The Meta Distributions of the SIR/SNR and Data Rate in Coexisting Sub-6GHz and Millimeter-wave Cellular Networks
aa r X i v : . [ ee ss . SP ] D ec The Meta Distributions of the
SIR / SNR and DataRate in Coexisting Sub-6GHz and Millimeter-waveCellular Networks
Hazem Ibrahim, Hina Tabassum, and Uyen T. Nguyen
Abstract
Meta distribution is a fine-grained unified performance metric that enables us to evaluate the reliability andlatency of next generation wireless networks, in addition to the conventional coverage probability. In this paper, usingstochastic geometry tools, we develop a systematic framework to characterize the meta distributions of the downlinksignal-to-interference-ratio (SIR)/signal-to-noise-ratio (SNR) and data rate of a typical device in a cellular networkwith coexisting sub-6GHz and millimeter wave (mm-wave) spectrums. Macro base-stations (MBSs) transmit onsub-6GHz channels (which we term “microwave” channels), whereas small base-stations (SBSs) communicate withdevices on mm-wave channels. The SBSs are connected to MBSs via a microwave ( µ wave) wireless backhaul. The µ wave channels are interference limited and mm-wave channels are noise limited; therefore, we have the meta-distribution of SIR and SNR in µ wave and mm-wave channels, respectively. To model the line-of-sight (LOS)nature of mm-wave channels, we use Nakagami-m fading model. To derive the meta-distribution of SIR/SNR, wecharacterize the conditional success probability (CSP) (or equivalently reliability) and its b th moment for a typicaldevice (a) when it associates to a µ wave MBS for direct transmission, and (b) when it associates to a mm-wave SBSfor dual-hop transmission (backhaul and access transmission). Performance metrics such as the mean and varianceof the local delay (network jitter), mean of the CSP (coverage probability), and variance of the CSP are derived.Closed-form expressions are presented for special scenarios. The extensions of the developed framework to the µ wave-only network or mm-wave only networks where SBSs have mm-wave backhauls are discussed. Numericalresults validate the analytical results. Insights are extracted related to the reliability, coverage probability, and latencyof the considered network. The authors are with the Department of Electrical Engineering and Computer Science, York University, Toronto, Ontario, M3J 1P3 Canada(e-mail:hibrahim,hina,[email protected]). Part of the manuscript was presented at the 2019 IEEE ICC: SAC Internet of Things Track [1].
Index Terms
5G Cellular networks, millimeter wave, meta distribution, reliability, latency, wireless backhaul, Nakagamifading, stochastic geometry.
I. I
NTRODUCTION T HE SUB-6GHZ SPECTRUM is running out of bandwidth to support a huge number of devicesin the cellular networks. Therefore, cellular operators of the upcoming 5G networks will tap intothe millimeter-wave (mm-wave) spectrum to use wider bandwidths. The mm-wave spectrum has widerbandwidths that can meet higher traffic demands and support data rates into the order of gigabits persecond. Mm-wave spectrum usage is one of the key enablers of 5G and beyond networks [2] and willcoexist with sub-6GHz frequencies [3], [4]. However, mm-wave transmissions are highly susceptibleto blockages and penetration losses; therefore the mm-wave spectrum will complement the sub-6GHzspectrum in 5G networks [5]–[8].In this article, we develop a framework to characterize the meta distributions of SIR/SNR as well as datarate in the coexisting sub-6GHz and mm-wave cellular network. We assume a two-tier network architectureas illustrated in Fig. 1. Tier 1 consists of macro base stations (MBSs) and tier 2 is composed of small basestations (SBSs). A MBS communicates with SBSs on backhaul links in the microwave spectrum. SBSscommunicate with devices on access links in the mm-wave spectrum. This scenario supports dual-hop communications between MBSs and devices. Devices can also communicate with MBSs via direct links in the microwave spectrum, as shown in Fig. 1.Given the above hybrid spectrum network architecture, it is crucial to develop new theoretic frameworksto characterize the performance of such networks. Within this context, we consider the use of metadistributions to study the performance of such hybrid spectrum networks.The meta distribution is first introduced by M. Haenggi [9] to provide a fine-grained reliability andlatency analysis of 5G wireless networks with ultra-reliable and low latency communication requirements[10], [11]. Meta distribution is defined as the distribution of the conditional success probability (CSP) of thetransmission link (also termed as link reliability ), conditioned on the locations of the wireless transmitters.The meta distribution provides answers to questions such as “
What fraction of devices can achieve x%transmission success probability? ” whereas the conventional success probability answers questions such as “What fraction of devices experience transmission success?” [9]. In addition to the standard coverage (or success) probability which is equivalent to the mean of CSP, the meta distribution can capture importantnetwork performance measures such as the mean of the local transmission delay, the variance of thelocal transmission delay (referred to as network jitter ), and the variance of the CSP which depicts thevariation of the devices’ performance from the mean coverage probability. Evidently, the standard coverageprobability provides limited information about the performance of a typical wireless network [12]–[14].In this article, we develop a novel stochastic geometry framework based on meta distributions to estimateand analyze the communication latency and reliability of devices in a coexisting sub-6GHz and mm-wavecellular network. (cid:68)(cid:349)(cid:272)(cid:396)(cid:381)(cid:449)(cid:258)(cid:448)(cid:286)(cid:3)(cid:17)(cid:258)(cid:272)(cid:364)(cid:346)(cid:258)(cid:437)(cid:367)(cid:3)(cid:62)(cid:349)(cid:374)(cid:364)(cid:68)(cid:373)(cid:882)(cid:449)(cid:258)(cid:448)(cid:286)(cid:3)(cid:4)(cid:272)(cid:272)(cid:286)(cid:400)(cid:400)(cid:3)(cid:62)(cid:349)(cid:374)(cid:364) (cid:68)(cid:349)(cid:272)(cid:396)(cid:381)(cid:449)(cid:258)(cid:448)(cid:286)(cid:3)(cid:24)(cid:349)(cid:396)(cid:286)(cid:272)(cid:410)(cid:3)(cid:62)(cid:349)(cid:374)(cid:364)(cid:18)(cid:381)(cid:396)(cid:286)(cid:3)(cid:69)(cid:286)(cid:410)(cid:449)(cid:381)(cid:396)(cid:364)(cid:24)(cid:437)(cid:258)(cid:367)(cid:882)(cid:44)(cid:381)(cid:393)(cid:3)(cid:100)(cid:396)(cid:258)(cid:374)(cid:400)(cid:373)(cid:349)(cid:400)(cid:400)(cid:349)(cid:381)(cid:374) (cid:94)(cid:349)(cid:374)(cid:336)(cid:367)(cid:286)(cid:882)(cid:44)(cid:381)(cid:393)(cid:3)(cid:100)(cid:396)(cid:258)(cid:374)(cid:400)(cid:373)(cid:349)(cid:400)(cid:400)(cid:349)(cid:381)(cid:374)(cid:68)(cid:373)(cid:882)(cid:449)(cid:258)(cid:448)(cid:286)(cid:3)(cid:4)(cid:272)(cid:272)(cid:286)(cid:400)(cid:400)(cid:3)(cid:62)(cid:349)(cid:374)(cid:364)
Fig. 1:
Coexisting sub-6GHz and mm-wave cellular networks.
A. Related Work
A variety of research works studied the coverage probability of mm-wave only cellular networks [15]–[17]. Di Renzo et al. [15] proposed a general mathematical model to analyze multi-tier mm-wave cellularnetworks. Bai et al. [16] derived the coverage and rate performance of mm-wave cellular networks. Theyused a distance dependent line-of-sight (LOS) probability function where the locations of the LOS andnon-LOS (NLOS) BSs are modeled as two independent non-homogeneous Poisson point processes, towhich different path loss models are applied. The authors assume independent Nakagami fading for eachlink. Different parameters of Nakagami fading are assumed for LOS and NLOS links. Turgut and Gursoy[17] investigated heterogeneous downlink mm-wave cellular networks consisting of K tiers of randomlylocated BSs where each tier operates in a mm-wave frequency band. They derived coverage probability forthe entire network using tools from stochastic geometry. They used Nakagami fading to model small-scalefading. Deng et al. [18] derived the success probability at the typical receiver in mm-wave device-to-device(D2D) networks. The authors considered Nakagami fading and incorporated directional beamforming.Some recent studies analyzed the coverage or success probability of coexisting µ wave and mm-wavecellular networks. A hybrid cellular network was considered by Singh et al. [19] to estimate the uplink- downlink coverage and rate distribution of self-backhauled mm-wave networks. Elshaer et al. [3] developedan analytical model to characterize decoupled uplink and downlink cell association strategies. The authorsshowed the superiority of this technique compared to the traditional coupled association in a network withtraditional MBSs coexisting with denser mm-wave SBSs. Singh et al. [19] and Elshaer et al. [3] modeledthe fading power as Rayleigh fading to enable better tractability.Compared to traditional coverage analysis conducted in [3], [16], [17], Deng and Haenggi [20] analyzedthe meta distribution of the SIR in mm-wave only single-hop D2D networks using the
Poisson bipolarmodel and simplified
Rayleigh fading channels for analytical tractability.
