IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018 3079

On Base Station Coordination in Cache- andEnergy Harvesting-Enabled HetNets:

A Stochastic Geometry StudyHuici Wu , Student Member, IEEE, Xiaofeng Tao, Senior Member, IEEE, Ning Zhang, Member, IEEE,

Danyang Wang, Shan Zhang, Member, IEEE, and Xuemin Shen, Fellow, IEEE

Abstract— In this paper, we study the performance of basestation (BS) coordination in heterogeneous networks (HetNets)with cache-enabled and renewable energy-powered small cellBSs (SBSs). Macrocell base stations (MBSs) provide basic cov-erage, while the SBSs, powered by harvested energy, conductcontent-aware coordinated transmission to provide high data rateand further improve the network coverage. Specifically, a jointtransmission strategy is performed based on the knowledge ofthe energy states and the cached contents of SBSs, along withthe awareness of the availability of channel resources and theaverage received signal strength (RSS) of the corresponding link.Stochastic geometry is applied to characterize the statistics ofthe cell load at MBSs and SBSs, as well as the aggregatedinformation and interference signal strength. Then, the averageuser capacity for the joint transmission is obtained. Additionally,the coverage probability is derived with gamma approximationfor the aggregated information and interference signal strength.Analytical results reveal that the average user capacity andcoverage probability can be maximized with optimal cache size,energy harvesting rate and cooperative RSS threshold. Finally,extensive numerical and simulation results are provided.

Index Terms— Joint transmission, cache, energy harvesting,stochastic geometry, gamma approximation.

I. INTRODUCTION

DRIVEN by mobile Internet-of-Things (IoT), social net-works as well as the proliferation of smart devices

Manuscript received August 22, 2017; revised October 31, 2017; acceptedDecember 23, 2017. Date of publication December 29, 2017; date of currentversion July 13, 2018. This work was supported in part by the NationalNatural Science Foundation for Distinguished Young Scholars of China underGrant 61325006, in part by the National Natural Science Foundation ofChina under Grant 61231009, in part by the 111 Project of China under theGrant B16006, and in part by the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC). The associate editor coordinating the review ofthis paper and approving it for publication was M. Di Renzo. (Correspondingauthor: Xiaofeng Tao.)

H. Wu and X. Tao are with the National Engineering Laboratory for MobileNetwork Technologies, Beijing University of Posts and Telecommunications,Beijing 100876, China (e-mail: dailywu@bupt.edu.cn; taoxf@bupt.edu.cn).

N. Zhang is with the Department of Computing Science, TexasA&M University-Corpus Christi, Corpus Christi, TX 78412 USA (e-mail:ning.zhang@tamucc.edu).

D. Wang is with the State Key Laboratory of Integrated Service Networks,School of Telecommunications Engineering, Xidian University, Xi’an 710071,China (e-mail: danyangwang@stu.xidian.edu.cn).

S. Zhang and X. Shen are with the Department of Electrical and ComputerEngineering, University of Waterloo, Waterloo, ON N2L 3G1, Canada (e-mail:s372zhan@uwaterloo.ca; sshen@uwaterloo.ca).

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

Digital Object Identifier 10.1109/TCOMM.2017.2788012

and new applications, mobile data traffic has been growingexplosively [1]. To support the ultra-high data volume, diverselow-power small cell base stations (SBSs), such as micro basestations (BSs), pico BSs, and femto BSs, are densely deployedin hotspots to provision high data rate, which results in theheterogeneous networks (HetNets) architecture [2].

With better coverage and higher throughput, HetNetsare expected to be the dominant network architecture inthe coming 5G era. However, the HetNets introduce seri-ous interference due to high frequency reuse among BSsacross tiers. Coordinated multi-point (CoMP) transmissionhas been emerged as a promising technology to alleviatethe cross tier interference and further harvest the benefits ofHetNets [3], [4]. But the coordination among multiple diverseBSs across tiers makes backhaul a bottleneck due to theheavy overhead signaling and information exchanges amongBSs [5], [6]. Recent studies have shown that the backhaulcongestions can be significantly eased by caching popularcontents at the network edge such as SBSs [7], [8]. Moreover,SBSs with caching can provide low-latency services for mobileusers (MUs) since it does not need to fetch the requestedcontents from core networks if a cache hit happens at SBSs.

Densely deployed SBS and large CoMP clustering can causehuge energy consumption [6], which brings heavy burden tonetwork operators and environment. Energy harvesting (EH),which can exploit green energy from ambient environmentsuch as wind, solar and terrestrial heat, has been lever-aged into wireless networks and attracted increasing atten-tions [9], [10]. It is of importance to study BS coordination incache- and EH-enabled HetNets, to provide practical insightsfor HetNet deployment and optimization. However, there aremany challenges. i) The intermittent energy arrival with EH,causing dynamic energy states at BSs, influencing the cell loaddistribution of BSs, and further affecting the activation of BSs.ii) The uncertainty in satisfying users’ content requests withcache, which causes mismatching of content requests at MUsand content caching at BSs. And iii) the various user qualityof service requirements with different kinds of BSs. These arekey factors that have great influence on the determination andformulation of cooperative BS clusters in HetNets with CoMP.

In this paper, a multi-tier orthogonal frequency divisionmultiple access (OFDMA) based HetNet is considered, wherethe macrocell BSs (MBSs) provide basic coverage for MUswhile the SBSs are coordinated to transmit the requested

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3080 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018

contents with harvested energy to provide high transmissionrate and further improve the network coverage. M0 orthog-onal channels are available at MBSs and partly reused andcoordinated at SBSs to serve MUs. A novel BS cooperativetransmission strategy is proposed among SBSs, based on theenergy states and the cached contents at SBSs, as well asthe availability of channel resources and the average received-signal-strength (RSS) of the corresponding link. With theproposed cooperative strategy, the cell load distributions areobtained by means of stochastic geometry. Then, the averageuser capacity is analytically derived based on the character-ization of the aggregated information and interference signalstrength by leveraging Laplace function. Moreover, the expres-sion for coverage probability is approximately derived withGamma approximation. Numerical and simulation results areprovided to validate the theoretical analysis and analyze theimpact of network parameters. The results reveal that optimalcooperative RSS thresholds exist such that the capacity andcoverage performance are maximized. The impacts of cachesize and energy harvesting rate on the coverage and capacityperformance depend on the cooperative RSS thresholds. Withhigh cooperative RSS threshold, the SBSs are light-loaded andthe capacity and coverage firstly improve with the increaseof the cache size, the maximum number of reused channelsand the energy harvesting rate at SBSs, and finally level offto optimum. For the scenario with a low cooperative RSSthreshold, the SBSs are heavy-loaded and the capacity andcoverage performance decrease with the increase of these SBSparameters. With an appropriate RSS threshold, the SBSs aremid-loaded and optimal cache size and energy harvesting rateexist such that the average capacity and coverage probabilityare maximized. However, the capacity and coverage perfor-mance firstly drop and then rise with the increase of themaximum number of channels reused at SBSs.

The remainder of this paper is organized as follows. Therelated work is presented in Section II. Section III introducesthe system model. In Section IV, the joint transmission strat-egy is described and the statistics of the cell load at MBSsand SBSs are characterized. Then, the coverage probabilityand average capacity are analyzed in Section V. Numerical andsimulation results are given in Section VI. Finally, Section VIIconcludes this work.

II. RELATED WORK

CoMP has been regarded as a key technique in wirelesscommunication system to improve network throughput, cell-edge performance, and spectral efficiency, etc., by turninginterfering signals into useful signals such that the inter-cellinterference is mitigated [11], [12]. With CoMP, the BSs areclassified into groups [13], cooperatively serving one or moreMUs. Joint transmission, one of the most advanced CoMPscenario, has been widely studied and analyzed in cellularnetworks [14]–[20]. Reference [14] introduced and analyzeda pairwise cooperation policy for the users at the cell borders.Reference [15] proposed a user-centric BS cooperation basedon the RSS threshold and analyzed the spectral efficiencyin K -tier HetNets. In [16], several heterogeneous BSs withthe strongest average RSS cooperatively serve a typical MU

by sorting all the BSs with their average RSS. FollowingNigam et al. [16], extended their work to the spatio-temporalcooperation, where a second transmission attempt occurs ifthe first transmission fails [17]. In [19], a location-aware BScooperation was proposed in a two-tier HetNets. As a furtherstep, [20] proposed a novel BS cooperation strategy with theconsideration of both the locations of MUs and the averageRSS in non-uniform HetNets.

