Modeling and Analysis of Point-to-Multipoint Millimeter-Wave ...1 Modeling and Analysis of...

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Modeling and Analysis of Point-to-MultipointMillimeter-Wave Backhaul Networks

Jia Shi, Member, IEEE, Lu Lv, Qiang Ni, Senior Member, IEEE, Haris Pervaiz, Member, IEEE,and Claudio Paoloni, Senior Member, IEEE

Abstract—A tractable stochastic geometry model is pro-posed to characterize the performance of the novel point-to-multipoint (P2MP) assisted backhaul networks with millimeter-wave (mmWave) capability. The novel performance analysis isstudied based on the general backhaul network (GBN) and thesimplified backhaul network (SBN) models. To analyze the signal-to-interference-plus-noise ratio (SINR) coverage probability ofthe backhaul networks, a range of the exact- and closed-formexpressions are derived for both the GBN and SBN models.With the aid of the tractable model, the optimal power controlalgorithm is proposed for maximizing the trade-off betweenenergy-efficiency (EE) and area spectral-efficiency (ASE) for themmWave backhaul networks. The analytical results of the SINRcoverage probability are validated, and they can match thoseobtained from Monte-Carlo experiments. Our numerical resultsfor ASE performance demonstrate the significant effectivenessof our P2MP architecture over the traditional point-to-point(P2P) setup. Moreover, our P2MP mmWave backhaul networksare able to achieve dramatically higher rate performance thanthat obtained by the ultra high frequency (UHF) networks.Furthermore, to achieve the optimal EE and ASE trade-off, themmWave backhaul networks should be designed to limit the linkdistances and line-of-sight (LOS) interferences, while optimizingthe transmission power.

I. INTRODUCTION

Over the past few years, global mobile traffic has an explo-sive rate increasing with 131% annual growth, and the peakdata rate has increased with 55% annual growth [1]. However,low frequency bands, such as ultra high frequency (UHF),have been heavily utilized and it is difficult to find sufficientfrequency bands in the sub 6 GHz range for 5G cellular net-works. In comparison, there are still a large number of unusedspectrum resources in the millimeter-wave (mmWave) bands,which may be of potential exploiting to future networks.Therefore, employing mmWave high frequency bands suchas 30-300 GHz and network densification are considered askey enablers to achieve high requirements of data rate, energyefficiency and spectrum efficiency for future 5G wirelessnetworks [2]–[8].

This work was supported by EU H2020 TWEETHER project under grantagreement number 644678, by the National Natural Foundation of China underGrant 61631015, by the Royal Society project under Grant IEC170324, andby the EPSRC under Grant EP/P015883/1. (Corresponding authors: QiangNi, Lu Lv.)

J. Shi and L. Lv are with the State Key Laboratory of IntegratedServices Networks, Xidian University, Xi’an, 710071, China (email: [email protected], [email protected]).

Q. Ni, H. Pervaiz are with the School of Computing and Commu-nications, Lancaster University, Lancaster LA1 4WA, UK (email: {q.ni,h.b.pervaiz}@lancaster.ac.uk).

C. Paoloni is with the Department of Engineering, Lancaster University,Lancaster LA1 4YW, UK (email: [email protected]).

A. Related Works

Increasing research efforts have been carried out to investi-gate the potential of mmWave cellular networks, such as easierdeployment, superior rate coverage and higher throughput forindoor/outdoor environment [9]–[15]. Most of the work in theliterature, such as [9]–[13], have mainly focused on utilizingmmWave spectrum on access networks for future wirelesscommunications. An initial theoretical study on the capacityand coverage of cellular networks using mmWave frequencybands has been addressed in [11]. MmWave cellular networks[12], [13] have been focused on the physical layer securityissues by exploiting the benefit from using directional anten-nas. More recently, some research attentions [16], [17] havebeen devoted to the ultra dense hybrid heterogeneous cellularnetworks with both UHF and mmWave BSs coexisting. Asproposed by [17], mmWave BSs only support downlink trans-mission whereas UHF BSs support both downlink and uplinktransmissions. However, the above studies have revealed thathigh path loss and severe penetration loss due to physicalblockages pose huge challenges to the implementation ofmmWave access networks, especially the physical limitationsof realizing mmWave terminals.

It is still very challenging to realize mmWave commu-nication in access networks due to the facts that, equip-ping high-cost mmWave antennas at mobile users, and highsignal overhead caused by beam training and tracking, aswell as diverse interference scenario. These issues will bealleviated in backhaul networks, which certainly provide abetter scenario for employing mmWave than access networks.Nevertheless, very limited research efforts have been devotedto mmWave backhaul networks. Recently, suggested by [18]–[20], mmWave communication is considered as a key enablerfor providing the wireless backhaul for outdoor small cellsdue to massive spectrum available providing high capacity,quick deployment, flexibility, and low cost in comparison tofibre. The studies of [19], [20] have proposed the optimalstrategies for small-cell BS deployment in order to designhigh-efficient mmWave backhaul networks. Furthermore, arange of researches in [14], [15], [21] have focused on jointlydesigning the access and backhaul networks with mmWavecapability. [14], [15] have studied the HetNet with mmWavecommunication for access and backhaul, and have investigatedmultihop routing schemes for the mmWave-based backhaulmesh. In [21], the authors have addressed the issue of man-aging radio resources for the access and backhaul links inthe small cell networks. Apparently, the current researches

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have barely analyzed the theoretical performances of mmWavebackhaul networks, especially by means of stochastic geome-try approach.

By exploiting a range of advantages, point-to-multipoint(P2MP) technique at mmWave will open new perspectivefor future backhaul networks. On one hand, P2MP approachcan increase link utilization thus achieving higher throughput,compared to traditional point-to-point (P2P). On the otherhand, P2MP approach requires much lower hardware cost fora single link than the P2P approach. In the literatures, P2MPtechnique has been exploited by various mmWave accessnetworks, such as [22], [23] investigated the P2MP assistedWLAN systems operating at 60 GHz. By contrast, the verylimited studies [24]–[26] have been devoted to investigatingP2MP technique assisted wireless backhaul networks. In [24],a TDM-based scheduling for in-band access and backhaullinks has been investigated to support point-to-multipoint andnon-LOS communication. The authors of [25] have proposeda novel two-tier backhaul networks, where P2MP techniqueis applied to UHF-based links, while however using P2Ptechnique for mmWave-based links. Further, in [26], theengineering challenges have been addressed for implementingthe Q-band backhaul link with P2MP.

B. Motivations and Contributions

So far, there is no satisfactory P2MP based backhaul so-lution in mmWave, which is urgently desired by theoreticalstudy and industry. In this trend, our EU funded TWEETHERproject1 is to set a milestone in the millimetre wave technologywith the realization of the first W-band (92-95GHz) wirelesssystem, supported by the high power of a new concept oftraveling wave tube [27]. The long range achieved by thetransmission hub proposes a credible solution for high capacitydensity and availability [28]. TWEETHER aims to realise themillimetre wave P2MP segment to link fibre, and sub-6 GHzdistribution for a full three segment hybrid network, that is themost cost-effective architecture to reach mobile or fixed finalindividual client.

To the best of our knowledge, there is no theoreticalresearch that investigates the performance of the mmWavebackhaul networks. Motivated by the lack of study and by ourTWEETHER project, in this paper we carry out the analyticalstudy for the P2MP aided mmWave backhaul network, andenable an energy and spectrum efficient backhaul architecturefor future. The main contributions can be summarized asfollows.• We propose the novel mmWave backhaul solution by

leveraging the P2MP technique in order to boost highcapacity of future wireless systems. A tractable stochasticgeometry model is proposed for analyzing the perfor-mance of the P2MP assisted mmWave backhaul networkconceived in this paper. Two different theoretical modelsare developed: one is the general P2MP assisted mmWavebackhaul network (GBN) model with the objective ofcomprehensively evaluating performance, and the other is

1TWEETHER stands for traveling wave tube based W-band wirelessnetworks with high data rate, distribution, spectrum and energy efficiency.

the simplified P2MP mmWave backhaul network (SBN)model for analyzing the dominant performance factorsincluding the LOS interference and path loss.

• We carry out novel performance analysis for both theGBN and SBN models, and derive the exact-form expres-sions for the SINR coverage probability, which can reflectcommunication reliability of desired link. Furthermore,by using the Gaussian-Chebyshev quadrature and theintelligent manipulations, the closed-form expressions ofthe SINR coverage probability can be obtained for theGBN and SBN models. The analytical results imply thatthe LOS interference will be a dominant effect on thecoverage rate performance of the backhaul networks.

• With the help of the stochastic geometry assisted tractablemodel, we investigate the trade-off between energy-efficiency (EE) and area spectral-efficiency (ASE) of ourP2MP mmWave backhaul networks. The original non-convex optimization problem is analyzed and solved bythe proposed algorithm, thereby resulting in the optimalpower control strategy. With the aid of this strategy, thebest trade-off between EE and ASE is obtained for ourmmWave backhaul networks.

• We carry out a range of performance evaluation for theP2MP mmWave backhaul networks. For the SINR cov-erage probability of the networks, the exact- and closed-form analytical results under the GBN and SBN modelsare validated and well agree with those obtained byMonte-Carlo simulations. According to the performanceevaluation, our proposed P2MP mmWave backhaul isshown to be a promising solution for future wirelesssystems, owing to the significant ASE performance gainbenefited from using P2MP technique over traditionalP2P, as well as the huge rate improvement by leveragingthe mmWave communications rather than conventionalUHF.

II. SYSTEM MODEL

A. Network Structure

We consider a point-to-multipoint (P2MP) mmWave back-haul network, which consists of multiple hubs (as macrobase stations) of each serving several small-cell base stations(SCBSs). Fig. 1 shows a simple example of our P2MPmmWave backhaul network model which has three neigh-boring hubs. Note that, Fig. 1(a) gives the schematic foran example of the mmWave backhaul network, and Fig.1(b) depicts the conceptual structure of P2MP architectureemployed by the example network.