B. Contributions
To the best of our knowledge, our work is the first to characterize the meta distributions of SIR/SNRand data rate for coexisting µ wave and mm-wave networks. Different from previous research in [3], [16],[17], [20], we develop a stochastic geometry framework that takes in consideration (i) coexistence of twodifferent network tiers with completely different channel propagation, interference, and fading models, (ii)dual-hop transmissions enabled by two different spectrums, one in each network tier, and (iii) Nakagami-mfading model with shape parameter m for LOS mm-wave channels. Nakagami-m fading is a generic andversatile distribution that includes Rayleigh distribution (typically used for non-LOS fading) as its specialcase when m = 1 and can well approximate the Rician fading distribution for ≤ m ≤ ∞ (typicallyused for LOS fading).We assume a hybrid spectrum network architecture described above and illustrated in Fig. 1. Sincemicrowave transmissions are interference limited and mm-wave transmission are noise limited , we studythe meta distributions of the SIR and SNR in µ wave and mm-wave channels, respectively. We alsocharacterize the meta distrubusiton of data rates. Our contributions and methodology include the following: • Different from existing works, we characterize the CSP (which is equivalent to reliability) of a typicaldevice and its b th moment when the device either associates to (1) µ wave MBS for direct transmissionor (2) mm-wave SBS for dual-hop transmission (access and backhaul transmission). Using the novelmoment expressions in the two scenarios, we derive a novel expression for the cumulative moment M b, T of the considered hybrid spectrum network. Given highly directional beams and high sensitivity to blockage, recent studies showed that mm-wave networks can be considered asnoise limited rather than interference limited [19], [21], [22]. • Using the cumulative moment M b, T , we characterize the exact and approximate meta distributions ofthe data rate and downlink SIR/SNR of a typical device. Since the expression of M b, T relies on aBinomial expansion of power b , the results for the meta-distribution requiring complex values of b are obtained by applying Newton’s Generalized Binomial Theorem. • We characterize important network performance metrics such as coverage probability, mean localdelay (which is equivalent to latency), and variance of the local delay (network jitter), using thederived cumulative moment M b, T . For metrics with negative values of b , we apply the binomialtheorem for negative integers. • To model the LOS nature of mm-wave, we consider the versatile
Nakagami-m fading channel model.To the best of our knowledge, the meta distribution for the Nakagami- m fading channel has not beeninvestigated yet. • We demonstrate the application of this framework to other specialized network scenarios where(i) SBSs are connected to MBSs via a mm-wave wireless backhaul and (ii) a network where alltransmissions are conducted in µ wave spectrum. Closed-form results are provided for special casesand asymptotic scenarios.We validate analytical results using Monte-Carlo simulations. Numerical results give valuable insightsrelated to the reliability, mean local delay, variance of CSP, and standard success probability of a device.For example, the mean local delay increases with the increasing density of SBSs in a µ wave-only network;however, it stays constant in a hybrid spectrum network. Moreover, the data rate reliability, i.e., the fractionof devices achieving a required data rate, increases as the number of antenna elements increases. We alsonote that as the number of antenna elements in a hybrid spectrum network increases, the reduction inthe variance of reliability is noticeable, which shows the importance of analyzing the higher moments ofthe CSP using the meta distribution. These insights would help 5G cellular network operators to find themost efficient operating antenna configurations for ultra-reliable and low latency applications. C. Outline of the Article
The remainder of the article is organized as follows. In Section II, we describe the system model andassumptions. In Section III, we provide mathematical preliminaries of the meta distribution. In Section IV,we characterize the association probabilities of a typical device and formulate the meta distribution ofthe SIR/SNR of a device in the hybrid spectrum 5G cellular networks. In Section V, we characterize the CSP and its b th moment for direct, access, and backhaul transmissions. Finally, we derive the exactand approximate meta distributions of the SIR/SNR and data rate in a hybrid spectrum network as wellas µ wave-only network in Section VII. Finally, Section VIII presents numerical results and Section IXconcludes the article. II. S YSTEM M ODEL AND A SSUMPTIONS
In this section, we describe the network deployment model (Section II-A), antenna model (SectionII-B), channel model (Section II-C), device association criteria (Section II-D), and SNR/SIR models foraccess and backhaul transmissions (Section II-E).
A. Network Deployment and Spectrum Allocation Model
We assume a two-tier cellular network architecture as shown in Fig. 1 in which the locations of theMBSs and SBSs are modeled as a two-dimensional (2D) homogeneous Poisson point process (PPP) Φ k = { y k, , y k, , ... } of density λ k , where y k,i is the location of i th MBS (when k = 1 ) or the i th SBS(when k = 2 ). Let the MBS tier be tier 1 ( k = 1 ) and the SBSs constitute tier 2 ( k = 2 ). Let D denotesthe set of devices. The locations of devices in the network are modeled as independent homogeneousPPP Φ D = { x , x , .... } with density λ D , where x i is the location of the i th device. We assume that λ D ≫ λ > λ as in [23]–[25]. We consider a typical outdoor device which is located at the origin and isdenoted by and its tagged BS is denoted by y k, , i.e., tagged MBS (when k = 1 ) or tagged SBS (when k = 2 ). All BSs in the k th tier transmit with the same transmit power P k in the downlink. A list of thekey mathematical notations is given in Table I.We assume that a portion ηW of the frequency band W is reserved for the access transmission andthe rest (1 − η ) W is reserved for the backhaul transmission, where W , and W denote the total available µ wave spectrum and mm-wave spectrum, respectively, and ≤ η ≤ . Determining the optimal spectrumallocation ratio η will be studied in our future work. B. Antenna Model
We assume that all MBSs are equipped with omnidirectional antennas with gain denoted by G o1 dB.We consider SBSs and devices are equipped with directional antennas with sectorized gain patterns as in TABLE I: Mathematical Notations
Notation Description Notation Description Φ k ; Φ D PPP of BSs of k th tier; PPP of devices λ k ; λ D Density of BSs of k th tier; density of devices P k Transmit power of BSs in k th tier B k Association bias for BSs of k th tier α , α ,L , α ,N Path loss exponent of MBS tier;LOS SBS; NLOS SBS G o omnidirectional antenna gain of µ wave MBSs G max ; G min ; θ a Main lobe gain; side lobe gain; and3 dB beamwidth for mm-wave SBS h l Gamma fading channel gain for mm-wave SBSs g Rayleigh fading channel gain m l Nakagami-m fading parameter where l ∈ { L, N } denotes LOS and NLOS transmission links p L ; p N Mm-wave blockage LOS probability; NLOS probability θ Predefined SIR/SNR threshold ¯ F P s ( x ) Meta distribution of SIR/SNR P s ( θ ) Conditional success probability (CSP) M b ( θ ) The b th moment of P s ( θ ) A ; A ,L ; A ,N Association Probability with µ wave MBS;LOS mm-wave SBS; NLOS mm-wave SBS [15], [20], [22] to approximate the actual antenna pattern. The sectorized gain pattern is given by: G a ( θ ) = G max a if | θ | ≤ θ a G min a otherwise , (1)where subscript a ∈ { , D} denotes for SBSs and devices, respectively. Considering a √N × √N uniformplanar square antenna array with N elements, the antenna parameters of a uniform planar square antennaarray can be given as in [20], i.e., G max a = N is the main lobe antenna gain, G min a = 1 / sin (cid:16) π √N (cid:17) isthe side lobe antenna gain, θ ∈ [ − π, π ) is the angle of the boresight direction, and θ a = √ √N is the mainlobe beam width. A perfect beam alignment is assumed between a device and its serving SBS [3] [16].The antenna beams of the desired access links are assumed to be perfectly aligned, i.e., the direction ofarrival (DoA) between the transmitter and receiver is known a priori at the BS and the effective gainon the intended access link can thus be denoted as G max2 G max D . This can be done by assuming that theserving mm-wave SBS and device can adjust their antenna steering orientation using the estimated anglesof arrivals. The analysis of the alignment errors on the desired link is beyond the scope of this work. C. Channel Model1) Path-Loss Model:
The signal power decay is modeled as L ( r ) = r α , where L ( r ) is the path lossfor a typical receiver located at a distance r from the transmitter and α is the path loss exponent (PLE).Let L ( r ) = k r , D k α denotes the path loss of a typical device associated with the MBS tier, where α isthe PLE. Similarly, L ( r ) = k r , D k α ,l denotes the path loss of a typical device associated with the SBS tier where α ,l = α ,L is the PLE in the case of LOS and α ,l = α ,N is the PLE in the case of NLOS.It has been shown that mm-wave LOS and NLOS conditions have markedly different PLEs [26]. Also,we consider the near-field path loss factor ζ = ( carrier wavelength π ) at 1 m [3], i.e., different path loss fordifferent frequencies at the reference distance.
2) Fading Model:
For outdoor mm-wave channels, we consider a versatile Nakagami-m fading channelmodel due to its analytical tractability and following the previous line of research studies [16]–[18], [27],[28]. Nakagami- m fading is a general and tractable model to characterize mm-wave channels. Also, inseveral scenarios, Nakagami-m can approximate the Rician fading which is commonly used to modelthe LOS transmissions but not tractable for meta distribution modeling [29], [30]. The fading parameter m l ∈ [1 , , ..., ∞ ) where l ∈ { L, N } denotes LOS and NLOS transmission links, respectively, and themean fading power is denoted by Ω l . The fading channel power h l follows a gamma distribution givenas f h l ( x ) = m mll x ml − Ω mll Γ( m l ) exp( − m l x Ω l ) , x > , where Γ( . ) is the Gamma function, m l is the shape (or fading)parameter, and m l Ω l is the scale parameter. That is, we consider h l ∼ Γ( m L , /m L ) for the LOS linksand h l ∼ Γ( m N , /m N ) for the NLOS links. Rayleigh fading is a special case of Nakagami- m for m L = m N = 1 . Due to the NLOS nature of µ wave channels, we assume Rayleigh fading with powernormalization, i.e., the channel gain g ( x , y ) ∼ exp(1) , is independently distributed with the unit mean.
3) Blockage Model for Mm-wave Access Links:
For mm-wave channels, LOS transmissions are vul-nerable to significant penetration losses [26]; thus LOS transmissions can be blocked with a certainprobability. Following [16], [27], [31], [32], we consider the actual LOS region of a device as a fixedLOS ball referred to as ”equivalent LOS ball”. For the sake of mathematical tractability, we consider adistance dependent blockage probability p ( r ) that a mm-wave link of length r observes, i.e., the LOSprobability p L ( r ) if the mm-wave desired link length is less than d and p N ( r ) otherwise. That is, SBSswithin a LOS ball of radius d are marked LOS with probability p L ( r ) , while the SBSs outside that LOSball are marked as NLOS with probability p N ( r ) . Note that we will drop the notation ( r ) in both p L ( r ) and p N ( r ) from this point onwards and we will use only p L and p N , respectively. D. Association Mechanism
Each device associates with either a MBS or a SBS depending on the maximum biased received powerin the downlink. The association criterion at the typical device can be written mathematically as follows: P k B k G k ζ k L k ( r ) − ≥ P j B j G j ζ j L min ,j ( r ) − , ∀ j ∈ { , } , j = k (2) where P ( · ) , B ( · ) , G ( · ) , and ζ ( · ) denote the transmission power, biasing factor, effective antenna gain, andnear-field path loss at 1 m of the intended link, respectively, in the corresponding tier (which is determinedby the index in the subscript). Let L min ,j ( r ) − be the minimum path loss of a typical device from a BSin the j th tier. When a device associates with a mm-wave SBS in tier-2, i.e., k = 2 , the antenna gain ofthe intended link is G = G max2 G max D , otherwise G = G o1 G D , where G o1 is defined as the omnidirectionalantenna gain of MBSs and G D is the device antenna gain while operating in µ wave spectrum. On theother hand, the SBS associates with a MBS offering the maximum received power in the downlink. E. SNR/SIR Models for Access and Backhaul Transmissions
The device associates to either a MBS for direct transmission or a SBS for dual-hop transmission. Thefirst link (backhaul link) transmissions occur on the µ wave spectrum between MBSs and SBSs and thesecond link (access link) transmissions take place in the mm-wave spectrum between SBSs and devices.Let θ denotes the predefined SIR threshold for SBSs in the backhaul link and θ D denotes the predefinedSIR/SNR threshold for devices. Throughout the paper, we use subscripts “ , ”, “ , D ”, “ , D ”, “ D ”, “ BH ”to denote backhaul link, access link, direct link, device, and backhaul, respectively.