With cache-enabled SBSs, [21] studied the outage and datarate performance in the small cell networks with stochasticgeometry. Reference [22] analyzed the throughput of Het-Nets with cache-enabled relays and device-to-device (D2D)pairs. To improve the network performance in cache-enablednetworks, [23] proposed an interference management schemewith the assistance of cache-enabled users. Reference [24] ana-lyzed energy efficiency (EE) of cache-enabled dense networkswhere cooperation is performed for downlink transmissions.A random caching based cooperative transmission was studiedin [25], where the successful transmission probability is ana-lyzed and optimized. Reference [26] considered the problemof optimal cache placement with the objective of maximizingthe cache hit probability. Cooperative caching is also anefficient way in increasing the probability of successful filediscovery. Reference [27] categorized the MUs into differentgroups according to their preferences and cooperatively sharetheir cached content among the group. Reference [28] char-acterized the coverage probability and area spectral efficiencyof cluster-centric content placement in a cache-enabled D2Dnetwork.

With EH employed in HetNets, [29] conducted a fun-damental analysis for EH-powered HetNets by means ofrandom walk theory and stochastic geometry. The availabilityregion where the BSs are in active states was characterized.With discrete-time Markov modeling of the energy statesof an EH-BS, the network throughput and EE of HetNetswere analyzed in [30]. Furthermore, [31] studied a power-availability-aware user association in the EH-enabled smallcell network. Joint transmission in wireless networks withEH-powered BSs or MUs were studied in [32] and [33].In [32], a fractional joint transmission strategy was proposedto eliminate interference. [33] investigated the green multi-cell cooperation in HetNets facilitated with hybrid energysources. The multicell cooperation and sleeping mechanismwere performed to deal with interference and save energy,respectively.

With cache and EH employed in HetNets, BS coordinationcan further improve the network coverage and throughout ina more efficient way. Performance analysis in cache-enabledand EH-powered HetNets with BS coordination is vitallyimportant to provide guidance for the practical deployment.However, existing studies either focused on the analysis ofBS coordination in HetNets without considering such prac-tical communication scenarios or studied cache-enabled andEH-powered wireless networks without considering BS coor-dination. In this paper, we studied the BS coordination inHetNets with cache-enabled and EH-powered SBSs. A noveljoint transmission strategy with the awareness of energy statesand cached contents at SBSs is performed and the network

WU et al.: ON BS COORDINATION IN CACHE- AND ENERGY HARVESTING-ENABLED HETNETS 3081

performance in terms of average user capacity and coverageprobability is analyzed in a stochastic geometry framework.

III. SYSTEM MODEL

A. Network Model

We consider a multi-tier OFDMA based HetNet, wherethe MBSs provide basic coverage for MUs and the cache-enabled and EH-powered SBSs are coordinated to transmit therequested contents. A (1 + K )-tier HetNet is assumed with onetier of MBSs overlaid with K tiers of SBSs. The distributionof MBSs are modeled as homogeneous PPP �0 ∈ R

2 withspatial intensity λ0. K tiers SBSs are deployed according toindependent PPP �k ∈ R

2 with intensity λk . The MUs are alsorandomly distributed as an independent PPP �u ∈ R

2 withintensity λu . Downlink joint transmission and performanceanalysis are performed and developed at a typical MU locatedat the origin based on Slivnyak¡¯s theorem [34].

M0 orthogonal channels with sub-bandwidth W are avail-able at MBSs, and are randomly and uniformly allocated tothe associated MUs within a cell. MUs are always associatedwith the nearest MBS to guarantee the basic coverage andconnectivity. The channel allocated to the typical MU isdefined as the reference channel. SBSs randomly reuse thechannels to serve MUs according to their requirements. Themaximum number of channels can be reused at an SBS in thek-th tier is Mk (≤ M0). The BSs allocate an exclusive channelto each associated MU, i.e., the MUs associated with a BS useorthogonal channels to avoid intra-cell interference. Therefore,the BS capacity, i.e., the maximum number of MUs that canbe simultaneously associated with a BS, is Mk , k = 0, . . . , K .

The transmission power of MBSs and SBSs, denoted asP(TX)

k , are uniformly allocated among the associated MUs.Signal propagation model is considered as a composite ofomni-directional path loss fading l (d) = d−α and small-scale fading, where d is the distance between transmitter andreceiver and α (α > 2) is the path loss exponent. For the small-scale fading, we consider independent quasi-static Rayleighfading where the fading stays the same within one data frameand varies independently over frames.

B. Energy Harvesting Model

The MBSs are powered by the power grid and thus havesufficient energy for transmission. The SBSs, equipped withenergy harvesting and storage modules, can harvest energyfrom environments. The SBSs in different tiers have differentharvesting capabilities, i.e., energy harvesting rate and energystorage capacity. For the SBSs in the k-th tier, the energyarrival process is assumed to be identically and independentlydistributed (i.i.d.) Bernoulli process with probability μk andthe amount of energy arrived in one time slot is denotedas �k . Let �k denote the required number of time slots toharvest sufficient energy for transmission, then the EH-SBSscan be silence mode without transmission if the harvestedenergy at SBSs is less than �k�k , or work mode withtransmission power P(TX)

k if the harvested energy at SBSs isabove �k�k .

TABLE I

KEY PARAMETERS AND NOTATIONS

C. Content Caching ModelDenote T as the number of multimedia contents. All the

contents, indexed as T = [1, 2, . . . , T ], are considered tobe with the same size S in bits and are ranked accordingto the request frequency. The lower indexed content has thehigher popularity. Denote

{fk,1, . . . , fk,T

}as the popularity

distribution of the contents in the k-th tier. The content popu-larity distribution, which is also known as the content requestprobability, is predicted with local or global measurements.Zipf distribution [22], [35] is a widely applied model with thefollowing distribution:

fk,t = 1/

tγk

T∑

i=11/

iγk

, t ∈ T , (1)

where γk (≥ 0) reflects the popularity distribution skewnessof the k-th tier. MBSs are connected to the core network viaoptical fiber with high capacity. SBSs are equipped with cachemodules and the cache size of SBSs in the k-th tier is Lk .Most Popular Cache (MPC) scheme is applied at SBS, i.e., theSBS in the k-th tier pre-caches the most Lk popular contentsand periodically updates the cached contents.1 MUs randomlyrequest the contents according to the popularity distribution.The key notations are summarized as in Table I.

IV. JOINT TRANSMISSION STRATEGY

AND STATISTICS OF CELL LOAD

In this section, a joint transmission strategy with the aware-ness of the energy state, the cached contents and the channelresources at SBSs, as well as the average RSS of the cor-responding link is proposed. Based on the joint transmissionstrategy, the statistics of cell load distributions of MBSs andSBSs are then investigated.

A. Joint Transmission StrategyAs mentioned in the network model, the typical MU is

connected to the nearest MBS with the reference channel toguarantee wireless connectivity. The SBSs in the k-th tier reusethe reference channel to cooperatively transmit the typicalMU’s requested contents. The joint transmission is performed

1Locally popular content caching is considered such that the SBSs in theproximity can better perform cooperation.

3082 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018

Fig. 1. Cooperative decision process of an SBS.

based on the energy states and the cached contents of SBSs,as well as the availability of channel resources and averageRSS of the corresponding link. Hence, an SBS participates inthe cooperation if the following four conditions are satisfied.i) Event E1: the SBS has sufficient energy for transmission;ii) Event E2: the requested contents of the typical MU havebeen cached by the SBS, i.e., cache hit happens; iii) EventE3: the SBS is under-loaded and meanwhile the referencechannel is not allocated to other associated MUs with the SBS;Lastly, iv) Event E4: the average RSS of the correspondinglink, i.e., P(TX)

k r−α , is above the predefined cooperative RSSthreshold τk , i.e., the SBS is located inside a circular area

centered at the typical MU with radius χk =(

P(TX)kτk

)1/α. The

cooperation decision process is illustrated in Fig. 1.From the MU’s point of view, the transmission strategy

performed in a user-centric manner. All the candidate SBSs(in the k-th tier) located inside B (o, χk) (E4) can jointly servethe typical UE if the conditions E1, E2 and E3 are satisfied.

