6360 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 9, SEPTEMBER 2018

Edge Caching in Dense Heterogeneous CellularNetworks With Massive MIMO-Aided

Self-BackhaulLifeng Wang , Member, IEEE, Kai-Kit Wong, Fellow, IEEE,

Sangarapillai Lambotharan , Senior Member, IEEE, Arumugam Nallanathan , Fellow, IEEE,

and Maged Elkashlan , Member, IEEE

Abstract— This paper focuses on edge caching in dense hetero-geneous cellular networks, in which small base stations (SBSs)with limited cache size store the popular contents, and mas-sive multiple-input multiple-output (MIMO)-aided macro basestations provide wireless self-backhaul when SBSs require thenon-cached contents. Our aim is to address the effects of cellload and hit probability on the successful content delivery (SCD)and present the minimum required base station density foravoiding the access overload in an arbitrary small cell andbackhaul overload in an arbitrary macrocell. The achievablerate of massive MIMO backhaul without any downlink channelestimation is derived to calculate the backhaul time, and thelatency is also evaluated in such networks. The analytical resultsconfirm that hit probability needs to be appropriately selectedin order to achieve SCD. The interplay between cache size andSCD is explicitly quantified. It is theoretically demonstrated thatwhen non-cached contents are requested, the average delay ofthe non-cached content delivery could be comparable to thecached content delivery with the help of massive MIMO-aidedself-backhaul, if the average access rate of cached content deliveryis lower than that of self-backhauled content delivery. Simulationresults are presented to validate our analysis.

Index Terms— Edge caching, dense small cell, massive MIMO,self-backhaul.

I. INTRODUCTION

A. Motivation and Background

NEW findings from Cisco [1] indicate that mobile videotraffic accounts for the majority of mobile data traffic.

To offload the traffic of the core networks and reduce the back-haul cost and latency, caching the popular contents at the edge

Manuscript received September 5, 2017; revised January 24, 2018 andApril 20, 2018; accepted July 10, 2018. Date of publication August 2, 2018;date of current version September 10, 2018. This work was supported by theU.K. Engineering and Physical Sciences Research Council (EPSRC) underGrant EP/M016005/1, Grant EP/M016145/2, and Grant EP/M015475/1. Theassociate editor coordinating the review of this paper and approving it forpublication was S. Pollin. (Corresponding author: Lifeng Wang.)

L. Wang and K.-K. Wong are with the Department of Electronic andElectrical Engineering, University College London, London WC1E 6BT, U.K.(e-mail: lifeng.wang@ucl.ac.uk; kai-kit.wong@ucl.ac.uk).

S. Lambotharan is with the School of Mechanical, Electrical and Manu-facturing Engineering, Loughborough University, Loughborough LE11 3TU,U.K. (e-mail: s.lambotharan@lboro.ac.uk).

A. Nallanathan and M. Elkashlan are with the School of ElectronicEngineering and Computer Science, Queen Mary University of London,London E1 4NS, U.K. (e-mail: a.nallanathan@qmul.ac.uk; maged.elkashlan@qmul.ac.uk).

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

Digital Object Identifier 10.1109/TWC.2018.2859936

of wireless networks becomes a promising solution [2]–[4].The latest 3GPP standard requires that the fifth generation (5G)system shall support content caching applications and opera-tors need to place the content caches close to mobile termi-nals [5]. In addition, the emerging radio-access technologiesand wireless network architectures provide edge caching withnew opportunities [6].

Recent works have focused on the caching design andanalysis in various scenarios. In [7], a probabilistic cachingmodel was considered in single-tier cellular networks and theoptimal content placement was designed to maximize the totalhit probability. In [8], a stochastic content multicast schedul-ing problem was formulated to jointly minimize the averagenetwork delay and power costs in heterogeneous cellularnetworks (HetNets), and a structure-aware optimal algorithmwas proposed to solve this problem. Caching cooperation inmulti-tier HetNets was studied in [9], where a low-complexitysuboptimal solution was developed to maximize the capacityin such networks. Caching in device-to-device (D2D) networkswas investigated in the literature such as [10], [11]. In [10],a holistic design on D2D caching at multi-frequency bandincluding sub-6 GHz and millimeter wave (mmWave) waspresented. In [11], the performance difference between maxi-mizing hit probability and maximizing cache-aided throughputin D2D caching networks was evaluated. The work of [12]showed that in multi-hop relaying systems, the efficiency ofcaching could be further improved by using collaborativecache-enabled relaying. Joint design of cloud and edge cachingin fog radio access networks were introduced in [13] and [14],where the popular contents were cached at the remote radioheads. However, prior works [7]–[13] did not present designand insights involving edge caching in the future dense/ultra-dense cellular networks (e.g., 5G) with backhaul limitations,where wireless self-backhauling shall be supported [4].

Cache-enabled small cell networks with stochastic modelshave been investigated in the literature such as [15]–[19].Cluster-centric caching with base station (BS) cooperationwas studied in [15], where the tradeoff between transmissiondiversity and content diversity was revealed. In [16] where itwas assumed that the intensity of BSs is much larger than theintensity of mobile terminals, two cache-enabled BS modeswere considered, namely always-on and dynamic on-off.The work of [17]–[19] concentrated on the cache-enabled

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WANG et al.: EDGE CACHING IN DENSE HETNETs WITH MASSIVE MIMO-AIDED SELF-BACKHAUL 6361

multi-tier HetNets. Specifically, [17] and [18] studied opti-mal content placement under probabilistic caching strategy,and [19] considered the joint BS caching and cooperation,in contrast to the single-tier case in [15]. However, [15]–[19]only aimed to maximize the probability that the requestedcontent is not only cached but also successfully delivered.In realistic networks, when the requested contents of usersare not cached at their associated BSs, they will be obtainedfrom the core network via wired/wireless backhaul, which alsoneeds to be studied in cache-enabled cellular networks.

In fact, existing contributions such as [20]–[22] havestudied the effects of backhaul on content delivery incache-enabled networks. The work of [20] considered thatnon-cached contents were obtained via backhaul, and adownlink content-centric sparse multicast beamforming wasproposed for the cache-enabled cloud radio access network(Cloud-RAN), to minimize the weighted sum of backhaul costand transmit power. In [21], the network successful contentdelivery consisting of cached content delivery and backhauledcontent delivery was studied. The optimization problem wasformulated to minimize the cache size under quality-of-serviceconstraint. The work of [22] analyzed the capacity scaling lawwhen there are limited number of wired backhaul in single-tiernetworks, and showed that cache size needs to be large enoughto achieve linear capacity scaling. However, none of [20]–[22]has studied the cache-enabled cellular networks with specifiedwireless backhaul transmission, such as massive multiple-inputmultiple-output (MIMO) aided self-backhaul.

B. Novelty and Contributions

In this paper, we focus on the edge caching in denseHetNets with massive MIMO aided self-backhaul, which hasnot been understood yet. Massive MIMO aided self-backhaulis motivated by the facts that it may not be feasible tohave optical fiber for every backhaul channel and massiveMIMO can support high-speed transmissions thanks to largearray gains and multiplexing gains [4]. Our contributions aresummarized as follows:

• In contrast to the prior works such as [15]–[22], we con-sider cache-enabled HetNets, in which randomly locatedsmall BSs (SBSs) cache finite popular contents, andmacro BSs (MBSs) equipped with massive MIMO anten-nas provide wireless backhaul to deliver the non-cachedrequested contents to the SBSs. Moreover, we also con-sider the resource allocation when multiple users requestthe contents from the same SBS, which has not beenstudied in a cache-enabled stochastic model.

