
(δlα
NLa
PANLa
)
are calculated by (17), (23) and (24).
V. OPTIMIZED CACHING PROBABILITIES
In dynamic on-off architecture, (12) shows that Pr(An) is afunction of the ratioQnλu/Snλs. Since the SBS density is muchhigher than the UE density in this architecture, i.e., λs >> λu,Pr(An) can be approximated as [29]
Pr (An) ≈ Qnλu
Snλs. (34)
Since both TU and UAV can be regarded as the typical user,the average SDP is given by
Pr =N∑
n=1
Qn
(λTU
λuPrt (Dn) +
λAU
λuPra (Dn)
). (35)
Consider that the SBS density approaches infinity. In thisscenario, the downlink transmission from the typical SBS tothe typical UE is dominantly characterized by the LoS path loss.In this context, we derive the asymptotic performance of Pr asfollows.
9168 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 9, SEPTEMBER 2019
Theorem 2: In a noise-free case where the SBS density goesto infinity, i.e., λs → +∞ and σ2 = 0, the average SDP isgiven by
Pr =N∑
n=1
QnSn
[∫ dT
0Gn exp
(−πSnλsr2)dr
+
∫ dA
0Hn exp
(−πSnλsr2)dr
], (36)
where
Gn=λTU2πλsr
λuexp
⎛
⎝N∑
i=1,i�=nQiB(r, h1)+QnC (r, h1)
⎞
⎠ , (37)
Hn=λAU2πλsr
λuexp
⎛
⎝N∑
i=1,i�=nQiE(r, h2)+QnF (r, h2)
⎞
⎠. (38)
In (37) and (38), B (r, h1), C (r, h1), E (r, h2) and F (r, h2)are given by (61), (62), (64) and (65) respectively in Appendix C.
Proof: See Appendix C. �From Theorem 1, Lemma 1 and Theorem 2, we show that
the SDPs of TUs and UAVs are correlated, i.e., the increase ofthe TU (or UAV) density degrades the SDP performance of allUEs, including TUs and UAVs. As a result, we cannot separatelymaximize the average SDP for each type of UEs. Instead, wejointly maximize the average SDP of TUs and UAVs.
According to Theorem 2, we can formulate the optimizationproblem of maximizing Pr as
P1 : maxSn|Nn=1
Pr
= maxSn|Nn=1
N∑
n=1
QnSn
[∫ dT
0Gn exp
(−πSnλsr2)dr
+
∫ dA
0Hn exp
(−πSnλsr2)dr
]
s.t.
N∑
n=1
Sn ≤ M
0 ≤ Sn ≤ 1, n = 1, . . . , N. (39)
Note that P1 is a non-convex optimization problem. To copewith this problem, we transform P1 into N sub-problems andsolve it in parallel. Adopting the partial Lagrangian, we have
L(S1, S2, . . . , SN , γ)
=
N∑
n=1
QnSn
[∫ dT
0Gn exp
(−πSnλsr2)dr
+
∫ dA
0Hn exp
(−πSnλsr2)dr
]+ γ
(N∑
n=1
Sn−M)
, (40)
where γM is constant and 0 ≤ Sn≤1, n = 1, . . . , N . In thefollowing, we opt to optimize each individual sub-problem.
Algorithm 1: Optimized Caching Probabilities in theDynamic On-Off Architecture.
1: Initial γ, ϕ, M and N2: repeat3: for n = 1 : 1 : N do4: Compute Sn from ∂L(Sn,γ)
∂Sn= 0 in (42)
5: if 0 ≤ Sn ≤ 1 then6: Set Sn = [Sn, 0, 1]7: else8: Set Sn = [0, 1]9: end if
10: Plug S ′n ∈ Sn into (41)
11: Choose Sn = argmaxS′n
L (S ′n, γ)
12: end for
13: Get S =N∑
i=1Si, then update γ
14: until S = M
First, the partial Lagrangian of the n-th sub-problem withrespect to Sn is given by
L (Sn, γ) =
∫ dT
0QnSnGn exp
(−πSnλsr2)dr
+
∫ dA
0QnSnHn exp
(−πSnλsr2)dr+γSn.
(41)
Second, by some mathematical manipulation, we have
∂L (Sn, γ)
∂Sn
=
∫ dT
0QnGn exp (−WSn)−WQnSnGn exp (−WSn)dr
+
∫ dA
0QnHnexp(−WSn)−WQnSnHn exp(−WSn)dr+γ,
(42)
where W = πλsr2.
Here, we discuss the key steps of optimizing the cachingprobabilities in Algorithm 1. With ∂L(Sn,γ)
∂Sn= 0, we can get the
extreme pointSn. LetSn = [Sn, 0, 1] if 0 ≤ Sn ≤ 1. Otherwise,let Sn = [0, 1]. Then, we identify the element in Sn that maxi-mizes the L (Sn, γ) in (41) as the optimized caching probabilitySn. After N iterations, we get
∑Ni=1 Si. If
∑Ni=1 Si = M ,
we obtain the optimized caching probability {S1, S2, . . . , SN}.Otherwise, the dual variable γ in (42) is updated byγi+1 = γi + (S1 + S2 + · · ·+ SN −M)ϕ, where ϕ is the stepsize.
The analytical results in Section V build upon an assumptionthat both UAVs and TUs have the same content request prob-ability. The following remark briefly discusses the case wherethe two tiers of UEs have different request probabilities.
