The meta distribution is first introduced by M. Haenggi  to provide a fine-grained reliability and latency analysis of fifth-generation and beyond (B5G) wireless networks with ultra-reliable and low latency communication (URLLC) requirements [2, 3]. Meta distribution is defined as the distribution of the conditional success probability (CSP) of the transmission 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 users can achieve x% transmission success probability?” whereas the conventional success probability answers questions such as “What fraction of users experiences transmission success?” . In addition to the standard coverage (or success) probability which is equivalent to the mean of CSP, the meta distribution can capture important network performance measures such as the mean of the local transmission delay, variance of the local transmission delay (referred to as network jitter), and variance of the CSP which depicts the variation of the users’ performance from the mean coverage probability. Evidently, the standard coverage probability provides limited information about the performance of a typical wireless network [4, 5, 6].
To further illustrate the significance of the meta distribution, let us consider 50% of the devices achieve 10% reliability and 50% achieve 99% reliability. Then, the standard mean coverage probability is 54.5%. On the other hand, if 100% of the devices achieve 54.5% reliability, the standard mean coverage probability is also 54.5%. However, the two scenarios are very different in terms of the user experiences. Cellular operators are typically interested in the performance of the “5% user percentile”, which is the performance level that 95% of the users achieve. The meta distribution directly reveals this information, while the standard coverage probability does not reveal any information about it.
Millimeter-wave (mm-wave) networks are among one of the key enablers of 5G/B5G networks  and will coexist with sub-6GHz frequencies [8, 9]. In this article, we develop a framework to characterize the meta distributions of SIR/SNR as well as data rate in the coexisting sub-6GHz (which we term “microwave” spectrum throughout the paper) and mm-wave cellular network. We consider a two-tier cellular network architecture where tier 1 consists of macro base stations (MBSs) operating on the microwave (wave) spectrum and tier 2 is composed of wireless-backhauled small base stations (SBSs) with mm-wave transmissions at access links, as illustrated in Fig. 1.
I-a Related Work and Motivation
A variety of research works studied the coverage probability of mm-wave only cellular networks [10, 11, 12]. For instance, Di Renzo et al.  proposed a general mathematical model to analyze multi-tier mm-wave cellular networks. Bai et al.  derived the coverage and rate performance of mm-wave cellular networks. They used a distance dependent line-of-sight (LOS) probability function where the locations of the LOS and non-LOS (NLOS) BSs are modeled as two independent non-homogeneous Poisson point processes, to which different path loss models are applied. The authors assume independent Nakagami fading for each link. Different parameters of Nakagami fading are assumed for LOS and NLOS links. Turgut and Gursoy  investigated heterogeneous downlink mm-wave cellular networks consisting of tiers of randomly located BSs where each tier operates in a mm-wave frequency band. They derived coverage probability for the entire network using tools from stochastic geometry. They used Nakagami fading to model small scale fading. Deng et al.  derived the success probability at the typical receiver in mm-wave device-to-device (D2D) networks. The authors modeled fading channel power as Nakagami fading and incorporated directional beamforming.
Some recent studies analyzed the success probability of coexisting wave and mm-wave cellular networks. A hybrid cellular network was considered by Singh et al. 
to estimate the uplink-downlink coverage and rate distribution of self-backhauled mm-wave networks. Elshaer et al. developed an analytical model to characterize decoupled uplink and downlink cell association strategies. The authors showed the superiority of this technique compared to the traditional coupled association in a network with traditional MBSs coexisting with denser mm-wave SBSs. Singh et al.  and Elshaer et al.  modeled the fading power as Rayleigh fading to enable better tractability.
Different from previous research in [12, 11, 8], Deng and Haenggi  recently studied the meta distribution of the SIR in mm-wave only single-hop D2D networks using the Poisson bipolar model and simplified Rayleigh fading channels for analytical tractability.