• We first derive the successful content delivery probabilitywhen the requested content is cached at the SBS. Themaximum small cell load is calculated, and the minimumrequired density of SBSs for avoiding access overloadis obtained. We show that hit probability needs to belower than a critical value, to guarantee successful cachedcontent delivery.

• We derive the successful content delivery probabilitywhen the requested content is not cached and has to beobtained via massive MIMO backhaul. We analyze the

Fig. 1. An illustration of cache-enabled heterogeneous cellular network withmassive MIMO backhaul.

massive MIMO backhaul achievable rate when downlinkchannel estimation is not necessary, to evaluate the back-haul transmission delay. The minimum required densityof MBSs for avoiding backhaul overload is obtained.We show that hit probability needs to be higher thana critical value, to guarantee successful self-backhauledcontent delivery.

• We analyze the effects of cache size on the successfulcontent delivery, and provide important insights on theinterplay between time-frequency resource allocation andcache size from the perspective of successful contentdelivery probability. We characterize the latency in termsof average delay in such networks. We confirm that whenthe requested contents are not cached, the average delayof the non-cached content delivery could be comparableto the cached content delivery with the assistance of mas-sive MIMO backhaul, if the average access rate of cachedcontent delivery is lower than that of self-backhauledcontent delivery.

II. NETWORK MODEL

As shown in Fig. 1, we consider a two-tier self-backhauledHetNet, in which each single-antenna SBS with finite cachesize can store popular contents to serve user equipment (UEs).Each massive MIMO aided MBS equipped with N antennashas access to the core network via optical fiber and deliversthe non-cached contents to the SBSs via wireless backhaul.UEs, SBSs, and MBSs are assumed to be distributed followingindependent homogeneous Poisson point processes (HPPPs)denoted by ΦU with the density λU, ΦS with the densityλS, and ΦM with the density λM, respectively. It is assumedthat UEs are associated with the SBSs that can provide themaximum average received power, which is also utilized in 4Gnetworks [6]. In addition, each channel undergoes independentand identically distributed (i.i.d.) quasi-static Rayleigh fading.

6362 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 9, SEPTEMBER 2018

A. Content Placement

Content placement mechanism is mainly designed based oncontent popularity [4]. We assume that there is a finite contentlibrary denoted as F := {f1, . . . , fj, . . . , fJ}, where fj is thej-th most popular content and the number of contents is J .The request probability for the j-th most popular content iscommonly-modeled by following the Zipf distribution [23]

aj = j−ς/∑J

m=1m−ς , (1)

where ς is the Zipf exponent to represent the popularityskewness [23]. Each content is assumed to be unit size andeach SBS can only cache L (L � J) contents. We employ theprobabilistic caching strategy [7], i.e., the probability that thecontent j is cached at an arbitrary SBS is qj(0 ≤ qj ≤ 1),and the sum of probabilities for all the contents being cachedat an arbitrary SBS should be less than the SBS’s cache size

(namelyJ∑

j=1

qj≤L) [7]. Note that based on the probabilistic

caching strategy, each SBS only stores L files from the contentlibrary for each caching realization, which is also illustratedin [7, Fig. 1].

B. Self-Backhaul Load

We assume that the access and backhaul links share thesame sub-6 GHz spectrum. The bandwidths allocated to theaccess and backhaul links are ηW and (1 − η) W , respec-tively, where η is the fraction factor and W is the systembandwidth. The number of UEs that is associated with anSBS is denoted by K . The UEs in the same small cell areserved in a time-division manner with equal-time sharing.Thus, the fraction of time-frequency resources allocated toeach access link is ηW/K during the cached content delivery.When an associated SBS does not cache the requested content,it has to be connected to an MBS that provides the strongestwireless backhaul link such that the requested content canbe obtained from core networks. Note that different SBSsmay be served by different MBSs. Let Sj (N � Sj) denotethe number of SBSs served by the j-th MBS (j ∈ ΦM) forwireless backhaul.

Hit probability characterizes the probability that a requestedcontent file is stored at an arbitrary SBS [4], and is calculated

as qhit =J∑

j=1

ajqj . The set of SBSs can be partitioned into

two independent HPPPs ΦaS and Φb

S based on the thinningtheorem [24], where Φa

S with the density λSqhit denotes thepoint process of SBSs with access links, and Φb

S with thedensity λS (1 − qhit) denotes the point process of SBSs withbackhaul links. Let ωb = λS (1 − qhit) /λM represent theaverage number of SBSs served by an MBS for wirelessbackhaul.

C. Resource Allocation Model

We consider the saturated traffic condition, i.e., all the SBSskeep active to serve their associated UEs.

1) Access: When the requested content is stored at a typicalSBS, the rate for a typical access link is given by

Ra =ηW

Klog2

(1 +

PahoL (|Xo|)∑

i∈ΦaS\{o}

PahiL (|Xo,i|)︸ ︷︷ ︸

Ia

+σ2a

), (2)

where Ia denotes the total interference power from other SBSs;Pa is the SBS’s transmit power; L (|X |) = β (|X |)−αa denotesthe path loss with frequency dependent constant value β,distance |X | and path loss exponent αa; ho ∼ exp(1) and |Xo|are the small-scale fading channel power gain and distancebetween the typical UE and its associated SBS, respectively;hi ∼ exp(1) and |Xo,i| are the small-scale fading interferingchannel power gain and distance between the typical UE andthe interfering SBS i ∈ Φa

S\ {o} (except the typical SBS o)respectively, and σ2

a is the noise power at the typical UE.2) Self-Backhaul: When the requested content is not stored

at SBSs, it is obtained through massive MIMO backhaul.For massive MIMO backhaul link, we consider that mas-sive MIMO enabled MBS adopts zero-forcing beamformingwith equal power allocation [25]. In such a time-divisionduplex (TDD) massive MIMO self-backhauled network, SBSswill not perform any channel estimation,1 and we will adoptan achievable backhaul transmission rate as confirmed in [29]and [30]. Based on the instantaneous received signal expres-sion in [29, eq. 6], given a typical distance |Yo| between thetypical SBS and its associated MBS, the instantaneous rate fora typical massive MIMO backhaul link is given by

Rb = (1 − η) W log2 (1 + SINRb) (3)

with SINRb, as shown at the bottom of the next page,where E {·} is the expectation operator. Here, Ib denotesthe total interference power from other MBSs; Pb is theMBS’s transmit power; L (|Y |) = β (|Y |)−αb denotes the pathloss with the distance |Y | and path loss exponent αb; go ∼Γ (N − So + 1, 1) is the small-scale fading channel powergain between the typical SBS and its associated MBS; gj ∼Γ (Sj , 1),2 and |Yo,j | are the small-scale fading interferingchannel power gain and distance between the typical SBS andinterfering MBS j, respectively, and σ2

b is the noise power atthe typical SBS.