1) Backhaul Transmission:
The
SIR of a typical SBS associated with a MBS can be modeled as:
SIR , = P r − α , g (0 , y , ) I , , (3)where I , denotes the backhaul interference received at a SBS from MBSs that are scheduled to transmiton the same resource block excluding the tagged MBS. Then, I , = P P i : y ,i ∈ Φ \{ y , } k y ,i k − α g (0 , y ,i ) .
2) Direct Transmission:
The
SIR of a typical device associated directly with a MBS is modeled as:
SIR , D = P r − α , D g (0 , y , ) I , D , (4)where I , D denotes the interference received at a typical device from MBSs excluding the tagged MBS.Then I , D can be calculated as: I , D = P P i : y ,i ∈ Φ \{ y , } k y ,i k − α g (0 , y ,i ) .
3) Access Transmission:
The SNR of a typical device associated with a mm-wave SBS is modeled as:
SNR , D = P G ζ k r , D k − α ,l h l (0 , y , ) σ , (5)where ζ is the near-field path loss at 1 m for mm-wave channels, and σ is the variance of the additivewhite Gaussian noise at the device receiver. Given highly directional beams and high sensitivity toblockage, recent studies showed that mm-wave networks are typically noise limited [19], [21], [22]. III. T HE M ETA D ISTRIBUTION : M
ATHEMATICAL P RELIMINARIES
In this section, we define the meta distribution of the SIR of a typical device and highlight exact andapproximate methods to evaluate the meta distribution.
Definition 1 (Meta Distribution of the SIR and CSP) . The meta distribution ¯ F P s ( x ) is the complementarycumulative distribution function (CCDF) of the CSP (or reliability) P s ( θ ) and given by [9]: ¯ F P s ( x ) ∆ = P ( P s ( θ ) > x ) , x ∈ [0 , , (6) where, conditioned on the locations of the transmitters and that the desired transmitter is active, the CSP P s ( θ ) of a typical device [9] can be given as P s ( θ ) ∆ = P (SIR > θ | Φ , tx ) where θ is the desired SIR . Physically, the meta distribution provides the fraction of the active links whose CSP (or reliability)is greater than the reliability threshold x . Given M b ( θ ) denotes the b th moment of P s ( θ ) , i.e., M b ( θ ) ∆ = E ( P s ( θ ) b ) , b ∈ C , the exact meta distribution can be given using the Gil-Pelaez theorem [33] as [9]: ¯ F P s ( x ) = 12 + 1 π Z ∞ ℑ (cid:0) e − jt log x M jt ( θ ) (cid:1) t d t, (7)where ℑ ( w ) is imaginary part of w ∈ C and M jt ( θ ) denotes the imaginary moments of P s ( θ ) , i.e., , j ∆ = √− . Using moment matching techniques and taking β ∆ = ( M ( θ ) − M ( θ ))(1 − M ( θ )) M ( θ ) − M ( θ ) , the meta distributionof the CSP can be approximated using the Beta distribution as follows: ¯ F P s ( x ) ≈ − I x (cid:18) βM ( θ )1 − M ( θ ) , β (cid:19) , x ∈ [0 , , (8)where M ( θ ) and M ( θ ) are the first and the second moments, respectively; I x ( a, b ) is the regularizedincomplete Beta function I x ( a, b ) ∆ = R x t a − (1 − t ) b − d t B ( a,b ) and B ( a, b ) is the Beta function.IV. T HE M ETA D ISTRIBUTION OF THE
SIR/SNR IN H YBRID S PECTRUM N ETWORKS
To characterize the meta distribution of the SIR/SNR of a typical device that can associate with eithera µ wave MBS with probability A or with a wireless backhauled mm-wave SBS with probability A , themethodology of analysis is listed as follows:1) Derive the probability of a typical device associating with µ wave MBSs A , LOS mm-wave SBSs A , L , and NLOS mm-wave SBSs A , N where A = A , L + A , N ( Section IV-A ).2) Formulate the meta distribution of the SIR/SNR of a device in the hybrid network ( ¯ F bP s, T ( x ) )considering the direct link and dual-hop link with wireless backhaul transmission ( Section IV-B ).
3) Formulate the CSP ( P s, T ( θ ) ) and its b th moment ( M b, T ) ( Section IV-B ).4) Derive the CSP at backhaul link P s, BH ( θ ) , CSP at access link P s, ( θ D ) , and CSP at direct link P s, ( θ D ) . Derive the b th moments of CSPs, i.e., M b, BH ( θ ) , M b, ( θ D ) , and M b, ( θ D ) for backhaullink, access link, and direct link transmissions, respectively ( Section V ).5) Obtain the meta distributions of SIR/SNR and data rate in hybrid spectrum network using Gil-Pelaezinversion and the Beta approximation (
Section VI ). A. Association Probabilities in Hybrid Spectrum Networks
In this subsection, we characterize the probabilities with which a typical device associates with µ waveMBSs ( A ) or mm-wave SBSs ( A ). The results are presented in the following. Lemma 1 (The Probability of Associating with mm-wave SBSs) . The probability of a typical device toassociate with a mm-wave SBS, using the association scheme in Eq. (2) , can be expressed as: A = 1 − πλ ˆ aα Z d α ,L H ( l ) e − πλ p L l α ,L d l + Z d α ,N d α ,L H ( l ) e − πλ p L d d l + Z ∞ d α ,N H ( l ) e − πλ (cid:20) ( p L − p N ) d + p N l α ,N (cid:21) d l ! , (9) where ˆ a ∆ = P B G ζ P B G ζ and H ( l ) ∆ = (cid:0) l ˆ a (cid:1) α − exp (cid:16) − πλ (cid:0) l ˆ a (cid:1) α (cid:17) . Subsequently, the probability of a deviceto associate with a µ wave MBS can be given as A = 1 − A . The conditional association probabilityfor a typical device to associate with SBS is as follows: ¯ A ( l ) = 1 − πλ ˆ aα H ( l ) e − πλ p L l α ,L + H ( l ) e − πλ p L d + H ( l ) e − πλ " ( p L − p N ) d + p N l α ,N ! , (10) subsequently, ¯ A ( l ) = 1 − ¯ A ( l ) .Proof. Using the approach in [3], we derive Lemma 1 in
Appendix A of our technical report [34]. (cid:4)
A closed-form expression of A can be derived for a case of practical interest as follows. Corollary 1.
When α = 4 , α ,L = 2 , and α ,N = 4 , then A can be given in closed-form as follows: A = e C (Φ[ √ C + p πλ p L d ] − Φ[ √ C ]) p p L λ / ˆ a + e − d πp L λ ( e − πλ √ d / ˆ a − e − πλ √ d / ˆ a ) πλ / a + e d π ( p N − p L ) λ − C √ d / ˆ a ) C / a , (11) where Φ( · ) is the error function, C = πλ ap L λ and C = π ( λ + √ ˆ ap N λ ) and A = 1 − A . It can be seen from
Corollary 1 that when the number of antenna elements N goes to infinity, i.e., G → ∞ , ˆ a → ∞ , then A can be simplified as A = Φ[ √ πλ p L d ] √ p L λ / ˆ a + e d π ( pN − pL ) λ C / a , which shows thatassociation probability to MBS will be very small. Similar insights can be extracted for other parameters.In order to derive the b th moment of CSP P s, ( θ D ) on an access link when a device associates witha SBS (the CSP will be discussed later in Lemma 4), we have to derive the probability of a device toassociate with LOS SBS A ,L and NLOS SBS A ,N which are defined follows. Lemma 2 (The Probability of Associating with LOS and NLOS mm-wave SBSs) . When a typical deviceassociates with the mm-wave SBS tier, this typical device has two possibilities to connect to (a) a LOS mm-wave SBS with association probability A ,L and (b) a NLOS mm-wave SBS with association probability A ,N which are characterized, respectively, as follows: A ,L = Z d α ,L ¯ A ,L ( l ,L ) d l ,L , A ,N = Z ∞ d α ,N ¯ A ,N ( l ,N ) d l ,N , (12) where ¯ A ,L ( l ,L ) and ¯ A ,N ( l ,N ) are the conditional probabilities with which a typical device may associateto the LOS and NLOS mm-wave SBSs, respectively, and are defined as follows: ¯ A ,L ( l ,L ) ∆ = 2 πλ p L α ,L l α ,L − ,L e − πλ (¯ al ,L ) α − πλ p L l α ,L ,L ! , ¯ A ,N ( l ,N ) ∆ = 2 πλ p N α ,N l α ,N − exp (cid:18) − πλ (¯ al ,N ) α − πλ (cid:2) p L d + p N ( l α ,N ,N − d ) (cid:3)(cid:19) d l ,N , where ¯ a ∆ = P B G ζ P B G ζ , ¯ A ( l ) = ¯ A ,L ( l ,N ) + ¯ A ,N ( l ,N ) and A = A ,L + A ,N .Proof. Using the approach in [17], we derive Lemma 2 in
Appendix B of our technical report [34]. (cid:4)
A case of interest is when the number of antenna elements at mm-wave SBSs increases asymptotically.In such a case, the LOS and NLOS association probabilities can be simplified as follows:
Corollary 2.