Remark 1: LetQTn andQA
n be the request probabilities of then-th file for TUs and UAVs respectively. From (34), the weightedsum request probability becomes Qn = λTU
λuQT
n + λAU
λuQA
n ,
SONG et al.: PROBABILISTIC CACHING FOR SMALL-CELL NETWORKS WITH TERRESTRIAL AND AERIAL USERS 9169
andPr =∑N
n=1 (QT
nλTU
λuPrt(Dn) +
QAnλAU
λuPra(Dn)). It can be
seen that the optimization of SDP in this case largely follows(36)–(42) in this Section. In addition, the SBSs are more likely tocache the file requested by the UEs with better channel quality,when the TUs and UAVs have the same request probability fordifferent files.
VI. ANALYSIS ON NETWORK PARAMETERS
UNDER UCS AND PCS
In this section, we analyze the performance limits of theaverage SDP for the UCS and PCS under a single-slope pathloss model [29], respectively, where the path loss of the channelfrom an SBS to an UE is modeled as l−α with α denoting thepath loss exponent. In a noise-free case, the average SDP is givenby (43), as shown at the bottom of this page.
Theorem 3: Consider a single-slope path loss model. Whenthe SBS density λs is large enough, the average SDP is given by
Pr =N∑
n=1
QnSn exp(−πh2λuF(δ,α))Sn+
λuλs
F(δ,α). (44)
For the PCS, Pr =∑M
n=1Qn exp(−πh2λuF(δ,α))
1+ λuλs
F(δ,α).
For the UCS, Pr =exp(−πh2λuF(δ,α))
1+NλuMλs
F(δ,α).
Proof: See Appendix D. �FromTheorem 3,Corollary 1 andCorollary 2 below show
the performance limits of the average SDPs under the PCS andUCS respectively.
Corollary 1: For λs → +∞, the performance limits of theaverage SDPs under the PCS and UCS are given by respectively,
Pr =
M∑
n=1
Qn exp(−πh2λuF(δ, α)
), for the PCS, (45)
Pr = exp(−πh2λuF(δ, α)
), for the UCS. (46)
Corollary 2: As the exponent β → +∞, the performanceslimits of the average SDPs for the PCS and UCS are given byrespectively,
Pr =exp
(−πh2λuF(δ, α))
1 + λu
λsF(δ, α)
, for the PCS (47)
Pr =exp
(−πh2λuF(δ, α))
1 + Nλu
MλsF(δ, α)
, for the UCS. (48)
Based on Theorem 3, Corollary 1 and Corollary 2, wehave the following remarks.
Remark 2: From Theorem 3, the average SDPs of both PCSand UCS increase as the SBS density λs increases. When λs →
TABLE IITHE NETWORK PARAMETERS FOR TUS AND UAVS
+∞, Corollary 1 shows that the UCS achieves a better averageSDP than the PCS. In particular, the SDP of the PCS is affectedby β and M , while it is not the case for the UCS.
Remark 3: From Theorem 3, as the cache sizeM increases,the average SDPs of both PCS and UCS increase, and the per-formance gap for the two strategies narrows down. In particular,both strategies achieve the same SDP when M = N .
Remark 4: From Theorem 3, the average SDPs of both PCSand UCS increase asβ gradually increases. In particular, the SDPof the PCS grows more rapidly. Based on Corollary 2, the PCSoutperforms the UCS when β → +∞.
Remark 5: From Theorem 3, the average SDPs of both PCSand UCS become smaller as the height difference h increases.
VII. NUMERICAL AND SIMULATION RESULTS
In this section, we use both numerical results and MonteCarlo simulation results to validate our analytical results. Inthe simulations, the performance is averaged over 105 networkdeployments, where in each deployment SBSs and UEs are ran-domly distributed according to HPPPs with different densities.According to the 3GPP recommendations [25], [31] and [35], weuse hBS = 10 m, P = 24 dBm, δ = −6 dB. Other parametersfor TUs and UAVs are listed in Table II. According to theapplicability range of UAV height in [25], the UAV height is setto 30m. More specifically, we first focus on the average SDPs ofdifferent caching strategies in the UD SCNs. Second, we furtherinvestigate the impacts of the key network parameters, i.e., theSBS density, the cache size of cache memory, the exponentof Zipf distribution and the height of UAVs on the averageSDP.
A. Impact of SBS Density
Fig. 2 compares the average SDPs Pr versus the SBS densityλs among the OCS, PCS and UCS for TUs and UAVs respec-tively. First, it can be seen that the numerical results match wellwith the simulation results in all scenarios. In the following, wefocus on the analytical results only. Second, Fig. 2 shows that theOCS always outperforms the other two caching strategies. Third,
Pr =
N∑
n=1
Qn
∫ ∞
0LIZ
(δlα
P
)f(r)dr =
N∑
n=1
Qn
∫ ∞
0exp
⎛
⎝−2πN∑
i=1,i�=n
Pr (Ai)Siλs
∫ ∞
0
u
1 + l−αδ−1(√
u2 + h2)α du
⎞
⎠
× exp
⎛
⎝−2πPr (An)Snλs
∫ ∞
r
u
1 + l−αδ−1(√
u2 + h2)α du
⎞
⎠ 2πSnλsr exp(−πSnλsr
2)dr (43)
9170 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 9, SEPTEMBER 2019
Fig. 2. The impacts of the SBS density λs on the Pr of TUs and the Pr ofUAVs with hAU = 30 m, N = 100, M = 10 and β = 1.0.