After obtaining the requested content via backhaul, the asso-ciated SBS delivers it to the corresponding UE. In this case,

1In TDD massive MIMO systems, downlink precoder is designed based onthe uplink channel estimation, thanks to channel reciprocity [26]. Moreover,when deploying massive number of antennas at MBS, wireless channelbehaves in a deterministic manner called “channel hardening,” and the effectof small-scale fading could be negligible [27]. Thus, in the consideredmassive MIMO backhaul scenario where MBSs and SBSs are usually still,the coherence time of backhaul channel will be much longer than ever before,and the time occupied by uplink channel estimation will be much lower.

It should be noted that when high-mobility UEs are served by TDD massiveMIMO BSs, downlink pilots may still be needed to estimate the fast-changingchannels [28].

2Γ (·, ·) is the upper incomplete gamma function [31, eq. (8.350)].

WANG et al.: EDGE CACHING IN DENSE HETNETs WITH MASSIVE MIMO-AIDED SELF-BACKHAUL 6363

the corresponding access-link rate is expressed as

Ra′ =(1 − η)W

K

× log2

(1 +

PahoL (|Xo|)∑

i′∈ΦbS\{o}

Pahi′L (|Xo,i′ |)︸ ︷︷ ︸

Ia′

+σ2a′

), (4)

where Ia′ is the total interference power, hi′ ∼ exp(1)and L (|Xo,i′ |) = β (|Xo,i′ |)−αa are the small-scale fadingchannel power gain and pathloss between the typical SBS andinterfering SBS i′ ∈ Φb

S\ {o}, respectively, and σ2a′ is the noise

power at the typical UE.From (3) and (4), we see that to reduce latency, massive

MIMO backhaul link needs to be of high-speed, which canbe achieved by using large antenna arrays at the MBS. In thefollowing section, we will further examine how much backhaultime is needed at an achievable backhaul rate.

III. CONTENT DELIVERY EFFICIENCY

In this paper, there are two cases for successful contentdelivery (SCD), i.e., 1) when the associated BS has cachedthe requested content, SCD occurs if the time for successfullydelivering Q bits will not exceed the threshold Tth; and2) when the requested content is not cached at the associatedBS and needs to be obtained via massive MIMO backhaul,SCD occurs if the total time for successfully delivering Q bitsto the UE is less than Tth.

A. Cached Content Delivery

Different from [15], [16], and [18] where it is assumed thateach small cell has only one active UE, we evaluate SCDprobability by considering multiple UEs served by an SBS, andanalyze the effect of resource allocation on SCD probability.We first have the following important theorem.

Theorem 1: When a requested content is stored at thetypical SBS, the SCD probability is derived as

ΨaSCD (Q, Tth) =

Kamax∑

k=1

PλUλS

(k), (5)

where PλUλS

(k) is the probability mass function (PMF)

that there are other k − 1 UEs (except typical UE)served by the typical SBS, and is given by PλU

λS

(k) =

γγ

(k−1)!Γ(k+γ)Γ(γ)

�λUλS

�k−1

�γ+

λUλS

�k+γ with γ = 3.5 [32]. In (5), K =

Kamax is the maximum load in a typical small cell, and can

be efficiently obtained by using Algorithm 1 to solve the

following equation

2Ka

maxQ

ηWTth+1 − 2

αa − 2χa

k (Kamax) =

1 − ε

qhitε, (6)

where χak (Ka

max) = 2F1

[1, 1 − 2

αa; 2 − 2

αa; 1 − 2

KamaxQ

ηWTth

],

2F1 [·, ·; ·; ·] is the Gauss hypergeometric function [31,eq. (9.142)]3, and ε is the predefined threshold, i.e., SCDoccurs when the probability that Ra is larger than Q

Tthis

above ε.

Algorithm 1 One-Dimension Search1: if t = 02: Initialize ϕ = 1−ε

qhitε, kl = 1, kh = 10 × λU

λS, and calculate

F l = 2klQ

ηW Tth+1−2

αa−2 2F1

[1, 1 − 2

αa; 2 − 2

αa; 1 − 2

klQηW Tth

]

and

Fh = 2khQ

ηW Tth+1−2

αa−2 2F1

[1, 1 − 2

αa; 2 − 2

αa; 1 − 2

khQηWTth

]

3: else4: While F l �= ϕ and Fh �= ϕ5: Let k = kl+kh

2 , and compute Fk.6: if Fk = ϕ7: The optimal k∗ is obtained, i.e., Ka

max =round (k∗).8: break9: elseif Fk < ϕ10: kl = k.11: else Fk > ϕ12: kh = k.13: end if14: end while15: end if

Proof: See Appendix A.It is implied from Theorem 1 that in the dense small

cell networks (i.e., interference-limited),4 the SCD probabilitydepends on the ratio of UE density to SBS density andhit probability given the time-frequency resource allocation.Based on Theorem 1, we have

Corollary 1: From (6), we see that to achieve the load K =Ka

max ≥ 1 in a small cell, the hit probability should satisfy

qhit ≤ min{

Ξa1 − ε

ε, 1

}, (7)

where Ξa =(

2Q

ηW Tth+1−2

αa−2 χak (1)

)−1

.

3In MATLAB R2015b software, hypergeom([a,b],c,z) is the Gauss hyper-geometric function 2F1 [a, b; c; z].

4The near-field pathloss exponent is assumed to be larger than 2 [4].

SINRb =PbSo

(E{√

go

})2L (|Yo|)

PbSo

(√go − E

{√go

})2L (|Yo|) +

∑

j∈ΦM\{o}

Pb

SjgjL (|Yo,j |)

︸ ︷︷ ︸Ib

+σ2b

,

6364 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 9, SEPTEMBER 2018

It is indicated from (7) that there is an upper-bound onthe hit probability, which can be explained by the fact thatwhen more UEs can obtain their requested contents fromtheir associated SBSs in dense cellular networks with large hitprobability, there will also be more interference from nearbySBSs that hinders the cached content delivery.

In realistic networks, there may be overload issues when thescale of small cells is not adequate to support large level ofconnectivity, which needs to be addressed. Therefore, given aspecified scale of UEs λU, we evaluate the minimum requiredscale of small cells as follows.

Corollary 2: To mitigate the harm of overloading, the min-imum required SBS density needs to satisfy

λS =

⎧⎪⎨

⎪⎩

λU

Kamax + 1

, if PλUλS

=Kamax+1

(Kamax + 1) ≤ ρ,

λU

μa, if PλU

λS=Ka

max+1(Ka

max + 1) > ρ,(8)

where μa ∈(0,

Kamaxγγ+1

]is the solution of

PλUλS

=μa(k = Ka

max + 1) = ρ with arbitrary small ρ > 0,

and can be easily obtained via one-dimension search, similarto Algorithm 1. Such network deployment given in (8) canguarantee PλU

λS

(k) ≤ ρ, ∀k > Kamax.

Proof: See Appendix B.From (8), we see that the minimum required density of

SBSs only depends on the maximum load of a small cell andthe density of UEs in dense cache-enabled cellular networks.

