## I Introduction

Network heterogeneity is at the heart of current 4G and upcoming 5G networks. A key consequence of the heterogeneous deployments is the emergence of different types of spatial couplings across the locations of BSs and users. Perhaps the most prominent one is the user-BS coupling, where the users tend to form spatial clusters or hotspots [1, 2, 3] and small cell BSs (SBSs) are deployed within these hotspots to provide additional capacity. Further, depending on the deployment objectives, the point patterns of BSs of a particular tier may exhibit some intra-tier coupling, such as clustering patterns for small cells [4, 5] and repulsive patterns for macrocells deployed under a minimum inter-site distance constraint. Further, inter-tier coupling may exist between the locations of BSs of different tiers, for instance, macrocells and small cells when the latter are deployed at the macrocell edge to boost cell edge coverage [6].

Not surprisingly, the HetNet simulation models used by the standardization bodies, such as the third generation partnership project (3GPP), are cognizant of the existence of this spatial coupling [6]. Unfortunately, this is not true for the stochastic geometry based analytical HetNet models, e.g., see [7, 8], which mostly still rely on the assumption that all network elements (BSs and users) are modeled as independent PPPs. That said, it has been recently shown that PCP-based models are well-suited to capture the aforementioned spatial coupling in a similar way as it is incorporated in 3GPP simulation models [1, 9, 10, 11]. Since PCPs are defined in terms of PPPs, they are also very amenable to mathematical analysis. A key prior work in this area is [1], which completely characterized the downlink coverage probability for a PCP-based HetNet model under max- based association scheme in which the typical user connects to the BS offering maximum instantaneous received [1]. However, the downlink analysis for the more practical association scheme in which the typical user connects to the BS offering the strongest received power requires a very different mathematical treatment and is still a key open problem. In this paper, we plug this knowledge gap by providing a complete characterization of coverage probability under this association model.

### I-a Background and related works

Since the use of PPPs for modeling HetNets is by now fairly well-known, we advise interested readers to refer to books, surveys and tutorials, such as [12, 13, 14, 15, 16] to learn more about this direction of research. Although sparse, there have been some works on modeling spatial coupling between BSs and users in random spatial models for cellular networks. In [17], user-BS coupling was introduced in a PPP-based single tier cellular network model by conditionally thinning the user PPP and biasing user locations towards the BS locations. Owing to the natural connection of the formation of hotspots to the clustering patterns of PCPs, PCP was used to model the user distributions in [18, 9, 10, 19], where coupling between the users and BS locations was introduced by placing the SBSs at the cluster centers (parent points) of the user PCP. Further, since intra-tier coupling can be either attractive or repulsive, no single point process is the best choice for capturing it. For modeling repulsions, Matérn hard-core process [20, 21], Gauss-Poisson process [22], Strauss hardcore process [23], Ginibre point process [24, 25], and more general determinantal point processes [26] have been used for BS distributions. On the other hand, for modeling attraction, PCP [11] and Geyer saturation process [23] have been proposed. For modeling inter-tier coupling, Poisson hole process (PHP) has been a preferred choice [27, 28], where the macro BSs (MBSs) are modeled as PPP and the SBSs are modeled as another PPP outside the exclusion discs (holes) centered at the MBS locations. Among these “beyond-PPP” spatial models of HetNets, the PCP has attracted significant interest because of its generality in modeling variety of user and BS configurations and its mathematical tractability [29, 1]. We now provide an overview of the existing work on PCP-based models for HetNets.