In Fig. 1(a), we assume that, all hubs are connected to corenetwork via fibre optic links. Each of the hubs can supporta cluster of SCBSs, and neighboring hubs’ service areas maygeographically overlap with each other. We assume that, themmWave backhaul transmissions from each hub to its SCBSsare based on time division multiple access (TDMA) scheme.The P2MP mmWave links are established for backhaul trans-mission from each hub to its SCBSs, as shown by Fig. 1(b),where a service area of a hub can be envisioned as a circle withmultiple sectors. Each of the hubs, seating at the origin of each

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circle, employs multiple sectored antennas of each servingthe SCBSs in the corresponding sector area. When TDMA isassumed, every sectored antenna of a hub can serve one SCBSwithin its sector area at a time. In that case, each hub cansimultaneously transmit information to multiple SCBSs, eachof which is supported by one sectored antenna. Compared totraditional point-to-point (P2P) approach, a P2MP is capableof increasing link utilization, and reducing communicationlatency as well as achieving higher network throughput [14].This is because that, our P2MP approach can allow a hub tosupport multiple links at a time, whereas the P2P can not.Furthermore, a P2MP approach demands lower hardware costfor a single link than a P2P one, thereby achieving a significantsaving for the total cost of ownership (TCO).

Without loss of generality, in this paper, each hub isassumed to have 6 sectored antennas. In order to achieve agood trade-off between interference avoidance and spectralefficiency, the frequency reuse pattern in Fig. 1(b) is employed,where the transmissions in every two neighboring sectors areallocated the orthogonal frequency bands f1 and f2, each ofwhich has a bandwidth B0. For theoretical study, the radiocharacteristic of a sectored antenna can be modeled as thebottom-right plot of Fig. 1(b), where a main lobe beam and aside lobe beam are labeled by light and dark colors, respec-tively. The beam pattern, Ψ(M,m, θ), can be characterizedby three values: main lobe gain M , side lobe gain m, andmain lobe beamwidth θ. The main lobe beamwidth of eachsectored antenna is constrained to be θ ≤ 120◦, so that inter-sector interference is minimized.

With the aid of stochastic geometry approach, a tractableanalytical network model is proposed for characterizing theperformance of our mmWave P2MP backhaul networks. With-out loss of generality, the backhaul network is modeled as onecircle, denoted by S with radius R, where the typical SCBSseats at the origin (0, 0) and its serving hub locates at (0, D0).The other hubs in the network are seen as interfering hubs ofthe typical SCBS. Known from [29], [30], the locations ofinterfering hubs distributed on the circle area can be modeledas a homogenous Poisson point process (PPP) Φ of intensityλ. The PPP assumption can be justified by the fact thatnearly any hub distribution in a 2-D plane results in a smallfixed SINR shift relative to the PPP [11]. Specifically, wewould like to clarify that, a homogeneous PPP is employedto approximate a non-homogeneous PPP or a binomial pointprocess (BPP), as long as the network size is sufficientlylarge (such that R ≥ 2km assumed in our simulation) sothat the boundary effect can be negligible. As an interestingfuture research, we will apply the BPP model to the mmWavebackhaul networks with limited size for precisely investigatingthe boundary effect. According to [31], a 2-D plane for thePPP assumption could be an infinite area or a finite area suchas a circle area assumed in this paper. Note that, we analyze theperformance for the typical SCBS while our results hold forother SCBSs in the P2MP mmWave backhaul network accord-ing to Slivnyak Theorem2 [31], [32]. By leveraging Slivnyak

2Slivnyak theorem: for a PPP Φ, conditioning on a point at x does notchange the distribution of the rest of the process, since the independenceamong all the points.

Core Network

Hub

Hub

SCBS

SCBS

user

user

Fibre Optics

Fibre Optics

P2MP mmWave links

P2MP mmWave links

Hub

SCBS

user

P2MP mmWave links

Fibre Optics

(a) Schematic of network model

f1

f2

f1

f1

f1

f2

f2

f2

f2

f2

f1

f1

f1f2

f2

f2

f1

f1

Sectored antenna Beam pattern

of a sectored

antenna

(b) P2MP architecture

Fig. 1. An example of P2MP aided mmWave backhaul network model.

theorem, it allows us to randomly select the typical SCBS inthe network, without changing its statistical properties. Notethat, we assume that our backhaul network has been optimized,where each SCBS is associated with the best hub and is ableto support its access links. In this paper we only focus onbackhaul transmission.B. Propagation and Blockage Model

Directional beamforming is assumed for each link, wheremain lobe beam is aligned towards the dominate propagationpath. Assume that, each sectored antenna of all hubs has thesame beam pattern of Ψ(Mt,mt, θt), and each SCBS hasthe same pattern of Ψ(Mr,mr, θr). Upon performing perfectbeam alignment, the effective antenna gain of the desiredcommunication link for the typical SCBS becomes

G0 = MtMr (1)

which is for analysis tractable. Owing to operating on thesame frequency band, the typical SCBS will suffer fromintra-hub interferences (IntraHI) caused by the serving huband inter-hub interferences (InterHI) caused by the interferinghubs. As shown by Fig. 1(b), the main lobe beams for thedesired link and its two IntraHI links are separated by at leastof 60◦. Therefore, for the desired link, the effective antennagain imposed by an IntraHI link can be modelled as

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G0,j = mtMr, ∀j ∈ {1, 2}, (2)

where subscript 0 denotes the serving hub, and subscript j isindex of an IntraHI sectored antenna.

By contrast, for the desired link, the effective antenna gainsimposed by the three InterHI links of an interfering hub arediscrete random variables according to

(Gi,j , ∀j ∈ {1, 2, 3}) = (Gi,1, Gi,2, Gi,3) =(MtMr,Mrmt,Mrmt), w.p. p1 = 3θtθr/4π

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(mtMr,mtMr,mtMr), w.p. p2 = (2π − 3θt)θr/4π2

(Mtmr,mtmr,mtmr), w.p. p3 = 3θt(2π − θr)/4π2

(mtmr,mtmr,mtmr), w.p. p4 = (2π − 3θt)(2π − θr)/4π2

(3)

where subscript i (i ∈ Φ) is index of an interfering hub, andsubscript j is index of an InterHI sectored antenna of the hub.

As illustrated by Fig. 1(b), there are three sectored antennasof an interfering hub using the frequency band same as thetypical link. As a result, the desired link has three InterHI linkscaused by each interfering hub, giving the effective antennagains Gi,1, Gi,2, Gi,3 in (3). They could have four differentgroups of values, which correspond to four different scenarios.Let us use the first scenario as an example to explain howto obtain the effective antenna gain and the correspondingprobability. In the first scenario, we have Gi,1 = MtMr,meaning that the main lobe beam of the desired link intersectswith the main lobe beam of InterHI link 1 initiated by hub i.Observed from the frequency reuse pattern assumed in Fig.1(b), for each hub, the main lobe beams of every two sectoredantennas operating on the same frequency band are separatedby at least 60◦. In that case, the main lobe beam of the desiredlink will only intersect with the side lobe beam of InterHI links2 and 3 of hub i, thereby giving that Gi,2 = Gi,3 = Mrmt.Further, the desired link’s sectored antenna has a main lobebeamwidth of θr, while each sectored antenna of hub i has abeamwidth θt of main lobe, in which θt, θr ∈ [0, 2π] followsuniform distribution. Hence, when considering the main lobebeam of the desired link intersect with any of the three InterHIlinks’ main lobe beam, we have the probability, given by

p1 = (θr2π

)× (θt2π

+θt2π

+θt2π

) =3θtθr4π2

. (4)

It is straightforward that the other three scenarios will havethe effective gains in (3), where the relating probabilities canbe deduced similar to that in (4).

When comparing (1) with (3), we readily know that InterHIwill be the dominant interference for the typical SCBS, whichis due to the following two reasons. First, when employingthe P2MP architecture in Fig. 1(b), the main lobe beam of thetypical link will be affected by the side lobe beams of the twoIntraHI links only. By contrast, the typical link could sufferfrom a strong interference caused by the main lobe beam of aInterHI link, as illustrated by the first case in (3). In general,sophisticated mmWave antenna design can enable a result ofMt >> mt, Mr >> mr. Hence, the InterHI gain will bemuch higher than the IntraHI gain. Second, in the mmWavebackhaul networks, the typical SCBS experiences two IntraHIlinks initiated from its serving hub only, while theoretically itcould have the InterHI caused by all the interfering hubs in thenetwork. Specifically, as the density of the hubs increases, the

number of InterHI will increase while the number of IntraHIwill not. Hence, InterHI will be the dominant effect.

The mmWave links between the typical SCBS and the hubscan be either line of sight (LOS) or non-line of sight (NLOS).Further, we assume that, the LOS probabilities for differentlinks are independent, and potential correlations of blockagesare ignored. Let ΦL and ΦN = Φ−ΦL be the point processesof LOS hubs and NLOS hubs, and are obtained by applyingindependent thinning on the PPP Φ using the LOS probabilitypL(x) to determine whether a link of length x is LOS or not[11]. The intensities of ΦL and ΦN are respectively determinedby pL(x)λ and (1 − pL(x))λ. The LOS probability functionpL(x) only depends on the length of the link, the distributionof the blockage process is stationary and isotropic.

The LOS probability function pL(x) is mainly related tostochastic blockage parameters, which are characterized bysome random distributions. In this paper, the blockages inthe networks are modeled by employing a Boolean modelof rectangles. In this case, the LOS probability function canbe expressed as pL(x) = e−βx, where β is a parametercharacterized by the statistics of densities and average sizesof blockages in the network. Therefore, we readily know thatthe probability pL(x) is a monotonically decreasing functionof distance x. The path loss exponent for each link is a discreterandom variable given by

αi =

{αL, w.p. pL(x)

αN , w.p. 1− pL(x). (5)

Furthermore, we denote the path loss intercept as κ =20 log10( 2πdref

φref) with dref = 1m and φref as the carrier wave-

length [33, Eq. (2.41)].