B. Self-Backhauled Content Delivery

1) Massive MIMO Backhaul: When the required content isnot stored at the typical SBS, SBS has to obtain it from thecore network via massive MIMO backhaul. Therefore, we needto evaluate the backhaul time for delivering the requestedcontent to the typical SBS. It should be noted that the loadin a macrocell will not change fast, in order to deliver therequested contents to the associated SBSs. Hence, given theload So in a typical macrocell, the achievable transmission ratefor a typical backhaul link is given by

Rb (So) = (1 − η) W

∫ ∞

rb

Cb (y)2πλMye−πλMy2

e−πλMr2b

dy, (9)

where Cb (y) = log2

(1+

PbSo

Ξ1(y)PbSo

Ξ2(y)+Ξ3(y)+σ2b

)with Ξ1 (y) =

L (y)(

Γ(N−So+ 32 )

Γ(N−So+1)

)2

, Ξ2 (y) = (N −So +1)L (y)−Ξ1, and

Ξ3 (y) = Pb2πλMβ y2−αb

αb−2 , and rb is the minimum distancebetween the typical MBS and its associated SBS. A detailedderivation of (9) is provided in Appendix C. Therefore,the time for delivering Q bits to the typical SBS via wirelessbackhaul is T1 = Q

Rb. When the number of antennas at the

MBS grows large, we have the following corollary.Corollary 3: For large N , the achievable transmission rate

for a typical backhaul link is tightly lower-bounded as

RLow

b (So) = (1−η)W log2

(1+Pbβ

N − So + 12

SoeΔ1−Δ2

),

(10)

where⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

Δ1 = −αbeπλMr2

b

(− Ei

(− r2bπλM

)

2+ e−r2

bπλM ln rb

),

Δ2 =∫ ∞

rb

ln(

Pbβ

2Soy−rb + Pb2πλMβ

y2−αb

αb − 2+ σ2

b

)

× 2πλMy

e−πλMr2be−πλMy2

dy,

in which Ei (z) is the exponential integral given by Ei (z) =− ∫∞

−ze−t

t dt [31]. Based on (10), the typical MBS’s requiredtime for delivering Q bits to its associated SBS satisfies

T1 ≤ Q(1 − η)−1W−1

log2

(1 + Pbβ

(N−So+ 12 )

SoeΔ1−Δ2

) . (11)

Proof: See Appendix D.It is explicitly shown from Corollary 3 that large number

of antennas and bandwidths are required, in order to signifi-cantly reduce the wireless backhaul delivery time. From (11),we see that the backhaul delivery time can at least be cutproportionally to 1/ log2 N .

In the self-backhauled networks, the number of SBSs beingsimultaneously served by an MBS for wireless backhaulshould not exceed the maximum value denoted by Smax,i.e., So ≤ Smax; otherwise high-speed massive MIMO aidedbackhaul transmission cannot be guaranteed. Hence, given theminimum required backhaul transmission rate Rmin

b , the max-imum backhaul load of a typical massive MIMO MBS isthe solution of Rb (Smax) = Rmin

b , which can be efficientlyobtained by using one-dimension search since Rb (So) is adecreasing function of So for large N , as suggested in Appen-dix D. After obtaining Smax, we can obtain the minimumnumber of massive MIMO aided MBSs that needs to bedeployed, in order to mitigate the backhaul overload.

Corollary 4: Similar to Corollary 2, the minimum requireddensity of MBSs is given by

λM =

⎧⎪⎨

⎪⎩

λS (1 − qhit)Smax + 1

, if Pωb=Smax+1 (Smax + 1) ≤ ρ,

λS (1 − qhit)μb

, if Pωb=Smax+1 (Smax + 1) > ρ,

(12)

where Pωb (�) = γγ

(−1)!Γ(+γ)Γ(γ)

(ωb)�−1

(γ+ωb)�+γ , μb ∈(0, Smaxγ

γ+1

]is

the solution of Pωb=μb (Smax + 1) = ρ with arbitrary smallρ > 0, and can be easily obtained via one-dimension search.

It is explicitly shown in (12) that higher hit probabilitycan significantly reduce the scale of MBSs because of lessbackhaul.

2) Access: After obtaining the required content via back-haul, the typical SBS transmits it to the associated UE. Thus,we have the following important theorem.

Theorem 2: When the required content is not stored at thetypical SBS and has to be obtained via massive MIMO self-backhaul, the SCD probability is derived as

ΨbSCD (Q, Tth) =

Kbmax∑

k=1

PλUλS

(k), (13)

WANG et al.: EDGE CACHING IN DENSE HETNETs WITH MASSIVE MIMO-AIDED SELF-BACKHAUL 6365

where Kbmax is the maximum number of UEs that a typical

small cell can serve when the typical UE’s content needs to beattained via backhaul, and Kb

max can be obtained by solvingthe following equation5

2Kb

maxQ

(1−η)W(Tth−T1)+1 − 2

αa − 2χb

k

(Kb

max

)=

1 − ε

(1 − qhit) ε(14)

with χbk

(Kb

max

)= 2F1

[1, αa−2

αa; 2αa−2

αa; 1 − 2

KbmaxQ

(1−η)W(Tth−T1)

],

and the minimum required SBS density for mitigating overloadis given from (8) by interchanging Ka

max → Kbmax.

Proof: See Appendix E.It is indicated from (14) that when a typical UE’s requested

content is not stored at the typical SBS, the number of UEsthat can be served by the typical SBS decreases with increasingbackhaul time. Based on Theorem 2, we have the followingcorollary

Corollary 5: From (14), we see that to achieve the loadK = Kb

max ≥ 1 in a small cell, the hit probability shouldsatisfy

qhit ≥[1 − Ξb

1 − ε

ε

]+

, (15)

where Ξb =

(2

Q

(1−η)W(Tth−T1)+1

−2αa−2 χb

k (1)

)−1

, and [x]+ =

max {x, 0}.From (15), we see that there is a lower-bound on the hit

probability, i.e., minimum cache capacity is demanded at theSBS, since more backhaul results in more interference, whichwill degrade the self-backhauled content delivery.

Corollary 6: After obtaining the maximum load Kbmax,

we can calculate the minimum required SBS density givenfrom (8) by interchanging Ka

max → Kbmax, to overcome

overload.Based on Theorem 1 and Theorem 2, the SCD probability

in dense cellular networks with massive MIMO self-backhaulfor a typical UE is calculated as

ΨSCD (Q, Tth)= qhitΨa

SCD (Q, Tth) + (1 − qhit)ΨbSCD (Q, Tth)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Kbmax∑

k=1

PλUλS

(k) + qhit

Kamax∑

k=Kbmax+1

PλUλS

(k),

Kamax ≥ Kb

max,Ka

max∑

k=1

PλUλS

(k) + (1 − qhit) ×Kb

max∑

k=Kamax+1

PλUλS

(k),

Kamax < Kb

max,

(16)

where Kamax and Kb

max are given by (6) and (14), respectively.The SCD probability given in (16) can be intuitively under-

stood based on the fact that when the small cell load is light,UEs’ requested contents can be successfully delivered whetherthey are cached or obtained from the core networks via

5It can be solved by following Algorithm 1.

massive MIMO backhaul. However, after a critical value ofcell load, UEs can only obtain their requested contents thatare cached by the SBSs or via backhaul, which dependson the maximum cell load in cached content delivery andself-backhauled content delivery cases.

IV. CONTENT PLACEMENT, CACHE SIZE AND LATENCY

In this section, we study the effects of content placementand cache size on the content delivery performance. Then,we evaluate the latency in such networks.