In this paper, we develop a stochastic geometry framework to analyze the meta distributions of the SIR/SNR and data rate in coexisting wave and mm-wave networks where a user can opportunistically associate to with wireless backhauled SBSs. The framework will enable cellular operators to analyze a wider range of system performance metrics including coverage probability, data rate, reliability, mean local delay, and network jitter. A summary of our contributions is listed herein:
We characterize the meta distribution of the data rate and SIR/SNR in a network where the MBSs operate on the wave spectrum while the SBSs communicate with users on the mm-wave spectrum. The SBSs are connected to MBSs via a wave wireless backhaul (Fig. 1). We term the coexisting wave and mm-wave network a hybrid spectrum network. The wave transmissions are interference limited and mm-wave transmission are noise limited111Given highly directional beams and high sensitivity to blockage, recent studies showed that mm-wave networks can be considered as noise limited rather than interference limited [16, 14, 17].; therefore, we have the meta-distribution of SIR and SNR in wave and mm-wave channels, respectively. To model the LOS nature of mm-wave transmissions, we consider the versatile Nakagami-m fading channel model.
We characterize the CSP (which is equivalent to reliability) and and its moment for two scenarios, i.e., (1) when a typical user associates with wave MBS for direct transmission and (2) when a typical user associates with mm-wave SBS for dual-hop transmission (access and backhaul transmission). Using the novel expressions of the moments in the aforementioned scenarios, we derive a novel expression for the cumulative moment of the considered hybrid spectrum network.
Using the cumulative moment , we characterize the exact and approximate meta distributions of the data rate and downlink SIR/SNR of the typical user. Since the expression of relies on a Binomial expansion of power , the results for the meta-distribution requiring complex values of are obtained by applying Newton’s Generalized Binomial Theorem.
Using the cumulative moment , we characterize important metrics such as coverage probability, mean local delay, variance of the local delay (network jitter), and variance of the reliability. For metrics requiring negative values of , we apply the Binomial Theorem for negative integers.
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 all transmissions are conducted in wave spectrum.
Closed-form results are provided for special cases and asymptotic scenarios.
We validate analytical results using Monte-Carlo simulations. Numerical results give valuable insights related to the performance metrics such as the reliability, mean local delay, variance of CSP, and standard success probability of a user. For example, the mean local delay increases with the SBS density in wave-only networks; whereas, it stays constant in the hybrid spectrum networks.
The remainder of the article is organized as follows. In Section II, we describe the system model and assumptions. In Section III, we provide mathematical preliminaries of the meta distribution. In Section IV, we characterize the association probabilities of a typical user and formulate the meta distribution of the SIR/SNR of a user in the hybrid spectrum network. In Section V, we characterize the CSP and its moment for direct, access, and backhaul transmissions. Finally, we derive the exact and approximate meta distributions of the SIR/SNR and data rate in a hybrid spectrum network as well as wave-only network in Section VI. Finally, Section VIII presents numerical results and Section IX concludes the article.
Ii System Model and Assumptions
In this section, we describe the network deployment model (Section II-A), antenna model (Section II-B), channel model (Section II-C), user association criteria (Section II-D), and SNR/SIR models for access and backhaul transmissions (Section II-E).
Ii-a Network Deployment and Spectrum Allocation Model
We assume a two-tier network architecture in which the locations of the MBSs and SBSs are modeled as a two-dimensional (2D) homogeneous Poisson point process (PPP) of density , where is the location of MBS (when ) or the SBS (when ). Let the MBS tier be tier 1 () and the SBSs constitute tier 2 (). Let denotes the set of users. The locations of users in the network are modeled as independent homogeneous PPP with density , where is the location of the user. We assume that as in [18, 19, 20]. We consider a typical outdoor user which is located at the origin and is denoted by and its tagged BS is denoted by , i.e., tagged MBS (when ) or tagged SBS (when ). All BSs in the tier transmit with the same transmit power in the downlink. A list of the key mathematical notations is given in Table I.
We assume that a portion of the frequency band is reserved for the access transmission and the rest is reserved for the backhaul transmission, where , and denote the total available wave spectrum and mm-wave spectrum, respectively, and . Determining the optimal spectrum allocation ratio will be studied in our future work.