C. SINR for Desired LinkWe assume that the small-scale fading pertaining each link

follows independent Nakagami-m fading, where the fadingparameters mL and mN for the LOS and NLOS are con-strained to be integer values for mathematical tractability. Inthe following, we refer h0 as the channel gain of the desiredlink, and hi, i ∈ S as the channel gain of InterHI link.For mathematical tractability, we assume that the small-scalefading gains are the same for the links from a hub to its SCBSswhich are served at the same time slot. The background noiseis characterized by a zero-mean, complex Gaussian randomvariable with variance N0. Based on the assumptions thusfar, the overall received SINR at the typical SCBS can beexpressed as

SINR =

PtG0|h0|2κD−α00∑2

j=1 PtG0,j |h0|2κD−α00 +

∑i>0,i∈Φ

∑3j=1 PtGi,j |hi|2κD

−αii +N0

(6)

where Pt is the transmit power of each sectored antennaof a hub, and Di is the distance from hub i to the typicalSCBS. G0, G0,j and Gi,j are defined in (1)–(3), respectively.Note that the SINR in (6) is a random variable, due to therandomness of the hub locations Di, small-scale fading hi, andthe effective antenna gain Gi,j . Note that in (6), the first termof the denominator denotes the IntraHI of the typical SCBS,while the second term of the denominator is the InterHI.D. Performance Metrics

For our P2MP mmWave backhaul network, we consider arange of performance metrics which include coverage proba-bility, area spectral efficiency (ASE), energy efficiency (EE),

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etc. The coverage probability is also referred to as the successprobability, which reflects the reliability of the link betweenthe typical SCBS and its serving hub. The definition of thecoverage probability for the typical SCBS can be expressedas [11], [34]

Pc(T ) = Pr(SINR > T

)(7)

where T denotes the SINR threshold. The coverage probabilitydefined in (7) can be further explained as follows: (i) theprobability that a randomly chosen SCBS in the network canachieve the target SINR T ; (ii) the average fraction of theSCBSs achieving the target SINR T at any given time; (iii)the average fraction of the network area that is in coverage atany given time. Therefore, the typical SCBS is said to be incoverage if it is able to connect at least one hub in the P2MPmmWave backhaul networks with SINR above threshold T .

Furthermore, the rate coverage probability Prc(Γ) is pro-vided to characterize the exact rate distribution in order toanalyze the rate performance of the networks. Given the SINRcoverage probability Pc(T ), the rate coverage probability canbe computed as

Prc(Γ) = Pr(Rate > Γ) = Pc(2Γ/B0 − 1) (8)

where Γ = B0 log2(1+T ) is the target rate requirement condi-tioned on the SINR threshold T . Note that, the rate coverageprobability enables to show the performance gain achievedby our mmWave backhaul networks over the traditional UHFnetworks.

Finally, we define another important performance metric,the EE of the P2MP mmWave backhaul network, as [34]

ηEE =B0ηASEPtot

=Pc(T )B0 log2(1 + T )

1εPt + Pcon

(9)

where ηASE describes the network capacity measured inb/s/Hz/m2, given by

ηASE = λNPc(T ) log2(1 + T ). (10)

Note that, N denotes the number of links supported by, i.e.the number of SCBSs served by each hub at each time slot, λdenotes the intensity of the hubs in our backhaul network andPtot = λN( 1

εPt+Pcon) is the total transmit power consumedby the network conditioned on the density of hubs. It should benoted that, the ASE characterizes the sum rate of the networkin a unit area normalized by bandwidth, and reflects thenetwork capacity according to various densification. Further,ε is the amplifier efficiency. We assume the same transmitpower Pt and constant circuit power consumption Pcon foreach sectored antenna.

III. PERFORMANCE ANALYSIS FOR GENERAL P2MPMMWAVE BACKHAUL NETWORKS

In this section, we analyze the SINR coverage probabilityfor the P2MP mmWave backhaul network under the GBNmodel. In this case, the LOS and NLOS probabilities ofcommunication links are modeled as the exponential functionsin [11]. When all links including desired link and interferinglinks experience the independent Nakagami-m fading, we firstderive the exact expressions for the SINR coverage probability.Further, with the aid of sophisticated approximation methods,we also obtain the closed-form expressions for the coverage

probability. In the end, when considering the special scenariothat all links experience independent Rayleigh fading, theclosed-form expressions for SINR coverage probability arederived.

Based on Total Probability Theorem, the SINR coverageprobability of typical SCBS is

Pc(T ) =∑

s∈{L,N}

Pr(SINR > T |α0 = αs

)Pr(α0 = αs) (11)

where Pr(SINR > T |α0 = αs) are the conditional probabili-ties on the event that the desired communication link is eitherLOS or NLOS, respectively. As discussed in Section II, theprobabilities of the desired link being LOS and NLOS can begiven by

Pr(α0 = αL) = e−βD0 and Pr(α0 = αN ) = 1− e−βD0 . (12)

Let us denote IΦ =∑i>0,i∈Φ

∑3j=1 PtGi,j |hi|2κD

−αii as

the total InterHI observed by the typical SCBS, which canbe easily known from (6). Then, the conditional probabilitiesPr(SINR > T |α0 = αl), s ∈ {L,N} can be written as

Pr(SINR > T |α0 = αs) =

∫ ∞0

Pr(|h0|2 > As(x+N0)

)fIΦ(x)dx,

(13)

which are obtained by assuming the condition: PtG0 >T∑2j=1 PtG0,j , otherwise, the probabilities in (13) become

zeros. Above, AL and AN can be given by

AL =TDαL

0

κ(PtG0 − T∑2j=1 PtG0,j)

,

AN =TDαN

0

κ(PtG0 − T∑2j=1 PtG0,j)

. (14)

In (13), fIΦ(x) represents the PDF of the InterHI for thetypical SCBS. Here, in order to derive the probabilities in(13), we introduce the following proposition.

Proposition 1: [11] The CDF of a Nakagami-m distributedrandom variable X with an integer parameter m can be closelyapproximated as

FX(x) ≈(1− e−ax

)m (15)

where a = m(m!)−1/m.Proof: See the proofs in [11, Lemma 6].

Therefore, with the aid of Proposition 1, we can rewrite theconditional probabilities Pr(SINR > T |α0 = αs) as

Pr(SINR > T |α0 = αs)

=1−∫ ∞

0

Pr(|h0|2 < As(x+N0)

)fIΦ(x)dx,

≈1−∫ ∞

0

(1− e−asAs(x+N0)

)mLfIΦ(x)dx. (16)

Upon applying some specific calculations on the equations,an exact expression of the coverage probability can be char-acterized in the following theorem.

Theorem 1: When assuming independent Nakagami-m fad-ing channels, the exact-form expression of the SINR coverageprobability Pc(T ) for the P2MP mmWave backhaul networksunder the GBN model can be obtained by (17).

Proof: See Appendix A.However, the exponent terms in (17) generally require

numerical evaluation of the integral. As shown in the above

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P (GBN)c (T ) ≈

mL∑r=1

(mL

r

)(−1)r+1e−raLALN0−βD0

4∏k=1

e−2πλpk

∫R0

(1−1/(1+

raLAL∑3j=1 PtQj,kκ

xαLmL)mL)e−βxxdx

4∏k=1

e−2πλpk

∫R0

(1−1/(1+


xαNmN)mN

)(1−e−βx)xdx

+

mN∑s=1

(mN

s

)(−1)s+1e−saNANN0(1− e−βD0)

4∏k=1

e−2πλpk

∫R0

(1−1/(1+

saNAN∑3j=1 PtQj,kκ

xαLmL)mL)e−βxxdx

4∏k=1

e−2πλpk

∫R0

(1−1/(1+


xαNmN)mN

)(1−e−βx)xdx (17)

equation, there are 32 number of the integral terms to computefor the best case of ML = MN = 1. As the values of ML

and MN get bigger, the number of the integral terms in theequation will significantly increases and, hence, it becomesmore challenging to analyze. To derive the accurate coverageprobability result based on (17), it demands extensive Monte-Carlo simulations. For this sake, we further derive the closed-form expression in the following theorem.

Theorem 2: When assuming independent Nakagami-m fad-ing channels, the closed-form expression of the SINR coverageprobability Pc(T ) for the P2MP mmWave backhaul networksunder the GBN model can be given by

P (GBN)c (T ) ≈mL∑r=1

(mL

r

)(−1)r+1e−raLALN0−βD0

4∏k=1

e−2πλpk(J1(k)+J2(k))

+

mN∑s=1

(mN

s

)(−1)s+1e−saNANN0(1− e−βD0)×

4∏k=1

e−2πλpk(J3(k)+J4(k)) (18)

where AL and AN are given by (14). In (18), J1(k)–J4(k)can be computed as

J1(k) ≈πR2U

U∑u=1

√1− θ2

uψue−βψu×

(1−

(1 +


ψαLu mL

)−mL), (19)

J2(k) ≈πR2U

U∑u=1

√1− θ2

uψu(1− e−βψu)×

(1−

(1 +


ψαNu mN

)−mN), (20)

J3(k) ≈πR2U

U∑u=1

√1− θ2

uψue−βψu×

(1−

(1 +


ψαLu mL

)−mL), (21)

J4(k) ≈πR2U

U∑u=1

√1− θ2

uψu(1− e−βψu)×

(1−

(1 +


ψαNu mN

)−mN), (22)

where aL = mL(mL!)−1/mL , aN = mN (mN !)−1/mN ,ψu = (θu+1)R

2 , and θu = cos( 2u−12U π) are the Gaussian-

Chebyshev nodes over interval [−1, 1]. The parameter U isGaussian-Chebyshev parameters, and it denotes the numberof series expansion terms for Gauss-Chebyshev quadrature.In addition, the parameter U relates to the trade-off betweenaccuracy and complexity for (18).

Proof: See Appendix B.To the best of our knowledge, stochastic geometry approach

based performance analysis on mmWave networks, the ex-isting literatures have derived the very complicated integralresults for coverage probability, which are unable to deriveuseful insights for theoretical studies and practical systemdesign. By contrast, in this paper, it is the first novel workthat derives the closed-form expressions of the SINR coverageprobability for the mmWave backhaul networks. In comparisonto the result in Theorem 1, the SINR coverage probabilityprovided by Theorem 2 significantly reduces the computa-tion complexity for evaluating the network performance. Theprobability result in Theorem 2 does not have the integralterms, and is easy to compute, due to the simple formswhich consist of power functions, fractional functions, andexponential functions.