in contrary to [32, Fig. 3] that uses the always-on architecture, weuse a dynamic on-off architecture where an SBS is only activewhen it is required to serve the UEs. We show that the SDPincreases with the increase of SBS density, while [32] showedthat the SDP first increases and then drops down with the increaseof SBS density. Fourth, we observe that the UCS can achievea good performance as long as λs is large enough, in spite ofa small caching probability. In this sense, it advocates cachingsome other files to further improve the saturated performance.These observations in Fig. 2 are in line with Remark 2. Thereasons are as follows:
1) when the SBS density is small, it is advisable to use thePCS because it smartly uses the limited number of SBSsto cache more popular files;
2) when the SBS density is large, the coverage probabil-ity becomes saturated [38]. To be specific, the coverageprobabilities are the same when λs = 105 SBSs/km2 orλs = 106 SBSs/km2. Hence, it is better to place the filesrandomly by the UCS than discarding less popular files bythe PCS.
Fig. 2(a) shows the Pr of TUs versus λs. As for the PCS, Princreases slowly with λs and becomes saturated when λs > 104
SBSs/km2. As for the UCS, Pr increases more rapidly than the
Fig. 3. The impacts of the cache size M on the average SDP with λs = 104
SBSs/km2, hAU = 30 m, N = 100 and β = 1.0.
PCS as λs goes up and becomes saturated when λs > 8 × 104
SBSs/km2. In addition, the PCS outperforms the UCS whenλs < 3 × 103 SBSs/km2, and the USC takes the lead when λs >3 × 103 SBSs/km2. The performance of the PCS is comparableto that of the OCS only when λs < 103 SBSs/km2. The sameobservation applies to the UCS when λs > 6 × 104 SBSs/km2.
Fig. 2(b) shows thePr of UAVs versus λs. This figure exhibitsthe similar observations to Fig. 2(a). As shown in Fig. 2(b), thePCS outperforms the UCS when λs < 1.4 × 104 SBSs/km2, andthe USC takes the lead when λs > 1.4 × 104 SBSs/km2. Theperformance of the PCS is comparable to that of the OCS onlywhen λs < 2 × 103 SBSs/km2. The same observation appliesto the UCS when λs > 6 × 104 SBSs/km2. As compared toFig. 2(a), it is observed that the average SDP of UAVs is shownto be worse than that of TUs. This is because the path loss ofUAVs is severer than that of TUs according to (26), (29) andTable II.
B. Impact of Cache Size
Fig. 3 compares the average SDPs Pr among the three strate-gies with the cache size M for TUs and UAVs respectively. Letλs = 104 SBSs/km2. First, we can see that the OCS exhibits abetter average SDP for TUs and UAVs than both the UCS andPCS. Second, it is observed thatPr increases monotonically withthe cache size. When M = 100, the three strategies reach thesame performance. These results are consistent with Remark 3.
Fig. 4 shows the average SDPs Pr versus the SBS densityλs with various cache sizes for TUs and UAVs, respectively.First, it can be seen that the OCS always outperforms both theUCS and PCS. Second, Pr of the PCS reaches the limit whenλs ≥ 104 SBSs/km2, and the performance limit becomes largerwith the increase of M . For TUs, the performance limit startswith 0.42 at M = 5 and goes up to 0.61 at M = 15. For UAVs,the performance limit increases from 0.37 at M = 5 to 0.50 atM = 15. Third, the Pr of the UCS reaches the limit when λs ≥5 × 105 SBSs/km2, the Pr of the UCS for TUs and UAVs keepsinvariant as M increases when λs ≥ 5 × 105 SBSs/km2. Theseobservations are line with Remark 2 and Remark 3. For TUs and
SONG et al.: PROBABILISTIC CACHING FOR SMALL-CELL NETWORKS WITH TERRESTRIAL AND AERIAL USERS 9171
Fig. 4. The impacts of the SBS density on the Pr of TUs and the Pr of UAVswith different cache sizes and hAU = 30 m, N = 100 and β = 1.0.
Fig. 5. The impacts of the exponent of Zipf distribution β on the average SDPwith λs = 104 SBSs/km2, hAU = 30 m, N = 100 and M = 10.
UAVs, the crossover point with the PCS and UCS achieving thesame Pr shifts to the left as M increases. This is because anincrease in the M will boost the average SDP given a fixed λs.
C. Impact of File Popularity Distribution
Fig. 5 compares the average SDPs versus the exponent ofZipf distribution β among the three strategies for TUs and UAVsrespectively. We set λs = 104 SBSs/km2. First, it can be seen
Fig. 6. The impacts of the SBS density on the Pr of TUs and the Pr of UAVswith different β and λs with hAU = 30 m, N = 100 and M = 10.
that Pr increases as β increases. Second, the average SDP ofthe UCS is independent of β, since it caches each file withequal probability in the always-on architecture [13]. However,in the dynamic on-off architecture, the performance of the UCSis slowly growing with β. According to (5) and (36), the changeof β leads to the change of Qn and eventually causes the changeof the average SDP. When the λs is large enough, the changesof β will not affect Pr of the UCS. Third, it can be seen that thePCS is worse than the UCS when β < 1.4 for TUs and β < 0.95for UAVs. As β gradually grows, the PCS becomes better. Thereason is that a few files dominate the requests and caching suchpopular files gives a large Pr as β becomes larger, since therequest probabilities of files are more unevenly distributed. Theaverage SDP of the PCS grows more rapidly with increasingβ, and the average SDP of the PCS is better than that of theUCS when β is large enough. These observations agree withRemark 4.