B. Statistics of Cell Load DistributionThe cell load of a BS is defined as the number of MUs that

are simultaneously associated with the BS.1) Cell Load Distribution of MBS: Define ρ0 (m) as prob-

ability that m MUs are simultaneously associated with anMBS. Since MUs are associated with the nearest MBSs,the resulting network topology of the macrocell network isPoisson Voronoi (PV) tessellation. The probability densityfunction (pdf ) of the cell size with PV tessellation is accuratelyapproximated by a Gamma distribution [36], [37]:

f (A) = (3.5λ0)3.5

� (3.5)A2.5 exp (−3.5λ0 A) , (2)

and probability that m0 MUs are located in the associationarea of an MBS is [38], [39]:

ρ0 (m0) = 3.53.5� (m0 + 3.5)

m0!� (3.5)

(λu/λ0)m0

(3.5 + λu

/λ0)m0+3.5

, (3)

Since the BS capacity of MBSs is M0, the maximum cell loadof an MBS is M0. The event that an MBS is fully loaded,

Fig. 2. Discrete-time Markov chain of the energy state at SBS.

i.e., m0 = M0, is equivalent to the event that more thanM0 MUs are located in the association area of the MBS butonly M0 MUs are served. Thus, the cell load distribution whenm0 < M0 is expressed as (3), and the probability for an MBSto be fully loaded is:

ρ0 (M0) =∞∫

0

∞∑

m=M0

(λu A)me−λu A

m! f (A) dA

(a)=∞∫

0

γ (M0, λu A)

� (M0)

(3.5λ0)3.5

� (3.5)A2.5 exp (−3.5λ0 A) dA

(b)= 3.53.5� (M0 + 3.5)

M0!� (3.5)

(λu/λ0)M0

(3.5 + λu

/λ0)M0+3.5

×2 F1

(

1, M0 + 3.5; M0 + 1; λu/λ0

3.5+λu/λ0

)

, (4)

where (a) follows the Erlang distribution and (b) followsfrom expression [41, eq. 6.455-2]. 2 F1 (a, b; c; z) is the hyper-geometric function. Note that if the BS capacity of MBSsis large enough to provide the basic connectivity for MUs,

i.e., M0λ0 � λu , thenM0−1∑

m0=0ρ0 (m0) ≈ 1 and ρ0 (M0) ≈ 0.

2) Cell Load Distribution of SBSs: According to the jointtransmission strategy, the serving area of a random SBS xi

k inthe k-th tier is a circular area with radius χk , i.e., B

(xi

k, χk).

Denote ρk (mk) as the probability that mk MUs are simultane-ously associated with an SBS in the k-th tier. With event E1,the SBSs without sufficient energy is zero-loaded. By mod-eling the energy state at SBS as Markov chain in Fig. 2, the

Markov chain is stationary with μk < �k

Mk∑

mk=1ρk (mk) and the

probability that an SBS has sufficient energy for transmissionis ωk = μk

�k

Mk∑

mk=1ρk(mk)

[30]. With E2 and (1), the cache hit

probability of the k-th tier is ηk =Lk∑

t=1fk,t . E3 poses a BS

capacity constraint on the cell load distribution at SBSs sincethe maximum number of MUs the SBS xi

k can simultaneouslyserved is Mk . With E4, only those MUs located in the circularserving area centered at an SBS have the opportunity tobe served by the SBS. When there are n candidate MUslocated inside the serving area of xi

k , the probability of mk

cache hits happening is a binomial distribution with parameters

WU et al.: ON BS COORDINATION IN CACHE- AND ENERGY HARVESTING-ENABLED HETNETS 3083

n and ηk , i.e.,

Pr (mk cache hits) = n!mk ! (n − mk)! (ηk)

mk (1 − ηk)n−mk . (5)

As a result, the probability that mk (< Mk) MUs are simulta-neously served by the SBS xi

k is:

ρk (mk)

= Pr{

mk MUs associated with xik

}

= Pr{≥ mk MUs inside B

(xi

k, χk

)& mk cache hits

}

(c)=∞∑

n=mk

⎡

⎢⎢⎣

(πχ2

k λu)n

exp(−πχ2

k λu)

n!× n!

mk ! (n − mk)! (ηk)mk (1 − ηk)

n−mk

⎤

⎥⎥⎦

(d)=(πχ2

k ηkλu

)mk exp(−πχ2

k ηkλu)

mk ! , (6)

where (c) holds due to the PPP distribution of MUs and thebinomial distribution of caching and (d) follows the Taylorexpansion of exponential distribution. The probability for anSBS to be fully loaded, i.e., mk = Mk , is

ρk (Mk) = Pr{

Mk MUs associated with xik

}

=∞∑

m=Mk

(πχ2

k ηkλu)m

exp(−πχ2

k ηkλu)

m! (7)

Combining with the cell load distribution, the probability foran SBS in the k-th tier to have sufficient energy can beexpressed as:

ωk = μk[1 − exp

(−πχ2k ηkλu

)]�k

. (8)

Note that with the Markov modeling of energy state ofSBSs, the Markov chain will not be stationary and the SBSwill always have enough energy for transmission if the energyarrival rate exceeds the energy consumption rate, i.e., μk ≥[1 − exp

(−πχ2k ηkλu

)]�k . Therefore, when the SBSs are non

zero-loaded with very high probability, the harvested energywill be consumed quickly and the probability ωk decreases.By setting a higher cooperative RSS threshold or a lowercache size at SBS, the probability that an SBS is zero-loaded increases and thus the stored energy at SBSs as wellas the probability that the SBSs have sufficient energy fortransmission increases. Overall, we can see from (6)-(8) thatthe cooperative RSS thresholds τk , the cache size L and themaximum number of channels Mk jointly influence the cellload distribution of SBSs, which further affects the energystates at SBSs.

V. ANALYSIS OF AVERAGE USER CAPACITY

AND COVERAGE PROBABILITY

In this section, the statistics of the aggregated informationand interference signal strength with the proposed joint trans-mission scheme are studied by means of Laplace function.Then, the average capacity achievable at the typical MU isobtained. Based on the statistics, the coverage probability is

approximately derived by leveraging Gamma approximationfor the distributions of the aggregated information and inter-ference signal strength.

A. Statistics of the Signaland Interference Strength

Denote x0 as the nearest MBS to the typical MU with dis-tance r0 = ‖x0‖, then the pdf of r0 is fr0 (r) = 2πλ0re−πλ0r2

.Denote Sk and Ik as the aggregated information and interfer-ence signal strength received from the SBSs in the k-th tier.With non-coherent transmission at the cooperative BSs, thereceived signal-to-interference-plus-noise ratio (SINR) at thetypical MU is

SINR =S0 +

K∑

k=1Sk

K1+K2∑

k=0Ik + σ 2

, (9)

where

S0 = P(TX)0

m0 + 1‖x0‖−αhx0 , m0 ≤ M0 − 1, (10)

and

I0 =∑

x∈�0\B(o,r0)

� (occupied)P(TX)

0

mx‖x‖−αhx , (11)

are the received information signal strength and aggregatedinterference strength from MBSs. m0 and mx are the numberof other MUs associated with the nearest MBS x0 and theinterfering MBS x , respectively. hx0 and hx are the channelfading, which are exponentially distributed with unit mean.� (occupied) is the indicator function, where � (occupied) = 1,if the MBS x have reused the reference channel to serve anMU, and � (occupied) = 0, otherwise. For an SBS located atxk ∈ �k ∩B (o, χk) and with mxk other associated MUs, it canjoin to transmit the typical MU’s data if it satisfies the coop-eration conditions E1, E2 and E3. Therefore, the aggregatedinformation signal strength received from the SBSs in thek-th tier is expressed as:

Sk =∑

xk∈�k∩B(o,χk)

� (E1&E2 & E3)P(TX)

k

mxk + 1‖xk‖−αhxk .

(12)

The SBSs in the k-th tier interfere with the typical MUif the reference channel is reused to serve other MUs. Thus,the aggregated interference strength from the k-th tier is

Ik =∑

xk∈�k

� (E1 & occupied)P(TX)

k

mxk

‖xk‖−αhxk (13)

In the following, we provide the Laplace function of Sk

and Ik to characterize the statistics of received informationand interference signal strength.

Lemma 1: The Laplace function of S0 conditioned on r0 is

LS0 (t|r0) = ρ0 (M0) +M0−2∑

m0=0

ρ0 (m0 + 1)

m0+1P(TX)

0

rα0

t + m0+1P(TX)

0

rα0

. (14)

3084 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018

Proof: Please refer to Appendix A-A.For an SBS in the k-th tier with mk (< Mk ) other associated

MUs, the reference channel is available to the typical MU ifnone of the mk MUs is served with the reference channel.Thus, the probability that the reference channel is available tothe typical MU is

(M0 − 1) . . . (M0 − mk)

M0 (M0 − 1) . . . (M0 − mk + 1)= M0 − mk

M0. (15)

When the reference channel is not available at the SBS or therequested contents is not cached, the SBS will not partici-pate in cooperation. Thus, the probability that an SBS, withsufficient energy, located inside B (o, χk) in the k-th tier canparticipate in transmitting the typical MU’s data is νk =ηk

Mk−1∑

mk=0ρk (mk + 1) M0−mk

M0.