In addition, according to Theorem 2, we are now able todeduce the following implications. In (18), the the first expo-nential terms, e−ALN0 and e−ANN0 , of the two summationsreflect the noise effects conditioned on the desired link isLOS and NLOS, respectively. Furthermore, observed from theproduct terms in (18), J1(k) and J3(k) correspond to the effectof the LOS interferences, while J2(k) and J4(k) refer to theeffect of the NLOS interferences. Note that, the interferenceeffects are evaluated by different antenna gains defined in (3).By comparing J1(k) (and J3(k)) with J2(k) (and J4(k)), wereadily find that, the LOS interferences have a much bigger ef-fect on the coverage probability than the NLOS interferences.When the LOS interference gets closer the typical SCBS, itcould be the dominant factor of the performance degradation.Furthermore, the accuracy of the closed-form expressions inTheorem 2 can be validated by comparing with the exact-form results in Theorem 1. Correspondingly, we derive thefollowing remark.

Remark 1: By employing the Gaussian-Chebyshev approxi-mation, the closed-form expression (18) for the coverage prob-ability generally provides a near-optimal approximation of theprobability in (17) when the Gaussian-Chebyshev parameterU is appropriately chosen such as U ≥ 10. Therefore, theanalytical results derived in (18) are more efficient to compute,

7

TABLE ICOMPARISON OF ANALYTICAL RESULTS BY THEOREM 1 AND THEOREM

2, WHERE D0 = 150, ML = 3 AND MN = 2.

SINR threshold (dB) T = −5 T = 5 T = 15 T = 25Pc(T ) in (17) 0.8038 0.7497 0.6762 0.3782

Pc(T ) in (18), U = 5 0.8244 0.7701 0.6968 0.3867Pc(T ) in (18), U = 8 0.8055 0.7594 0.6808 0.3798Pc(T ) in (18), U = 10 0.8046 0.7515 0.6801 0.3799Pc(T ) in (18), U = 15 0.8042 0.7507 0.6777 0.3791Pc(T ) in (18), U = 20 0.8039 0.7499 0.6767 0.3793Pc(T ) in (18), U = 25 0.8039 0.7498 0.6765 0.3787Pc(T ) in (18), U = 30 0.8039 0.7498 0.6764 0.3787

instead of numerical evaluation of the integral in (17).

Table I provides the numerical comparison of the coverageprobabilities characterized by the integral result of Theorem1 and the closed-form result of Theorem 2, where the mainparameters can be found in the beginning of Section VI. Notethat, all the values in Table I are analytical results, whichare all based on using the approximated Nakagami-m CDFintroduced in Proposition 1. Furthermore, the results of thesecond row in the table are obtained by Theorem 1 withoutusing Gaussian-Chebyshev approximation, while the resultsof the other rows are derived by Theorem 2 with employingGaussian-Chebyshev approximation. In order to show theapproximation tightness of Proposition 1, we compare theresults derived by Theorem 1 (i.e. the second row of TableI) with the Monte-Carlo simulation results in Fig. 2 whichcan be used to reflect the coverage probability based onthe exact Nakagami-m CDF. In Table I, it is observed thatthe approximation by Gaussian-Chebyshev quadrature withU ≥ 10 agrees well with the exact integral expression,thus validating the efficiency and accuracy of the closed-form expression. This can be understood by the fact that theapproximation of Gaussian-Chebyshev quadrature is shown tobe exact when using more points U [36]. Furthermore, wenote that the closed-form result slightly deviates from the exactintegral result when U is relatively small, such as U = 5. Wealso observe that the difference between our closed-form resultand the integral result becomes slightly more fluctuating whenthe target SINR increases. Therefore, the Gaussian-Chebyshevquadrature parameters should be properly selected to yield thenear-optimal approximation.

Known from (6), the SINR gets larger as the fading con-dition of interfering links becomes worse, while it does notmonotonically decrease as the fading condition of interferinglinks become worse due to the existence of the first term inthe denominator. Hence, it is worth investigating the SINRcoverage probability for the special scenario, when all thelinks including desired link and interfering links experienceindependent Rayleigh fading. The following corollary givesthe SINR coverage probability of the typical SCBS whenassuming the special case.

Corollary 1: When assuming independent Rayleigh fadingchannels, which corresponds to a rich scattering environment,the closed-form expression of the coverage probability Pc(T )

for the dipole model can be given by

P (GBN)c (T ) = e−ALN0−βD0

4∏k=1

e−2πλpk(J1(k)+J2(k))

+ e−ANN0(1− e−βD0)

4∏k=1

e−2πλpk(J3(k)+J4(k)),

(23)

where J1(k)–J4(k) can be found as

J1(k) ≈πR2U

U∑u=1

√1− θ2

uψue−βψu

×ALκ3∑j=1

PtQj,k/(ψu +ALκ

3∑j=1

PtQj,k), (24)

J2(k) ≈πR2U

U∑u=1

√1− θ2

uψu(1− e−βψu)

×ALκ3∑j=1

PtQj,k/(ψu +ALκ

3∑j=1

PtQj,k), (25)

J3(k) ≈πR2U

U∑u=1

√1− θ2

uψue−βψu

×ANκ3∑j=1

PtQj,k/(ψu +ANκ

3∑j=1

PtQj,k), (26)

J4(k) ≈πR2U

U∑u=1

√1− θ2


×ANκ3∑j=1

PtQj,k/(ψu +ANκ

3∑j=1

PtQj,k). (27)

In above, ψu = (θu+1)R2 and θu = cos( 2u−1

2U π), while AL,

AN can be found in (14). Note that, U is the parametersdenoting the number of series expansion terms for Gauss-Chebyshev quadrature.

Proof: See Appendix C.In Corollary 1, we readily obtain the observations similar

to those in Theorems 1 and 2. Known from (23)–(27), J1(k)and J2(k) reflect the LOS interference, while J3(k) and J4(k)reflect the NLOS interference. Interestingly, when D0 ≥ 10and αN − αL ≥ 2, we have that J1(k) >> J3(k), J2(k) >>J4(k) regardless of other parameters of interest. Combiningthis observation and those from Theorem 2, we are able todeduce the following lemma.

Lemma 1: Regardless of desired link being LOS or NLOS,the LOS interference is the dominant effect on the SINRcoverage probability for general P2MP mmWave backhaulnetworks.

IV. COVERAGE PROBABILITY ANALYSIS FOR SIMPLIFIEDP2MP

MMWAVE BACKHAUL NETWORKS

In this section, we specialize our analysis to the simplifiedP2MP mmWave backhaul network (SBN) model, in orderto characterize the key factors of performance degradation.Known from Lemma 1, the LOS interference has a much

8

larger affect on the network performance in comparison to theNLOS interference. Further, the studies in [9], [10] concludedthe simple fact that, the influence of path loss and LOSpath on the various performance metrics is found to bealmost deterministic in practical mmWave backhaul networks.Motivated by the above findings, our SBN model are proposedto ignore both the impact of the small-scale fading and theNLOS path, so that the performance analysis can be carriedout more concisely, while providing more useful insights fromboth theoretical and practical aspects.

Let us first introduce the system model of the simplifiedbackhaul networks and make the relating assumptions. Firstof all, we propose to simplify the analysis by approximatinga general LOS probability function pL(x) by a step function.In this regard, we define the function pL(x) as

pL(x) =

{1, for 0 < x < Rb

0, for x ≥ Rb(28)

where Rb is the radius of LOS circle. In general, the LOSprobability of a link is taken to be one within the circle areawith radius Rb, and the probability becomes zero outside theLOS circle. The above assumption allows us to investigate howthe LOS interference affects the performance of our mmWavebackhaul networks. In addition, such a simplification alsoenables us to efficiently analyze the ASE and EE trade-offin Section V. In the following, we state the above assumptionin detail.

Assumption 1: In the context of the SBN model, all theinterfering hubs in a circle area with a radius of Rb (whichsatisfies Rb ≤ R) have LOS links observed by the typicalSCBS. By contrast, the NLOS hubs outside the circle area areignored. This can be validated by the fact that the path loss ofNLOS component will be much more insignificant than thatof the LOS component, especially when the exponent αN forNLOS case is large [37], [38]. Therefore, the performance ofthe desired link is mainly limited by the LOS interferers.

Remark 2: Based on the above assumption, we readilyknow that the point process ΦL for the LOS interfering hubsis obtained by thinning the PPP Φ using the LOS probabilitygiven by (28). The average number of the LOS interferinghubs observed by the typical SCBS should be: NL = λπR2

b ,according to Cambell’s formula [31].

Furthermore, in the following, we introduce another impor-tant assumption for the mmWave backhaul networks under theSBN model.

Assumption 2: In the context of the SBN model, all commu-nication links are free of small-scale fading, since the signalpower from a mmWave LOS transmitter is found to be almostdeterministic in measurements [9].

Based on the above assumptions, the received SINR of thetypical SCBS under the simplified P2MP mmWave backhaulnetworks can be expressed as

SINR =

PtG0κD−αL0∑2

j=1 PtG0,jκD−αL0 +

∑i>0,i∈ΦL

∑3j=1 PtGi,jκD

−αLi +N0

.

(29)

We should note that, the analysis below is based on the factthat assuming D0 < Rb, which means that the desired link isguaranteed to be LOS link.

TABLE IIACCURACY VALIDATED BY USING DIFFERENT VALUES OF V , WHERE

D0 = 150 AND Rb = 450.

SINR threshold (dB) T = −5 T = 5 T = 15 T = 25Pc(T ) by simulations 0.9948 0.9423 0.7010 0.4802Pc(T ) in (30), V = 1 0.9739 0.8707 0.6583 0.3674Pc(T ) in (30), V = 3 0.9916 0.9197 0.6835 0.4320Pc(T ) in (30), V = 5 0.9934 0.9323 0.6896 0.4559Pc(T ) in (30), V = 8 0.9944 0.9410 0.6968 0.4753Pc(T ) in (30), V = 10 0.9947 0.9419 0.6998 0.4783

Let us now analyze the SINR coverage probability of thetypical SCBS for the P2MP mmWave backhaul networksunder the SBN case. Considering the specific assumptions 1and 2, we can derive the following theorem characterizing theSINR coverage probability.