When the number of antenna elements at mm-wave SBSs increases, i.e.,
N → ∞ , α = 4 , α ,L = 2 , and α ,N = 4 , then ¯ a → . The association probabilities can be given in closed-form as follows: A ,L =1 − e − πp L d λ , A ,N = e d π ( − p L + p N ) λ (1 − πp N d λ F [1; 2; πp N d λ ]) , where F [ a ; b ; z ] is the Kummer Confluent Hypergeometric function. An interesting insight from
Corollary 2 can be seen when the intensity of SBSs λ → ∞ or d is large,the probability of association to LOS SBSs A ,L becomes almost 1. On the other hand, when λ → or d is small, F [ a ; b ; 0] = 1 thus A ,N becomes almost 1. B. Formulation of the Meta distribution, CSP and its b th Moment in the Hybrid Network
When a device associates with a mm-wave SBS, the overall CSP depends on the CSPs of the SIRand SNR on both the backhaul link and the access link, respectively. On the other hand, when a deviceassociates to MBS the CSP depends on the SIR of the direct link. It is thus necessary to formulate therelationship between the meta distribution, CSP, and its b th moment in the considered hybrid network asfollows. Lemma 3 (Meta Distribution of the Typical device in the Hybrid Network) . The combined meta distri-bution of the SIR/SNR in the hybrid spectrum network can be characterized as follows: ¯ F P s, T ( x ) = 12 + 1 π Z ∞ ℑ (cid:0) e − jt log x M jt, T ( · ) (cid:1) t d t, (13) where M jt, T ( θ ) can be characterized by deriving the b th moment of the P s, T ( · ) . M b, T ( · ) = M b, Dual − Hop + M b, Single − Hop ( a ) = E Φ [ ¯ A ( l ) P bs, Dual − Hop ( θ )] + E Φ [ ¯ A ( l ) P bs, ( θ D )] , ( b ) = E Φ (cid:20) ¯ A ( l )( P s, BH ( θ ) P s, ( θ D )) b (cid:21) + E Φ (cid:20) ¯ A ( l ) P bs, ( θ D ) (cid:21) , ( c ) = E Φ (cid:20) P s, BH ( θ ) b (cid:21) E Φ (cid:20) ¯ A ( l ) P s, ( θ D ) b (cid:21) + E Φ (cid:20) ¯ A ( l ) P bs, ( θ D ) (cid:21) , ( d ) = E Φ (cid:20) P s, BH ( θ ) b (cid:21) E Φ (cid:20) ( ¯ A ,L ( l ,L ) + ¯ A ,N ( l ,N )) P s, ( θ D ) b (cid:21) + E Φ (cid:20) ¯ A ( l ) P bs, ( θ D ) (cid:21) , ( e ) = M b, BH ( θ ) M b, ( θ D ) | {z } Device Associated with SBS + M b, ( θ D ) | {z } Device Associated with MBS , (14) where M b, Dual − Hop is the b th moment of the SIR/SNR when a device associates to mm-wave SBS fordual-hop transmission and M b, Single − Hop is the b th moment of the SIR when a device associates to MBSfor direct transmission. After reformulation, we define M b, BH ( θ ) as the unconditional b th moment ofthe backhaul SIR, M b, ( θ D ) as the unconditional b th moment of the SNR at access link when a deviceassociates to mm-wave SBS, and M b, ( θ D ) as the unconditional b th moment of the SIR at direct linkwhen a device associates to µ wave BS. Note that P s, ( θ D ) ∆ = P (SIR , D > θ D | Φ , tx ) denotes the CSPof device over the direct link, P s, BH ( θ ) ∆ = P (SIR , > θ | Φ , tx ) denotes the CSP at backhaul link, and P s, ( θ D ) ∆ = P (SNR , D > θ D | Φ , tx ) denotes the CSP for the access link transmission.Proof. Step (a) follows from the fact that the b th moment of the SIR or SNR of a device associated totier i can be defined as M ( i ) b = E [ ¯ A i M b | i ] where ¯ A i is the conditional association probability to tier i and M b | i = P bs,i is the conditional b th moment of the SIR or SNR in tier i . In our case, we have ¯ A ( l ) The b th moment of a random variable X is the expected value of random variable to the power b , i.e., E [ X b ] . which is the conditional association probability to mm-wave SBS where l ∈ { L, N } since a device canassociate to either LOS or NLOS mm-wave SBS. The step (b) follows from the fact that the CSP of thedual-hop transmission depends on the CSP of access and backhaul link; therefore, we have a product ofthe access and backhaul CSPs, i.e., P s, BH ( θ ) P s, ( θ D ) that are independent random variables. There is nocorrelation since µ wave backhaul does not interfere with mm-wave transmissions. The step (c) followsfrom the fact if X and Y are independent then E [( XY ) b ] = E [ X b ] E [ Y b ] . Finally, the step (d) follows fromthe definition of ¯ A ( l ) in Lemma 2 and the step (e) follows by applying the definition of moments. (cid:4)
In the next section, we derive the CSP of access, backhaul, and direct links along with their respective b th moments, as needed in Lemma 4 to characterize the overall moment as well as the meta distribution.V. C
HARACTERIZATION OF THE
CSP
S AND M OMENTS
In this section, we derive the CSPs P s, BH ( θ ) , P s, ( θ D ) , P s, ( θ D ) and the b th moments M b, BH ( θ ) , M b, ( θ D ) , and M b, ( θ D ) for backhaul link, access link, and direct link, respectively. A. CSP and the b th Moment - Access Link
We condition on having a device at the origin which becomes a typical device. The CSP of a typicaldevice at the origin associating with the mm-wave SBS-tier (when k = 2 ) can be described as follows: P s, ( θ D ) = p L P s, ,L ( θ D ) + p N P s, ,N ( θ D ) . (15)The CSP of the SNR of a device on the access link with LOS can be defined by substituting SNR , D defined in Eq. (5) into Definition 1 as follows: P s, ,L ( θ D ) = P h L (0 , y , ) > θ D r α ,L , D σ P G | Φ , Φ , tx ! ( a ) = 1 − γ (cid:16) m L , m L Ω L ν L (cid:17) Γ( m L ) ( b ) = Γ (cid:16) m L , m L Ω L ν L (cid:17) Γ( m L ) , (16)where (a) follows from the definition of ν L ∆ = θ D r α ,L , D σ P G and the fact that the channel gain h L (0 , y , ) isa normalized gamma random variable and γ ( ., . ) is the lower incomplete gamma function and Γ( s ) = γ ( s, x ) + Γ( s, x ) , where Γ( ., . ) is the upper incomplete gamma function. Similarly, CSP of the SNR onthe access link for NLOS case can be given as follows: P s, ,N ( θ D ) = Γ (cid:16) m N , m N Ω N ν N (cid:17) Γ( m N ) , (17) where ν N ∆ = θ D r α ,N , D σ P G . As such, the b th moment of the CSP on the access link for the typical devicewhen it is served by the mm-wave SBS tier is given by the following: Lemma 4.
The b th moment of the SNR at an “access link” when a device associates with a mm-waveSBS can be characterized as follows: M b, ( θ D ) = b X k =0 (cid:18) bk (cid:19) ( − k p bL m L k X ¨ k =0 (cid:18) m L k ¨ k (cid:19) ( − ¨ k Z d α ,L e − ζ L ¨ k ¨ ν L l ¯ A ,L ( l ,L ) + p bN m N k X ¨ k =0 (cid:18) m N k ¨ k (cid:19) ( − ¨ k Z ∞ d α ,N e − ζ N ¨ k ¨ ν N l ¯ A ,N ( l ,N ) , (18) where ¯ A ,L ( l ,L ) and ¯ A ,N ( l ,N ) are given in Lemma 2 , ζ L ∆ = m L ( m L !) − /m L , ν L ∆ = θ D r α ,L , D σ P G , ζ N ∆ = m N ( m N !) − /m N , and ν N ∆ = θ D r α ,N , D σ P G , ¨ ν L ∆ = ν L r α ,L = ν L l = θ D σ P G and ¨ ν N ∆ = ν N r α ,N = ν N l = θ D σ P G .Proof. See
Appendix C . (cid:4) For α = 4 , α ,L = 2 , and α ,N = 4 , we can get M b, ( θ D ) in closed-form using Corollary 1 . Also, forscenarios where
N → ∞ , α = 4 , α ,L = 2 , and α ,N = 4 , then ¯ a → . Also, ¨ v L → and ¨ v N → , wecan get M b, ( θ D ) in closed-form using Corollary 2 . B. CSP and Moment - Backhaul Link
For the backhaul link, we condition on having a SBS at the origin which becomes the typical SBS.Using the expression of
SIR , in Eq. (3) the CSP of the backhaul link P s, BH ( θ ) can be given as: P s, BH ( θ ) = P (cid:18) g (0 , y , ) > θ r α , P I , | Φ , Φ , tx (cid:19) ( a ) = E (cid:20) exp( − θ r α , X i : y ,i ∈ Φ \{ y , } k y ,i k − α g (0 , y ,i )) (cid:21) , = Y y ,i ∈ Φ \{ y , } E (cid:20) exp (cid:0) − θ r α , k y ,i k − α g (0 , y ,i ) (cid:1) (cid:21) ( b ) = Y y ,i ∈ Φ \{ y , }
11 + θ (cid:16) r , k y ,i k (cid:17) α . (19)where (a) follows from the Rayleigh fading channel gain g (0 , y , ) ∼ exp(1) and (b) is found by takingthe expectation with respect to g (0 , y ,i ) . The b th moment of the CSP on the backhaul link is given as: M b, BH ( θ ) = E (cid:20) P s, BH ( θ ) b (cid:21) = E (cid:20) Y y ,i ∈ Φ \{ y , } (cid:16) θ (cid:16) r , k y ,i k (cid:17) α (cid:17) b (cid:21) , ( a ) = (cid:18) Z (cid:18) − θ r α ) b (cid:19) r − dr (cid:19) − = 1 F ( b, − α ; 1 − α ; − θ ) , (20)where (a) follows from the probability generating functional (PGFL) of PPP, i.e., G R [ f ] ∆ = E Q x ∈R f ( x ) = R (1 − f ( x )) x − dx . [35, lemma 1] and F ( ., . ; . ; . ) represents Gauss‘ Hyper-geometric function. C. CSP and Moment - Direct Link
Using the expression of
SIR , D in Eq. (4), we calculate the CSP of the direct link P s, ( θ D ) as follows: P s, ( θ D ) = P (cid:18) g (0 , y , ) > θ D r α , D P I , D | Φ , Φ , tx (cid:19) ( a ) = E (cid:20) exp − θ D r α , D X i : y ,i ∈ Φ \{ y , } k y ,i k − α g (0 , y ,i ) (cid:21) , = Y y ,i ∈ Φ \{ y , } E (cid:20) exp (cid:16) − θ D r α , D k y ,i k − α g (0 , y ,i ) (cid:17) (cid:21) ( b ) = Y y ,i ∈ Φ \{ y , }
11 + θ D (cid:16) r , D k y ,i k (cid:17) α , (21) where (a) follows from the channel gain g (0 , y , ) ∼ exp(1) and is independently exponentially distributedwith unit mean and (b) is obtained by taking the expectation with respect to g (0 , y ,i ) . While taking theassociation probabilities into consideration, the b th moment of the CSP P s, ( θ D ) of the typical devicewhen it is served by a µ wave MBS is characterized in the following lemma. Lemma 5 (The b th moment of the CSP ( P s, ( θ D ) ) when a device associates with a MBS) . The b th momentof the CSP experienced by a device, when the device associates with a MBS, can be characterized asfollows: M b, ( θ D ) = 2 πλ ˆ aα ( Z d α ,L H ( l ) exp (cid:18) − πλ p L l α ,L (cid:19) d l + Z d α ,N d α ,L H ( l ) exp (cid:0) − πλ p L d (cid:1) d l + Z ∞ d α ,N H ( l ) exp (cid:18) − πλ [ p L d + p N (cid:18) l α ,N − d (cid:19)(cid:19) d l ) × exp − λ πl α α Z (cid:20) − θ D v ) b (cid:21) v α +1 d v , (22) Proof.