Lemma 2: The Laplace function of the aggregated informa-tion signal strength Sk is

LSk (t) = exp

⎧⎪⎪⎨

⎪⎪⎩

−πλkωkχ2k ηk

Mk−1∑

mk=0

ρk (mk + 1)

× M0 − mk

M0

(1 − Z1

(mk + 1

tτk

))

⎫⎪⎪⎬

⎪⎪⎭, (16)

where

Z1 (x) = 2x

α + 2× 2 F1

(1, 1 + 2

α; 2 + 2

α; −x

), (17)

Proof: Please refer to Appendix A-B.The MBSs randomly and uniformly distribute the channels

to the associated MUs. Thus, the probability that an MBSwith m0 other associated MUs reuses the reference channeland interferes with the typical MU is m0

M0.

Lemma 3: The Laplace function of the aggregated interfer-ence strength from MBSs is

LI0 (t|r0)=exp

⎧⎨

⎩−πλ0r2

0

M0∑

m0=1

ρ0 (m0)m0

M0Z2

(m0rα

0

t P(TX)0

)⎫⎬

⎭,

(18)

where

Z2 (x) = 2x−1

α − 2× 2 F1

(1, 1 − 2

α; 2 − 2

α; −x−1

). (19)

Proof: Please refer to Appendix A-C.For the interfering SBSs in the k-th tier with enough energy,

they interfere with the typical MU if they have served otherMUs with the reference channel. The probability that thereference channel is reused at an SBS with mk associatedMU is

mk (M0 − 1) . . . (M0 − mk + 1)

M0 (M0 − 1) . . . (M0 − mk + 1)= mk

M0. (20)

Lemma 4: The Laplace function of the aggregated interfer-ence strength from the SBSs in the k-th tier is

LIk (t) = exp

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

−2πωkλk

αM0

(t P(TX)

k

)2/α×

B

(2

α, 1 − 2

α

) Mk∑

mk=1

ρk (mk) m1−2/αk

⎫⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎭

(21)

where B( 2

α , 1 − 2α

)is the Beta function.

Proof: Please refer to Appendix A-D.From the cell load distribution (6) and the Laplace func-

tion (16), we can see that with the decrease of the cooperativeRSS threshold (i.e., with the expansion of the cooperativearea), the number of candidate cooperative SBSs as well as theprobability of heavy-loaded SBSs increase, while the proba-bilities νk and ωk decrease, which results in a tradeoff betweenthe number of candidate SBSs and the cooperation probabilityof a candidate SBS. Similarly, with the increasing of thecache size at SBSs, there is a tradeoff between the cache hit

probability ηk and the joint probability ωk

Mk−1∑

mk=0ρk (mk + 1).

For the aggregated interference strength Ik , a tradeoff betweenthe probability for having sufficient energy ωk and the inter-

fering probabilityMk∑

mk=1ρk (mk) exists with the varying of the

cooperative RSS threshold and the cache size. To investigatethe influence of network parameters, we then theoreticallyanalyze the average user capacity and coverage probability tostudy the capacity and coverage performance of the HetNets.

B. Average Capacity Analysis

When the cooperative BSs adopts appropriate adaptivemodulation and coding, the average Shannon capacity of themultiple-input single-output channel between the cooperativeBSs and the typical MU is

Rc = WE [ln (1 + SINR)] , (22)

in nats/s.Theorem 1: In the cache-enabled EH-powered HetNets

with the performed BS cooperation strategy, the average usercapacity is

Rc = W

∞∫

0

⎛

⎝e−σ 2t

t

K∏

k=1

LIk (t)

∞∫

0

(LI0 (t|r )

×(

1 − LS0 (t|r)

K∏

k=1

LSk (t)

))

fr0 (r) dr

)

dt (23)

Proof: Applying the result as in [41], the average capacityper frequency band (also known as spectral efficiency) at thetypical MU is expressed as:

E [ln (1 + SINR)]

= Er

⎡

⎢⎢⎢⎣

∞∫

0

K∏

k=0LIk (t)

(1 −

K∏

k=0LSk (t)

)

te−σ 2t dt

⎤

⎥⎥⎥⎦

(e)=∞∫

0

∞∫

0

⎛

⎜⎜⎜⎜⎜⎝

LI0 (t|r)

K∏

k=1

LIk (t) − LI0 (t|r)

×LS0 (t|r)

K∏

k=1

LIk (t) LSk (t)

⎞

⎟⎟⎟⎟⎟⎠

e−σ 2t

tdt fr0 (r) dr

(24)

where (e) holds due to the independence of the variables.Substituting (24) into (22), the average capacity (23) isobtained.

WU et al.: ON BS COORDINATION IN CACHE- AND ENERGY HARVESTING-ENABLED HETNETS 3085

E [Sk] = 2πλkωkηk P(TX)k

M0 (α − 2)

(1 − χ2−α

k

) Mk−1∑

mk=0

ρk (mk + 1)M0 − mk

mk + 1,

V [Sk] =πλkωkηk

(P(TX)

k

)2

2M0 (α − 1)

(1 − χ2−2α

k

) Mk−1∑

mk=0

ρk (mk + 1)M0 − mk

(mk + 1)2 . (25)

E [I0|r0] = r2−α0

2πλ0 P(TX)0

M0 (α − 2)(1 − ρ0 (0)) , V [I0|r0] = r2−2α

0

πλ0

(P(TX)

0

)2

2 (α − 1) M0

M0∑

m=1

ρ0 (m)

m. (26)

E [Ik] = 2πλkωk P(TX)k

M0 (α − 2)(1 − ρk (0)) , V [Ik] =

πλkωk

(P(TX)

k

)2

2 (α − 1) M0

Mk∑

mk=1

ρk (mk)

mk. (27)

C. Coverage Probability Analysis

When the cooperative BSs serve the typical MU at a fixedrate, outage can occur and the typical MU cannot reliablydecode the signals if the received SINR at the typical MU isbelow the predefined threshold. Thus, we define the coverageprobability, also known as successful transmission probabil-ity, to measure the coverage performance of the HetNets.Mathematically, the coverage probability is the complemen-tary cumulative distribution function (CCDF) of the receivedSINR, i.e.,

Pcov (β) = Pr [SINR ≥ β] , (28)

where β is the predefined coverage threshold.Theorem 2: In the cache-enabled and EH-powered HetNets

with the performed BS cooperation, the coverage probabilityof a typical MU is approximately expressed as:

Pcov (β)

≈M0−2∑

m0=0

ρ0 (m0 + 1)

∞∫

0

exp

{

−r2K∑

k=0

λkuk

}

fr0 (r) dr

+M0−2∑

m0=0

ρ0 (m0 + 1)

∞∫

0

� (κS + κZ ) ϕκS

� (κS) � (κZ )

×[

1

κZ

(

Z3 (1) − Z3

(

1 + βθZ (m0 + 1)

P(TX)0 r−α

))

+ 1

κS

(

Z4 (ϕ) − Z4

(

ϕ + βθZ (m0 + 1)

P(TX)0 r−α

))]

fr0 (r) dr,

(29)

where u0 and uk are expressed as:

u0 = π

M0∑

m0=1

m0

M0ρ0 (m0)Z2

(m0

β (m0 + 1)

),

uk = 2πωkβ2/α(m0 + 1)2/α

αM0

(P(TX)

k

P(TX)0

)2/α

× B

(2

α, 1 − 2

α

) Mk∑

mk=1

ρk (mk) m1−2/αk , (30)

respectively. ϕ is defined as ϕ = βθZθS

. κS, θS and κZ , θZ areexpressed as (43) and (46) in Appendix B with the mean andvariance of Sk and Ik shown as (25)-(27) at the top of thepage. Z3 (x) and Z4 (x) are respectively expressed as:

Z3 (x) = 2 F1

(1, κS + κZ ; κZ + 1; x

(x+ϕ)

)

(x + ϕ)κS+κZ, (31)

and

Z4 (x) =2 F1

⎛

⎝1, κS + κZ ; κS + 1;x− βθZ (m0+1)

P(TX)0 r−α

01+x

⎞

⎠

(1 + x)(κS+κZ ). (32)

Proof: Please refer to Appendix B.Although (23) and (29) are not in closed-form, the Laplace

functions can be easily calculated by MatLab, and thereforethe coverage probability and average user capacity can benumerically evaluated. Moreover, from the above analysis, wecan see that for a dominant cooperating or interfering SBS(i.e., the SBSs located inside B (o, χk).) with uniform powerallocation, the average cooperating transmission power is

ωkηkP(TX)

k

M0

Mk−1∑

m=0

ρk (m + 1)M0 − m

m + 1, (33)

and the average interfering transmission power is

ωkP(TX)

k

M0

Mk∑

m=1

ρk (m). (34)

Thus, there exists a tradeoff between the average cooperatingtransmission power and the average interfering transmissionpower at the dominant SBSs. The cooperative RSS thresholds,the cache size at SBSs, and the energy harvesting capabilityare key parameters that influence the average cooperating andinterfering transmission power.