A. Content Placement and Cache Size

As shown in (16), hit probability plays an important role incontent delivery. Since hit probability depends on the cachesize and content placement, SBSs with large storage capacitycan cache more popular contents, to avoid frequent backhauland reduce backhaul cost and latency. Therefore, higher hitprobability is meaningful to reduce the network’s operationaland capital expenditures (OPEX, CAPEX). Given the SBS’scache size, different content placement strategies may resultin various hit probability, and caching the most popular con-tents (MPC) can achieve the highest hit probability, which iscommonly-considered in the literature involving edge cachingsuch as [13] and [33]. Therefore, we consider MPC cachingand analyze the appropriate cache size in such networks.Considering the fact that for large J with MPC caching,qhit =

∑Lj=1 aj ≈ (

LJ

)1−ς, we have

Corollary 7: Given Tth−T1Tth

≤ η1−η (i.e., more time-

frequency resources are allocated to the cached contentdelivery), the SCD probability is

ΨSCD (Q, Tth) ≈Kb

max∑

k=1

PλUλS

(k), (17)

and it is an increasing function of the cache size, if the cache

size L ∈[J([

1 − Ξb1−ε

ε

]+) 11−ς

, J(

12

) 11−ς

]and the mini-

mum SBS density satisfies the condition given in Corollary 6;Given Tth−T1

Tth> η

1−η , the SCD probability is

ΨSCD (Q, Tth) ≈Ka

max∑

k=1

PλUλS

(k), (18)

if L ∈[J(

12

) 11−ς ,

(min

{Ξa

1−εε , 1

}) 11−ς

], and the minimum

SBS density satisfies the condition given in Corollary 4.Proof: See Appendix F.

The above corollary provides some important insights intothe interplay between time-frequency resource allocation andcache size in cache-enabled dense cellular networks withmassive MIMO backhaul, which plays a key role in the contentdelivery performance.

B. Latency

To evaluate the latency in such networks, we consider theaverage delay for successfully obtaining the requested contentin such networks. It should be noted that when the small

6366 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 9, SEPTEMBER 2018

TABLE I

SIMULATION PARAMETERS

cells are overloaded, UEs may suffer longer delay. There aremany approaches to solve the overload issue such as deployingenough small cells following the rule of Corollary 2 andCorollary 6 or advanced multi-antenna techniques. Moreover,it may be more lightly loaded in realistic small cell net-works [34]. For tractability, we assume that the load of a smallcell will not exceed its maximum load Kmax. As suggestedin [35], the average delay for requesting a content from atypical small cell can be expressed as

D =Kmax∑

k=1

PλUλS

(k)(

qhitQ

E {Ra}

+ (1 − qhit)(T1 +

Q

E {Ra′}))

, (19)

where T1 is the massive MIMO backhaul time detailed inSection III-B, and E {Ra} and E {Ra′} are the average accessrate of the cached and self-backhauled content delivery, respec-tively, which are given by

⎧⎪⎨

⎪⎩

E {Ra} =∫ ∞

0

ϕ (x, qhit, η)dx,

E {Ra′} =∫ ∞

0

ϕ (x, 1 − qhit, 1 − η)dx,(20)

where ϕ (x, θ1, θ2) =(

1 + θ12

kxθ2 W

+1−2αa−2 χ (k)

)−1

with

χ (k) = 2F1

[1, 1 − 2

αa; 2 − 2

αa; 1 − 2

kxθ2 W

]is the comple-

mentary cumulative distribution function of the Ra or Ra′ ,respectively, which is obtained by using the approachin Appendix A.

Given the hit probability, i.e., the cache size is fixed,the spectrum fraction η = ηo for meeting E {Ra} = E {Ra′}can be easily obtained by using one-dimension search, consid-ering the fact that E {Ra}−E {Ra′} is an increasing functionof η.

Corollary 8: When η < ηo, the average delay ofself-backhauled content delivery could be lower than cachedcontent delivery if massive MIMO antennas meet

N ≥(

2Θ(ηo) − 1PbβeΔ1−Δ2

+ 1)

So − 12

(21)

with Θ (ηo) = (1−ηo)−1W−1E{Ra}E{Ra′}

E{Ra′}−E{Ra} , for a specified typi-cal backhaul load So.

Fig. 2. The complementary cumulative distribution function (CCDF) of theRa: Q

Tth= 1 Mbps, λU = 3 × 10−4 m−2, λS = 10−4 m−2, η = 0.5, and

Cache Size = 3 × 103.

The proof of Corollary 8 can be easily obtained byconsidering T1 ≤ Q

E{Ra}−Q

E{Ra′} for η < ηo and Corollary 3.

It is implied from Corollary 8 that for the case of requestingnon-cached contents, the average delay of the non-cachedcontent delivery via massive MIMO backhaul could be com-parable to that of the cached content delivery, if the averageaccess rate of cached content delivery is lower than that ofself-backhauled content delivery.

V. SIMULATION RESULTS

In this section, simulation results are presented to validatethe prior analysis and further shed light on the effects of keysystem parameters including cell load, cache size, BS density,and massive MIMO antennas on the performance. The basicsimulation parameters are shown in Table I.

A. Cached Content Delivery

In this subsection, we illustrate the cell load, SCD proba-bility, and minimum required SBS density when the requestedcontent is cached at the associated SBS.

Fig. 2 shows the complementary cumulative distributionfunction (CCDF) of the rate Ra for different number of UEs

WANG et al.: EDGE CACHING IN DENSE HETNETs WITH MASSIVE MIMO-AIDED SELF-BACKHAUL 6367

Fig. 3. The SCD probability: QTth

= 1 Mbps, λU = 3 × 10−4 m−2,

λS = 10−4 m−2, and η = 0.5.

served in a small cell. The analytical maximum cell load Kamax

for different CCDF thresholds are obtained from (6), whichhas a precise match with the Monte Carlo simulations. TheCCDF is a decreasing function of number of UEs served in asmall cell, since resources allocated to each UE become lesswhen serving more UEs.

Fig. 3 shows the SCD probability when the requestedcontent is cached at the associated SBS, based on Theorem 1and Fig. 2. The stair-like curves are induced by the fact thatthe SCD probability given by (5) is a discrete function ofmaximum cell load. We see that for fixed cache size, the SCDprobability decreases when the system requires higher SCDthreshold ε, since higher ε reduces the level of maximumallowable cell load, as suggested in Fig. 2. Moreover, for agiven ε, the SCD probability decreases with increasing thecache size. The reason is that hit probability increases withincreasing the cache size, i.e., UEs are more likely to obtainthe requested contents cached by their associated SBSs, whichresults in more interference at the same frequency band andreduces the maximum allowable cell load.

Fig. 4 shows the minimum required SBS density to avoidthe overload issue given the UE density λU. Without loss ofgenerality, we assume that the maximum allowable load of asmall cell is Ka

max = 5 in this figure (note that for specifiedsystem performance requirement, the maximum small cellload is obtained from (6), as illustrated in Fig. 2.). Thenumerical result precisely matches with the analysis shownin Corollary 2. We see that when the probability that morethan Ka

max UEs need to be served in a small cell is not largerthan ρ = 0.1, the minimum required SBS density satisfiesλUλS

= Kamax + 1 = 6, as confirmed in (8). When the system

requires lower ρ = 0.1 (i.e., lower overload probability.),the density ratio λU

λSin such networks decreases, which means

that more SBSs need to be deployed.