See
Appendix D . (cid:4) Note that R (cid:20) − θ D v ) b (cid:21) v α d v is independent of l , thus where N → ∞ or α = 4 , α ,L = 2 , and α ,N = 4 , then we can get a closed-form for the three integral over l using Corollary 1 and
Corollary 2 . D. Combined b th Moment of the CSP in Hybrid Networks
After substituting the values of M b, BH ( θ ) , M b, ( θ D ) , and M b, ( θ D ) in Eq. (20), Eq. (18), and Eq. (22),respectively into the total meta distribution for the entire network in Eq. (14), we get the b th moment ofthe CSP at a typical device as follows: M b, T = 1 F ( b, − α ; 1 − α ; − θ ) × (cid:26) b X k =0 (cid:18) bk (cid:19) ( − k p bL m L k X ¨ k =0 (cid:18) m L k ¨ k (cid:19) ( − ¨ k Z d α ,L e − ζ L ¨ k ¨ ν L l ¯ A ,L ( l ,L )+ p bN m N k X ¨ k =0 (cid:18) m N k ¨ k (cid:19) ( − ¨ k Z ∞ d α ,N e − ζ N ¨ k ¨ ν N l ¯ A ,N ( l ,N ) (cid:27) + M b, ( θ D ) , (23) In the next section, we use the combined b th moment in (23) to compute the meta distributions ofSIR/SNR and data rate using Gil-Pelaez inversion and the Beta approximation. VI. C
OMPUTING THE M ETA D ISTRIBUTIONS AND S PECIAL C ASES
In this section, we compute the meta distribution of SIR/SNR using Gil-Pelaez inversion and betaapproximation by applying the derived result of M b, T . Special cases where b = 1 provides the standardcoverage probability and b = − provides the mean local delay are discussed. Further, we show how toevaluate the data rate meta distribution from the derived framework. A. Computing the Meta Distribution of SIR/SNR
Technically, substituting b = jt in (23), we should obtain the imaginary moments M jt, T . However, sincethe expression of M jt, T relies on a Binomial expansion of power b , the results cannot be obtained directlythrough substitution. Therefore, we apply Newton’s generalized binomial theorem given as follows: Definition 2.
Isaac Newton‘s generalized binomial theorem is to allow real exponents other than non-negative integers, i.e., imaginary exponent r , as (cid:0) rk (cid:1) = r ( r − ... ( r − k +1) k ! = ( r ) k k ! , where ( . ) k is the Pochhammersymbol, which stands here for a falling factorial. Applying
Definition 2 in step (e) of
Appendix C , we then obtain the expression for M jt, T as follows: M jt, T = 1 F ( jt, − α ; 1 − α ; − θ ) × (cid:26) p jtL ∞ X k =0 ( jt ) k k ! ( − k m L k X ¨ k =0 (cid:18) m L k ¨ k (cid:19) ( − ¨ k Z d α ,L e − ζ L ¨ k ¨ ν L l ¯ A ,L ( l ,L )+ p jtN ∞ X k =0 ( jt ) k k ! ( − k m N k X ¨ k =0 (cid:18) m N k ¨ k (cid:19) ( − ¨ k Z ∞ d α ,N e − ζ N ¨ k ¨ ν N l ¯ A ,N ( l ,N ) (cid:27) + M jt, ( θ D ) , (24) The imaginary moments can be substituted in the Gil-Pelaez inversion theorem as in
Definition 1 to obtain ¯ F P s , T . Furthermore, we follow [9], [13], [36] to approximate the meta distribution by a Betadistribution by matching the first and second moments, which are easily obtained from the general result inEq. (23) by substituting b = 1 and b = 2 to get M , T and M , T , respectively. Taking β ∆ = ( M , T − M , T )(1 − M , T ) M , T − M , T ,the meta distribution using beta approximation can be given as follows: ¯ F P s, T ( x ) ≈ − I x (cid:18) βM , T − M , T , β (cid:19) , x ∈ [0 , , (25) B. Mean and Variance of the Local Delay
The mean local delay is the mean number of transmission attempts, i.e., re-transmissions, needed tosuccessfully transmit a packet to the target receiver. The mean local delay M − , T which is the − st momentof the CSP of a typical device should be calculated by substituting b = − in Eq. (23). However, since the expression of M b, T relies on a Binomial expansion of power b , the results cannot be obtained directlythrough substitution. Therefore, we apply Binomial theorem for the negative integers as follows: Definition 3.
The Binomial theorem for a negative integer power n can be given [37] as ( x + y ) n = P ∞ k =0 ( − k (cid:0) − n + k − k (cid:1) y n − k x k , Applying
Definition 3 in step (e) of
Appendix C , we then obtain the expression for M − , T as follows: M − , T = 1 F ( − , − α ; 1 − α ; − θ ) × (cid:26) p − L ∞ X k =0 m L k X ¨ k =0 (cid:18) m L k ¨ k (cid:19) ( − ¨ k Z d α ,L e − ζ L ¨ k ¨ ν L l ¯ A ,L ( l ,L )+ p − N ∞ X k =0 m N k X ¨ k =0 (cid:18) m N k ¨ k (cid:19) ( − ¨ k Z ∞ d α ,N e − ζ N ¨ k ¨ ν N l ¯ A ,N ( l ,N ) dl ,N (cid:27) + M − , ( θ D ) , (26) Remark:
In order to better characterize the fluctuation of the local delay, the variance of the localdelay also referred to as network jitter can be given by
N J = M − , T − M − , T . C. The Meta Distribution of the Data Rate in Hybrid Spectrum Networks
Let T denote the data rate (in bits/sec) of the typical device on a specific transmission link whichis a random variable and is defined as R = W log (1 + SIR) using Shannon capacity. Using the metadistribution of the SIR, the meta distribution of the data rate can be derived to present the fraction of activedevices in each realization of the point process that have a data rate R greater than T with probabilityat least x , i.e., devices data rate reliability threshold. That is, first deriving the CSP of the data rate asfollows: P [ R > T | Φ , tx ] = P [ W log (1 + SIR) > T | Φ , tx ] = P [SIR > T W − | Φ , tx ] , (27)where P s (2 T W − ∆ = P (SIR > T W − | Φ , tx ) denote the CSP of the device data rate over single link.Finally, deriving the b th moment of the CSP of the data rate and applying Gil-Pelaez inversion we canobtain the meta distribution of the data rate. Corollary 3.
Similar to the meta distribution of the SIR/SNR derived in Lemma 3 and conditioned onthe location of the point process, we derive the meta distribution of the data rate in hybrid 5G cellularnetworks, using the moment Q b of the conditional data rate as follows: Q b ( T ) = E [ ¯ A ( l ) P (cid:16) P s, BH (2 T BH(1 − η ) W − P s, (2 T W − > x (cid:17) ] + E [ ¯ A ( l ) P ( P s, (2 T ηW − > x )] , = M b, BH (cid:16) T BH(1 − η ) W − (cid:17) M b, (cid:16) T W − (cid:17) + M b, (cid:16) T ηW − (cid:17) , (28) where P s, (2 T ηW − ∆ = P (SIR , D > T ηW − | Φ , tx ) , P s, BH (2 T BH(1 − η ) W − ∆ = P (SIR , > T BH(1 − η ) W − | Φ , tx ) ,and P s, (2 T W − ∆ = P (SNR , D > T W − | Φ , tx ) denotes the CSP of the device data rate at the direct,backhaul, and access link, respectively. In the following section, we discuss the application of this framework in two scenarios (i) µ wave onlynetwork and (ii) mm-wave backhauls and microwave access links.VII. E XTENSIONS OF T HE M ODEL TO O THER N ETWORK A RCHITECTURES
The framework discussed above can be flexibly applied to different network architectures. In this sectionwe discuss how to extend the framework to two other network architectures: 1) both tiers operating inthe sub-6GHz (microwave) spectrum as in traditional cellular networks; and 2) the two tiers operating intwo millimeter-wave spectrums which are orthogonal to each other. Due to space limitation, we provideonly general directions of how to extend the earlier framework to these two other network architectures.
A. The Meta Distribution of the SIR in Microwave-only Cellular Networks
We characterize the meta distribution of the downlink
SIR attained at a typical device in a µ wave-only cellular network, i.e., the access and backhaul links of SBSs operate in the µ wave frequency. Adevice associates with either a serving MBS for direct transmissions (when k = 1 ) or a SBS for dual-hoptransmissions (when k = 2 ), depending on the biased received signal power criterion. MBSs and SBSs areassumed to operate on orthogonal spectrums; thus, there is no inter-tier interference. On the other hand,each SBS associates with a MBS based on the maximum received power at the SBS. The associationcriterion for a typical device can be described as follows [38]: P k B k (min i k y k,i − x k ) − α k ≥ P j B j (min i ′ k y j,i ′ − x k ) − α j , ∀ j (29)where k . k denotes the Euclidean distance. A typical device associates with a serving node (given by Eq.(29))), which is termed the tagged SBS. For the sake of clarity, we define ˆ P jk ∆ = P j P k , ˆ B jk ∆ = B j B k , ˆ λ jk ∆ = λ j λ k .As derived in [38], the conditional association probability for the typical device connecting to the k th tier(conditional over the desired link distance r D ,k ) is as follows: P ( n = k | r D ,k ) = Y j = k e − πλ j ( ˆ P jk ˆ B jk ) /αj r , (30) where n denotes the index of the tier associating with the typical device. We calculate the CSP P s, ′ ( θ D ) (when k = 2 ) of the access link operating in the µ wave band as follows: P s, ′ ( θ D ) = P (cid:18) g (0 , y , ) > θ D r α , D P I , D | Φ , Φ , tx (cid:19) , ( a ) = E (cid:20) exp − θ D r α , D X i : y ,i ∈ Φ \{ y , } k y ,i k − α g (0 , y ,i ) (cid:21) ( b ) = Y y ,i ∈ Φ \{ y , }
11 + θ D (cid:16) r , D k y ,i k (cid:17) α . (31)where (a) follows from the channel gain g (0 , y , ) ∼ exp(1) and is independently exponentially distributedwith unit mean and (b) is obtained by taking the expectation with respect to g (0 , y ,i ) . Lemma 6.
Using Eq. (21) and Eq. (31), we calculate a general expression for the b th moment of theCSP on direct link M b, k’ (when k = 2 ) and the b th moment of the CSP at access link (when k = 1 ) as: M b, k’ = 1 P j = k ˆ λ jk ( ˆ P jk ˆ B jk ) /α j + F ( b, − α k ; 1 − α k ; − θ D ) . (32) Proof.