In [30, 27], the authors assumed that the SBSs in a two-tier HetNet are distributed as a PCP and derived downlink coverage probability assuming that the serving BS is located at a fixed distance from the receiver, which circumvented the need to consider explicit cell association. While this setting provides useful initial insights, the analysis cannot be directly extended to incorporate realistic cell association rules, such as -power based association [31]. The primary challenge in handling cell association in a stochastic geometry setting is to jointly characterize the serving BS distance and the interference field. This challenge does not appear when BS is assumed to lie at a fixed distance from the receiver, such as in [30, 27]. It is worth noting that the distance of the serving BS from the receiver under -power based association can be evaluated using contact distance distributions of PCPs, which have recently been characterized in [32, 33]. However, these alone do not suffice because we need to jointly characterize the serving BS distance and the interference field, which is much more challenging. This is the main reason why this problem has remained open for several years. In [34], the authors characterized handoff rates for a typical user following an arbitrary trajectory in a HetNet with PPP and PCP-distributed BSs. Although the setup is very similar to this paper, the metric considered in [34] did not require the characterization of coverage probability. In [35], the authors developed analytical tools to handle this correlation and derived coverage probability when the BSs are modeled as a Matérn cluster process (MCP). However, the analysis is dependent on the geometrical constructions which is very specific to an MCP and is hence not directly applicable to a general PCP. The coverage analysis for a general PCP was provided in [11] where analytical tractability was preserved by assuming that the SBSs to be operating in a closed access mode, i.e, a user can only connect to a SBS of the same cluster. In [1], we have provided a comprehensive coverage analysis for the unified HetNet model under -based association strategy with threshold greater than unity. However, the analysis cannot be directly extended to the power-based association setup because of the fundamental difference in the two settings from the analytical perspective. Recently, in [36], the open problem of the coverage analysis for a single tier cellular network under -power connectivity with PCP-distributed BSs and independent user locations was solved by expressing coverage probability as a sum-product functional over the parent PPP of the BS PCP. Motivated by this novel analytical approach, in this paper we provide the complete coverage analysis for -power based association strategy for the unified HetNet model introduced in [1].

### I-B Contributions

We derive the coverage probability of a typical user of a unified -tier HetNet in which the spatial distributions of BS tiers are modeled as PCPs and BS tiers are modeled as PPPs (). The PCP assumption for the BS tier introduces spatial coupling among the BS locations. We consider two types of users in this network, Type 1: users having no spatial coupling with the BSs, and Type 2: users whose locations are coupled with the BS locations. For Type 1, the user locations are modeled as a stationary point process independent of the BS point processes. For Type 2, the coupling between user and BS locations is incorporated by modeling the user locations as a PCP with each user cluster sharing the same cluster center with a BS cluster. The key contributions are highlighted next.

Exact coverage probability analysis. Assuming that a user connects to the BS offering the maximum average received power, we provide an exact analysis of coverage probability for a typical user which is an arbitrarily selected point from the user point process. The key enabler of the coverage probability analysis is a fundamental property of PCP that conditioned on the parent PPP, the PCP can be viewed as an inhomogeneous PPP, which is a relatively more tractable point process compared to PCP. Using this property, we condition on the parent PPPs of all the BS PCPs and derive the conditional coverage probability. Finally, while deconditioning over the parent PPPs, we observe that the coverage probability can be expressed as the product of PGFLs and sum-product functionals of the parent PPPs. This analytical formulation of coverage probability in terms of known point process functionals over the parent PPPs is the key contribution of the paper and yields an easy-to-compute expression of coverage probability under

-power based association. We then specialize the coverage probability for two instances when the PCPs associated with the BSs are either (i) Thomas cluster process (TCP), where the offspring points are normally distributed around the cluster center, or (ii) MCP, where the offspring points are distributed uniformly at random within a disc centered at the cluster center.

System-level insights. Using the analytical results, we study the impact of spatial parameters such as cluster size, average number of points per cluster and BS density on the coverage probability. Our numerical results demonstrate that the variation of coverage probability with cluster size has conflicting trends for Types 1 and 2: for Type 1, coverage decreases as cluster size increases, and for Type 2, coverage increases as cluster size decreases. As cluster size increases, the coverage probabilities under Type 1 and Type 2 approach the same limit which is the well-known coverage probability of the PPP-based -tier HetNet [31], but from two opposite directions. Our numerical results demonstrate that the impact of the variation of cluster size on coverage probability is not as prominent in Type 2 as in Type 1.

## Ii System Model

### Ii-a PCP Preliminaries

Before we introduce the proposed PCP-based system model for -tier HetNet, we provide a formal introduction to PCP.

###### Definition 1 (Poisson Cluster Process).

A PCP in can be defined as:

(1) |

where is the parent PPP with intensity and denotes the offspring point process corresponding to a cluster center at where

is an independently and identically distributed (i.i.d.) sequence of random vectors with probability density function (PDF)

. The number of points in is denoted by , where .Index set of all BS tiers () | |
---|---|

Index set of all BS tiers modeled as PCP and PPP | |

The point process of the BS tier, , which is a PCP. | |

The point process of the BS tier, , which is a PPP. | |

Transmit power of a BS in | |

Path-loss exponent () | |

Coverage threshold of | |

Conditional PDF of distance of a point of () | |

from origin given its cluster center is located at () | |

PDF and CDF of contact distance of | |

Notation. While we reserve the symbol to denote any point process, to indicate whether it is a PCP or PPP we specify the parameters in parentheses accordingly, i.e., denotes a PCP according to Definition 1 and denotes a PPP with intensity .