B. Massive MIMO Backhaul Transmission

In this subsection, we focus on the massive MIMO backhaulachievable rate, which determines the amount of backhaul time

Fig. 4. The minimum required SBS density for avoiding overloading.

Fig. 5. Backhaul achievable rate: λM = 10−5 m−2, η = 0.5 andrb = 5 m.

when an SBS obtains the requested content from its associ-ated MBS. Note that the macrocell load and minimum requiredMBS density have been studied in Section III-B, which aresimilar to Theorem 1 and Corollary 2, and numerical resultscan be easily obtained by following Figs. 2 and 4.

Fig. 5 shows the backhaul achievable rate for differentmacrocell load and massive MIMO antennas. The analyticalexact and lower-bound curves are obtained from (9) and (10),respectively, which tightly matches with the simulated exactcurves. We see that the backhaul achievable rate decreaseswhen macrocell load increases, since each SBS will obtain lesstransmit power and array gains. Adding more massive MIMOantennas improves the achievable rate because of larger arraygains.

C. Latency

In this subsection, we evaluate the average delay intwo scenarios: 1) The requested content is cached at the

6368 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 9, SEPTEMBER 2018

Fig. 6. Average delay: Q = 1 Gbit, λU = 3 × 10−4 m−2, λS = 10−4

m−2, λM = 10−5 m−2, N = 128, So = 10, K = 5, η = 0.45, andrb = 5 m.

associated SBS; and 2) the requested content is not cachedand needs to be obtained via massive MIMO backhaul.

Fig. 6 shows the average delay for different cache size. Theanalytical curves are obtained based on the average rate givenby (20). We see that the average delay for cached content deliv-ery is lower than that of the self-backhauled content delivery.The average delay for cached content delivery increases withincreasing the cache size. In contrast, the average delay forself-backhauled content delivery decreases with increasing thecache size. The reason is that larger cache size results in higherhit probability, and more SBSs can provide cached contentdelivery. This results in more inter-SBS interference over thefrequency band allocated to the cached content delivery, andless inter-SBS interference over the frequency band allocatedto the self-backhauled content delivery. In addition, the contentdelivery time of massive MIMO backhaul T1 is much lowerthan that of the access.

VI. CONCLUSION

We have studied content delivery in cache-enabled HetNetswith massive MIMO backhaul. In such networks, the suc-cessful content delivery probability involving cached contentdelivery and non-cached content delivery via massive MIMObackhaul was analyzed. The effects of hit probability, UE andSBS densities on the performance were addressed. Partic-ularly, we provided the minimum required SBS and MBSdensities for avoiding overloading. The derived results demon-strated that hit probability needs to be properly determined,in order to achieve successful content delivery. The interplaybetween cache size and time-frequency resource allocationswas quantified from the perspective of successful contentdelivery probability. The latency was characterized in termsof average delay in this networks. It was proved that whenUEs request non-cached contents, the average delay of thenon-cached content delivery could be comparable to that ofthe cached content delivery with the help of massive MIMOaided self-backhaul in some cases.

APPENDIX APROOF OF THEOREM 1

Based on (2), SCD probability is calculated as

ΨaSCD (Q, Tth) = Pr

(Ra ≥ Q

Tth

)

= EK

{Pr

(Ra ≥ Q

Tth|K = k

)

︸ ︷︷ ︸Λ(k)

}

=∑

k=1

PλUλS

(k) Λ (k), (A.1)

where PλUλS

(k) is the probability mass function (PMF) of the

number of other k − 1 UEs (except typical UE) served bythe typical SBS, and Λ (k) is the conditional SCD probabilitygiven K = k. According to [36], PλU

λS

(k) can be calculated as

PλUλS

(k) =γγ

(k − 1)!Γ (k + γ)

Γ (γ)

(λUλS

)k−1

(γ + λU

λS

)k+γ, (A.2)

where γ = 3.5 [32]. Given K = k, Λ (k) is calculated as

Λ (k) = Pr(

Ra ≥ Q

Tth

)

= E|Xo|

{Pr

(PahoL (|Xo|)

Ia + σ2a

≥2kQ

ηW Tth − 1)}

=∫ ∞

0

Pr(

PahoL (x)Ia + σ2

a

≥2kQ

ηW Tth − 1)

︸ ︷︷ ︸Υ1(x)

f|Xo| (x) dx,

(A.3)

where f|Xo| (x) = 2πλSx exp(−πλSx2

)is the probability

density function (PDF) of the distance between the typicalUE and its associated SBS, and Υ1 (x) is the conditional SCDprobability given K = k and |Xo| = x. Considering the factthat dense cellular network is interference-limited in practice,the effect of noise power on the performance is negligible.As such, we can evaluate Υ1 (x) as

Υ1 (x)

= EΦaS

{exp

(−2

kQηW Tth − 1PaL (x)

Ia

)}

(a)= exp

⎛

⎝−2πλSqhit

∫ ∞

x

(2

kQηWTth − 1

)xαar1−αa

1 +(2

kQηWTth − 1

)xαar−αa

dr

⎞

⎠

= exp(− 2πλSqhit

x2

αa − 2

(2

kQηW Tth − 1

)

× 2F1

[1, 1 − 2

αa; 2 − 2

αa; 1 − 2

kQηW Tth

]), (A.4)

where step (a) is obtained by using the generating functionalof the PPP [37]. By substituting (A.4) into (A.3), Λ (k) can

WANG et al.: EDGE CACHING IN DENSE HETNETs WITH MASSIVE MIMO-AIDED SELF-BACKHAUL 6369

be derived in closed-form as

Λ (k) =1

1 + qhit2

kQηWTth

+1−2αa−2 χa

k (k), (A.5)

where χak (k) = 2F1

[1, 1 − 2

αa; 2 − 2

αa; 1 − 2

kQηWTth

]. Based

on (A.5), the maximum load Kamax of a typical small cell is

given by

Λ (k)|k=Kamax

= ε, (A.6)

where ε is the threshold that SCD occurs when Λ (k) ≥ ε.Although the closed-form solution with respect to (w.r.t.) k =Ka

max of (A.6) is unfeasible, it can be efficiently obtained byusing one-dimension search as detailed in Algorithm 1 dueto the fact that Λ (k) is a decreasing function of k. The SCDprobability in (A.1) is rewritten as

ΨaSCD (Q, Tth) =

Kamax∑

k=1

PλUλS

(k), (A.7)

where PλUλS

(k) and Kamax are defined by (A.2) and (A.6),

respectively, and the proof of Theorem 1 is completed.

APPENDIX BPROOF OF COROLLARY 2

After obtaining Kamax, we can find out how many small

cells are sufficient to serve a specified scale of UEs λU,since serving larger than Ka

max UEs in a small cell cannotachieve SCD. Assuming that PλU

λS

(Kamax + 1) = ρ with

arbitrary small ρ > 0, we need to guarantee PλUλS

(k) ≤ ρ,

∀k > Kamax, in order to avoid content delivery failure resulting

from overloading. Let

PλUλS

(k + 1)

PλUλS

(k)=

(1 +

γ

k

) λUλS

γ + λUλS

≤ 1, k ≥ Kamax + 1.