Fig. 6 shows the average SDPs versus λs and β for TUs andUAVs, respectively. First, the results with a fixed β or a fixed λs
are consistent with that in Fig. 2 and Fig. 5 respectively. Second,as λs increases, we need to increase value of β to meet the sameperformance of both the PCS and UCS. For example, in Fig. 6(a),the value of β is 1.5 when λs = 104 and it becomes 2.2 when
9172 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 9, SEPTEMBER 2019
Fig. 7. The impacts of the UAV height hAU on the Pr of TUs/UAVs withN = 100, M = 10 and β = 1.0.
λs = 105. In Fig. 6(b), the value of β is 0.9 when λs = 104 andit becomes 1.5 when λs = 105.
D. Impact of UAV Height
Fig. 7 depicts the impacts of the UAV height on the Pr ofTUs and the Pr of UAVs among the three strategies. Considerthat λs = 104 SBSs/km2 and λs = 105 SBSs/km2 respectively.First, we observe that the change of the UAV height affects thePr of UAVs but has no performance impact on TUs. Second,we can see that Pr of UAVs decreases monotonically as theheight of UAVs increases, which is consistent with Remark 5.Third, similar to Fig. 2, the average SDP of the PCS almostremains the same when λs varies from 104 SBSs/km2 to 105
SBSs/km2, while the average SDP of the UCS increases overthe same range of λs. When λs = 104 SBSs/km2, the PCS isbetter than the UCS. When hAU < 50m, the UCS is better thanthe PCS when λs = 105 SBSs/km2. However, the PCS is betterthan the UCS when hAU > 50 m. This is because the path lossof UAV is generally large such that λs is not dense enough tosupport the average SDP of the UCS.
VIII. CONCLUSION
In this paper, we have developed an optimized probabilisticsmall-cell caching strategy for small-cell networks with TUs andUAVs to maximize the average SDP. Our analytical results haveshown that the OCS can achieve a better average SDP than thePCS and UCS. Moreover, we have further analyzed the impactsof key parameters on the average SDP of TUs and UAVs andobtained the following valuable insights verified by the extensivesimulation results:
1) The average SDP increases as either of the SBS density, thecache size, or the exponent of Zipf distribution increases.With the increase of the UAV height, the average SDP ofUAVs decreases.
2) When the SBS density λs is relatively small, the PCSachieves a better average SDP than the UCS. As the densityincreases, the performance of the UCS gradually improvesand outperforms that of the PCS.
3) When the exponent of Zipf distribution β is relativelysmall, the UCS outperforms the PCS. As β increases, the
performance of the PCS gradually improves and surpassesthat of UCS.
4) When the cache size M is equal to the popular files N , theaverage SDP of the PCS, UCS, and OCS converges, sinceeach SBS caches all N files with probability of 1.
Going forward, several directions deserve further investiga-tion. First, the uplink performance of the UAVs in the proposedcaching network is yet to be analyzed. As shown in this paper,establishing the analytical results of SBSs as interferers is al-ready non-trivial for performance analysis, the introduction ofterrestrial users and UAVs as interferers in the uplink will makethe analysis even more challenging. Second, it is of interest toconsider that each SBS uses the coded caching strategy to furtherenhance the performance of the OCS by exploring the advan-tages of prefetching coded files over the uncoded placement inthis paper.
APPENDIX APROOF OF THEOREM 1
In order to evaluate Pr (Dn) in Theorem 1, the first key stepis to calculate the PDFs for the events that the typical UE isassociated with an SBS under an LoS path or a NLoS path, andthe second key step is to calculate Pr (SINR > δ) for the LoSand NLoS cases conditioned on distance r. Given the piece-wisepath loss model presented in (3), we have
Pr (Dn) =
∫ ∞
0Pr (SINR > δ) f(r, h)dr
=
∫ ∞
0Pr
[Pζ(r, h)g
σ2 + IZ> δ
]f(r, h)dr
=
∫ d1(h)
0Pr
[PζL1 (r, h)g
σ2 + IZ> δ
]fL
1 (r, h)dr
+
∫ d1(h)
0Pr
[PζNL
1 (r, h)g
σ2 + IZ> δ
]fNL
1 (r, h)dr
+· · ·
+
∫ ∞
dK−1(h)
Pr
[PζLK(r, h)g
σ2 + IZ> δ
]fLK(r, h)dr
+
∫ ∞
dK−1(h)
Pr
[PζNL
K (r, h)g
σ2 + IZ> δ
]fNLK (r, h)dr
=
K∑
k=1
(∫ dk(h)
dk−1(h)
Pr
[PζLk (r, h)g
σ2 + IZ>δ
]fLk (r, h)dr
+
∫ dk(h)
dk−1(h)
Pr
[PζNL
k (r, h)g
σ2 + IZ>δ
]fNLk (r, h) dr
)
,
(49)
where fLk (r, h) and fNL
k (r, h) are the PDFs of LoSpath and NLoS path respectively. Moreover, let TL
k
=∫ dk(h)
dk−1(h)Pr[
PζLk (r,h)g
σ2+IZ> δ]fL
k (r, h) dr and TNLk =
∫ dk(h)
dk−1(h)
Pr[PζNL
k (r,h)g
σ2+IZ>δ]fNL
k (r, h) dr respectively. Therefore, we
have Pr(Dn) =∑K
k=1
(TLk + TNL
k
).
SONG et al.: PROBABILISTIC CACHING FOR SMALL-CELL NETWORKS WITH TERRESTRIAL AND AERIAL USERS 9173
In the following, we discuss how to obtain fLk (r, h) and
fNLk (r, h).