Theorem 3: Given that all links experiencing no fading,the closed-form expression of the SINR coverage probabilityPc(T ) for the simplified P2MP mmWave backhaul networkscan be obtained by

P (SBN)c (T ) ≈

V∑v=1

(V

v

)(−1)v+1e−vηALN0

4∏k=1

e−2πλpkJ(k), (30)

where η = V (V !)−1/V . In (30), J(k) can be found by

J(k) =πRb2W

W∑w=1

√1− θ2

wψw

(1− e

−vηAL

∑3j=1 PtQj,kκ

ψαLw

), (31)

where θw = cos( 2w−12W π), and ψw = (θw+1)Rb

2 . The pa-rameters V and W are Gaussian-Chebyshev parameters, andthey denote the number of series expansion terms for Gauss-Chebyshev quadrature.

Proof: See Appendix D.Remark 3: It is important to point out that the closed-form

expression derived by Theorem 3 generally provides an exactlower bound of the coverage probability for the simplifiedbackhaul networks under the dipole scenario. The expressionin (30) gives a near-optimum approximation for the coverageprobability when more and more terms are employed, i.e. whenV ≥ 5.

In Table II, it provides the numeric comparison of thetheoretical analysis derived by Theorem 3 and the corre-sponding monte-carlo simulation results (obtained by 105

realizations). Observed from the table, the analytical resultsalways get closer to the simulation ones as the value of Vincreases. Furthermore, when the SINR threshold becomeslower, the analytical results have better approximation. Forvarious cases considered, the differences between the analysisand simulation are always ignorable, especially when V is big,such as V ≥ 5. Hence, the above observations from Table IIvalidate Remark 3.

V. EE AND ASE TRADE-OFF

Based on the analysis in Section III, we study the trade-offbetween the EE and ASE achieved by our P2MP mmWavebackhaul networks. For the sake of analytical tractable, weassume all the hubs have the same transmit power Pt for eachsector antenna. In this section, for theoretical study we aim tofind the best strategy to control transmit power, so that the EEand ASE trade-off can be maximized.

9

Known from the analysis in Section III, increasing thetransmit power Pt will increase the SINR coverage probabilityof the P2MP mmWave backhaul networks, while increasingthe total power consumption. Hence, to maximize the ASE,it is optimal to use the maximum transmit power availablefor each sector antenna of each hub. By contrast, the EEperformance is not monotonically increasing with the transmitpower Pt. Therefore, we motivate to maximize the EE andASE trade-off for the P2MP mmWave backhaul networks. Inthe following, we formulate the optimization problem P0 as

P0 : maxPt

ηEE (32)

subject to ηASE ≥ η(min)ASE (33)

λN(1

εPt + Pcon) ≤ P (max)

tot (34)

where ηASE and ηEE is obtained by substituting the SINRcoverage probabilities in Section III into (10) and (9). Inproblem P0, we define the final solution denoted by P ∗t .Constraint (33) assumes that the ASE of the network has tosatisfy the minimum required ASE which is η(min)

ASE . In (34),it constrains the maximum available power for the network tobe P (max)

tot .Problem P0 is a nonlinear fractional programming problem,

which is very difficult to solve. In order to find promisingsolution to problem P0, we propose to apply the dinkelbachmethod [39] to this problem. With the aid of introducing anew variable q, we can convert the original objective functionto

F (q) = B0ηASE(Pt)− qλN(1

εPt + Pcon), Pt ∈ A, (35)

where A = {Pt | ηASE ≥ η(min)ASE , λN( 1

εPt + Pcon) ≤P

(max)tot }. Known from [39], the optimal solution P ∗t can

be found by finding the optimal q∗ that satisfies F (q∗) =0. When updating variable q iteratively, such as qn+1 =B0ηASE(P ∗t,n)/λN( 1

εP∗t,n +Pcon) for the (n+ 1)th iteration,

F (q∗) will converge to 0 and the final solution of Pt can beobtained, which is proved in [39]. To solve problem P0, wecan iteratively solve the converted optimization problem of P0,given the variables Pt,n and qn for the nth iteration.

P1 : maxPt,n

F (qn, Pt,n) = B0ηASE(Pt,n)− qnλN(1

εPt,n + Pcon)

(36)subject to (33) and (34).

After converting to the optimization problem of P1, we canhave the following lemma.

Lemma 2: For our P2MP mmWave backhaul networks un-der the SBN, each sub-problem during an iteration under theconverted problem of P1 is a concave problem, and the valueof objective function F (qn, Pt,n) converges to 0.

Proof: See Appendix E.For the P2MP mmWave backhaul networks, we propose the

novel power control algorithm, given by Algorithm 1, whichaims to maximize the trade-off between the EE and ASE.

VI. NUMERICAL SIMULATION

In this section, we first show and discuss the numericalresults for the SINR coverage probability of both the GBNand SBN models. Then we evaluate the ASE performance ofthe backhaul networks. Finally, we present the EE and ASE

Algorithm 1:Algorithm 1:Algorithm 1: Proposed Power Control Algorithm1: Initialization: (1) Set index n = 1, (2) Set maximum

iteration number nmax, (3) Set ε > 0, (4) Set qn = 0;2: whilewhilewhile n <= nmax3: (A): Solve optimization problem P1(qn) by interior

point method;4: (B): Derive the solution of ith iteration: P ∗t,n;5: (C): Compute the value of the objective function

F (qn, P∗t,n);

6: ififif F (qn, P∗t,n) > ε thenthenthen

7: Update qn ← B0ηASE(P ∗t,n)/λ( 1εP ∗t,n + Pcon);

8: Set n← n+ 1;9: Go back to (A);10: elseelseelse11: Output:Output:Output: the optimal variable P ∗t = P ∗t,n;12: endendend13: endendend

TABLE IIISIMULATION PARAMETERS FOR MMWAVE BACKHAUL NETWORK.

Parameter Value Parameter Valuefc 94 GHz B0 500 MHzR 2 km σ2 -124 dBm/HzmL 3 mN 2αL 2 αN 4Mt 15 dB mt -15 dBMr 3 dB mr -3 dBλ 4× 10−6 hubs/m2 β 0.008

trade-off for our backhaul networks. Note that, we assumethat the mmWave backhaul networks operate at W band,and each of the two frequency bands f1 and f2 has theavailable bandwidth of B0. For theoretical study, we assumethat the noise power of each link is assumed to be the same,which is σ2 (dBm/Hz). Further, each link in the network havethe independent probability of being LOS and NLOS, andexperience independent fading. In the following figures, themain simulation parameters are summarized in Table III ifthey are not specified.

A. SINR Coverage Probability Performance

In this section, we provide the numerical results to validatethe SINR coverage probability derived for the GBN andSBN models in Sections III and IV, respectively. Further,when considering the various simulation scenarios, we com-prehensively evaluate the SINR coverage probability of ourmmWave networks, and discuss the implications on practicalnetwork design. Note that, we assume that, for the GBNmodel, the communication links experience either independentNakagami-m fading or independent Rayleigh fading. To guar-antee high-accuracy, all the Monte-Carlo simulation results areobtained from at least 105 times of realizations.

Fig. 2 compares the analytical SINR coverage probabilityresults with those obtained by Monte-Carlo simulations forthe backhaul networks under the GBN. Seen from the figure,our analytical results perfectly match the simulations under thevarious scenarios considered, which proves that the analyticalSINR distribution for the GBN model is very accurate. InFig. 2, when the desired link length, i.e. that between thetypical SCBS and the serving hub, becomes bigger, the SINR

10

-5 0 5 10 15 20 25

SINR Threshold (dB)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9S

INR

Co

ve

rag

e P

rob

ab

ility

General mmWave Backhaul NetworksNaka,analytic

Rayl,analytic

Naka,D0=100,sim.

Naka,D0=150,sim.

Naka,D0=200,sim.

Rayl,D0=100,sim.

Rayl,D0=150,sim.

Rayl,D0=200,sim.

-5 -4 -3 -2 -1

0.75

0.8

0.85

0.9

Fig. 2. Coverage probability against SINR threshold T for the mmWaveP2MP backhaul networks, validation of analytical results for the GBN modelwhen all the links experience Nakagami-m fading and Rayleigh fadingrespectively.

coverage probability decreases, and dramatically drops downin high SINR threshold region. This observation shows thequality of the mmWave backhaul links high depends on thelink length. For supporting relatively long distance trans-mission, it demands high-quality practical implementation,including optimization of hub location, proper antenna arraysetup, optimization of available radio resources. Further, thefigure also shows that, for medium and high SINR thresholds,the coverage probability is lower bounded by the specificscenario that all links experience independent Rayleigh fading.By contrast, observed from the enlarged subplot of Fig. 2,when the SINR threshold is low and distance D0 increases,the coverage probability for Rayleigh fading becomes slightlyhigher than that for Nakagami-m fading. Furthermore, thetightness of Proportion 1 can be evaluated by observing theresults in Table I and Fig. 2. Seen from the second row ofTable I, we readily know the coverage probability Pc(T ) inTheorem 1, which employs Proportion 1 while without usingthe Gaussian-Chebyshev approximation. When comparing theresults of Table I with the corresponding ones of Fig. 2, it findsthat there are only up to 0.13% deviation for −5 ≤ T ≤ 10dBregion and up to 0.20% deviation for 10 < T ≤ 25dBregion, respectively. Hence, this observation shows that theapproximation introduced by Proportion 1 is relatively tight.From the above observations, we imply that the link reliabilityof the mmWave backhaul network will have more significanteffect by fast fading as the link of the communication distanceincreases.

Next, we validate the analytical results of the SINR coverageprobability for the network under the SBN model in Fig. 3.According to the comparison, we prove that, the analyticalresults perfectly match the simulations under most cases.Nevertheless, there is only a small deviation (less than 6%)between the analytical results and the simulations, when theradius Rb is small. This is because our analysis slightly over-estimates interference effect on the reliability of the backhaullinks under the simplified network model. It is worth noting

SINR Threshold (dB)-5 0 5 10 15 20 25

SIN

R C

overa

ge P

robabili

ty

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Simplified mmWave Backhaul Networks (SBN)

analyticD

0=100,R

b=300,sim.

D0=150,R

b=300,sim.