(B.1)

We can intuitively interpret (B.1) based on the fact that giventhe maximum load Ka

max, the probability that serving morethan Ka

max UEs should be lower when adding more UEs.From (B.1), we get λU

λS≤ Ka

max + 1 such that PλUλS

(k) ≤ ρ,

∀k > Kamax. Then, we need to solve PλU

λS

(Kamax + 1) = ρ

w.r.t. λUλS

under the constraint λUλS

≤ Kamax +1. The first-order

partial derivative of PλUλS

(k) w.r.t. λUλS

is

∂PλUλS

(k)

∂ λUλS

=γγΓ (k + γ)(k − 1)!Γ (γ)

(λU

λS

)k−2 (γ +

λU

λS

)−(k+γ+1)

×(

(k − 1) γ − (γ + 1)λU

λS

). (B.2)

From (B.2), we see that for k = Kamax+1,

∂PλUλS

∂λUλS

≥ 0 as λUλS

∈(0,

Kamaxγγ+1

], and

∂PλUλS

∂λUλS

< 0 as λUλS

∈(

Kamaxγγ+1 , Ka

max + 1].

Therefore, the minimum required density of SBSs satisfies

λU

λS=

⎧⎨

⎩(Ka

max + 1) , if PλUλS

=Kamax+1

(Kamax + 1) ≤ ρ,

μa, if PλUλS

=Kamax+1

(Kamax + 1) > ρ,

(B.3)

where μa ∈(0,

Kamaxγγ+1

]is the solution of

PλUλS

=μa(Ka

max + 1) = ρ, and can be easily obtained by using

one-dimension search approach, since PλUλS

=μa(Ka

max + 1)

is an increasing function of μa as μa ∈(0,

Kamaxγγ+1

]. Thus,

we obtain the minimum required SBS density, in order toavoid overloading.

APPENDIX CDETAILED DERIVATION OF (9)

Since the typical SBS is associated with the nearest MBS,the PDF of the typical communication distance is

f|Yo| (y) =2πλMy

e−πλMr2be−πλMy2

, y ≥ rb, (C.1)

where rb is the minimum distance between the typical MBSand its associated SBS. According to (3) and [29] and [30],the achievable transmission rate can be written as

Rb = (1 − η)WE|Yo|

{log2

(1 +

PbSo

Ξ1

PbSo

Ξ2 + Ξ3 + σ2b

)}

= (1 − η)W

∫ ∞

rb

Cb (y) f|Yo| (y) dy, (C.2)

where Cb (y) = log2

(1 +

PbSo

Ξ1(y)PbSo

Ξ2(y)+Ξ3(y)+σ2b

)with Ξ1(y) =

L (y)(E{√

go

})2, Ξ2 (y) = L (y) var

{√go

},6 and Ξ3 (y) =

E|Yo|=y {Ib}.We first calculate Ξ1 as

Ξ1 (y) = L (y)(∫ ∞

0

√x

xN−Soe−x

Γ (N − So + 1)dx

)2

= L (y)

(Γ(N − So + 3

2

)

Γ (N − So + 1)

)2

. (C.3)

Then, Ξ2 is given by

Ξ2 (y) = L (y) E {go} − Ξ1 = (N − S + 1)L (y) − Ξ1.

(C.4)

By using the Campbell’s theorem [24], Ξ3 is obtained as

Ξ3 (y) =Pb

SjE {gj} 2πλMβ

∫ ∞

y

t1−αbdt

= Pb2πλMβy2−αb

αb − 2. (C.5)

By substituting (C.3), (C.4) and (C.5) into (C.2), we obtain (9).

6var {·} is the variance operator.

6370 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 9, SEPTEMBER 2018

APPENDIX DPROOF OF COROLLARY 3

According to the Stirling’s formula, i.e., Γ (x + 1) ≈(xe

)x √2πx as x → ∞, we have

Ξ1(y) ≈ L(y)

⎛

⎜⎝

(N−So+

12

e

)N−So+ 12√

2π(N − So + 1

2

)

(N−S

e

)N−So√

2π (N − So)

⎞

⎟⎠

2

≈ L (y)N − So + 1

2

e

(1 +

12 (N − So)

)2(N−So)

(a)≈(

N − So +12

)L (y), (D.1)

when the number of antennas at the MBS grows large. Notethat step (a) is obtained by the fact that

(1 + 1

x

)x ≈ e asx → ∞. Thus, Ξ2 (y) = L(y)

2 . By using Jensen’s inequal-ity [38], we derive a tight lower-bound on the achievabletransmission rate (C.2) as

RLow

b = (1 − η) W log2

(1 +

Pb

SoeΔ1−Δ2

), (D.2)

where⎧⎨

⎩

Δ1 = E|Yo| {ln Ξ1},Δ2 = E|Yo|

{ln

(Pb

SoΞ2 + Ξ3 + σ2

b

)}.

(D.3)

For large N , based on (D.1), Δ1 can be asymptotically derivedas

Δ1 ≈ ln(

N − So +12

)+ E {ln L (y)}

= ln(

N − So +12

)+ ln (β)

− αb

e−πλMr2b

(− Ei

(− r2bπλM

)

2+ e−r2

bπλM ln rb

)

︸ ︷︷ ︸Δ1

,

(D.4)

where Ei (z) is the exponential integral given by Ei (z) =− ∫∞

−ze−t

t dt. Then, Δ2 can be asymptotically calculated as

Δ2 =∫ ∞

rb

ln(

Pb

SoΞ2 (y) + Ξ3 (y) + σ2

b

)f|Yo| (y) dy

≈∫ ∞

rb

ln(

Pbβ

2Soy−rb + Pb2πλMβ

y2−αb

αb − 2+ σ2

b

)

× 2πλMy

e−πλMr2be−πλMy2

dy︸ ︷︷ ︸

Δ2

. (D.5)

Substituting (D.4) and (D.5) into (D.2), we obtain (10).Considering the fact that T1 = Q

Rb≤ Q

RLowb

, we obtain T1 ≤Q

(1−η)W

(log2

(1 +

Pbβ(N−So+ 12 )

SoeΔ1−Δ2

))−1

, which con-

firms the Corollary 3.

APPENDIX EPROOF OF THEOREM 2

Based on (4), SCD probability is given by

ΨbSCD (Q, Tth) = Pr

(Ra′ >

Q

Tth − T1

)

=∑

k≥1

PλUλS

(k) Λbk, (E.1)

where PλUλS

(k) is given by (A.2), and Λbk is the conditional

SCD probability given K = k. Similar to (A.3), Λbk is

calculated as

Λbk = Pr

(Pa′hoL (|Xo|)

Ia′ + σ2a′

>2kQ

(1−η)W(Tth−T1) − 1)

=1

1 + (1 − qhit) 2

kQ

(1−η)W(Tth−T1)+1

−2αa−2 χb

k

, (E.2)

where χbk = 2F1

[1, 1 − 2

αa; 2 − 2

αa; 1 − 2

kQ

(1−η)W(Tth−T1)

].

Like (A.6), the maximum load Kbmax of a typical small cell is

the solution of Λ (k)|k=Kbmax

= ε. Then, the SCD probabilityis obtained as (13).