A PCP can be viewed as a collection of offspring process translated by for each . Then the sequence of points is conditionally i.i.d. with PDF . Note that the conditional distribution of the point coordinates given its cluster center at is equivalent to translating a cluster centered at the origin to . For a PCP, the following result can be established.

###### Proposition 1.

Conditioned on the parent point process , is an inhomogeneous PPP with intensity

(2) |

###### Proof:

While one can prove this result for a more general setting of Cox processes (see [37]), we prove this result for PCP for completeness as follows. Let be the random counting measure associated with the point process . Then, for a Borel set , where is the Borel -algebra on ,

is a random variable denoting the number of points of

falling in . First it is observed that for , since the probability generating function (PGF) of isHere follows from the fact that the offspring points are i.i.d. around the cluster center at , and

are obtained by using the PGF of Bernoulli and Poisson distributions, respectively. Hence

. Now, conditioned on the PGF of is expressed as:where follows from the fact that conditioned on , is sequence of i.i.d. offspring point processes, is obtained by substituting the PGF of . Hence it is observed that . Thus is an inhomogeneous PPP with intensity measure . ∎

### Ii-B -tier HetNet Model

We assume a -tier HetNet where BSs of each tier are distributed as a PPP or PCP. Let and denote the index sets of the BS tiers which are modeled as PCP and PPP, respectively, with and . We denote the point process of the BS tier as , where is either a PCP i.e. () where is the parent PPP or a PPP (). Also define . Each BS of transmits at constant power . We assume that the users are distributed according to a stationary point process . We now consider two types of users in the network.

Type 1: No user-BS coupling.

The first type of users are uniformly distributed over the network, such as, the pedestrians and users in transit, and their locations are independent of the BS locations. There is no restriction on the distribution of these users as long as the distribution is stationary. For instance, one way of modeling the locations of these users is to assume that they are distributed as a homogeneous PPP.

Type 2: User-BS coupling. The second type of users are assumed to form spatial clusters (also called user hotspots) and their locations can be modeled as a PCP [9, 1]. When the users are clustered, we also assume that one BS tier (say, the tier, ) is deployed to serve the user hotspots, thus introducing coupling between and . In other words, and are two PCPs having same parent PPP . Hence conditioned on , and are (conditionally) independent but not identically distributed. This assumption is motivated by the way SBSs are placed at higher densities in the locations of user hotspots in 3GPP simulation models of HetNets [4, 5].

###### Remark 1.

For Type 1, can be any general stationary point process including PPP and for Type 2, we do not specify of , since does not appear explicitly in the coverage analysis. However, these specifications of are required when one has to characterize other metrics like BS load and rate coverage probability [38, 39]. Further, the coverage probability analysis that follows can be extended for a general user distribution which is the superposition of PPP and PCPs along similar lines to [9].

In Fig. 1, we provide an illustration of the system model. For both types of user distributions, we perform our analysis for a typical user which corresponds to a point selected uniformly at random from . Since is stationary, the typical user is assumed to be located at the origin without loss of generality. For Type 1, since and , are independent, the selection of the typical user does not bias the distribution of . However, for Type 2, the selection of the typical user affects the BS point process due to the existence of user-BS coupling. We assume that the typical user belongs to a cluster centered at . By construction, is always conditioned to have a cluster centered at . Hence, the typical user will see the palm version of which, by Slivnyak’s theorem, is equivalent to the superposition of and where and are independent. For Type 2, we modify as . Consequently, the underlying parent point process is modified as . The BS cluster is termed as the representative cluster.

Considering the typical user at origin, the downlink power received from a BS at is expressed as , where and denote the small scale fading and path-loss exponent, respectively. We assume that each link undergoes Rayleigh fading, hence is a sequence of i.i.d. random variables with . The user connects to the BS providing the maximum received power averaged over fading. Thus, if is the location of the serving BS, then, , where is the location of candidate serving BS in . Note that the candidate serving BS in is the BS in which is geometrically closest to the user. We first define the association event corresponding to the tier as the event that the serving BS belongs to , denoted as . Conditioned on , the experienced by the typical user is

(3) |

where is the thermal noise power. We define the coverage probability as the probability of the union of mutually exclusive coverage events

(4) |

where we call the term under the summation as the tier coverage probability which is the joint probability of the events and . Here is the threshold for the tier required for successful demodulation and decoding of the received signal.