Define BLk as the event that the signal comes from
the k-th piece LoS path. By definition, fLk (r, h) = fk|BL
k(r, h
∣∣BL
k
)Pr[BL
k
], where Pr
[BL
k
]= PrLk (r, h) according to
(3) and fk|BLk
(r, h
∣∣BL
k
)jointly characterize the following in-
dependent sub-events:1) For the typical UE, its serving SBS bo exists with the
horizontal distance r from the UE, and the correspondingunconditional PDF of r is 2πrλ [26].
2) The probability that the LoS SBS bo in event BLk has a
better link to the typical UE than any other LoS SBSsis [31]
pLk (r, h) = exp
(−∫ r
0PrL(r, h) 2πλudu
). (50)
3) The probability that the LoS SBS bo in event BLk has a
better link to the typical UE than any other NLoS SBSsis [31]
pNLk (r, h)=exp
(−∫ r1
0
(1−PrL(r, h)
)2πλudu
),
(51)where r1 = arg
r1
{ζNL(r1, h) = ζLk (r, h)
}.
With reference to [31], we obtain
fk|BLk
(r, h
∣∣BLk
)= pNL
k (r, h)pLk (r, h)2πrλ. (52)
Thus, fLk (r, h) for n-th tier can be written as
fLk(r, h) = exp
(−∫ r1
02πSnλs
(1−PrLk (u, h)
)udu
)
× exp
(−∫ r
02πSnλsPr
Lk (u, h)udu
)
× PrLk (r, h)2πrSnλs, dk−1(h)<r≤dk(h). (53)
In a similar way, fNLk (r, h) for n-th tier can be written as
fNLk (r, h)= exp
(−∫ r2
02πSnλsPr
Lk (u, h)udu
)
× exp
(−∫ r
02πSnλs
(1 − PrLk (u, h)
)udu
)
× (1−PrLk (r, h)
)2πrSnλs, dk−1(h)<r≤dk(h),
(54)
where r2 = argr2
{ζL(r2, h) = ζNL
k (r, h)}
.
APPENDIX BPROOF OF LEMMA 1
Given IZ = IZ1 + IZ2, we have
LIZ
(δ
Pζk(r, h)
)= EIZ
[exp
(− δIZPζk(r, h)
)]
=EIZ1
[exp
(− δIZ1
Pζk(r, h)
)]
× EIZ2
[exp
(− δIZ2
Pζk(r, h)
)]. (55)
Since the distribution of the SBSs in the i-th tier is viewedas a thinned HPPP φi with density of Siλs, for the interferencefrom the i-th tier, we have
EIZ1
[exp
(− δIZ1
Pζk(r, h)
)]
= Egu,u
⎡
⎢⎣
∏
u∈∑Ni=1,i�=n φi
exp(−ζk(r, h)
−1δguζ(u, h))⎤
⎥⎦
= exp
⎛
⎝−N∑
i=1,i�=n
2πPr (Ai)Siλs
×∫ ∞
0
(
1 − 1
1 + ζk(r, h)−1δζ (u, h)
)
udu
⎞
⎠ . (56)
For LoS or NLoS signal,
∫ ∞
0
(
1 − 1
1 + ζ{L,NL}k (r, h)−1δζ(u, h)
)
udu
=
∫ ∞
0
PrL(u, h)u
1 + ζ{L,NL}k (r, h)(δζL(u, h))−1
du
+
∫ ∞
0
[1 − PrL(u, h)
]u
1 + ζ{L,NL}k (r, h)(δζNL(u, h))−1
du. (57)
Likewise, for the interference from the n-th tier, we have
EI2
[exp
(− δIZ2
Pζk(r, h)
)]
= exp (−2πPr (An)Snλs
×∫ ∞
r
(
1 − 1
1 + ζk(r, h)−1δζ (u, h)
)
udu
⎞
⎠ . (58)
For LoS or NLoS signal,
∫ ∞
r
(
1 − 1
1 + ζ{L,NL}k (r, h)−1δζ(u, h)
)
udu
=
∫ ∞
{r,r2}
PrL(u, h)u
1 + ζ{L,NL}k (r, h)(δζL(u, h))−1
du
+
∫ ∞
{r1,r}
[1 − PrL(u, h)
]u
1 + ζ{L,NL}k (r, h)(δζNL(u, h))−1
du. (59)
APPENDIX CPROOF OF THEOREM 2
Consider that r→0 and neglecting noise. When λs→+∞,the average SDP of TUs over all possible N files is given by
Prt(D) =
N∑
n=1
QnPrt (Dn)
=
N∑
n=1
Qn
∫ dT
0LIZ
(δlα
Lt
PALt
)
ft (r, h1) dr
9174 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 68, NO. 9, SEPTEMBER 2019