VI. NUMERICAL RESULTS

In this section, we provide numerical and simulation resultsto evaluate the average user capacity and coverage probabilityin a two-tier HetNet, where the MBSs are overlaid by one tierof cache-enabled and EH-powered SBSs. Simulation results

3086 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018

TABLE II

SIMULATION PARAMETERS

Fig. 3. Coverage probability with respect to the coverage threshold β, whenτ1 = −60dBm.

Fig. 4. Average capacity and coverage probability with respect to τ1.

are obtained with Monte Carlo methods in a circular area withradius 5km. The detailed simulation parameters are given asin Table II unless otherwise specified.

Firstly, the simulation and numerical results are comparedin Fig. 3. The gap between the analytical and the simulationresult is due to the Gamma approximation for the distributionsof aggregated information and interference signal strength.

A. The Impact of Cooperative RSS ThresholdFig. 4 shows the impact of the cooperative RSS thresh-

olds (τ1). It can be seen that when τ1 approximately equalsto or less than −90 dBm, the SBSs are full-loaded with

Fig. 5. Average capacity and coverage probability with respect to M1.

probability 1 and almost zero-loaded when τ1 approachesto or above 0 dBm. Moreover, optimal cooperative RSSthreshold exists such that the average user capacity and cov-erage probability are maximized. This is due to the tradeoffbetween the cooperative probability of an SBS and the numberof candidate SBSs, as discussed in Sec. V-A. With low τ1, thearea for candidate SBSs is large and the load at SBSs withsufficient energy is heavy, which results in a low probabilityof sufficient energy at SBSs. As a result, very few SBSs canparticipate in the cooperation while the SBSs with sufficientenergy generate interference with almost probability 1, whichcauses low data rate and coverage probability. As τ1 increases,although the area for the candidate cooperative SBSs shrinksand the number of the candidate SBSs decreases, the probabil-ity that an SBS is cooperative and the probability of sufficientenergy increases. Hence, the aggregated information signalstrength firstly increases and then decreases with the increaseof τ1. Similarly, the interfering probability of an SBS reducessharply with the increase of τ1 and thus the average aggregatedinterference strength drops rapidly. As a result, the averageSINR received at the typical MU firstly rises and then dropsand there is an optimal τ1 maximizing the average capacity.

Combining with Eqs. (25)-(27), we can see that the varianceof SINR changes with the increase of τ1. When τ1 is largeenough, the SBSs neither join cooperation nor interfere withthe typical MU and the resulting SINR is almost SINR =

S0I0+σ 2 with very high probability. In such cases, the variance of

SINR is quite small. Increasing τ1, the average cell load at theSBSs varies from M1 to 0 and the number of cooperative SBSsfirstly increases and then decreases, leading to the varianceof SINR an peak function with respect to τ1. Combiningwith the result for the average SINR, the coverage probabilityPcov (β) = Pr {SINR > β} will have a minimum value if thecoverage threshold is small, as show in Fig. 4.

B. The Impact of Channel ReuseTo investigate the influence of the channel reuse among

tiers, Fig. 5 shows the average capacity and coverage probabil-ity with respect to the maximum number of channels reused atSBS, i.e., M1. With fixed cooperative RSS threshold and cache

WU et al.: ON BS COORDINATION IN CACHE- AND ENERGY HARVESTING-ENABLED HETNETS 3087

Fig. 6. Average capacity and coverage probability with respect to L1.

size at SBSs, the probability of zero-loading at SBSs as wellas the probability for having sufficient energy at SBSs is deter-mined. If the average cooperating transmission power (33) isabove the interfering transmission power (34) at a dominantSBS, the network performance will be improved. As shownin Fig. 5, with a low threshold τ1, e.g., τ1 = −70 dBm, thezero-load probability is very small (e.g., ρ1 (0) = 0.057) ifmore than one channel is reused at the SBS. In this case, theaggregated interference strength (I1) dominates the aggregatedinformation signal strength (S1) when M1 increases from0 to 1, and the performance decreases. However, S1 increaseswith M1 while I1 almost keeps unchanged, and the per-formance improves. With an appropriate cooperative RSSthreshold, e.g., τ1 = −50 dBm, the zero-load probability isρ1 (0) = 0.75. In this case, the signal strength S1 alwaysdominates the interference strength I1 and the average capacityas well as the coverage performance improves with M1.

C. The Impact of Cache Size

Fig. 6 shows the impact of the cache size of SBSs. It canbe seen that an optimal cache size maximizing the averageuser capacity and coverage probability exists with appropriatecooperative RSS threshold. When τ1 is too small, e.g., τ1 =−90 dBm, the SBSs will be full-loaded if cache hit happensand (34) > (33) +δ2 always establishes. As a result, I1 alwaysdominates S1 and the network performance decreases. Whenτ1 = −70 dBm, the SBSs are mid-loaded. (34) < (33) +δholds when the cache size L1 is below certain value and (34)> (33) +δ holds when L1 is above the value, which resultsin the optimal cache size at SBSs. When the cooperativeRSS threshold is τ1 = −50 dBm or more high, the SBSsare light-load and (34) < (33) +δ always follows. In thiscase, the network performance improves with L1 and finallylevels off.

D. The Impact of Energy Harvesting

Fig. 7 shows the impact of the energy harvesting rate μ.It can be seen that the network performance decreases when

2δ is a compensation for the interfering SBSs except the dominantinterfering SBSs.

Fig. 7. Average capacity and coverage probability with respect to μ.

τ1 = −70 dBm and levels off to maximum when τ1 =−30 dBm. Moreover, there exists an optimal energy harvestingrate when τ1 = −50 dBm. As we can see from (8) that withfixed τ1 and L1, the probability for having sufficient energyat an SBS is determined by μ. With heavy-loaded SBSs (e.g.,τ1 = −70 dBm), the SBSs can hardly join cooperation whilethey introduce severe interference with more energy harvested.With mid-loaded SBSs (e.g., τ1 = −50 dBm), the capacity andcoverage firstly increases because the increased informationsignal strength dominates the interference strength. However,due to the limited number of cooperative SBSs, the perfor-mance finally decreases due to the increasing of interferencestrength. With light-loaded SBSs (e.g., τ1 = −30 dBm), thenetwork performance improves due to the limited numberof interfering SBSs and levels off because the cooperativeSBSs tend to have sufficient energy with probability 1, evenincreasing the energy harvesting rate.

VII. CONCLUSION

In this paper, we have proposed a joint transmission strategyfor the cache-enabled and renewable energy-powered HetNets,to provide high transmission data rate and further improve thenetwork coverage performance. Furthermore, we have theoret-ically analyzed the coverage probability and average capacitywith stochastic geometry approach and Gamma approxima-tion. Numerical results have been provided to validate thetheoretical analysis and revealed that there exists an optimalcooperative RSS threshold for maximizing the coverage andcapacity performance. Moreover, with an appropriate coop-erative RSS threshold around the optimal cooperative RSSthreshold, the SBSs are mid-loaded. In such a scenario, i) theoptimal cache size and energy harvesting rate exist in maxi-mizing the average capacity and coverage probability, and ii)the average capacity and coverage probability firstly decreaseand then increase with the increase of the maximum number ofchannels available at SBSs, and finally level off to optimumperformance. These results provide insights for the deploy-ment and operation of cache-enabled and energy harvesting-powered SBSs in HetNets. The proposed BS coordinationstrategy can be widely applied in practice, especially for the

3088 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018

energy-constraint and backhaul-limited networks, to enhancethe coverage and service quality for MUs, in a cost-effectiveway. For the future work, we will consider more realisticchannel models [42] and investigate the optimal cooperativeRSS threshold and cache size at SBSs, to further enhance theuser experience and improve the coverage performance.

APPENDIX APROOF OF LEMMAS

A. Proof of Lemma 1

The distribution of S0 conditioned on r0 and m0 is

fS0 (s|r0, m0) = m0 + 1

P(TX)0

rα0 fh

(m0 + 1

P(TX)0

rα0 s

)

.

Then, the Laplace function of S0 conditioned on r0 is obtainedby applying the Laplace transform, i.e.,

LS0 (t|r0)

= ES0

(e−t S0

)

( f )= ρ0 (M0)+∞∫

0

e−stM0−2∑

m0=0

ρ0 (m0+1)m0+1

P(TX)0

rα0 e

− m0+1

P(TX)0

rα0 s

ds

= ρ0 (M0) +M0−2∑

m0=0

ρ0 (m0 + 1)

m0+1P(TX)

0

rα0

t + m0+1P(TX)

0

rα0

(35)

where (f ) is the result of the fact that S0 = 0 when the nearestMBS is full-loaded and will not serve the typical MU.