See
Appendix E . (cid:4) Note that Lemma 6 is novel and different from [39] where we derive the b th moment of CSP fororthogonal spectrum two tier network while the work in [39] is done for shared spectrum tiers.Similarly, the moment of the CSP of a typical device with offloading biases is defined as follows: M b, T = M b, dual-hop | {z } Dual-hop transmission + M b, ′ ( θ D ) | {z } Direct transmission ( a ) = M b, BH ( θ ) M b, ′ ( θ D ) + M b, ′ ( θ D ) , (33)where M b, BH ( θ ) , M b, ′ ( θ D ) , and M b, ′ ( θ D ) are defined in Eq. (20), Eq. (32) (when k = 2 ), and Eq. (32)(when k = 1 ), respectively. The step (a) follows from the similar approach as taken in Lemma 4 . M b, dual-hop = E (cid:20) P s, BH ( θ ) b × Y j = k e − πλ j ( ˆ P jk ˆ B jk ) /αj r P s, ′ ( θ D ) b (cid:21) , ( a ) = E (cid:20) P s, BH ( θ ) b (cid:21)| {z } M b, BH ( θ ) (Backhaul link ) E (cid:20) Y j = k e − πλ j ( ˆ P jk ˆ B jk ) /αj r P s, ′ ( θ D ) b (cid:21)| {z } M b, ( θ D ) (access link) , ( b ) = 1 F ( b, − α ; 1 − α ; − θ ) × λ ( ˆ P ˆ B ) /α + F ( b, − α ; 1 − α ; − θ D ) , (34) where (a) follows from the independence between the location of the MBSs and SBSs. In step (b) wesubstitute M b, BH ( θ ) from Eq. (20) and M b, ( θ D ) into Eq. (32) when k = 2 . By substituting Eq. (34) andEq. (32) (when k = 1 ) in Eq. (33), we get the b th moment M b, T . Finally, by substituting M b, T in Eq. (33)into either Eq. (13) or Eq. (8), we get the meta distribution of the SIR . B. Extensions to Millimeter-wave Backhauls Networks
The proposed framework can be extended to a scenario where the backhaul and access transmissionsare conducted on orthogonal mm-wave spectrums. Note that Eq. (3) will be changed similar to Eq. (5).Then, only the first term, M b, BH ( θ ) in the main Eq. (14) of our model that characterizes the moment ofthe CSP in the backhaul will be re-defined as M b, BH ( θ ) = E [ P bs, ( θ )] .The framework can also be extended to a scenario where the backhaul transmissions are conducted onthe mm-wave spectrum and the access links of SBSs operate on µ -wave. In this case, we will need touse the results in Section VII.A while redefining the term M b, BH ( θ ) as M b, BH ( θ ) = E [ P bs, ( θ )] in (34).VIII. N UMERICAL R ESULTS AND D ISCUSSIONS
We present the simulation parameters in Section VIII-A. Then, we validate our numerical results usingMonte-Carlo simulations in Section VIII-B. In Section VIII-B, we use the developed analytical models toobtain insights related to the meta distribution of the SIR/SNR of a typical device, mean and variance ofthe success probability, transmission delay, and the reliability of a typical device in the downlink direction.
A. Simulation Parameters
Unless otherwise stated, we use the following simulation parameters throughout our numerical results.The transmission powers of MBSs and SBSs in the downlink are P = 50 Watts and P = 5 Watts,respectively. The size of the simulated network is km × km. We assume that the density of MBSsis λ = 2 MBSs/km and the density of SBSs is λ = 70 SBSs/km . The offloading biases for the MBSsand the SBSs are B = B = 1 , respectively. The PLE for MBSs is set to α = 4 and for mm-wave SBSs, α ,L = 2 in the case of LOS and α ,N = 4 in the case of NLOS. The network downlink bandwidth is 100MHz for µ wave MBSs and 1 GHz for mm-wave SBSs with channel frequency 28 GHz. The LOS (NLOS)states are modeled by large (small) values of m , i.e., m L = 2 and m N = 1 [17]. SBSs number of antennaelements is N = 10 . The receiver noise is calculated as [19] σ = − dBm/Hz + 10 log ( W ) + 10 dB,where W = 1 GHz is bandwidth allocated to the mm-wave SBSs. The antenna gains of MBSs are G o1 = 0 dB and devices directional antenna gain is G max D = 10 dB. B. Numerical Results and Discussions1) Association Probability:
Fig. 2 illustrates the accuracy of association probabilities in a hybridspectrum network, derived in
Lemma 1 and
Lemma 2 , as a function of λ by showing a comparison with Monte-Carlo simulations. We notice from Fig. 2 that by increasing the density of the mm-wave SBSs λ ,the probability of association with mm-wave LOS SBSs A ,L increases which confirms the insights from Corollary 1 and
Corollary 2 . The reason is the increasing number of SBSs per unit area within the LOSball will favour the device association towards LOS SBSs and reduces the chances of associating withNLOS SBSs. The addition of A + A ,L + A ,N = 1 is equal to unity for different densities of SBSs λ .Note that the probability of associating with µ wave MBSs is minimal due to a higher path-loss exponentand NLOS omnidirectional transmissions from MBSs.
2) The Meta Distribution of the SIR/SNR:
In Fig. 3, we validate our analytical results for the metadistribution of the SIR/SNR of a typical device in a hybrid spectrum network through simulations. Fig. 3also depicts the probability of achieving reliability x , i.e., x % fraction of devices can achieve their qualityof service for θ ∈ { , , . } dB. From Fig. 3, we note that about 18% of the devices (when θ = 10 ),51% of devices (when θ = 1 ), and 96% of devices (when θ = 0 . ) have success probabilities equal to . . SBSs/Km ( ) A ss o c i a t i on p r obab ili t y A (MBS Assoc. - Analysis)A (LOS SBS Assoc. Analysis)A (NLOS SBS Assoc. Analysis)A Total (Analysis)Simulation
Fig. 2:
Association probabilities as a function of λ for thehybrid spectrum network when λ = 2 MBSs/km , B = B = 1 , and d = 200 m. x (Reliability Threshold) - F p ( x ) ( M e t a D i s t r i bu t i on ) Beta Approx. ( =10)Beta Approx. ( =1)Beta Approx. ( =0.1)Simulation
Fig. 3:
The meta distribution vs. reliability threshold x for θ = θ D = θ =
10, 1, and 0.1 for the hybrid spectrum networkwhen B = B = 1 , and d = 200 m.
3) Coverage and Variance as a Function of SIR/SNR Threshold in Hybrid Spectrum Networks:
Fig.4 illustrates the standard success probability M , T and its variance M , T − M , T as a function of targetSIR/SNR threshold θ of devices in a hybrid spectrum network. As we can see in Fig. 4 that the simulationresults match the analytical results, however the slight gap is due to the Alzer’s inequality considered in Appendix C . This gap will be zero when Nakagami fading turns into Rayleigh fading as shown in the next figure. By examining Fig. 4, a numerical evaluation shows that the variance is maximized at θ = − dB where the success is M , T = 0 . . For moderate values of θ , there is a trade-off between maximizingcoverage or reducing variance because the variance first increases and then decreases while the coverageprobability is monotonically decreasing. For higher values of θ , lower coverage probabilities have lowervariance so its a low-reliability regime where more devices’ performances are spread around low coverageprobability. As such, the low values of θ provides a higher reliability regime.Fig. 5 illustrates the standard success probability M , T and the variance as a function of θ with Rayleighfading (i.e., m L = m N = 1 ). As we can see in Fig. 5 that the simulation results closely match the analyticalresults. The reason is that the approximation of the incomplete Gamma function (also referred to as Alzer’sinequality) becomes exact when m L becomes equal to unity. Subsequently, this figure explains the reasonfor the gap between the simulation and the analytical curves in Fig. 4. -20 -15 -10 -5 0 5 10 15 20 SIR/SNR Threshold [dB] C o v e r age P r obab ili t y ( M ) V a r i an c e Coverage Probability (Simulation)Coverage Probability (Analysis)Variance (Simulation)Variance (Analysis)
Fig. 4:
Coverage probability M , T and variance M , T − M , T as a function of θ considering Nakagami-m fading when B = B = 1 and d = 200 m. -20 -15 -10 -5 0 5 10 15 20 SIR/SNR Threshold [dB] C o v e r age P r obab ili t y V a r i an c e Coverage Prob. (Analysis)Coverage Prob. (Sim.)Variance (Analysis)Variance (Simulation)
Fig. 5:
Coverage probability M , T and variance M , T − M , T as a function of θ considering Rayleigh fading (i.e., m L = m N = 1 , when B = B = 1 and d = 200 m.
4) Coverage and Variance as a Function of the Number of Antenna Array Elements in Hybrid SpectrumNetworks:
Fig. 6 depicts the coverage probability and variance as a function of θ considering the numberof antenna array elements as N = 10 , 20, and 30 to show the effect of higher directional antenna gains.The general trends for the coverage probability and its variance are found to be the same as in previousfigures. The main observation is that although the coverage enhancement is not significant with increasingantenna elements, the reduction in the variance is noticeable which supports higher directional antennagains and the importance of analyzing the higher moments of the CSP.
5) Coverage and Variance as a Function of B in µ wave-only Networks: In Fig. 7, we study the effectof offloading devices from the MBS tier to the SBSs tier in terms of the coverage probability (which is themean reliability) and the variance of the CSP (or reliability). By offloading devices from the MBS tier tothe SBSs tier when B = 30 , the coverage probability M , T suffers due to the dual-hop transmission effectin wireless backhauled SBSs; however the variance of the results reduces which is a novel and positiveinsight. Another observation is that the variance of the CSP in µ wave-only network is high compared tothe hybrid network. This can be shown by comparing points V = (1 , . in Fig. 6 and V = (4 , . in Fig. 7, for the case of B = B = 1 . We noticed that the variance has decreased from 0.19 to 0.1when the SBS antenna array size is increased to N = 20 . This implies that the hybrid spectrum networkoutperforms the µ wave-only network due to the directional antenna gains. -20 -15 -10 -5 0 5 10 15 20 SIR/SNR Threshold [dB] C o v e r age P r obab ili t y V a r i an c e N=10N=20N=30N=10N=20N=30 V =(1,0.1) N=10,20,30N=10,20,30
Fig. 6:
Coverage probability M , T and variance M , T − M , T as a function of N for hybrid spectrum network when B = B = 1 , and d = 200 m. -20 -15 -10 -5 0 5 10 15 20 SIR/SNR Threshold [dB] C o v e r age P r obab ili t y V a r i a n ce Variance (B =1, B =30)Variance (B =1, B =1)Coverage Prob. (B =1, B =30)SimulationCoverage Prob. (B =1, B =1) V =(4,0.19) Fig. 7:
Coverage probability M , T and variance M , T − M , T as a function of θ for µ wave-only network when α = α =4 , B = 1 , and B = 1 and 30.