=
N∑
n=1
Qn
{∫ dT
02πSnλsr exp
(−πSnλsr2)
×exp
⎛
⎝N∑
i=1,i�=n
QiB (r, h1) +QnC (r, h1)
⎞
⎠ dr
⎫⎬
⎭, (60)
where
B (r, h1) =
− 2πλu
⎡
⎢⎣∫ dT
0
u
1 + ζLt (r, h1) (δALt )
−1(√
u2 + h21
)αLt
×(
1 −√u2 + h2
1
l0
)
du
+
∫ dT
0
u
1 + ζLt (r, h1) (δANLt )
−1(√
u2 + h21
)αNLt
×√u2 + h2
1
l0du
+
∫ ∞
dT
u
1+ζLt (r, h1) (δANLt )
−1(√
u2 + h21
)αNLt
du
⎤
⎥⎦ , (61)
C (r, h1) =
− 2πλu
⎡
⎢⎣∫ dT
r
u
1 + ζLt (r, h1) (δALt )
−1(√
u2 + h21
)αLt
×(
1 −√u2 + h2
1
l0
)
du
+
∫ dT
r1
u
1 + ζLt (r, h1) (δANLt )
−1(√
u2 + h21
)αNLt
×√u2 + h2
1
l0du
+
∫ ∞
dT
u
1+ζLt (r, h1) (δANLt )
−1(√
u2 + h21
)αNLt
du
⎤
⎥⎦ . (62)
Consider that r→0 and neglecting noise. When λs→+∞,the average SDP of UAVs over all possible N files is given by
Pra(D) =N∑
n=1
QnPra (Dn)
=
N∑
n=1
Qn
⎧⎨
⎩
∫ dA
02πSnλsr exp
(−πSnλsr2)
×exp
⎛
⎝N∑
i=1,i�=n
QiE (r, h2) +QnF (r, h2)
⎞
⎠ dr
⎫⎬
⎭, (63)
where
E (r, h2) =
−2πλu
⎡
⎢⎣∫ dA
0
⎛
⎜⎝
1
1 + ζLa (r, h2) (δALa )
−1(√
u2 + h22
)αLa
⎞
⎟⎠udu
+
∫ ∞
dA
⎛
⎜⎝
1
1 + ζLa (r, h2) (δALa )
−1(√
u2 + h22
)αLa
⎞
⎟⎠
×(dAu
+ exp
(−u
p1
)(1 − dA
u
))udu
+
∫ ∞
dA
⎛
⎜⎝
1
1 + ζLa (r, h2) (δANLa )−1
(√u2 + h2
2
)αNLa
⎞
⎟⎠
×(
1 − dAu
− exp
(−u
p1
)(1 − dA
u
))udu
], (64)
F (r, h2) =
−2πλu
⎡
⎢⎣∫ dA
r
⎛
⎜⎝
1
1 + ζLa (r, h2) (δALa )
−1(√
u2 + h22
)αLa
⎞
⎟⎠udu
+
∫ ∞
dA
⎛
⎜⎝
1
1 + ζLa (r, h2) (δALa )
−1(√
u2 + h22
)αLa
⎞
⎟⎠
×(dAu
+ exp
(−u
p1
)(1 − dA
u
))udu
+
∫ ∞
dA
⎛
⎜⎝
1
1 + ζLa (r, h2) (δANLa )−1
(√u2 + h2
2
)αNLa
⎞
⎟⎠
×(
1 − dAu
− exp
(−u
p1
)(1 − dA
u
))udu
]. (65)
Overall, the average SDP of TUs and UAVs is
Pr =
N∑
n=1
Qn
(λTU
λuPrt (Dn) +
λAU
λuPra (Dn)
)
=
N∑
n=1
QnSn
[∫ dT
0Gn exp
(−πSnλsr2)dr
+
∫ dA
0Hn exp
(−πSnλsr2)dr
], (66)
where
Gn=λTU2πλsr
λuexp
⎛
⎝N∑
i=1,i�=n
QiB (r, h1) +QnC (r, h1)
⎞
⎠ ,
(67)
SONG et al.: PROBABILISTIC CACHING FOR SMALL-CELL NETWORKS WITH TERRESTRIAL AND AERIAL USERS 9175
Hn=λAU2πλsr
λuexp
⎛
⎝N∑
i=1,i�=n
QiE (r, h2) +QnF (r, h2)
⎞
⎠ .
(68)
APPENDIX DPROOF OF THEOREM 3
According to (43), when the interference come from the n-thtier IZ2, we have
exp
⎛
⎝−2πPr (An)Snλs
∫ ∞
r
u
1 + l−αδ−1(√
u2 + h2)α du
⎞
⎠
= exp
(
−πPr (An)Snλsδ2α l2
2α
∫ ∞
δ−1
z2α −1
1 + zdz
)
= exp
(−πPr (An)Snλsl
2 2δα− 2 2F1
×(
1, 1 − 2α; 2 − 2
α;−δ
)), (69)
where z = δ−1l−α(√
u2 + h2)α
and 2F1 (·) denotes the hyper-
geometric function.When the interference come from other tiers IZ1, we have
exp
⎛
⎝−2πN∑
i=1,i�=n
Pr (Ai)Siλs
∫ ∞
0
u
1+l−αδ−1(√
u2+h2)α du
⎞
⎠
= exp
⎛
⎝−π
N∑
i=1,i�=n
Pr (Ai)Siλsl2 2δα− 2 2
×F1
(1, 1 − 2
α; 2 − 2
α;−δlα
hα
)⎞
⎠ . (70)
Overall, we have
Pr =
N∑
n=1
Qn
∫ ∞
0πSnλs exp
(−πλsr2Sn
−πN∑
i=1,i �=n
Pr(Ai)Siλsl2 2δα− 2 2F1
(1, 1 − 2
α; 2 − 2
α;−δl−α
hα
)
−πPr (An)Snλsl2 2δα− 2 2F1
(1, 1 − 2
α; 2 − 2
α;−δ
))dr2.