B. Proof of Lemma 2

With the definition of Laplace transform, the Laplace func-tion of Sk is

LSk (t)

= ESk

[e−t Sk

]

(g)= exp

⎧⎨

⎩−2πλkωk

χk∫

0

(

1−Eh

[

e−�(E2&E3)t

P(TX)k

mk +1 r−αh

])

rdr

⎫⎬

⎭

(h)= exp

⎧⎨

⎩−2πλkωk

χk∫

0

⎛

⎝νk − ηk

Mk−1∑

mk=0

M0 − mk

M0

ρk (mk + 1)

1 + tP(TX)

kmk+1r−α

⎞

⎟⎠ rdr

⎫⎪⎬

⎪⎭

= exp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

−πλkχ2k ηkωk

Mk−1∑

mk=0

ρk (mk + 1)M0 − mk

M0×

⎛

⎜⎜⎝

1 − 2

α + 2

mk + 1

tτk×

2 F1

(1, 1 + 2

α; 2 + 2

α; −mk + 1

tτk

)

⎞

⎟⎟⎠

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

(36)

where (g) follows the probability generation function ofPPP [34] and (h) holds because the SBS participates in

cooperation with probability νk and h ∼ exp {1}. Thus,

Eh

[

e−t

P(TX)k

mk +1 r−αh

]

= 1

1 + tPtx

kmk+1r−α

.

C. Proof of Lemma 2

The interfering macro BSs are located outside B (o, r0).

LI0 (t|r0)

= EI0

⎡

⎢⎣e

−t∑

x∈�0\B(o,r0 )�(occupied )

P(TX)0mx

‖x‖−αhx

⎤

⎥⎦

(i)= exp

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

−2πλ0

∞∫

r0

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 − ρ0 (0) −M0∑

m0=1

ρ0 (m0)

×⎛

⎜⎝

m0

M0

1

1+ t P(TX)0m0

r−α

+1− m0

M0

⎞

⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

rdr

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎭

= exp

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

−πλ0r20

M0∑

m0=1

ρ0 (m0)m0

M0

2

(α − 2)

t P(TX)0

m0rα0

×2 F1

(

1, 1 − 2

α; 2 − 2

α; − t P(TX)

0

m0rα0

)

⎫⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎭

(37)

where (i) follows the fact that the MBS is zero-loaded withprobability ρ0 (0) and the MBS with m0 associated MUs reusethe reference channel with probability m0

M0.

D. Proof of Lemma 3

For the interference caused by PG-SBS, the Laplace func-tion of Ik is (38), as shown at the top of the next page, where(j) is the result of the following integral.

2

∞∫

0

⎛

⎜⎝1 − 1

1 + t P(TX)kmk

r−α

⎞

⎟⎠ rdr

u=(

mk

t P(TX)k

)2/α

r2

=(

t P(TX)k

mk

)2/α ∞∫

0

(

1 − uα2

1 + uα2

)

du

v=uα2= 2

α

(t P(TX)

k

mk

)2/α

B

(2

α, 1 − 2

α

)(39)

APPENDIX BPROOF OF THEOREM 2

According to the definition of the coverage probability,Pcov (β) is expressed as:

Pcov (β) = Pr

[

S0 +K∑

k=1

Sk ≥ β

(

I0 +K∑

k=1

Ik + σ 2

)]

. (40)

WU et al.: ON BS COORDINATION IN CACHE- AND ENERGY HARVESTING-ENABLED HETNETS 3089

LIk (t) = E

⎡

⎢⎣e

−t∑

xk∈�k

�(E1&occupied)P

(TX)kmxk

‖xk‖−αhxk

⎤

⎥⎦

= exp

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

−2πλkωk

∞∫

0

⎛

⎜⎜⎜⎜⎜⎜⎜⎝

1 − ρk (0) −Mk∑

mk=1

ρk (mk)

×⎛

⎜⎝

mk

M0

1

1 + t P(TX)kmk

r−α

+ 1 − mk

M0

⎞

⎟⎠

⎞

⎟⎟⎟⎟⎟⎟⎟⎠

rdr

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

( j )= exp

⎧⎨

⎩−2πλk

αM0

(t P(TX)

k

) 2α

B

(2

α, 1 − 2

α

) Mk∑

mk=1

ρk (mk) m1−2/αk

⎫⎬

⎭(38)

Defining S =K∑

k=1Sk and Z = I0 +

K∑

k=1Ik , then

Pcov (β) ≈ Pr [S0 ≥ β Z ] + Pr [S0 < β Z , S ≥ β Z − S0]

= Pr [S0 ≥ β Z ] + Pr [S0 < β Z , S ≥ β Z]

+ Pr [S0 < β Z , β Z − S0 < S ≤ β Z] (41)

The first item Pr [S0 ≥ β Z] can be easily calculated with theLaplace functions of Ik , i.e.,

Pr [S0 ≥β Z](k)=

M0−2∑

m0=0

ρ0 (m0 + 1)

×∞∫

0

K∏

k=0

LIk

(β (m0+1) rα

P(TX)0

)

fr0 (r)dr , (42)

where (k) follows the exponential distribution of hx0 . Since itis complex to get closed-form expression for the pdf of Sk

and Ik by inverse Laplace transform. we apply the Gammadistribution [43] to approximate the distribution for S and Z .It is proved in [45] and [46] that second-order moment match-ing of Gamma is sufficient to approximate the distribution ofsums of Gammas. Thus, the Gamma distribution with the samefirst and second order moments as S has the following shapeparameter κS and scale parameter θS .

κS =

(K∑

k=1E [Sk ]

)2

K∑

k=1V [Sk ]

, θS =

K∑

k=1V [Sk]

K∑

k=1E [Sk ]

, (43)

where E [Sk] and V [Sk] are the mean and variance of Sk . Themean of Sk can be obtained with (2-19) in [34]:

E [Sk]

= E

⎡

⎣∑

xk∈�k∩B(0,χk )

� (E1&E2&E4)P(TX)

k

mxk + 1‖xk‖−αhxk

⎤

⎦

= 2πλkωkηk P(TX)k

M0

χ2−αk −d2−α

0

2 − α

Mk−1∑

mk=0

ρk (mk +1)M0−mk

mk + 1,

(44)

where d0 = 1 is the reference distance at which the path lossfading is 1. With the result (2-21) in [34], the variance of Sk

is

V [Sk ] = E(

h2) Mk−1∑

mk=0

ρk (mk + 1)M0 − mk

M0

(P(TX)

k

mk + 1

)2

× 2πλkωkηk

χk∫

d0

r1−2αdr

=πλkωkηk

(P(TX)

k

)2

M0

χ2−2αk − d2−2α

0

2 − 2α

×Mk−1∑

mk=0

ρk (mk + 1)M0 − mk

(mk + 1)2 . (45)

Similarly, the shape and scale parameters of the Gammadistribution with the same first and second order moments asZ are expressed as:

κZ =

(E [I0] +

K∑

k=1E [Ik ]

)2

V [I0] +K∑

k=1V [Ik]

, θZ =V [I0] +

K∑

k=1V [Ik ]

E [I0] +K∑

k=1E [Ik]

,

(46)

where the mean and variance of I0 conditioned on r0 are

E [I0|r0] = E

⎡

⎣∑

x∈�0\x0

1 (occupied)P(TX)

0

mx‖x‖−αhx

⎤

⎦

= 2πλ0

M0∑

m=1

ρ0 (m)P(TX)

0

M0

r2−α0

α − 2, (47)

and

V [I0|r0] = r2−2α0

2α − 2

πλ0

(P(TX)

0

)2

M0

Mk∑

mk=1

ρ0 (m)

m. (48)

The mean and variance of Ik are obtained as

E [Ik ] = 2πλkωk P(TX)k

M0

d2−α0

α − 2

Mk∑

mk=1

ρk (mk), (49)

3090 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018

and

V [Ik] = d2−2α0

2α − 2

πλkωk

(P(TX)

k

)2

M0

Mk∑

mk=1

ρk (mk)

mk. (50)

With the Gamma approximation of S and Z , the probabilityPr [S0 < β Z , S ≥ β Z] is approximately obtained as:

Pr [S0 < β Z , S ≥ β Z ]

= Em0,r0

⎧⎪⎨

⎪⎩

∞∫

0

βz∫

0

∞∫

βz

fS (s) ds fS0 (x) dx fI (z) dz

⎫⎪⎬

⎪⎭

(l)= Em0,r0

⎧⎨

⎩

∞∫

0

�(κS, βz

θS

)

� (κS)