D0=150,R

b=450,sim.

D0=150,R

b=600,sim.

D0=200,R

b=450,sim.

Fig. 3. Coverage probability against SINR threshold T for mmWave P2MPbackhaul networks, validation of analytical results for the SBN model withvarious Rb values, illustration for the effect of LOS interference on the linkreliability.

the following important observations when comparing Fig. 3with Fig. 2. First, the networks under the SBN always achievea higher SINR coverage probability than the networks underthe GBN when the target SINR is small. This implies that, thefast fading and NLOS interference will have a big impact onthe network performance when link reliability requirement ishigh. Second, we observe that, as the length of the desiredlink increases, the performance gap between the SBN andGBN at the low SINR threshold region becomes bigger. Thisobservation indicates that, the effect of NLOS interference cannot be ignored especially when the length of communicationlink is long. Third, the results for the SBN can match withthose for the GBN model in high SINR threshold region whenthe radius Rb is properly selected, such as Rb = 450 forD0 = 150. Also, the SINR coverage probability significantlyreduces as radius Rb increases. From the above observations,we may conclude the reliability of backhaul link is heavilyaffected by neighbouring LOS interfering links especiallywhen high target SINR is required.

In Fig. 4, we evaluate the SINR coverage probability againstintensity value λ for the P2MP mmWave backhaul networks.Once again, the figure shows the analytical results of the SINRcoverage probability match with the simulations regardless ofnetwork density. Observed from the figure, the reliability ofmmWave backhaul links can be highly affected by interfer-ence, and the network exhibits strong interference, when thedensity of hubs increases. In Fig. 4, under the GBN model,the performance of Rayleigh fading is approaching that of theNakagami-m when the network density increases. This impliesour backhaul networks with relatively high density can beevaluated by ignoring small-scale fading effect. By comparingthe results of the GBN with those of the SBN, it derivesthe observations and conclusions same as those made fromFig. 3. Furthermore, Fig. 5 is shown in order to investigatehow the network performance is affected by the network size.Observed from the figure, the coverage probability for the caseD0 = 100 will be decreased by less than 1% when the radius

11

10-6 10-5 10-4

(hubs/m2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1S

INR

Co

ve

rag

e P

rob

ab

ility

mmWave Backhaul Networks, D0=150, T=15 dB

GBN,naka,analytic

GBN,rayl,analytic

SBN,analytic

naka,sim.

rayl,sim.

Rb=300,sim.

Rb=450,sim.

Rb=600,sim.

Fig. 4. Coverage probability against intensity λ for mmWave backhaulnetworks, validation of analytical results, performance comparison betweenthe GBN model and SBN model, the networks trend to be interference limitedas hub density increases.

-5 0 5 10 15 20 25

SINR Threshold (dB)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

SIN

R C

ove

rag

e P

rob

ab

ility

General mmWave Backhaul Networks

D0=100

D0=150

R=500

R=800

R=1500

R=2000

R=2500

Fig. 5. Coverage probability against SINR threshold T for mmWave P2MPbackhaul networks, evaluation of network size R, the interference outside0.8km can be ignored.

R is bigger than 0.8km, while that for the case D0 = 150will deviate up to 3% only at high SINR threshold regionwhen R ≥ 0.8km. Moreover, it is also seen that, the curveswill become almost the same for the cases of R ≥ 2km. Theabove observations imply that, the interference outside 0.8kmis very weak, and the network size of R ≥ 2km is sufficientlylarge for theoretical study, so that the boundary effect can beignored.

According to the observations in this section, we concludethat, in contrast to mmWave access networks, the reliabilityof our backhaul mmWave links is more likely dominatedby interference (including both LOS and NLOS interference)rather than thermal noise especially when the link lengthbecomes larger. Owing to this, we urge caution that for theargument that mmWave networks are all noise limited.

B. ASE, EE and Rate Performance

In this section, we investigate and evaluate the ASE andrate performance of the P2MP mmWave backhaul networksunder both the GBN and SBN models. A range of numericalresults are provided to validate the advantages of the P2MParchitecture employed by our networks over the traditionalP2P approach used by the existing networks. To make afair comparison, we assume the total transmit power and theother power consumption at each hub is the same for boththe P2MP and P2P cases. Furthermore, we investigate theperformance gain achieved by our proposed networks withmmWave capability over the ones with traditional UHF linksonly.

Fig. 6 compares the ASE performance of the P2MP ap-proach aided mmWave backhaul networks with that of theP2P approach aided networks, when varying the values ofSINR threshold and intensity λ, respectively. Note that, in thefigure we assume all links experience independent Nakagami-m fading in the context of GBN model. In Fig. 6(a), itclearly observes that the novel P2MP approach can achieve asignificantly higher ASE, i.e. average network capacity, thanP2P approach for all different SINR thresholds, despite theP2P approach is free of InterHI. This performance gain, thatis up to 800%, is due to the fact that multiple SCBSs areserved by a hub during each time slot. From Fig. 6(a), italso observes that, for the P2MP mmWave backhaul networks,the optimum ASE is achieved at lower SINR threshold whenhubs are located more closer to their SCBSs. Furthermore,for the SBN model, the optimum ASE becomes smaller, andit is obtained at lower SINR threshold as Rb decreases. Incontrast, Fig. 6(b) shows the effect of intensity λ on theASE performance of the backhaul networks, and demonstratesthe trade-off between ASE and link reliability (reflected bySINR coverage probability). It is clearly shown that the P2MPapproach can significantly outperform the P2P approach, interms of the optimum ASE. According to comparison, theoptimum ASE for the SBN always exists at less dense scenariothan that for the GBN. This implies that network capacity arehighly affected by strong interference from neighbouring LOSlinks. Therefore, from Fig. 6, we may conclude that, to designpractical mmWave backhaul networks, it is very important tofind the best trade-off between network capacity required andlink reliability target, which demands efficiently mitigatinginterference, carefully managing resource, and optimal hubsdeployment.

Fig. 7 investigates the EE performance of the mmWavebackhaul networks with the aid of the P2MP approach as wellas the networks with the traditional P2P approach. Note that,for ensuring a fair comparison, we assume that the total powerconsumed by each hub is the same for both the P2MP andP2P approaches. Regardless of the GBN and SBN models, weclearly observe that the P2MP approach derives significantlybetter EE performance than the P2P, especially when thelength of the typical link D0 is small. Further, we can seethat, there is also a trade-off between the achievable EE andthe link reliability. Obviously, to achieve the best trade-off, thenovel P2MP approach requires much smaller link reliability

12

-5 0 5 10 15 20 25

SINR Threshold (dB)

0

0.4

0.8

1.2

1.6

2

2.4

2.8A

SE

(b

ps/H

z/m

2)

10-4 mmWave Backhaul Networks

P2MP,GBN,naka,D0=100



P2MP,GBN,rayl

P2MP,SBN,D0=150,R

b=300

P2MP,SBN,D0=150,R

b=600

P2P,GBN,naka

P2P,SBN,D0=150,R

b=300

(a) ASE vs T

10-6 10-5 10-40

0.5

1

1.5

2

2.5

AS

E (

bp

s/H

z/m

2)

10-4 mmWave Backhaul Networks, D0=150, T=15dB

P2MP,GBN,naka

P2MP,SBN,Rb=300

P2MP,SBN,Rb=600

P2P,GBN,naka

P2P,SBN,Rb=300

P2P,SBN,Rb=600

(b) ASE vs λ

Fig. 6. Trade-off between ASE and SINR threshold T as well as that between ASE and intensity λ for mmWave backhaul networks, ASE performanceadvantage of novel P2MP over traditional P2P when assuming the same total power consumption.

-5 0 5 10 15 20 25

SINR Threshold (dB)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

EE

(b

ps/W

/m2)

109 mmWave Backhaul Networks

P2MP,GBN,D0=100

P2MP,GBN,D0=150

P2P,GBN,D0=100

P2P,GBN,D0=150

P2MP,SBN,D0=100,R

b=300

P2MP,SBN,D0=100,R

b=600

P2P,SBN,D0=100,R

b=300

P2P,SBN,D0=100,R

b=600

Fig. 7. Trade-off between EE and SINR threshold T for mmWave backhaulnetworks, EE performance advantage of novel P2MP technique over tradi-tional P2P one when assuming the same total power consumption, illustrationfor the effect of LOS interference on EE performance.

than the P2P approach, which enables a lower complexity ofreceiver design for decoding. The other observations in Fig. 7are similar to those in Fig. 6, and similar conclusions can bederived. Therefore, we may conclude that, our P2MP aidedmmWave backhaul networks not only achieve significantlyhigher ASE, but also obtain much better EE, compared tothe networks with the P2P approach.

Next, we compare the rate coverage probability of ourmmWave backhaul networks with that of UHF networks inFig. 8. Note that we assume the UHF network is operatedat 2.4 GHz with B0 = 50 MHz available. For the sake oftheoretical study, we further assume that the UHF networkscan achieve the same antenna gains as those of the mmWavenetworks at both hubs and SCBSs. To capture the effectof LOS and NLOS transmissions, we use αL = 2 andαN = 3.75 for the UHF networks. Observed from Fig. 8,the rate performance of backhaul networks with mmWavecapability is significantly improved in comparison with the

106 107 108 109

Rate requirement (bps)

0.4

0.5

0.6

0.7

0.8

0.9

1

Ra

te c

ove

rag

e p

rob

ab

ility

Backhaul Networks

mmWave,GBN,D0=100

mmWave,GBN,D0=150

mmWave,GBN,D0=200

mmWave,SBN,D0=150,R

b=300

mmWave,SBN,D0=150,R

b=600

UHF,GBN

Fig. 8. Rate coverage probability against minimum rate requirementfor backhaul networks, rate performance gain achieved by mmWave aidednetworks over UHF based networks, rate performance comparison betweenthe GBN model and SBN model.

traditional UHF networks, which is benefited from largebandwidth provided by mmWave communication. Also, wecan clearly see the mmWave networks are able to supportover several Gbps data links when assuming unit transmitpower. Furthermore, as known, the achievable antenna gain isrelated to G = 4πAe/λ

2f , where Ae is the aperture of antenna.