6) Mean Local Delay ( µ wave vs mm-wave SBSs): Fig. 8 depicts the mean local delay experienced bya typical device as a function of the SBSs density λ in a hybrid spectrum network. The mean local delayis the mean number of transmission attempts to successfully transmit a packet. The mean local delayincreases by increasing λ . After the SBS density reaches λ = 20 SBSs/km , the mean local delay staysconstant at value 1.11. This result can be intuitively explained as follows. When the mm-wave SBS densityis low, the typical device has a higher probability to connect to a MBS, i.e., the mean local delay of thenetwork results from only one hop communication (from the MBS to the device). However, when the λ increases, the typical device has a higher probability to connect to a mm-wave SBS, i.e., the networklocal delay results from two hops communication (from the MBS to the SBS then from the SBS to the device). Furthermore, the beamforming high directional gain steerable antennas will push more devicesto associate with SBSs thus a higher network delay is observed. Fig. 9 shows that, all else being equal,the mean local delay of the hybrid spectrum network is lower than that of the µ wave-only network. SBS/KM ( ) M ean Lo c a l de l a y Mean Local delay G =0 dBMean Local delay G =10 dBMean Local delay G =20 dB Fig. 8:
Mean local delay M − , T as a function of λ for thehybrid spectrum network when λ = 2 MBS/Km , B = 1 , B = 10 , α = 4 , d = 200 m, and θ = θ D = θ = -10 dB.
10 20 30 40 50 60 70 80 90 100 Density M ean Lo c a l de l a y ( M - ) Mean Local Delay ( = =4)Mean Local Delay ( = =3) Fig. 9:
Mean local delay M − , T as a function of λ for the µ wave-only network when λ = 2 MBS/KM , B = 1 and B = 10 , α = α = 3 and 4, and θ = θ D = θ = -10 dB. Fig. 9 depicts the mean local delay for a µ wave-only network as a function of λ . When λ increasesthe mean local delay of the total network increases again due to the increase in interference which is notthe case in the hybrid spectrum network. The network mean local delay in the case of α = α = 3 ishigher than that in the case of α = α = 4 due to higher path loss degradation for higher PLEs.
7) The Meta Distribution of the Achievable Data Rate in Hybrid Spectrum Networks:
Fig. 10 depictsthe meta distribution of the data rate in hybrid spectrum networks as a function of reliability x for differentnumber of antenna elements N = 10 , 20, 40, and 50 with rate threshold T = 1 Gbps. As shown in Fig. 10,the fraction of devices achieving a required rate increases as the number of antennas elements increases.In other words, increasing the number of antenna elements of SBSs has a positive effect on the achievablerate and its meta distribution. This insight helps 5G cellular network operators to find the most efficientoperating antenna configuration to achieve certain reliability for certain 5G applications.
8) The Meta Distribution in a Microwave-only Network:
In Fig. 11, we validate our analysis bydepicting the exact (Gil-Pelaez) meta distribution in a µ wave-only network defined in Eq. (13), andthe beta approximation for the meta distribution defined in Eq. (8). Our simulation result provides anexcellent match for a wide range of θ values and this validates the correctness of our analytical model.Fig. 11 also serves as an illustration of the meta distribution of the SIR of a typical device in a µ wave-only network. We note that about 23% of devices (when θ = 10 ), 72% of devices (when θ = 1 ), and 98% ofdevices (when θ = 0 . ) have reliability, i.e., success probability, equal to . . x (Reliability Threshold) - F R () ( D a t a r a t e M e t a D i s t r i bu t i on ) N=10N=20N=40N=50N=10,20,40,50
Fig. 10:
Meta distribution of the achievable data rate asa function of reliability x for different number of antennaelements N with rate threshold T = 1 Gbps. x (Reliability Threshold) - F p ( x ) ( M e t a D i s t r i bu t i on ) SimulationsGil-Pelaez ( =10)Gil-Pelaez ( =1)Gil-Pelaez ( =0.1)Beta Approx.
Fig. 11:
The meta distribution as a function of reliability x for θ = θ D = θ =
10, 1, and 0.1 for SBSs in a µ wave-onlynetwork when B = B = 1 and α = α = 4 . IX. C
ONCLUSION
This paper characterizes the meta distributions of the SIR/SNR and data rate of a typical device in ahybrid spectrum network and µ wave-only network. The meta distribution is evaluated first by formulatingand then characterizing the moments of the CSP of a typical device in the hybrid network. Importantperformance metrics such as the mean local delay, coverage probability, network jitter, and variance ofthe CSP (or reliability) are studied. Numerical results demonstrate the significance of evaluating the metadistribution which requires a systematic evaluation of the generalized moment of order b that helps inevaluating network metric such as coverage probability when b = 1 , mean local delay when b = − ,network jitter using b = − and b = − , etc. Numerical results provide valuable insights related to thereliability and latency of the hybrid spectrum network and µ wave-only network. These insights wouldhelp 5G cellular network operators to find the most efficient operating antenna configuration to achievecertain reliability for certain 5G applications.A PPENDIX
C: P
ROOF OF L EMMA b th moment of the CSP of a typical device served by the mm-wave SBS can be derived as: M b, ( θ D ) = E l (cid:20) P ( n = 2 | L ,min = l ) | {z } ¯ A ( l ) P s, ( θ D ) b (cid:21) = E l (cid:20) ¯ A ( l ) ( p L P s, ,L ( θ D ) + p N P s, ,N ( θ D )) b (cid:21) , ( a ) = E l " ¯ A ( l ) p L Γ (cid:16) m L , m L Ω L ν L (cid:17) Γ( m L ) + p N Γ (cid:16) m N , m N Ω N ν N (cid:17) Γ( m N ) b , = E l " ( ¯ A ,L ( l ,L ) + ¯ A ,N ( l ,N )) × p L Γ (cid:16) m L , m L Ω L ν L (cid:17) Γ( m L ) + p N Γ (cid:16) m N , m N Ω N ν N (cid:17) Γ( m N ) b , ( b ) = E l " ¯ A ,L ( l ,L ) p L Γ (cid:16) m L , m L Ω L ν L (cid:17) Γ( m L ) b + E l " ¯ A ,N ( l ,N ) p N Γ (cid:16) m N , m N Ω N ν N (cid:17) Γ( m N ) b , ( c ) = E l " ¯ A ,L ( l ,L ) p bL − γ (cid:16) m L , m L Ω L ν L (cid:17) Γ( m L ) b + E l " ¯ A ,N ( l ,N ) p bN − γ (cid:16) m N , m N Ω N ν N (cid:17) Γ( m N ) b , ( d ) ≈ E l " ¯ A ,L ( l ,L ) p bL (cid:0) − [1 − e − ζ L ν L ] m L (cid:1) b + E l " ¯ A ,N ( l ,N ) p bN (cid:0) − [1 − e − ζ N ν N ] m N (cid:1) b , ( e ) = E l " ¯ A ,L ( l ,L ) p bL b X k =0 (cid:18) bk (cid:19) (cid:0) − [1 − e − ζ L ν L ] m L (cid:1) k + E l " ¯ A ,N ( l ,N ) p bN b X k =0 (cid:18) bk (cid:19) (cid:0) − [1 − e − ζ N ν N ] m N (cid:1) k , ( f ) = E l " ¯ A ,L ( l ,L ) p bL b X k =0 m L k X ¨ k =0 (cid:18) bk (cid:19)(cid:18) m L k ¨ k (cid:19) ( − ¨ k + k e − ζ L ν L ¨ k + E l " ¯ A ,N ( l ,N ) p bN b X k =0 m N k X ¨ k =0 (cid:18) bk (cid:19)(cid:18) m N k ¨ k (cid:19) ( − ¨ k + k e − ζ N ν N ¨ k , where (a) follows from substituting the value of P s, ,L ( θ D ) and P s, ,N ( θ D ) from Eq. (16) and Eq. (17),respectively, (b) follows from l = r α ,L , D and l = r α ,N , D and the considered blockage model where p L = 1 when mm-wave intended link distance r , D < d and p N = 1 when mm-wave intended link distance r , D > d , (c) follows from Γ( s ) = γ ( s, x ) + Γ( s, x ) , (d) follows from the CDF of Gamma random variablewhich can be tightly upper bounded by γ (cid:16) m L , mL Ω L ν L (cid:17) Γ( m L ) < [1 − e − ζ L ν L ] m L [40], where ζ L ∆ = m L ( m L !) − /m L , ν L ∆ = θ D r α ,L , D σ P G , ζ N ∆ = m N ( m N !) − /m N , and ν N ∆ = θ D r α ,N , D σ P G [16]. The steps in (e) and (f) are done byfollowing the binomial expansion theorem. Finally, the Lemma 4 follows from de-conditioning on l andusing the definitions ¨ ν L ∆ = ν L r α ,L , D = ν L l = θ D σ P G and ¨ ν N ∆ = ν N r α ,N , D = ν N l = θ D σ P G .A PPENDIX
D: P
ROOF OF L EMMA b th moment of the CSP of a typical device when associated to µ wave MBS is derived as follows: M b, ( θ D ) = E l (cid:20) P ( n = 1 | L ,min = l ) | {z } ¯ A ( l ) P s, ( θ D ) b (cid:21) ( a ) = E l (cid:20) ¯ A ( l ) Y y ,i ∈ Φ \{ y , } (cid:16) θ D (cid:16) r , D k y ,i k (cid:17) α (cid:17) b (cid:21) , ( b ) = E l (cid:20) ¯ A ( l ) exp Z ∞ r − λ π (cid:20) − (cid:16) θ D (cid:16) ry (cid:17) α (cid:17) b (cid:21) y d y (cid:21) ( c ) = E l (cid:20) ¯ A ( l ) exp Z ∞ l α − λ π (cid:20) − (cid:16) θ D l y α (cid:17) b (cid:21) y d y , ( d ) = E l (cid:20) ¯ A ( l ) exp Z − λ π (cid:20) − θ D v ) b (cid:21) v − y α d v ! (cid:21) , ( e ) = E l (cid:20) ¯ A ( l ) exp − λ πl α α Z (cid:20) − θ D v ) b (cid:21) v α +1 d v , where (a) follows from taking expectation over l = r α and considering the conditional associationprobability for the typical device connecting to the MBSs tier given in Lemma (1) and substituting thevalue of P s, ( θ D ) from Eq. (21). In step (b) we apply PGFL of the PPP [41, Chapter 4]. Step (c) followsfrom averaging over l . In step (d), we use the change of variable v = l y α , dy = − α l y − α − dv = − α v − ydv ,when y = l α → v = 1 and when y = ∞ → v = 0 and we swap the integral limits and multiply by − ,(e) follows from y = l α /v α and doing some mathematical manipulations.A PPENDIX