(71)
When the SBS density is large enough, we rewrite (71) as
Pr =
N∑
n=1
Qn
∫ ∞
0πSnλs exp
(
−πλsr2Sn
−π
N∑
i=1
Pr (Ai)Siλsl2 2δα− 2 2F1
(1, 1− 2
α; 2− 2
α;−δ
))
dr2
=N∑
n=1
QnSn exp(−πλsh
2 ∑Ni=1 Pr (Ai)SiF(δ, α)
)
Sn +∑N
i=1 Pr (Ai)SiF(δ, α),
(72)
where F (δ, α) = 2δα−2 2F1
(1, 1 − 2
α ; 2 − 2α ;−δ
).
Substituting (34) into (72), we have
Pr =
N∑
n=1
QnSn exp(−πλsh
2 ∑Ni=1
Qiλu
SiλsSiF(δ, α)
)
Sn +∑N
i=1Qiλu
SiλsSiF(δ, α)
=
N∑
n=1
QnSn exp(−πh2λuF(δ, α)
)
Sn + λu
λsF(δ, α)
. (73)
For the PCS, we have
Pr =M∑
n=1
Qn exp(−πh2λuF(δ, α)
)
1 + λu
λsF(δ, α)
. (74)
For the UCS, we have
Pr =
N∑
n=1
QnMN exp
(−πh2λuF(δ, α))
MN + λu
λsF(δ, α)
=exp
(−πh2λuF(δ, α))
1 + Nλu
MλsF(δ, α)
. (75)
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Fei Song received the M.S. degree from Xi’an Poly-technic University, Xi’an, China, in 2017. FromSeptember 2017 to now, she is working toward thePh.D. degree with the School of Electronic and Op-tical Engineering, Nanjing University of Science andTechnology, Nanjing, China. Her research interestsinclude small-cell caching and network slicing.
Jun Li (M’09–SM’16) received the Ph.D. degreein electronic engineering from Shanghai Jiao TongUniversity, Shanghai, China, in 2009. From January2009 to June 2009, he worked with the Departmentof Research and Innovation, Alcatel Lucent ShanghaiBell as a Research Scientist. From June 2009 to April2012, he was a Postdoctoral Fellow with the Schoolof Electrical Engineering and Telecommunications,the University of New South Wales, Australia. FromApril 2012 to June 2015, he was a Research Fellowwith the School of Electrical Engineering, the Uni-
versity of Sydney, Australia. From June 2015 till now, he is a Professor with theSchool of Electronic and Optical Engineering, Nanjing University of Science andTechnology, Nanjing, China. His research interests include network informationtheory, channel coding theory, wireless network coding, resource allocations incellular networks.
Ming Ding (M’12–SM’17) received the B.S. andM.S. degrees (with first class Hons.) in electron-ics engineering from Shanghai Jiao Tong University(SJTU), Shanghai, China, and the Ph.D. degree in sig-nal and information processing from SJTU, in 2004,2007, and 2011, respectively. From April 2007 toSeptember 2014, he worked with Sharp Laboratoriesof China in Shanghai, China as a Researcher/SeniorResearcher/Principal Researcher. He also served asthe Algorithm Design Director and Programming Di-rector for a system-level simulator of future telecom-
munication networks in Sharp Laboratories of China for more than seven years.He is currently a Senior Research Scientist with Data61, CSIRO, Sydney, NSW,Australia. He has authored more than 90 papers in the IEEE journals andconferences, all in recognized venues, and about 20 3GPP standardization con-tributions, as well as a Springer book Multi-point Cooperative CommunicationSystems: Theory and Applications. Also, he holds 16 US patents and co-inventedanother 100+ patents on 4G/5G technologies in CN, JP, EU, etc. He is currentlyan Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. Be-sides, he is or has been Guest Editor/Co-Chair/Co-Tutor/TPC member of severalIEEE top-tier journals/conferences, e.g., the IEEE JOURNAL ON SELECTED
AREAS IN COMMUNICATIONS, the IEEE COMMUNICATIONS MAGAZINE, and theIEEE Globecom Workshops, etc. He was the Lead Speaker of the industrialpresentation on unmanned aerial vehicles in IEEE Globecom 2017, which wasawarded as the Most Attended Industry Program in the conference. Also, he wasawarded in 2017 as the Exemplary Reviewer for the IEEE TRANSACTIONS ON
WIRELESS COMMUNICATIONS.
SONG et al.: PROBABILISTIC CACHING FOR SMALL-CELL NETWORKS WITH TERRESTRIAL AND AERIAL USERS 9177
Long Shi (S’10–M’13) received the Ph.D. degree inelectrical engineering from the University of NewSouth Wales, Sydney, NSW, Australia, in 2012. Hewas a Visiting Student with the Chinese Universityof Hong Kong and University of Delaware in 2010and 2011, respectively. From 2013 to 2016, he wasa Postdoctoral Fellow with the Institute of NetworkCoding, Chinese University of Hong Kong. From2014 to 2017, he was a Lecturer with the Collegeof Electronic and Information Engineering, NanjingUniversity of Aeronautics and Astronautics. He is
a currently a Postdoctoral Research Fellow with the Singapore University ofTechnology and Design. His research interests include wireless caching, wirelessnetwork coding, and channel coding.