|;||PPP of BSs of tier; PPP of users||;||Density of BSs of tier; density of users|
|Transmit power of BSs in tier||Association bias for BSs of tier|
|Path loss exponent of MBS tier; LOS SBS; NLOS SBS||omnidirectional antenna gain of wave MBSs|
|;;||Main lobe gain; side lobe gain; and 3 dB beamwidth for mm-wave SBS||Gamma fading channel gain for mm-wave SBSs|
|Rayleigh fading channel gain||Nakagami-m fading parameter where denotes LOS and NLOS transmission links|
|;||Mm-wave blockage LOS probability; NLOS probability||Predefined SIR/SNR threshold|
|Meta distribution of SIR/SNR||Conditional success probability (CSP)|
|The moment of||;;||Association Probability with wave MBS; LOS mm-wave SBS; NLOS mm-wave SBS|
Ii-B Antenna Model
We assume that all MBSs are equipped with omnidirectional antennas with gain denoted by dB. We consider SBSs and users are equipped with directional antennas with sectorized gain patterns as in [17, 10, 15] to approximate the actual antenna pattern. The sectorized gain pattern is given by:
where subscript denotes for SBSs and users, respectively. Considering a uniform planar square antenna array with elements, the antenna parameters of a uniform planar square antenna array can be given as in , i.e., is the main lobe antenna gain, is the side lobe antenna gain, is the angle of the boresight direction, and is the main lobe beam width. A perfect beam alignment is assumed between a user and its serving SBS  .The antenna beams of the desired access links are assumed to be perfectly aligned, i.e., the direction of arrival (DoA) between the transmitter and receiver is known a priori at the BS and the effective gain on the intended access link can thus be denoted as . This can be done by assuming that the serving mm-wave SBS and user can adjust their antenna steering orientation using the estimated angles of arrivals. The analysis of the alignment errors on the desired link is beyond the scope of this work.
Ii-C Channel Model
Ii-C1 Path-Loss Model
The signal power decay is modeled as , where is the path loss for a typical receiver located at a distance from the transmitter and is the path loss exponent (PLE). Let denotes the path loss of a typical user associated with the MBS tier, where is the PLE. Similarly, denotes the path loss of a typical user associated with the SBS tier where is the PLE in the case of LOS and is the PLE in the case of NLOS. It has been shown that mm-wave LOS and NLOS conditions have markedly different PLEs . Also, we consider the near-field path loss factor at 1 m , i.e., different path loss for different frequencies at the reference distance.
Ii-C2 Fading Model
For outdoor mm-wave channels, we consider a versatile Nakagami-m fading channel model due to its analytical tractability and following the previous line of research studies [11, 12, 22, 23, 13]. Nakagami-m fading is a general and tractable model to characterize mm-wave channels. Also, in several scenarios, Nakagami-m can approximate the Rician fading which is commonly used to model the LOS transmissions but not tractable for meta distribution modeling [24, 25]. The fading parameter where denotes LOS and NLOS transmission links, respectively, and the mean fading power is denoted by . The fading channel power
follows a gamma distribution given as, , where is the Gamma function, is the shape (or fading) parameter, and is the scale parameter. That is, we consider for the LOS links and for the NLOS links. Rayleigh fading is a special case of Nakagami-m for . Due to the NLOS nature of wave channels, we assume Rayleigh fading with power normalization, i.e., the channel gain , is independently distributed with the unit mean.
Ii-C3 Blockage Model for Mm-wave Access Links
For mm-wave channels, LOS transmissions are vulnerable to significant penetration losses ; thus LOS transmissions can be blocked with a certain probability. Following [11, 26, 22, 27], we consider the actual LOS region of a user as a fixed LOS ball referred to as ”equivalent LOS ball”. For the sake of mathematical tractability, we consider a distance dependent blockage probability that a mm-wave link of length observes, i.e., the LOS probability if the mm-wave desired link length is less than and otherwise. That is, SBSs within a LOS ball of radius are marked LOS with probability , while the SBSs outside that LOS ball are marked as NLOS with probability . Note that we will drop the notation in both and from this point onwards and we will use only and , respectively.
Ii-D Association Mechanism
Each user associates with either a MBS or a SBS depending on the maximum biased received power in the downlink. The association criterion at the typical user can be written mathematically as follows:
where , , , and denote the transmission power, biasing factor, effective antenna gain, and near-field path loss at 1 m of the intended link, respectively, in the corresponding tier (which is determined by the index in the subscript). Let be the minimum path loss of a typical user from a BS in the tier. When a user associates with a mm-wave SBS in tier-2, i.e., , the antenna gain of the intended link is , otherwise , where is defined as the omnidirectional antenna gain of MBSs and is the user antenna gain while operating in wave spectrum. On the other hand, the SBS associates with a MBS offering the maximum received power in the downlink.