E: P
ROOF OF L EMMA b th moment of the CSP P s,k ( θ D ) of thetypical device when it is served by the k th tier is given as follows: M b, k’ ( θ D ) = E r k, D (cid:20) P ( n = k | r k, D ) P s,k ′ ( θ D ) b (cid:21) , ( a ) = E r k, D (cid:20) Y j = k e − πλ j ( ˆ P jk ˆ B jk ) /αj r × Y y k,i ∈ Φ k \{ y k, } (cid:16) θ D (cid:16) r k, D k y k,i k (cid:17) α k (cid:17) b (cid:21) , ( b ) = E r k, D (cid:20) Y j = k e − πλ j ( ˆ P jk ˆ B jk ) /αj r × exp Z ∞ r k, D − λ k π (cid:20) − (cid:16) θ D (cid:16) r k, D y (cid:17) α k (cid:17) b (cid:21) y d y (cid:21) , ( c ) = Z ∞ λ k πre − λ k πr e − P j = k λ j ( ˆ P jk ˆ B jk ) /αj πr × exp Z ∞ r − λ k π (cid:20) − (cid:16) θ D (cid:16) ry (cid:17) α k (cid:17) b (cid:21) y d y d r, ( d ) = Z ∞ e − q e − q P j = k ˆ λ jk ( ˆ P jk ˆ B jk ) /αj × exp − q Z (cid:20) − θ D v α k ) b (cid:21) v − d v ! d q, ( e ) = Z ∞ e − q e − q P j = k ˆ λ jk ( ˆ P jk ˆ B jk ) /αj × exp − q Z ∞ (cid:20) − (cid:0) θ D u − α k / (cid:1) b (cid:21) d u ! d q, ( f ) = Z ∞ e − q e − q P j = k ˆ λ jk ( ˆ P jk ˆ B jk ) /αj × exp (cid:18) − q (cid:20) F ( b, − α k ; 1 − α k ; − θ D ) − (cid:21)(cid:19) d q, = 1 P j = k ˆ λ jk ( ˆ P jk ˆ B jk ) /α j + F ( b, − α k ; 1 − α k ; − θ D ) . where (a) follows from considering the conditional association probability for the typical device connectingto the k th tier given in Eq. (30). In step (b), we apply PGFL of the PPP [41, Chapter 4]. Step (c)follows from averaging over r k, D , step (d) is by using variable substitution q = πλ k r and v = r/y . Instep (e), we perform variable substitution v = u ( ˆ P jk ˆ B jk ) − /α j and step (f) follows from the fact that F ( b, − α ; 1 − α ; − θ ) ≡ R ∞ (1 − θh − α/ ) b ) d h .R EFERENCES [1] H. Ibrahim, H. Tabassum, and U. T. Nguyen, “Meta distribution of SIR in dual-hop Internet-of-Things (IoT) networks,” IEEEInternational Conference on Communications (ICC) , May 2019.[2] 3GPP, “Study on new radio (NR) access technology - physical layer aspects,”
TR 38.802 (Rel. 14) , 2017.[3] H. Elshaer, M. N. Kulkarni, F. Boccardi, J. G. Andrews, and M. Dohler, “Downlink and uplink cell association with traditionalmacrocells and millimeter wave small cells,”
IEEE Transactions on Wireless Communications , vol. 15, no. 9, pp. 6244–6258, 2016.[4] O. Semiari, W. Saad, M. Bennis, and M. Debbah, “Integrated millimeter wave and sub-6 GHz wireless networks: A roadmap for jointmobile broadband and ultra-reliable low-latency communications,”
IEEE Wireless Communications , 2019.[5] H. Ji, S. Park, J. Yeo, Y. Kim, J. Lee, and B. Shim, “Ultra-reliable and low-latency communications in 5G downlink: Physical layeraspects,”
IEEE Wireless Communications , vol. 25, no. 3, pp. 124–130, 2018.[6] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. Soong, and J. C. Zhang, “What will 5G be?”
IEEE Journal onselected areas in communications , vol. 32, no. 6, pp. 1065–1082, 2014.[7] P. Wang, Y. Li, L. Song, and B. Vucetic, “Multi-gigabit millimeter wave wireless communications for 5G: From fixed access tocellular networks,”
IEEE Communications Magazine , vol. 53, no. 1, pp. 168–178, 2015.[8] M. Polese, M. Giordani, M. Mezzavilla, S. Rangan, and M. Zorzi, “Improved handover through dual connectivity in 5G mmwavemobile networks,”
IEEE Journal on Selected Areas in Communications , vol. 35, no. 9, pp. 2069–2084, 2017.[9] M. Haenggi, “The meta distribution of the SIR in Poisson bipolar and cellular networks,”
IEEE Transactions on WirelessCommunications , vol. 15, no. 4, pp. 2577–2589, 2016.[10] M. Bennis, M. Debbah, and H. V. Poor, “Ultrareliable and low-latency wireless communication: Tail, risk, and scale,”
Proceedingsof the IEEE , vol. 106, no. 10, pp. 1834–1853, 2018.[11] S. S. Kalamkar and M. Haenggi, “Per-link reliability and rate control: Two facets of the SIR meta distribution,”
IEEE WirelessCommunications Letters , 2019.[12] M. Salehi, A. Mohammadi, and M. Haenggi, “Analysis of D2D underlaid cellular networks: SIR meta distribution and mean localdelay,”
IEEE Transactions on Communications , vol. 65, no. 7, pp. 2904–2916, 2017.[13] M. Salehi, H. Tabassum, and E. Hossain, “Meta distribution of sir in large-scale uplink and downlink NOMA networks,”
IEEETransactions on Communications , 2018.[14] N. Deng and M. Haenggi, “The energy and rate meta distributions in wirelessly powered D2D networks,”
IEEE Journal on SelectedAreas in Communications , vol. 37, no. 2, pp. 269–282, 2019.[15] M. Di Renzo, “Stochastic geometry modeling and analysis of multi-tier millimeter wave cellular networks,”
IEEE Transactions onWireless Communications , vol. 14, no. 9, pp. 5038–5057, 2015.[16] T. Bai and R. W. Heath, “Coverage and rate analysis for millimeter-wave cellular networks,”
IEEE Transactions on WirelessCommunications , vol. 14, no. 2, pp. 1100–1114, 2015.[17] E. Turgut and M. C. Gursoy, “Coverage in heterogeneous downlink millimeter wave cellular networks,”
IEEE Transactions onCommunications , vol. 65, no. 10, pp. 4463–4477, 2017.[18] N. Deng, Y. Sun, and M. Haenggi, “Success probability of millimeter-wave D2D networks with heterogeneous antenna arrays,” in
Wireless Communications and Networking Conference (WCNC) . IEEE, 2018, pp. 1–5.[19] S. Singh, M. N. Kulkarni, A. Ghosh, and J. G. Andrews, “Tractable model for rate in self-backhauled millimeter wave cellularnetworks,”
IEEE Journal on Selected Areas in Communications , vol. 33, no. 10, pp. 2196–2211, 2015.[20] N. Deng and M. Haenggi, “A fine-grained analysis of millimeter-wave device-to-device networks,”
IEEE Trans. Commun , vol. 65,no. 11, pp. 4940–4954, 2017. [21] A. Ghosh, T. A. Thomas, M. C. Cudak, R. Ratasuk, P. Moorut, F. W. Vook, T. S. Rappaport, G. R. MacCartney, S. Sun, and S. Nie,“Millimeter-wave enhanced local area systems: A high-data-rate approach for future wireless networks,” IEEE Journal on SelectedAreas in Communications , vol. 32, no. 6, pp. 1152–1163, 2014.[22] J. G. Andrews, T. Bai, M. N. Kulkarni, A. Alkhateeb, A. K. Gupta, and R. W. Heath, “Modeling and analyzing millimeter wavecellular systems,”
IEEE Transactions on Communications , vol. 65, no. 1, pp. 403–430, 2017.[23] H. Ibrahim, H. ElSawy, U. T. Nguyen, and M.-S. Alouini, “Modeling virtualized downlink cellular networks with ultra-dense smallcells,” in
IEEE International Conference on Communications (ICC) , 2015, pp. 5360–5366.[24] H. Ibrahim, W. Bao, and U. T. Nguyen, “Data rate utility analysis for uplink two-hop internet-of-things networks,”
IEEE Internet ofThings Journal , 2018.[25] H. Ibrahim, H. ElSawy, U. T. Nguyen, and M.-S. Alouini, “Mobility-aware modeling and analysis of dense cellular networks withC-plane/U-plane split architecture,”
IEEE Transactions on Communications , vol. 64, no. 11, pp. 4879–4894, 2016.[26] G. R. MacCartney, J. Zhang, S. Nie, and T. S. Rappaport, “Path loss models for 5G millimeter wave propagation channels in urbanmicrocells.” in
Globecom , 2013, pp. 3948–3953.[27] T. A. Khan, A. Alkhateeb, and R. W. Heath, “Millimeter wave energy harvesting,”
IEEE Transactions on Wireless Communications ,vol. 15, no. 9, pp. 6048–6062, 2016.[28] W. Yi, Y. Liu, and A. Nallanathan, “Modeling and analysis of D2D millimeter-wave networks with poisson cluster processes,”
IEEETransactions on Communications , vol. 65, no. 12, pp. 5574–5588, 2017.[29] M. Nakagami, “The m-distribution: A general formula of intensity distribution of rapid fading,” in
Statistical methods in radio wavepropagation . Elsevier, 1960, pp. 3–36.[30] M. K. Simon and M.-S. Alouini,
Digital communication over fading channels . John Wiley & Sons, 2005, vol. 95.[31] K. Venugopal, M. C. Valenti, and R. W. Heath, “Device-to-device millimeter wave communications: Interference, coverage, rate, andfinite topologies,”
IEEE Transactions on Wireless Communications , vol. 15, no. 9, pp. 6175–6188, 2016.[32] T. Bai, R. Vaze, and R. W. Heath, “Analysis of blockage effects on urban cellular networks,”
IEEE Transactions on WirelessCommunications , vol. 13, no. 9, pp. 5070–5083, 2014.[33] J. Gil-Pelaez, “Note on the inversion theorem,”
Biometrika
IEEE Transactionson Wireless Communications , vol. 15, no. 3, pp. 2130–2143, 2016.[36] Y. Wang, M. Haenggi, and Z. Tan, “The meta distribution of the SIR for cellular networks with power control,”
IEEE Transactionson Communications , vol. 66, no. 4, pp. 1745–1757, 2018.[37] M. Kronenburg, “The binomial coefficient for negative arguments,” arXiv preprint arXiv:1105.3689 , 2011.[38] H.-S. Jo, Y. J. Sang, P. Xia, and J. G. Andrews, “Heterogeneous cellular networks with flexible cell association: A comprehensivedownlink SINR analysis,”
IEEE Transactions on Wireless Communications , vol. 11, no. 10, pp. 3484–3495, 2012.[39] Y. Wang, M. Haenggi, and Z. Tan, “Sir meta distribution of k-tier downlink heterogeneous cellular networks with cell range expansion,”
IEEE Transactions on Communications , 2018.[40] H. Alzer, “On some inequalities for the incomplete gamma function,”
Mathematics of Computation of the American MathematicalSociety , vol. 66, no. 218, pp. 771–778, 1997.[41] M. Haenggi,