Hence, the achievable antenna gain will be increased by 20 dBwhen increasing carrier frequency λ2

f ten-fold [13]. In caseof operating at low gain, the antenna size for P2MP can befurther reduced, with very low footprint. To maintain the samesize of antenna aperture, mmWave backhaul networks willobtain more orders of magnitude rate improvement over UHFbackhaul networks. Further, by comparing the results of GBNmodel with those of the SBN, we observe that mmWave back-haul communications exhibit interference limited performanceif the neighbouring LOS interfering links are not properlymanaged.

13

5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7x 10

9 mmWave Backhaul Networks, EE and ASE trade−off, T=15dB

Power (dBm) for minimum ASE requirement

Opt

imum

EE

(bp

s/W

/m2 )

Exhaustive SearchProposed,D

0=100,R

b=300

Proposed,D0=150,R

b=300

Proposed,D0=150,R

b=450

Proposed,D0=150,R

b=600

Proposed,D0=200,R

b=450

(a) Optimum EE vs η(min)ASE

−5 0 5 10 15 20 250

1

2

3

4

5

6

7x 10

9 mmWave Backhaul Networks

SINR Threshold (dB)

Optim

um

EE

(bps/W

/m2)

D0=100,R

b=300

D0=100,R

b=600

D0=150,R

b=300

D0=300,R

b=600

D0=200,R

b=300

D0=200,R

b=600

(b) Optimum EE vs T

Fig. 9. optimum EE performance against minimum ASE requirement for mmWave backhaul networks, performance trade-off between between optimum EEand SINR threshold T , validation of proposed power control algorithm.

C. Energy Efficiency and ASE Trade-off

In this section, we investigate the trade-off between EE andASE for the mmWave backhaul networks. Due to the lack ofspace, we focus on the performance evaluation for the SBNmodel. The similar observations can be obtained for the GBNmodel.

Fig. 9 invesigates the performance trade-off between EEand ASE, as well as shows the trade-off between EE and linkreliability. In the figure, the optimum EE is derived by solvingthe optimization problem in P0. In Fig. 9 (a), it evaluatesthe optimum EE performance subject to various minimumASE, which, known by (52), is monotonically increasing withtransmit power Pt. From the figure, we can clearly observethe results obtained by the proposed approach agree withthose obtained by the exhaustive search, which validates high-efficiency of our method given in Algorithm 1. Further, therealways exits a trade-off between the optimum EE and ASErequirement for each case, and the best trade-off can enablehigher ASE requirement when D0 decreases or when theradius of LOS circle Rb becomes smaller. This observationimplies that, the mmWave backhaul networks become lessenergy efficient demanding more transmit power consumptionwhen the required network capacity gets higher. In Fig. 9(b), we study the effect of SINR threshold on the EE perfor-mance when fixing the minimum ASE η

(min)ASE = 0.4 × 10−4

bps/Hz/m2. Observed from Fig. 9 (b), for each scenario thereis also a trade-off between EE and link reliability (i.e. SINRcoverage probability). Moreover, it is seen that the peak EEincreases as the length of the desired link D0 gets smallerand as the radius of LOS circle Rb decreases. Whereas,in order to achieve the peak EE, it demands higher linkreliability when Rb becomes smaller. This observation reflectsthe fact that, the EE performance of the mmWave backhaulnetworks can be significantly degraded by low link reliabilityand strong neighbouring LOS interfering links. From the aboveobservations, we can conclude that, to achieve the best EEand ASE trade-off, the mmWave backhaul networks should

be designed to limit the communication links, and avoid LOSinterferences, while optimizing transmission power.

VII. CONCLUSION

We have proposed and developed the novel P2MP mmWavebackhaul networks, which are modeled by the tractablestochastic geometry approach for performance analysis. Tocomprehensively evaluate the network performance, we haveproposed two different models including the GBN and SBN.For both the models, a range of exact- and closed-form expres-sions for the SINR coverage probability have been derived.With the help of the stochastic geometry based tractablemodel, we have developed the optimal power control algorithmfor maximizing the trade-off between EE and ASE perfor-mance of our backhaul networks. The analytical results ofthe SINR coverage probability have been validated by Monte-Carlo simulations. Both the analytical and simulation resultshave shown that the LOS interference should be avoided ormitigated for our backhaul network to achieve a good cov-erage probability. The simulation results have shown that, toachieve the best EE and ASE trade-off, the mmWave backhaulnetworks should be designed to limit the link distances andLOS interferences, while optimizing the transmission power.Furthermore, according to the performance evaluation, weconclude that our proposed P2MP mmWave backhaul is apromising solution for future wireless systems.

APPENDIX A: PROOF TO THEOREM 1

In order to derive the exact expression of the coverage prob-ability, we first give the details of derivations for Pr(SINR >T |α0 = αL). As known, the binomial theorem is given by(x− y)K =

∑Ki=0

(Ki

)(−1)(K−i)xiyK−i.

Upon applying the binomial theorem to (16), Pr(SINR >T |α0 = αL) can be rewritten as

14

Pr(SINR > T |α0 = αL)

=

mL∑r=1

(mL

r

)(−1)r+1EIΦ

[e−raLAL(IΦ+N0)

](a)=

mL∑r=1

(mL

r

)(−1)r+1e−raLALN0EIΦL

[e−raLALIΦL

]× EIΦN

[e−raLALIΦN

](37)

where step (a) is due to the fact that the effect of potentialblockages is to thin the original PPP IΦ for interfere hubsinto two independent PPPs IΦL and IΦN , respectively. Notethat, EIΦL

[e−raLALIΦL

]for the LOS interference is Laplace

functional of IΦL [31], given by

EIΦL

[e−raLALIΦL

]= E|hi,j |2

[EDi

[EGi,j

[e−raLALIΦL

]]](b)= EDi

4∑k=1

pkE|hi,j |2

e− raLAL∑i>0,i∈ΦL

∑3j=1 PtQi,j,kκ|hi,j |

2

DαLi

(c)= e

−λ∑4k=1 pk

∫ 2π0

∫R0

(1−E|h|2

[e−raLAL

∑3j=1 PtQj,kκ|h|

2

xαL])pL(x)xdxdθ

(d)=

4∏k=1

e−2πλpk

∫R0

(1−E|h|2

[e−raLAL

∑3j=1 PtQj,kκ|h|

2

xαL])e−βxxdx

(e)=

4∏k=1

e−2πλpk

∫R0

(1−1/(1+


xαLmL)mL

)e−βxxdx

.

(38)

In (38), step (b) follows the total probability theorem, whereGi,j = Qi,j,k given by (3). Further, step (c) is obtained bycomputing the Laplace transform function of the PPP ΦL. Weshould note that, in steps (c)-(e), we have Qj,k = Qi,j,k bydropping subscript i for the sake of notational convenience.Moreover, step (d) is derived with the aid of using the polarcoordinates in our stochastic geometry based network model.At last, step (e) can be obtained by using the probabilitygenerating functional [31].

Similar to the derivations in (38), we can obtain the ex-pression of EIΦN

[e−raNANIΦN

]for NLOS interfering links.

Upon substituting the expression of (38) for LOS case and theone for NLOS case into (37), we can obtain

Pr(SINR > T |α0 = αL) =

mL∑r=1

(mL

r

)(−1)r+1×

4∏k=1

e−2πλpk

∫R0

(1−1/(1+


xαLmL)mL

)e−βxxdx

e−raLALN0

×4∏k=1

e−2πλpk

∫R0

(1−1/(1+


xαNmN)mN

)(1−e−βx)xdx

.

(39)

Applying the procedures similar to (37)-(39), we areable to derive the expression for the conditional probabilityPr(SINR > T |α0 = αN ) as

Pr(SINR > T |α0 = αN ) =

mN∑s=1

(mN

r

)(−1)s+1×

4∏k=1

e−2πλpk

∫R0

(1−1/(1+


xαLmL)mL

)e−βxxdx

e−saNANN0

×4∏k=1

e−2πλpk

∫R0

(1−1/(1+


xαNmN)mN

)(1−e−βx)xdx

.

(40)Therefore, Theorem 1 is proved when substituting (39), (40)

and (12) into (11).

APPENDIX B: PROOF TO THEOREM 2Let us first discuss the derivation procedures for the con-

ditional probability Pr(SINR > T |α0 = αL), which can berewritten as


mL∑r=1

(mL

r

)(−1)r+1e−raLALN0

×4∏k=1

e−2πλpkJ1(k)4∏k=1

e−2πλpkJ2(k). (41)

In (41), J1(k) and J2(k) are given by

J1(k) =

∫ R

0

(1− 1/(1 +


xαLmL)mL

)e−βxxdx ,

(42)

J2(k) =

∫ R

0

(1− 1/(1 +


xαNmN)mN

)(1− e−βx)xdx .

(43)

In order to derive the expression for the SINR coverageprobability in Theorem 2, we need to approximate the integralterms in (42) by closed-from expressions. To simplify theanalysis, here we use the Gaussian-Chebyshev quadrature tofind the closed-form expressions. As known, the Gaussian-Chebyshev quadrature can be expressed by [35, eq. (25.4.38)]∫ 1

−1

f(x)(1−x2)−1/2dx =π

K

K∑i=1

f(xi) +π

22K−1

f (2K)(ξ)

(2K)!(44)

for some ξ ∈ [−1, 1], with quadrature nodes: xi =cos( 2i−1

2K π), i = 1, · · · ,K.When employing the above approximation, we can replace

variable x in (44) by (ψu+1)R2 , and then multiply the weight

coefficient (1− ψu)−1/2. Hence, J1(k) is approximated by

J1(k) ≈πR2U

U∑u=1

√1− θ2

uψue−βψu

×(

1−(

1 +raLAL

∑3j=1 PtQj,kκ

ψαLu mL

)−mL). (45)

Similarly, we can obtain an approximated expression forJ2(k) as

J2(k) ≈πR2U

U∑u=1

√1− θ2


×(

1−(

1 +raLAL

∑3j=1 PtQj,kκ

ψαNu mN

)−mN). (46)

By substituting (45), (46) into (41), a closed-form ofPr(SINR > T |α0 = αL) is obtained.