(

1 − e− m0+1

P(TX)0 r−α

0

βz)

fZ (z) dz

⎫⎬

⎭

(m)= Em0,r0

{� (κS + κZ ) ϕκS Z3 (1)

� (κS) � (κZ + 1)

}

− Em0,r0

{� (κS + κZ ) ϕκS

� (κS) � (κZ + 1)Z3

(

1 + βθZ (m0 + 1)

P(TX)0 r−α

0

)}

(51)

where ϕ = βθZθS

and Z3 (x) is expressed as (31).(l) and (m) follow the Gamma distribution of S, Z andthe exponential distribution of hx0 . Similarly, the probabilityPr [S0 < β Z , β Z − S0 < S ≤ β Z] can be obtained as:

Pr [S0 < β Z , β Z − S0 < S ≤ β Z]

(n)= Em0,r0

⎧⎪⎨

⎪⎩

∞∫

0

βz∫

0

βz∫

βz−s

fS0 (x) dx fS (s) ds fZ (z) dz

⎫⎪⎬

⎪⎭

= Em0,r0

{� (κS + κZ ) ϕκS

� (κZ ) � (κS + 1)Z4 (ϕ)

}

− Em0,r0

{� (κS + κI ) ϕκS

� (κS + 1) � (κZ )Z4

(

ϕ + βθZ (m0 + 1)

P(TX)0 r−α

0

)}

(52)

where (n) is the result of the area transformation of the doubleintegral and Z4 (x) is expressed as (32). Combining (41), (42),(51) and (52), the coverage probability can be approximatedas (29).

REFERENCES

[1] N. Zhang, S. Zhang, S. Wu, J. Ren, J. W. Mark, and X. Shen, “Beyondcoexistence: Traffic steering in LTE networks with unlicensed bands,”IEEE Wireless Commun., vol. 23, no. 6, pp. 40–46, Dec. 2016.

[2] N. Zhang, N. Cheng, A. Gamage, K. Zheng, J. W. Mark, and X. Shen,“Cloud assisted HetNets toward 5G wireless networks,” IEEE Commun.Mag., vol. 53, no. 6, pp. 59–65, Jun. 2015.

[3] Q. C. Li, H. Niu, A. T. Papathanassiou, and G. Wu, “5G networkcapacity: Key elements and technologies,” IEEE Veh. Technol. Mag.,vol. 9, no. 1, pp. 71–78, Mar. 2014.

[4] D. Marabissi, G. Bartoli, R. Fantacci, and M. Pucci, “An optimizedCoMP transmission for a heterogeneous network using eICIC approach,”IEEE Trans. Veh. Technol., vol. 65, no. 10, pp. 8230–8239, Oct. 2016.

[5] D. Lee et al., “Coordinated multipoint transmission and reception inLTE-advanced: Deployment scenarios and operational challenges,” IEEECommun. Mag., vol. 50, no. 2, pp. 148–155, Feb. 2012.

[6] S. Bassoy, H. Farooq, M. A. Imran, and A. Imran, “Coordinated multi-point clustering schemes: A survey,” IEEE Commun. Surveys Tuts.,vol. 19, no. 2, pp. 743–764, 2nd Quart., 2017.

[7] G. Paschos, E. Bastug, I. Land, G. Caire, and M. Debbah,“Wireless caching: Technical misconceptions and business barriers,”IEEE Commun. Mag., vol. 54, no. 8, pp. 16–22, Aug. 2016.

[8] I. Atzeni, M. Maso, I. Ghamnia, E. Bastug, and M. Debbah. (May 2017).“Flexible cache-aided networks with backhauling.” [Online]. Available:https://arxiv.org/abs/1705.08576

[9] S. Zhang, N. Zhang, S. Zhou, J. Gong, Z. Niu, and X. Shen,“Energy-aware traffic offloading for green heterogeneous networks,”IEEE J. Sel. Areas Commun., vol. 34, no. 5, pp. 1116–1129, May 2016.

[10] Z. Zheng, X. Zhang, L. X. Cai, R. Zhang, and X. Shen, “Sustainablecommunication and networking in two-tier green cellular networks,”IEEE Wireless Commun., vol. 21, no. 4, pp. 47–53, Apr. 2014.

[11] H. Holma and A. Toskala, LTE-Advanced: 3GPP Solution forIMT-Advanced. Hoboken, NJ, USA: Wiley, 2012.

[12] Y. Li, L. Bai, C. Chen, Y. Jin, and J. Choi, “Successive orthogonalbeamforming for cooperative multi-point downlinks,” IET Commun.,vol. 7, no. 8, pp. 706–714, May 2013.

[13] X. Tao, X. Xu, and Q. Cui, “An overview of cooperative communica-tions,” IEEE Commun. Mag., vol. 50, no. 6, pp. 65–71, Jun. 2012.

[14] F. Baccelli and A. Giovanidis, “A stochastic geometry framework foranalyzing pairwise-cooperative cellular networks,” IEEE Trans. WirelessCommun., vol. 14, no. 2, pp. 794–808, Feb. 2015.

[15] W. Nie, F.-C. Zheng, X. Wang, W. Zhang, and S. Jin, “User-centric cross-tier base station clustering and cooperation in heterogeneous networks:Rate improvement and energy saving,” IEEE J. Sel. Areas Commun.,vol. 34, no. 5, pp. 1192–1206, May 2016.

[16] G. Nigam, P. Minero, and M. Haenggi, “Coordinated multipointjoint transmission in heterogeneous networks,” IEEE Trans. Commun.,vol. 62, no. 11, pp. 4134–4146, Nov. 2014.

[17] G. Nigam, P. Minero, and M. Haenggi, “Spatiotemporal cooperation inheterogeneous cellular networks,” IEEE J. Sel. Areas Commun., vol. 33,no. 6, pp. 1253–1265, Jun. 2015.

[18] R. Tanbourgi, S. Singh, J. G. Andrews, and F. K. Jondral, “A tractablemodel for noncoherent joint-transmission base station cooperation,”IEEE Trans. Wireless Commun., vol. 13, no. 9, pp. 4959–4973,Sep. 2014.

[19] A. H. Sakr and E. Hossain, “Location-aware cross-tier coordi-nated multipoint transmission in two-tier cellular networks,” IEEETrans. Wireless Commun., vol. 13, no. 11, pp. 6311–6325,Nov. 2014.

[20] H. Wu, X. Tao, N. Li, and J. Xu, “Coverage analysis for CoMP in two-tier HetNets with nonuniformly deployed femtocells,” IEEE Commun.Lett., vol. 19, no. 9, pp. 1600–1603, Sep. 2015.

[21] E. Bastug, M. Bennis, M. Kountouris, and M. Debbah, “Cache-enabledsmall cell networks: Modeling and tradeoffs,” EURASIP J. WirelessCommun. Netw., vol. 2015, no. 1, p. 41, Jan. 2015.

[22] C. Yang, Y. Yao, Z. Chen, and B. Xia, “Analysis on cache-enabledwireless heterogeneous networks,” IEEE Trans. Wireless Commun.,vol. 15, no. 1, pp. 131–145, Jan. 2016.

[23] Y. Liu, C. Yand, Y. Yao, B. Xia, Z. Chen, and X. Li, “Interferencemanagement in cache-enabled stochastic networks: A content diversityapproach,” IEEE Access, vol. 5, no. 4, pp. 1609–1617, Feb. 2017.

[24] J. Liu and S. Sun, “Energy efficiency analysis of cache-enabled coop-erative dense small cell networks,” IET Commun., vol. 11, no. 4,pp. 477–482, Mar. 2017.

[25] W. Wen, Y. Cui, F.-C. Zheng, S. Jin, and Y. Jiang. (Jan. 2017).“Random caching based cooperative transmission in heterogeneouswireless networks.” [Online]. Available: https://arxiv.org/abs/1701.05761

[26] B. Blaszczyszyn and A. Giovanidis, “Optimal geographic caching incellular networks,” in Proc. IEEE Int. Conf. Commun. (ICC), Jun. 2015,pp. 3358–3363.

[27] Y. Guo, L. Duan, and R. Zhang, “Cooperative local caching underheterogeneous file preferences,” IEEE Trans. Commun., vol. 656, no. 1,pp. 444–457, Jan. 2017.

[28] M. Afshang, H. S. Dhillon, and P. H. J. Chong, “Fundamentals of cluster-centric content placement in cache-enabled device-to-device networks,”IEEE Trans. Commun., vol. 64, no. 6, pp. 2511–2526, Jun. 2016.

[29] H. S. Dhillon, Y. Li, P. Nuggehalli, Z. Pi, and J. G. Andrews,“Fundamentals of heterogeneous cellular networks with energy harvest-ing,” IEEE Trans. Wireless Commun., vol. 13, no. 5, pp. 2782–2797,May 2014.