## Iii Coverage Probability Analysis

We begin our coverage analysis by first conditioning on every parent PPP, i.e., . Following Proposition 1, will be inhomogeneous PPP with intensity . A slightly different situation occurs for in Type 2. However, once is modified by adding a point at to the original parent PPP, again becomes an inhomogeneous PPP with intensity .

Notation. To denote the distance of a point , we use and interchangeably.

### Iii-a Contact Distance Distribution

First we will derive the distribution of , , or the distance distribution of the candidate serving BS of tier . Since is the nearest BS to the typical user which is at origin, the distribution of is the same as the contact distance distribution of , denoted as . Let and denote the PDF and CDF of the distance of a randomly selected point of () given its cluster center is located at . Before presenting the contact distance distributions, we observe the following property of the conditional distance distribution.

###### Lemma 1.

If the offspring points are isotropically distributed around the cluster center i.e., the radial coordinates of the offspring points with respect to the cluster center have the joint PDF , where is the marginal PDF of the radial coordinate, then, and . That is, the conditional distance distribution depends only on the magnitude of .

###### Proof:

Let is the location of the point whose cluster center is located at . Then,

Here denotes a disc of radius centered at the origin. Differentiating with respect to ,

(5) |

Since the region of the integral over is the perimeter of the disc , it is independent of the choice of .∎

We now characterize the PDF of .

###### Lemma 2.

For , conditioned on , the PDF and CDF of are given as:

(6) |

and

(7) |

###### Proof:

For , the CDF of is

The last step is due the fact that integrating a joint PDF of polar coordinates over is equivalent to integrating the marginal PDF of radial coordinate over . The final result in (6) can be obtained by differentiating the CDF with respect to . ∎

Note that one can obtain the PDF of the contact distance of PCP by deconditioning over [36]. This is an alternative approach to the one presented in [32, 33] for the derivation of contact distance distribution of PCP.

When is a PPP, i.e., , the distribution of is the well-known Rayleigh distribution, given by:

(8) |

### Iii-B Association Probability and Serving Distance Distribution

We define association probability to the tier as . The association probability is derived as follows.

###### Lemma 3.

Conditioned on , the association probability to the tier is given by:

(9a) | |||

(9b) |

where .

###### Proof:

See Appendix -A. ∎

We now derive the PDF of the conditional serving distance i.e. given and .

###### Lemma 4.

The PDF of conditioned on association to the tier and is given as:

(10a) | |||

(10b) |

###### Proof:

See Appendix -B. ∎

### Iii-C Coverage Probability

Before deriving the main results on coverage probability, we first introduce PGFL and sum-product functional of a point process which will be appearing repeatedly into the coverage analysis.

###### Definition 2 (Pgfl).

PGFL of a point process is defined as: , where is measurable.

###### Lemma 5.

When is a PPP, the PGFL is given as [37]:

(11) |

When and is homogeneous, then (11) becomes:

(12) |

###### Definition 3 (Sum-product functional).

Sum-product functional of a point process is defined in this paper as , where and are measurable.

For a PPP, the sum-product functional is given by the following Lemma.

When , , and is homogeneous, then (13) becomes:

(14) |

Before providing the final expression of coverage probability, we provide an important intermediate expression of the conditional tier coverage probability given all parent point processes. In fact, a key contribution of this paper, as will be evident in sequel, is to show that this conditional coverage probability can be factored as a product of standard functionals (such as PGFL and sum-product functional) of the parent PPPs.

###### Lemma 7.

The tier coverage probability given is given by

(15a) | |||

(15b) |

where

(16) |

and, , where is the Gauss hypergeometric function [31].

###### Proof:

See Appendix -C. ∎

Looking closely at the expressions of the tier coverage probability given , we find two terms: (i) in (15a) and (15b), and (ii) in (15a). These two terms are respectively product and sum-product over all points of which will be substituted by the PGFL and sum-product functional of while decondtioning over .

###### Remark 2.

Until this point, all results were conditioned on . Remember that in Section II-B, we introduced two types of spatial interaction between the users and BSs. Since, by construction, these two types differ only in for some , we were able to treat Type 1 and Type 2 within the same analytical framework. We now present the final expression of for the two types explicitly by deconditioning the conditional coverage probability over in the following Theorems.

###### Theorem 1 (Type 1).

The coverage probability is given as:

(17) |

where, the tier coverage probability,

(18a) | |||

(18b) |

###### Proof:

When , we get from (15a),

This step is enabled by the assumption that -s are independent . The final expression is obtained by substituting the PGFL of for from (12) and the sum-product functional of from (14). The final expression follows from some algebraic simplifications. Now, for , from (15b), we get

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