Feng Shu (M’09) received the B.S., M.S., and Ph.D.degrees from Fuyang teaching College, Fuyang,China, XiDian University, Xian, China, and SoutheastUniversity, Nanjing, China, in 1994, 1997, 2002, re-spectively. From September 2009 to September 2010,he was a Visiting Post-doctor with the University ofTexas at Dallas. In October 2005, he joined the Schoolof Electronic and Optical Engineering, Nanjing Uni-versity of Science and Technology, Nanjing, China,where he is currently a Professor and Supervisor ofPh.D and graduate students. He is also with Fujian
Agriculture and Forestry University and awarded with Mingjian Scholar ChairProfessor and Fujian hundred-talent plan in Fujian Province. His research inter-ests include wireless networks, wireless location, and array signal processing.He is currently an Editor for the journal IEEE ACCESS. He has published morethan 200 in archival journals with more than 50 papers on the IEEE Journals and80 SCI-indexed papers. He holds six Chinese patents.
Meixia Tao (S’00–M’04–SM’10–F’19) received theB.S. degree in electronic engineering from FudanUniversity, Shanghai, China, in 1999, and the Ph.D.degree in electrical and electronic engineering fromthe Hong Kong University of Science and Technol-ogy, Hong Kong, in 2003. She is currently a Pro-fessor with the Department of Electronic Engineer-ing, Shanghai Jiao Tong University, Shanghai, China.Prior to that, she was a Member of Professional Staffwith the Hong Kong Applied Science and TechnologyResearch Institute during 2003–2004, and a Teaching
Fellow then an Assistant Professor with the Department of Electrical andComputer Engineering, National University of Singapore from 2004 to 2007.Her research interests include wireless caching, edge computing, physical layermulticasting, and resource allocation.
Dr. Tao was a Member of the Executive Editorial Committee of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS. She was on the editorial boardof the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (2007–2011),and the IEEE TRANSACTIONS ON COMMUNICATIONS (2012–2018), the IEEECOMMUNICATIONS LETTERS (2009–2012), and the IEEE WIRELESS COMMU-NICATIONS LETTERS (2011–2015). She was as Symposium Oversight Chair ofthe IEEE ICC 2019, Symposium Co-Chair of the IEEE GLOBECOM 2018,the TPC Chair of the IEEE/CIC International Conference on Communicationsin China (ICCC) 2014, and Symposium Co-Chair of the IEEE ICC 2015. Sheis the recipient of the IEEE Marconi Prize Paper Award in 2019, the IEEEHeinrich Hertz Award for Best Communications Letters in 2013, the IEEE/CICICCC Best Paper Award in 2015, and the International Conference on WirelessCommunications and Signal Processing Best Paper Award in 2012. She was alsothe recipient of the IEEE ComSoc Asia-Pacific Outstanding Young Researcheraward in 2009.
Wen Chen (M’03–SM’11) is a tenured Profes-sor with the Department of Electronic Engineering,Shanghai Jiao Tong University, Shanghai, China,where he is also the Director with the Institute forSignal Processing and Systems. He has authored 97papers in the IEEE journals and more than 120 papersin IEEE conferences. His research interests includemultiple access, coded cooperation, green heteroge-neous networks, and vehicular communications. Hewas the recipient of the Ariyama Memorial ResearchAward in 1997; and the JSPS Fellowship and PIMS
Fellowship, in 1999 and 2001, respectively. He was the recipient of the Honorsof New Century Excellent Scholar in China in 2006 and the Pujiang ExcellentScholar in Shanghai in 2007. He was also the recipient of the Best ServiceAward of the China Institute of Electronics, Shanghai Master Thesis SupervisionAward, and the Best Paper Award of the Chinese Information Theory Societyin 2013, the best presentation paper awards of the Chinese Information TheorySociety in 2014, the Innovate 5G Competition Award and the IEEE WCSPBest Paper Award in 2015, the IEEE APCC Keynote Speaker Appreciationin 2016, the Excellent Ph.D. Thesis Supervision Award of Chinese Instituteof Communications in 2017, the IEEE VTS Chapter-of-the-Year Award andthe IEEE ComSoc Distinguished Lecturer in 2018, the Excellent ResearcherAward of Chinese Institute of Electronics in 2019. He is the Chair of the IEEEVTS Shanghai Chapter, the Editor for the IEEE TRANSACTIONS ON WIRELESS
COMMUNICATIONS, the IEEE TRANSACTIONS ON COMMUNICATIONS, and theIEEE ACCESS.
H. Vincent Poor (S’72–M’77–SM’82–F’87) re-ceived the Ph.D. degree from Princeton University,Princeton, NJ, USA, in 1977. From 1977 until 1990,he was on the faculty of the University of Illinois atUrbana-Champaign. Since 1990, he has been on thefaculty at Princeton, where he is currently the MichaelHenry Strater University Professor of Electrical En-gineering. During 2006 to 2016, he was as Deanof Princetons School of Engineering and AppliedScience. He has also held visiting appointments atseveral other universities, including most recently at
Berkeley and Cambridge. His research interests include information theory andsignal processing, and their applications in wireless networks, energy systemsand related fields. Among his publications in these areas is the recent bookMultiple Access Techniques for 5G Wireless Networks and Beyond. (Springer,2019).
Dr. Poor is a Member of the National Academy of Engineering and theNational Academy of Sciences, and is a Foreign Member of the ChineseAcademy of Sciences, the Royal Society, and other national and internationalacademies. Recent recognition of his work includes the 2017 IEEE AlexanderGraham Bell Medal, the 2019 ASEE Benjamin Garver Lamme Award, a D.Sc.honoris causa from Syracuse University awarded in 2017, and a D.Eng. honoriscausa from the University of Waterloo awarded in 2019. He is currently servingas a Distinguished Lecturer for the IEEE Vehicular Technology Society.