[30] P.-S. Yu, J. Lee, T. Q. S. Quek, and Y.-W. P. Hong, “Traffic offloading inheterogeneous networks with energy harvesting personal cells-networkthroughput and energy efficiency,” IEEE Trans. Wireless Commun.,vol. 15, no. 2, pp. 1146–1161, Feb. 2016.

WU et al.: ON BS COORDINATION IN CACHE- AND ENERGY HARVESTING-ENABLED HETNETS 3091

[31] F. Parzysz, M. Di Renzo, and C. Verikoukis, “Power-availability-awarecell association for energy-harvesting small-cell base stations,” IEEETrans. Wireless Commun., vol. 16, no. 4, pp. 2409–2422, Apr. 2017.

[32] J. Gong, S. Zhou, and Z. Zhou, “Networked MIMO with fractionaljoint transmission in energy harvesting systems,” IEEE Trans. Commun.,vol. 64, no. 8, pp. 3323–3336, Aug. 2016.

[33] Y.-H. Chiang and W. Liao, “Green multicell cooperation in hetero-geneous networks with hybrid energy sources,” IEEE Trans. WirelessCommun., vol. 15, no. 12, pp. 7911–7925, Dec. 2016.

[34] F. Baccelli and B. Blaszczyszyn, Stochastic Geometry and WirelessNetworks: Theory, vol. 1. Hanover, MA, USA: Now Publishers, 2009.

[35] X. Wang, M. Chen, T. Taleb, A. Ksentini, and V. C. M. Leung, “Cachein the air: Exploiting content caching and delivery techniques for 5Gsystems,” IEEE Commun. Mag., vol. 52, no. 2, pp. 131–139, Feb. 2014.

[36] J.-S. Ferenc and Z. Néda, “On the size distribution of poisson Voronoicells,” Phys. A, Statist. Mech. Appl., vol. 385, no. 2, pp. 519–529, 2007.

[37] T. Kiang, “Random fragmentation in two and three dimensions,”Zeitschrift Astrophys., vol. 64, pp. 433–439, Jun. 1966.

[38] H. S. Dhillon and J. G. Andrews, “Downlink rate distribution inheterogeneous cellular networks under generalized cell selection,” IEEEWireless Commun. Lett., vol. 3, no. 1, pp. 42–45, Feb. 2014.

[39] M. Di Renzo, W. Lu, and P. Guan, “The intensity matching approach:A tractable stochastic geometry approximation to system-level analysisof cellular networks,” IEEE Trans. Wireless Commun., vol. 15, no. 9,pp. 5963–5983, Sep. 2016.

[40] S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products.6th ed. New York, NY, USA: Academic, 2000.

[41] M. Di Renzo, A. Guidotti, and G. E. Corazza, “Average rate of down-link heterogeneous cellular networks over generalized fading channels:A stochastic geometry approach,” IEEE Trans. Commun., vol. 61, no. 7,pp. 3050–3071, Jul. 2013.

[42] L. Bai, L. Zhu, J. Choi, F. Liu, and Q. Yu, “Cooperative transmissionover Rician fading channels for geostationary orbiting satellite colloca-tion system,” IET Commun., vol. 11, no. 4, pp. 538–547, Mar. 2017.

[43] R. W. Heath, M. Kountouris, and T. Bai, “Modeling heterogeneousnetwork interference using poisson point processes,” IEEE Trans. SignalProcess., vol. 61, no. 16, pp. 4114–4126, Aug. 2013.

[44] R. W. Heath, Jr., T. Wu, Y. H. Kwon, and A. C. K. Soong, “MultiuserMIMO in distributed antenna systems with out-of-cell interference,”IEEE Trans. Signal Process., vol. 59, no. 10, pp. 4885–4899, Oct. 2011.

Huici Wu received the B.S. degree in informationengineering from the Communication University ofChina, Beijing, China, in 2013. She is currentlypursuing the Ph.D. degree in communications andinformation systems with the Beijing Universityof Posts and Telecommunications, Beijing, China.From 2016 to 2017, she was visiting the Broad-band Communications Research Group with theDepartment of Electrical and Computer Engineering,University of Waterloo, Waterloo, ON, Canada. Herresearch interests are in the area of wireless commu-

nications and networks, with current emphasis on the cooperation and physicallayer security in heterogeneous networks.

Xiaofeng Tao (SM’13) received the B.S. degreein electrical engineering from Xi’an Jiaotong Uni-versity, Xi’an, China, in 1993, and the M.S.E.E.and Ph.D. degrees in telecommunication engineeringfrom the Beijing University of Posts and Telecom-munications (BUPT), Beijing, China, in 1999 and2002, respectively.

He was the Chief Architect of the ChineseNational FuTURE Fourth-Generation (4G) TDDworking group from 2003 to 2006, established the4G TDD CoMP trial network in 2006, and a Visiting

Professor with Stanford University, Stanford, CA, USA, from 2010 to 2011.He is currently a Professor with the BUPT and a fellow of the Institutionof Engineering and Technology. He is currently the inventor or co-inventorof 50 patents and the author or co-author of 120 papers in 4G and beyond 4G.

Ning Zhang (M’15) received the Ph.D. degreefrom the University of Waterloo in 2015. He iscurrently an Assistant Professor with the Departmentof Computing Science, Texas A&M University-Corpus Christi. Before that, he was a Post-DoctoralResearch Fellow with the BBCR laboratory, Uni-versity of Waterloo. His current research interestsinclude next generation wireless networks, softwaredefined networking, vehicular networks, and physi-cal layer security. He was a co-recipient of the BestPaper Award at the IEEE GLOBECOM 2014 andthe IEEE WCSP 2015.

Danyang Wang received the B.S. degree in commu-nications engineering from Xidian University, Xi’an,China, in 2012. He is currently pursuing the Ph.D.degree with the State Key Laboratory of IntegratedService Networks, School of TelecommunicationsEngineering, Xidian University, Xi’an. His researchinterests include cognitive radio networks, coopera-tive spectrum sensing, and NOMA.

Shan Zhang (S’13–M’16) received the B.S. degreefrom the Department of Information, Beijing Insti-tute Technology, Beijing, China, and the Ph.D.degree from the Department of Electronic Engi-neering, Tsinghua University, in 2011 and 2016,respectively. She is currently a Post-Doctoral Fellowwith the Department of Electronic and ComputerEngineering, University of Waterloo, Waterloo, ON,Canada. Her research interests include resource andtraffic management for green communication, intel-ligent vehicular networking, and software defined

networking. He was a recipient of the Best Paper Award at the Asia-PacificConference on Communication in 2013.

Xuemin (Sherman) Shen (M’97–SM’02–F’09)received the B.Sc. degree from Dalian MaritimeUniversity, China, in 1982, and the M.Sc. and Ph.D.degrees from Rutgers University, Middlesex County,NJ, USA, in 1987 and 1990, respectively, all inelectrical engineering. He is currently a UniversityProfessor and the Associate Chair for GraduateStudies, Department of Electrical and ComputerEngineering, University of Waterloo, Waterloo, ON,Canada. He research focuses on resource manage-ment, wireless network security, social networks,

smart grid, and vehicular ad hoc and sensor networks. He was an electedmember of the IEEE ComSoc Board of Governor, and the Chair of Distin-guished Lecturers Selection Committee. He was a recipient of the ExcellentGraduate Supervision Award in 2006, and the Premiers Research ExcellenceAward in 2003 from the Province of Ontario, Canada. He served as theTechnical Program Committee Chair/Co-Chair for the IEEE Globecom16,Infocom14, the IEEE VTC10 Fall, and Globecom07, the Symposia Chairfor the IEEE ICC10, the Tutorial Chair for the IEEE VTC11 Spring, andthe IEEE ICC08, the General Co-Chair for ACM Mobihoc15, Chinacom07,and QShine06, the Chair for the IEEE Communications Society TechnicalCommittee on Wireless Communications, and P2P Communications and Net-working. He also serves/served as the Editor-in-Chief for the IEEE INTERNET

OF THINGS JOURNAL, the IEEE Network Magazine, Peer-to-Peer Networkingand Application, and the IET Communications; a Founding Area Editor forthe IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS; an AssociateEditor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, Com-puter Networks, and the ACM/Wireless Networks, etc.; and the Guest Editorfor the IEEE JSAC, the IEEE WIRELESS COMMUNICATIONS, the IEEECommunications Magazine, and ACM Mobile Networks and Applications, etc.He is currently a registered Professional Engineer of Ontario, Canada, anEngineering Institute of Canada Fellow, a Canadian Academy of EngineeringFellow, a Royal Society of Canada Fellow, and a Distinguished Lecturer ofthe IEEE Vehicular Technology Society and Communications Society.