APPENDIX FPROOF OF COROLLARY 7

Based on (6) and (14), we see that Kamax ≥ Kb

max

if Tth−T1Tth

≤ η1−η and qhit ≤ 1

2 . In this case, UE’srequested contents are more likely to be delivered via mas-sive MIMO self-backhaul. As such, based on (16), theSCD probability can be approximated as ΨSCD (Q, Tth) ≈Kb

max∑k=1

PλUλS

(k) given in (17). Considering the fact that qhit =(

LJ

)1−ς ≥ [1 − Ξb

1−εε

]+in Corollary 5 and qhit =(

LJ

)1−ς ≤ 12 , the corresponding cache size is obtained as

L ∈[J([

1 − Ξb1−ε

ε

]+) 11−ς

, J(

12

) 11−ς

]. Moreover, increas-

ing the cache size boosts the hit probability, and thusenhances the maximum cell load Kb

max for self-backhauledcontent delivery. The reason is that more SBSs can deliverthe cached contents, which reduces inter-cell interference inself-backhauled content delivery.

Likewise, Kamax < Kb

max if Tth−T1Tth

> η1−η and

qhit > 12 , and we can obtain (18) and the corresponding cache

size L ∈[J(

12

) 11−ς ,

(min

{Ξa

1−εε , 1

}) 11−ς

].

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Lifeng Wang (M’16) is currently a ResearchAssociate with the Department of Electronic andElectrical Engineering, University College London.His research interests include massive MIMO,millimeter-wave communications, edge comput-ing/caching, V2X networks, HetNets, physical layersecurity, and wireless energy harvesting. He receivedthe Exemplary Editor Certificate of the IEEE COM-MUNICATIONS LETTERS in 2016 and 2017, respec-tively. He serves as a Review Editor for the IEEECOMMUNICATIONS LETTERS.

Kai-Kit Wong (M’01–SM’08–F’16) received theB.Eng., M.Phil., and Ph.D. degrees in electricaland electronic engineering from The Hong KongUniversity of Science and Technology, Hong Kong,in 1996, 1998, and 2001, respectively. After gradua-tion, he took up academic and research positions atThe University of Hong Kong, Lucent Technologies,Bell-Labs, Holmdel, the Smart Antennas ResearchGroup, Stanford University, and the University ofHull, U.K. He is currently the Chair in WirelessCommunications at the Department of Electronic

and Electrical Engineering, University College London, U.K.His current research interests include 5G and beyond mobile communi-

cations, including topics such as massive MIMO, full-duplex communica-tions, millimeter-wave communications, edge caching and fog networking,physical layer security, wireless power transfer and mobile computing, V2Xcommunications, and course cognitive radios. There are also a few otherunconventional research topics that he has set his heart on, including, forexample, fluid antenna communications systems and remote ECG detection.He was a co-recipient of the 2013 IEEE Signal Processing Letters Best PaperAward and the 2000 IEEE VTS Japan Chapter Award at the IEEE VehicularTechnology Conference in Japan in 2000, and a few other international bestpaper awards.

He is a fellow of IET and is also on the editorial board of severalinternational journals. He has been serving as a Senior Editor for theIEEE COMMUNICATIONS LETTERS since 2012 and the IEEE WIRELESS

COMMUNICATIONS LETTERS since 2016. He had also previously servedas an Associate Editor for the IEEE SIGNAL PROCESSING LETTERS from2009 to 2012 and an Editor for the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS from 2005 to 2011. He was also a Guest Editor forthe IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS SpecialIssue on Virtual MIMO in 2013 and is currently a Guest Editor for IEEEJOURNAL ON SELECTED AREAS IN COMMUNICATIONS Special Issue onPhysical Layer Security for 5G.

6372 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 9, SEPTEMBER 2018

Sangarapillai Lambotharan (SM’06) received thePh.D. degree in signal processing from Imperial Col-lege London, U.K., in 1997. He was a Post-DoctoralResearch Associate with Imperial College Londonuntil 1999. He was a Visiting Scientist with theEngineering and Theory Centre, Cornell University,NY, USA, in 1996. From 1999 to 2002, he was withthe Motorola Applied Research Group, U.K., wherehe investigated various projects, including physicallink layer modeling and performance characteriza-tion of GPRS, EGPRS, and UTRAN. He was with

King’s College London and Cardiff University as a Lecturer and a SeniorLecturer, respectively, from 2002 to 2007. He is currently a Professor ofdigital communications and the Head of the Signal Processing and NetworksResearch Group, Wolfson School of Mechanical, Electrical and ManufacturingEngineering, Loughborough University, U.K. His current research interestsinclude 5G networks, MIMO, radars, smart grids, machine learning, andnetwork security. He has authored over 200 journal and conference articlesin these areas.

Arumugam Nallanathan (S’97–M’00–SM’05–F’17) was with the Department of Informatics,King’s College London, from 2007 to 2017, wherehe was a Professor of wireless communications from2013 to 2017 and has been a Visiting Professorsince 2017. He was an Assistant Professor with theDepartment of Electrical and Computer Engineering,National University of Singapore, from 2000 to2007. He has been a Professor of wireless communi-cations and the Head of the Communication SystemsResearch Group, School of Electronic Engineering

and Computer Science, Queen Mary University of London, since 2017.He has published nearly 400 technical papers in scientific journals andinternational conferences. His research interests include 5G wireless networks,Internet of Things, and molecular communications. He was a co-recipientof the Best Paper Awards presented at the IEEE International Conferenceon Communications 2016, the IEEE Global Communications Conference2017, and the IEEE Vehicular Technology Conference 2018. He is an IEEEDistinguished Lecturer. He was selected as a Web of Science Highly CitedResearcher in 2016.

Maged Elkashlan (M’06) received the Ph.D. degreein electrical engineering from The University ofBritish Columbia, Canada, in 2006. From 2007 to2011, he was with the Wireless and Network-ing Technologies Laboratory, Commonwealth Scien-tific and Industrial Research Organization, Australia.During this time, he held an adjunct appointmentat the University of Technology Sydney, Australia.In 2011, he joined the EECS, Queen Mary Uni-versity of London, U.K. where he is currently anAssociate Professor with the School of Electronic

Engineering and Computer Science. He also holds visiting faculty appoint-ments at the University of New South Wales, Australia, and the BeijingUniversity of Posts and Telecommunications, China. His research interestsinclude communication theory, wireless communications, and statistical sig-nal processing for distributed data processing, heterogeneous networks, andmassive MIMO.

Dr. Elkashlan received the Best Paper Award at the IEEE International Con-ference on Communications in 2014 and 2016, the International Conferenceon Communications and Networking in China (CHINACOM) in 2014, and theIEEE Vehicular Technology Conference (VTC-Spring) in 2013. He currentlyserves as an Editor for the IEEE TRANSACTIONS ON WIRELESS COM-MUNICATIONS, the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY,and the IEEE COMMUNICATIONS LETTERS. He also serves as a LeadGuest Editor for the Special Issue on Green Media: The Future of WirelessMultimedia Networks of the IEEE Wireless Communications Magazine andthe Special Issue on Millimeter Wave Communications for 5G of the IEEECommunications Magazine, and a Guest Editor for the Special Issue on EnergyHarvesting Communications of the IEEE Communications Magazine and theSpecial Issue on Location Awareness for Radios and Networks of the IEEEJOURNAL ON SELECTED AREAS IN COMMUNICATIONS.