Ii-E SNR/SIR Models for Access and Backhaul Transmissions
The user associates to either a MBS for direct transmission or a SBS for dual-hop transmission. The first link (backhaul link) transmissions occur on the wave spectrum between MBSs and SBSs and the second link (access link) transmissions take place in the mm-wave spectrum between SBSs and users. Let denotes the predefined SIR threshold for SBSs in the backhaul link and denotes the predefined SIR/SNR threshold for users. Throughout the paper, we use subscripts “”, “”, “”, “”, “” to denote backhaul link, access link, direct link, user, and backhaul, respectively.
Ii-E1 Backhaul Transmission
The of a typical SBS associated with a MBS can be modeled as:
where denotes the backhaul interference received at a SBS from MBSs that are scheduled to transmit on the same resource block excluding the tagged MBS. Then,
Ii-E2 Direct Transmission
The of a typical user associated directly with a MBS is modeled as:
where denotes the interference received at a typical user from MBSs excluding the tagged MBS. Then can be calculated as:
Ii-E3 Access Transmission
The SNR of a typical user associated with a mm-wave SBS is modeled as:
where is the near-field path loss at 1 m for mm-wave channels, and is the variance of the additive white Gaussian noise at the user receiver. Given highly directional beams and high sensitivity to blockage, recent studies showed that mm-wave networks are typically noise limited [16, 14, 17].
Iii The Meta Distribution: Mathematical Preliminaries
In this section, we define the meta distribution of the SIR of a typical user and highlight exact and approximate methods to evaluate the meta distribution.
Definition 1 (Meta Distribution of the SIR and CSP).
where, conditioned on the locations of the transmitters and that the desired transmitter is active, the CSP of a typical user  can be given as where is the desired .
Physically, the meta distribution provides the fraction of the active links whose CSP (or reliability) is greater than the reliability threshold . Given denotes the moment of , i.e., , , the exact meta distribution can be given using the Gil-Pelaez theorem  as :
where is imaginary part of and denotes the imaginary moments of , i.e., , . Using moment matching techniques and taking
, the meta distribution of the CSP can be approximated using the Beta distribution as follows:
where and are the first and the second moments, respectively; is the regularized incomplete Beta function and is the Beta function.
Iv The Meta Distribution of the SIR/SNR in Hybrid Spectrum Networks
To characterize the meta distribution of the SIR/SNR of a typical user that can associate with either a wave MBS with probability or with a wireless backhauled mm-wave SBS with probability , the methodology of analysis is listed as follows:
Derive the probability of a typical user associating with wave MBSs , LOS mm-wave SBSs , and NLOS mm-wave SBSs where (Section IV-A).
Formulate the meta distribution of the SIR/SNR of a user in the hybrid network () considering the direct link and dual-hop link with wireless backhaul transmission (Section IV-B).
Formulate the CSP () and its moment (Section IV-B).
Derive the CSP at backhaul link , CSP at access link , and CSP at direct link . Derive the moments of CSPs, i.e., , , and for backhaul link, access link, and direct link transmissions, respectively (Section V).
Obtain the meta distributions of SIR/SNR and data rate in hybrid spectrum network using Gil-Pelaez inversion and the Beta approximation (Section VI).
Iv-a Association Probabilities in Hybrid Spectrum Networks
In this subsection, we characterize the probabilities with which a typical user associates with wave MBSs () or mm-wave SBSs (). The results are presented in the following.
Lemma 1 (The Probability of Associating with mm-wave SBSs).
The probability of a typical user to associate with a mm-wave SBS, using the association scheme in Eq. (2), can be expressed as:
where and . Subsequently, the probability of a user to associate with a wave MBS can be given as . The conditional association probability for a typical user to associate with SBS is as follows:
A closed-form expression of can be derived for a case of practical interest as follows.
When , , and , then can be given in closed-form as follows:
where is the error function, and and .
It can be seen from Corollary 1 that when the number of antenna elements goes to infinity, i.e., , , then can be simplified as which shows that association probability to MBS will be very small. Similar insights can be extracted for other parameters.
In order to derive the moment of CSP on an access link when a user associates with a SBS (the CSP will be discussed later in Lemma 4), we have to derive the probability of a user to associate with LOS SBS and NLOS SBS which are defined follows.