Given the desired link is NLOS, the conditional probabilityPr(SINR > T |α0 = αN ) can be derived similar to that

15

for (41). Therefore, combining the above results with (11),and after some algebraic simplifications, we prove Theorem 2straightforwardly.

APPENDIX C: PROOF TO COROLLARY 1

The SINR coverage probability of the typical SCBS canbe evaluated by (11), which needs to derive the expressionsfor Pr(SINR > T |α0 = αL) and Pr(SINR > T |α0 = αN ),conditioned on the desired link is either LOS or NLOS.

When considering Rayleigh fading scenario, we can applythe CDF of FY (y) = 1−e−y (for Rayleigh distribution) to thederivation of the probabilities Pr(SINR > T |α0 = αL) andPr(SINR > T |α0 = αN ), and follow the approach similarto that in Appendix A. Then, upon employing the Gaussian-Chebyshev approximation used in Appendix B, the closed-form expressions for the probabilities can be obtained. In alittle more detail, the derivation for the probability Pr(SINR >T |α0 = αL) can be described as


∫ ∞0

(|h0|2 > AL(x+N0)

)fIΦ(x)dx

= e−aLALN0EIΦL

[e−aLALIΦL

]EIΦN

[e−aLALIΦN

]= e−aLALN0

4∏k=1

e−2πλpk∫R0

(1−1/(1+

aLAL∑3j=1 PtQj,kκ

xαL))e−βxxdx

×4∏k=1

e−2πλpk∫R0

(1−1/(1+

aLAL∑3j=1 PtQj,kκ

xαN))

(1−e−βx)xdx

= e−aLALN0

4∏k=1

e−2πλpk(J1(k)+J2(k)). (47)

where J1(k) and J2(k) are expressed by (24) and (25).Similarly, the closed-form expression for the probabilityPr(SINR > T |α0 = αN ) can be given by

Pr(SINR > T |α0 = αN )

=

∫ ∞0

Pr(|h0|2 > AN (x+N0)

)fIΦ(x)dx

= e−aNANN0

4∏k=1

e−2πλpk(J3(k)+J4(k)). (48)

where J3(k) and J4(k) can be found in (26) and (27). Finally,when substituting the results of (47) and (48) into (11) basedon total probability theorem, Corollary 1 is proved.

APPENDIX D: PROOF TO THEOREM 3Using the SINR expression for the SBN model given by

(29), the SINR coverage probability can be computed as

Pc(T ) = Pr(κPtD

−αL0 (G0 − T

2∑j=1

G0,j) > T (IΦL +N0))

(a)≈ Pr

(ζ > AL(IΦL +N0)

)(b)≈ 1− EIΦL

[(1− e−ηAL(IΦL

+N0))N]=

V∑v=1

(V

v

)(−1)v+1e−vηALN0EIΦL

[e−vηALIΦL

](49)

where IΦL =∑i>0,i∈ΦL

∑3j=1 PtGi,jκD

−αLi is the interfer-

ence power observed by the typical SCBC. In (a), the dummyvariable ζ is a normalized gamma variable with parameterV , and the approximation in (a) follows from the fact that a

normalized Gamma distribution converges to identity when itsparameter goes to infinity, i.e. limv→∞

vvxv−1e−vx

Γ(v) = δ(x−1),where δ(x) is the Dirac delta function. The results in Theorem3 generally provide a close approximation of Pc(T ) whenenough terms are used, as this can be validated in remark3.

Next, we focus on the computation of EIΦL[e−vηALIΦL

],

given byEIΦL

[e−vηALIΦL

](c)= e

−λ∑4k=1 pk

∫ 2π0

∫Rb0

(1−e−vηAL

∑3j=1 PtQj,kκ

xαL

)xdxdθ

=

4∏k=1

e−2πλpk

∫Rb0

(1−e−vηAL

∑3j=1 PtQj,kκ

xαL

)xdx

(d)≈

4∏k=1

e−2πλpkJ(k) (50)

where step (c) is due to computing the Laplace functionalof the PPP IΦL , and step (d) is obtained by applying theGaussian-Chebyshev quadrature, and similar derivation pro-cedures can be found in Appendix B. In addition, J(k) isgiven by

J(k) =πRb2W

W∑w=1

√1− θ2

wψw

(1− e

−vηAL

∑3j=1 PtQj,kκ

ψαLw

)(51)

where θw = cos( 2w−12W π), ψw = (θw+1)Rb

2 , and W denotesan accuracy-complexity trade-off parameter. Combining (51)with (50), Theorem 3 is proved straightforwardly.

APPENDIX E: PROOF TO LEMMA 2According to the analysis in Section III, we can summarize

the expressions for the SINR coverage probabilities of the SBNmodel, and rewrite the ASE as

ηASE(Pt,n) =λN log2(1 + T )

[Z∑r=1

Xre− ϕrPt,n(G0−T

∑2j=1

G0,j)

](52)

where the expressions for ϕr, Xr, Θr can be easily knownfrom (30) in Theorems 3. To guarantee the problem of P1

concave, it needs to prove that the objective function and theinequality constraint functions are all concave.

Let us now prove that the objective function F (qn, Pt,n) ofproblem P1 is concave. Note that, we can rewrite the objectivefunction as F (qn, Pt,n) = F1−F2, where F2 is linear functionknown from (36). Hence, we only need to prove that F1 isconcave. After some calculations, the second derivative of F1can be given by

d2F1

dP 2t,n

=λN log2(1 + T )

P 4t,n(G0 − T

∑2j=1 G0,j)2

Z∑r=1

Xrφre− ϕrPt,n(G0−T

∑2j=1

G0,j)

×

[φr − 2Pt,i(G0 − T

2∑j=1

G0,j)

]

=λN log2(1 + T )

P 4t,n(G0 − T

∑2j=1 G0,j)2

Z∑r=1

f(r) . (53)

It readily finds the first fraction part in (53) is non-negative.We find that f(r = 1) < 0 & f(r = 1) << f(r ≥ 1),∀r ≤ Z,when applying the parameters in Section VI to function f(r).Hence, the second derivative of F1 is non-positive and hence,the objective function F (qi, Pt,i) is concave.

16

Similarly, when substituting (52) into the constraint of(33), we derive the expressions for the constraint, whichis concave. In addition, constraint (34) is a linear function.Hence, the optimization problem in P0 is a concave problem.Furthermore, the objective function F (qi, Pt,i) for problem P1

will converge to 0, and the proof for this convergence is in[39].

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17

Jia Shi Dr. Jia Shi received both his MSc. andPh.D degrees from University of Southampton, UK,in 2010 and 2015, respectively. He was a researchassociate with Lancaster University, UK, during2015-2017. Then, he became a research fellow with5GIC, University of Surrey, UK, from 2017 to 2018.Since 2018, he joined Xidian University, China, andnow is an Associate Professor in State Key Lab.of Integrated Services Networks (ISN). His currentresearch interests include mmWave communications,artificial intelligence, resource allocation in wireless

systems, covert communications, physical layer security, cooperative commu-nication, etc.

Lu Lv received the B.Eng degree in Telecommu-nications Engineering from Hebei Normal Univer-sity in 2013. He is currently working toward thePh.D. degree in Telecommunications Engineering atXidian University. From March 2016 to September2016, he was a visiting Ph.D. student at LancasterUniversity, UK. Since December 2016, he has beenwith the University of Alberta, Canada, as a jointPh.D. student sponsored by the China ScholarshipCouncil (CSC). His current research interests focuson cooperative relaying, physical layer security, non-

orthogonal multiple access, and mmWave communications.

Qiang Ni (M’04–SM’08) received the B.Sc., M.Sc.,and Ph.D. degrees from the Huazhong University ofScience and Technology, China, all in engineering.He is currently a Professor and the Head of theCommunication Systems Group, School of Comput-ing and Communications, Lancaster University, Lan-caster, U.K. His research interests include the area offuture generation communications and networking,including green communications and networking,millimeter-wave wireless communications, cognitiveradio network systems, non-orthogonal multiple ac-

cess (NOMA), heterogeneous networks, 5G and 6G, SDN, cloud networks,energy harvesting, wireless information and power transfer, IoTs, cyber phys-ical systems, machine learning, big data analytics, and vehicular networks. Hehas authored or co-authored over 200 papers in these areas. He was an IEEE802.11 Wireless Standard Working Group Voting Member and a contributorto the IEEE Wireless Standards.

Haris Pervaiz (S’09-M’09) received the M.Sc. de-gree in information security from the Royal Hol-loway University of London, Egham, U.K., in 2005,and the Ph.D. degree from the School of Com-puting and Communication, Lancaster University,Lancaster, U.K., in 2016. Currently, he is workingas an Assistant Professor (Lecturer) at School ofComputing and Communications (SCC), LancasterUniversity, UK. From 2017 to 2018, he was aResearch Fellow with the 5G Innovation Centre,University of Surrey, Guildford, U.K. From 2016 to

2017, he was an EPSRC Doctoral Prize Fellow with the School of Computingand Communication, Lancaster University. His current research interests in-clude green heterogeneous wireless communications and networking, 5G andbeyond, millimeter wave communication, and energy and spectral efficiency.Dr. Pervaiz is an Associate Editor of the IEEE Access, an Associate Editorof Internet Technology Letters (Wiley) and Editorial board of EmergingTransactions on Telecommunications Technologies (Wiley).

Claudio Paoloni received the degree cum Laudein Electronic Engineering from the University ofRome “Sapienza”, Rome, Italy, in 1984. Since 2012,he has been a Professor of Electronics with theDepartment of Engineering, Lancaster University,Lancaster, U.K. Since 2015, he has been the Head ofEngineering Department. His research is focussed onmicrowaves and millimetre waves technologies forvacuum electronics and wireless communications.He is author of more than 200 publications. He iscoordinator of the European Commission Horizon

2020 project TWEETHER “Traveling wave tube based W-band wirelessnetwork for high data rate, distribution, spectrum and energy efficiency” and ofthe Horizon 2020 ULTRAWAVE project “Ultra-capacity wireless layer beyond100 GHz based on millimeter wave Traveling Wave Tubes” for millimetrewaves high capacity wireless networks. He is IEEE Senior Member. He isChair of the IEEE Electron Device Society Vacuum Electronics TechnicalCommittee.

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