Lemma 2 (The Probability of Associating with LOS and NLOS mm-wave SBSs).
When a typical user associates with the mm-wave SBS tier, this typical user has two possibilities to connect to (a) a LOS mm-wave SBS with association probability and (b) a NLOS mm-wave SBS with association probability which are characterized, respectively, as follows:
where and are the conditional probabilities with which a typical user may associate to the LOS and NLOS mm-wave SBSs, respectively, and are defined as follows:
where , and .
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:
When the number of antenna elements at mm-wave SBSs increases, i.e., , , , and , then . The association probabilities can be given in closed-form as follows:
where is the Kummer Confluent Hypergeometric function.
An interesting insight from Corollary 2 can be seen when the intensity of SBSs or is large, the probability of association to LOS SBSs becomes almost 1. On the other hand, when or is small, thus becomes almost 1.
Iv-B Formulation of the Meta distribution, CSP and its Moment in the Hybrid Network
When a user associates with a mm-wave SBS, the overall CSP depends on the CSPs of the SIR and SNR on both the backhaul link and the access link, respectively. On the other hand, when a user associates to MBS the CSP depends on the SIR of the direct link. It is thus necessary to formulate the relationship between the meta distribution, CSP, and its moment in the considered hybrid network as follows.
Lemma 3 (Meta Distribution of the Typical User in the Hybrid Network).
The combined meta distribution of the SIR/SNR in the hybrid spectrum network can be characterized as follows:
where can be characterized by deriving the moment of the 222The moment of a random variable
moment of a random variableis the expected value of random variable to the power , i.e., ..
where is the moment of the SIR/SNR when a user associates to mm-wave SBS for dual-hop transmission and is the moment of the SIR when a user associates to MBS for direct transmission. After reformulation, we define as the unconditional moment of the backhaul SIR, as the unconditional moment of the SNR at access link when a user associates to mm-wave SBS, and as the unconditional moment of the SIR at direct link when a user associates to wave BS. Note that denotes the CSP of user over the direct link, denotes the CSP at backhaul link, and denotes the CSP for the access link transmission.
Step (a) follows from the fact that the moment of the SIR or SNR of a user associated to tier can be defined as where is the conditional association probability to tier and is the conditional moment of the SIR or SNR in tier . In our case, we have which is the conditional association probability to mm-wave SBS where since a user can associate to either LOS or NLOS mm-wave SBS. The step (b) follows from the fact that the CSP of the dual-hop transmission depends on the CSP of access and backhaul link; therefore, we have a product of the access and backhaul CSPs, i.e., that are independent random variables. There is no correlation since wave backhaul does not interfere with mm-wave transmissions. The step (c) follows from the fact if and are independent then . Finally, the step (d) follows from the definition of in Lemma 2 and the step (e) follows by applying the definition of moments. ∎
In the next section, we derive the CSP of access, backhaul, and direct links along with their respective moments, as needed in Lemma 4 to characterize the overall moment as well as the meta distribution.
V Characterization of the CSPs and Moments
In this section, we derive the CSPs and the moments , , and for backhaul link, access link, and direct link, respectively.
V-a CSP and the Moment - Access Link
We condition on having a user at the origin which becomes a typical user. The CSP of a typical user at the origin associating with the mm-wave SBS-tier (when ) can be described as follows:
The CSP of the SNR of a user on the access link with LOS can be defined by substituting defined in Eq. (5) into Definition 1 as follows:
where (a) follows from the definition of and the fact that the channel gain is a normalized gamma random variable and is the lower incomplete gamma function and , where is the upper incomplete gamma function. Similarly, CSP of the SNR on the access link for NLOS case can be given as follows:
where . As such, the moment of the CSP on the access link for the typical user when it is served by the mm-wave SBS tier is given by the following:
The moment of the SNR at an “access link” when a user associates with a mm-wave SBS can be characterized as follows:
where and are given in Lemma 2, , , , and , and .
See Appendix C. ∎
For , , and , we can get in closed-form using Corollary 1. Also, for scenarios where , , , and , then . Also, and , we can get in closed-form using Corollary 2.
V-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 in Eq. (3) the CSP of the backhaul link can be given as:
where (a) follows from the Rayleigh fading channel gain and (b) is found by taking the expectation with respect to . The moment of the CSP on the backhaul link is given as: