# Neveu’s Exchange Formula for Analysis of Wireless Networks With Hotspot Clusters

- Department of Mathematical and Computing Science, Tokyo Institute of Technology, Tokyo, Japan

Theory of point processes, in particular Palm calculus within the stationary framework, plays a fundamental role in the analysis of spatial stochastic models of wireless communication networks. Neveu’s exchange formula, which connects the respective Palm distributions for two jointly stationary point processes, is known as one of the most important results in the Palm calculus. However, its use in the analysis of wireless networks seems to be limited so far and one reason for this may be that the formula in a well-known form is based upon the Voronoi tessellation. In this paper, we present an alternative form of Neveu’s exchange formula, which does not rely on the Voronoi tessellation but includes the one as a special case. We then demonstrate that our new form of the exchange formula is useful for the analysis of wireless networks with hotspot clusters modeled using cluster point processes.

## 1 Introduction

Spatial stochastic models have been widely accepted in the literature as mathematical models for the analysis of wireless communication networks, where irregular locations of wireless nodes, such as base stations (BSs) and user devices, are modeled using spatial point processes on the Euclidean plane (see, e.g., (Baccelli and Błaszczyszyn, 2009a; Baccelli and Błaszczyszyn, 2009b; Haenggi and Ganti, 2009; Haenggi, 2013; Mukherjee, 2014; Błaszczyszyn et al., 2018) for monographs and (Andrews et al., 2016; ElSawy et al., 2017; Hmamouche et al., 2021; Lu et al., 2021) for recent survey and tutorial articles). In such analysis of wireless networks, the theory of point processes, in particular Palm calculus within the stationary framework, plays a fundamental role. Neveu’s exchange formula, which connects the respective Palm distributions for two jointly stationary point processes, is known as one of the most important results in the Palm calculus. However, its use in the analysis of wireless networks seems to be limited so far and one reason for this may be that the formula in a well-known form is based upon the Voronoi tessellation [see, e.g., (Baccelli et al., 2020, Section 6.3)]. In this paper, we present an alternative form of Neveu’s exchange formula, which does not rely on the Voronoi tessellation but includes the one as a special case, and then demonstrate that it is useful for the analysis of spatial stochastic models based on cluster point processes.

A cluster point process represents a state such that there exist a large number of clusters consisting of multiple points and is used to model the locations of wireless nodes in an (urban) area with a number of hotspots. Indeed, many researchers have adopted the cluster point processes in their models of various wireless networks such as ad hoc networks (Ganti and Haenggi, 2009), heterogeneous networks (Chun et al., 2015; Suryaprakash et al., 2015; Saha et al., 2017, 2018; Afshang and Dhillon, 2018; Saha et al., 2019; Yang et al., 2021), device-to-device (D2D) networks (Afshang et al., 2016), wireless powered networks (Chen et al., 2017), unmanned aerial vehicle assisted networks (Turgut and Gursoy, 2018), and so on. In this paper, we focus on so-called stationary Poisson-Poisson cluster processes (PPCPs) [see, e.g., (Błaszczyszyn and Yogeshwaran, 2009; Miyoshi, 2019)] and apply the new form of the exchange formula to the analysis of stochastic models based on them.

We first use the exchange formula for the Palm characterization, where we derive the intensity measure, the generating functional and the nearest-neighbor distance distribution for a stationary PPCP under its Palm distribution. Although these results are known in the literature [see, e.g., (Baudin, 1981; Ganti and Haenggi, 2009)], we here give them simple and unified proofs using the new form of the exchange formula. We next consider some applications to wireless networks modeled using stationary PPCPs, where we examine the problems of coverage and device discovery in a D2D network. The coverage analysis of a D2D network model based on a cluster point process was considered in (Afshang et al., 2016), where a device communicates with another device in the same cluster. In contrast to this, we assume here that a device receives messages from the nearest transmitting device, which is possibly in a different cluster because clusters may overlap in space. For this model, we derive the coverage probability using the exchange formula. On the other hand, in the problem of device discovery, transmitting devices transmit broadcast messages and a receiving device can detect the transmitters if it can successfully decode the broadcast messages. Such a problem was studied in (Hamida et al., 2008; Baccelli et al., 2012; Kwon and Choi, 2014) when the devices are located according to a homogeneous Poisson point process (PPP) and in (Kwon et al., 2020) when the devices are located according to a Ginibre point process [see, e.g., (Miyoshi and Shirai, 2014, 2016), for the Ginibre point process and its applications to wireless networks]. We consider the case where the devices are located according to a stationary PPCP and derive the expected number of transmitting devices discovered by a receiving device. We should note that Neveu’s exchange formula is also introduced in a more general form in (Last and Thorisson, 2009; Last, 2010; Gentner and Last, 2011), so that the form presented in the paper may be within its scope. Nevertheless, we see in the rest of the paper that our new form would be valuable and could spread the application fields of the exchange formula.

The rest of the paper is organized as follows. The new form of Neveu’s exchange formula is derived in the next section, where the relations with the existing forms are also discussed. In Section 3, the exchange formula is applied to the Palm characterization of a stationary PPCP, where alternative proofs of the intensity measure, the generating functional and the nearest-neighbor distance distribution under the Palm distribution are given. In Section 4, some applications to wireless network models are examined, where for a D2D network model based on a stationary PPCP, the coverage probability and the expected number of discovered devices are derived using the exchange formula. The results of numerical experiments are also presented there. Concluding remarks are provided in Section 5.

## 2 Neveu’s Exchange Formula

In this section, we discuss point processes on the *d*-dimensional Euclidean space *σ*-field on *δ*_{x} denotes the Dirac measure with mass at *θ*_{t}: Ω → Ω is *θ*_{t}◦*θ*_{u} = *θ*_{t+u} for *θ*_{0} is the identity for *B*)◦*θ*_{t} = Φ(*θ*_{t}(*ω*), *B*) = Φ(*ω*, *B* + *t*) = Φ(*B* + *t*) for *ω* ∈ Ω, *X*_{n}◦*θ*_{t} = *X*_{n’} − *t*. Under the assumption of the

Let *λ*_{Φ} and *λ*_{Ψ}, respectively. Thus, Φ and Ψ are jointly stationary in probability *X*_{0} = 0 on {Φ({0}) = 1}, and this is also the case for Ψ; that is, *Y*_{0} = 0 on {Ψ({0}) = 1}. To present an alternative form of Neveu’s exchange formula, we introduce a family of shift operators *S*_{t}, *η* on *S*_{t}*η*(*B*) = *η*(*B* + *t*) for *S*_{t} on the point process *S*_{t}, *h* on *S _{t}h*(

*x*) =

*h*(

*x*+

*t*) for

Theorem 1. For the two jointly stationary point processes *,*

1) *,* *.*

2)

where *X*_{0} = 0 on {Φ({0}) = 1}*.*

**Proof. **As with the proof of the exchange formula in (Baccelli et al., 2020, Theorem 6.3.7), we start our proof with the mass transport formula [see, e.g., (Baccelli et al., 2020, Theorem 6.1.34)]; that is, for any measurable function *ξ*:

Let *ξ*(*y*) = *W*◦*θ*_{y} Ψ_{0}({*y*}) on {Φ({0}) = 1}. Then, the left-hand side of Eq. 2 becomes

On the other hand, the right-hand side of Eq. 2 is reduced to

where the first equality follows from *ω* ∈ {Ψ({0}) = 1} is shifted to location − *x* on the shifted sample *θ*_{x}(*ω*) for *X*_{n}, of Φ such that its mark *Y*_{n,k}, satisfying *X*_{n} + *Y*_{n,k} = 0 on {Ψ({0}) = 1}. The proof is completed.

Remark 1: Let *W* ≡ 1 in (Eq. 1)*.* Then, we have _{n} in Theorem 1 has finite points. In the form of the exchange formula in (Baccelli et al., 2020, Theorems 6.3.7 and 6.3.19), the point process Ψ is partitioned by the Voronoi tessellation for Φ*,* which corresponds to a special case of (Eq. 1) such that *V*_{Φ}(*X*_{n}) denotes the Voronoi cell of point *X*_{n} of Φ. The condition in (Baccelli et al., 2020) such that there are no points of Ψ on the boundaries of Voronoi cells *V*_{Φ}(*X*_{n})*,* *,*

## 3 Applications to Cluster Point Processes

In this section, we demonstrate that Neveu’s exchange formula (Eq. 1) in Theorem 1 is useful to characterize the Palm distribution of stationary cluster point processes. A cluster point process is, roughly speaking, constructed by placing point processes (usually with finite points), called offspring processes, around respective points of another point process, called a parent process, and is used to represent a state such that there exist a large number of clusters consisting of multiple points (see Figure 1). In particular, we focus here on a stationary PPCP described next.

**FIGURE 1**. A sample of a 2-dimensional cluster point process (represents the points of the cluster point process and represents the points of the parent process).

### 3.1 Poisson-Poisson Cluster Processes

Let

The PPCP Ψ constructed as above is stationary since the parent process Φ is stationary and the offspring processes Ψ_{n}, *λ*_{Φ}, and Ψ_{n}, *Λ*_{o} = *μ Q*, where *μ* is a positive constant and *Q* is a probability distribution on *μ*, so that the intensity of Ψ is equal to *λ*_{Ψ} = *λ*_{Φ}*μ*, and offspring points are scattered on *Q* independently of each other. We further assume that *Q* is diffuse; that is, *Q*({*x*}) = 0 for any *X*_{n} for *Q* is an isotropic normal distribution, then the obtained PPCP is called the Thomas point process. On the other hand, when *Q* is the uniform distribution on a fixed ball centered at the origin, then the result is called the Matérn cluster process. Note that the PPCP Ψ and its parent process Φ fulfill the conditions of Theorem 1.

### 3.2 Characterization of Palm Distribution

For a stationary point process Ψ, let Ψ^{!}≔Ψ − *δ*_{0} on the event {Ψ({0}) = 1}, which is referred to as the reduced Palm version of Ψ.

Lemma 1. For the stationary PPCP Ψ described in Section 3.1, the intensity measure of the reduced Palm version Ψ^{!} (with respect to the Palm distribution) is given by

where |⋅| denotes the Lebesgue measure on *Q*^{−}(*B*) = *Q*(−*B*) with − *B* = { − *x*: *x* ∈ *B*} for *.*

**Proof. **Since the offspring processes Ψ_{n},

Taking the expectation with respect to

where *λ*_{Ψ} = *λ*_{Φ}*μ* is used in the second equality, the third equality follows because, for any *X*_{n}◦*θ*_{y} = *X*_{n’} − *y* on {Φ({0}) = 1}, and in the last equality, we apply Campbell’s formula [see, e.g., (Last and Penrose, 2017, Proposition 2.7) or (Baccelli et al., 2020, Theorem 1.2.5)] for Ψ_{0}. For the expectation in the last expression above, Slivnyak’s theorem, Campbell’s formula for Φ and then Fubini’s theorem yield

which completes the proof.

Remark 2. The second term on the right-hand side of (Eq. 4) is of course equal to μ∫Q(B + y) Q(dy). We adopt the form in Lemma 1 due to its interpretability. Since Q is the distribution for the position of an offspring point viewed from its parent, Q^{−} represents the distribution for the location of the parent of the offspring point at the origin on the event {Ψ({0}) = 1}. On the other hand, μ Q(B − y) gives the expected number of offspring points falling in *.* In other words, the second term on the right-hand side of (Eq. 4) represents the expected number of offspring points falling in *B* among the cluster which is given to have one point at the origin. Since the first term on the right-hand side of (Eq. 4) is equal to

Proposition 1. For the stationary PPCP ^{!} (with respect to the Palm distribution) is given by

for any measurable function *h:* *, where*

and _{1} whose parent is shifted to

Note that in Proposition 1 above, _{1} are PPPs, respectively (see, e.g., (Last and Penrose, 2017, Exercise 3.6), or [Baccelli et al., 2020, Corollary 2.1.5)]. The relation (Eq. 5) is derived in (Ganti and Haenggi, 2009, Lemma 1), to which we give another proof using the exchange formula in Theorem 1.

**Proof. **As stated in the proof of Lemma 1, once the parent process

where the generating functional of a PPP is applied in the second equality. Taking the expectation with respect to

where Campbell’s formula for Ψ_{0} is applied in the last equality. By Slivnyak’s theorem and the stationarity for Φ, we have

Remark 3. The right-hand side of (Eq. 5) is given as the generating functional *Q*^{−} is the distribution of the location of the parent point of the offspring at the origin on the event {Ψ({0}) = 1}, this integral term represents the generating functional of the cluster which is given to have a point at the origin. In other words, Proposition 1 implies that, for a stationary PPCP, its Palm version is obtained by the superposition of the original stationary version and an additional independent offspring process whose parent is placed such that it has an offspring point at the origin. This observation is already found in, e.g., (Saha et al., 2019) and is also interpreted such that a point *Q*^{−} and the Palm version of Φ at *z* is then obtained as Φ + *δ*_{z} by Slivnyak’s theorem, which works as a parent process of the Palm version of Ψ. Proposition 1 indeed supports this interpretation.

### 3.3 Nearest-Neighbor Distance Distributions

For a stationary point process Ψ on *Y*_{*} denote the nearest point of Ψ^{!} from the origin. Then, the nearest-neighbor distance distribution for Ψ is defined as the probability distribution for ‖*Y*_{*}‖ with respect to

Proposition 2. For the stationary PPCP Ψ described in Section 3.1, the complementary nearest-neighbor distance distribution is given by

where *b*_{0}(*r*) denotes a *d*-dimensional ball centered at the origin with radius *r.*

**Proof. **As with the proof of Proposition 1, we consider the conditional probability given the parent process

where the second equality follows from Slivnyak’s theorem and the third does because Ψ^{!} is conditionally an inhomogeneous PPP with the intensity measure

Remark 4. In (Eq. 8), the term

## 4 Applications to Wireless Networks With Hotspot Clusters

In this section, we apply Theorem 1 to the analysis of a D2D network with hotspot clusters modeled using a stationary PPCP. We here suppose *d* = 2, but unless otherwise specified, the discussion holds for *d* ≥ 2 theoretically.

### 4.1 Model of a Device-To-Device Network

Wireless devices are distributed on *p* ∈ (0, 1) or in receiving mode with probability 1 − *p* independently of the others (half duplex with random access). Devices in the transmission mode transmit signals but can not receive ones, whereas the devices in the receiving mode can receive signals but can not transmit ones. We assume that all transmitting devices transmit signals with identical transmission power (normalized to one) and share a common frequency spectrum. The path-loss function representing attenuation of signals with distance is given by *ℓ* satisfying *ℓ*(*r*) ≥ 0, *r* > 0, and *ϵ* > 0. We further assume that all wireless links receive Rayleigh fading effects while we ignore shadowing effects. We focus on the device at the origin, referred to as the typical device, under the condition of {Ψ({0}) = 1} and examine whether the typical device can decode messages from other transmitting devices. Let *H*_{m} denote a random variable representing the fading effect on signals transmitted from the device at *H*_{m},

where *N* denotes a constant representing noise at the origin. We suppose that the typical device can successfully decode a message from the device at *θ* > 0.

### 4.2 Coverage Analysis

We here suppose that a device in the receiving mode communicates with the nearest device in transmission mode. The probability that the typical device can successfully decode a message from its partner is called the coverage probability and is given by

where 1 − *p* on the right-hand side indicates that the typical device must be in the receiving mode and the sum over *θ*. We now suppose that the point process Ψ representing the locations of devices is given as a stationary PPCP studied in Section 3. Then, *λ*_{Φ}, whereas the offspring processes *pμQ*.

Theorem 2. For the model of a D2D network described in Section 4.1 with the devices deployed according to a stationary PPCP in Section 3.1, the coverage probability is given by

where *Q*^{−} is given in Lemma 1 and

Before proceeding on the proof of Theorem 2, we give an intuitive interpretation to the result of it. First, as stated in the preceding section, *Q*^{−} denotes the distribution for the location of the parent point of the typical device at the origin. Thus, *pμI*_{1,θ}(*t*) and *pμI*_{2,θ}(*t*) in (Eq. 12) represent the cases where the typical device, whose parent is located at *pμQ*(d*y* − *t*) in *I*_{1,θ}(*t*) and is from one with *pμQ*(d*y* − *x*) in *I*_{2,θ}(*t*), where *x* is also sampled from a homogeneous PPP with intensity *λ*_{Φ}. Moreover, *E*_{θ}(*y*) represents the effect from other clusters which are neither the one having the typical device nor the one having its communication partner at *y*. Finally, *C*_{θ}(*y*, *x*) represents the effect of the cluster with the parent point at *y*.

**Proof. **Similar to the proof of Proposition 1, once the parent process *H*_{m},

where *1*_{A} denotes the indicator function for set *A* and we use *a* ≥ 0 and

Furthermore, the generating functional of a PPP applying to the above expectation yields

Plugging this into (Eq. 14), taking the expectation with respect to

where we note the existence of *X*_{0} = 0 on {Φ({0}) = 1} in the second equality and apply Campbell’s formula in the third equality. Noting that *X*_{0} = 0 on {Φ({0}) = 1}, we separate the expectation in (Eq. 15) into

and consider the two terms on the right-hand side of (Eq. 16) one by one. For the first term, the generating functional of a PPP yields (1st term of (Eq. 16))

On the other hand, applying Campbell’s formula and the generating functional for Φ to the second term on the right-hand side of (Eq. 16), we have (2nd term of (Eq. 16))

Finally, plugging (Eqs 17, 18) into (Eq. 16), and then into (Eq. 15), we have (Eq. 12) and the proof is completed.When *d* = 2 and the distribution *Q* for the locations of offspring points depends only on the distance; that is, *Q*(d*y*) = *f*_{o}(‖*y*‖) d*y* for

Corollary 1. When *d* = 2 and *Q*(d*y*) = *f*_{o}(‖*y*‖) d*y,*

where

**Proof. **Since the distribution *Q* depends only on the distance, it holds that *Q*^{−}(d*t*) = *Q*(d*t*) = *f*_{o}(‖*t*‖) d*t*,

where the polar coordinate conversion is applied in the second equality and

Therefore, we have

Plugging this into (Eq. 21), we have (Eq. 19) and the proof is completed.

### 4.3 Device Discovery

We next consider the problem of device discovery. Devices in the transmission mode transmit broadcast messages, whereas a device in the receiving mode can discover the transmitters if it can successfully decode the broadcast messages. When a device in the receiving mode receives the signal from one transmitting device, the signals from all other transmitting devices work as interference. Then, the expected number of transmitting devices discovered by the typical device is represented by

Proposition 3. Consider the D2D network model described in Section 4.1 with the devices deployed according to a stationary PPCP given in Section 3.1. Then, the expected number *z*‖ > ‖*y*‖ in (Eq. 13) by *d* = 2 and *Q*(d*y*) = *f*_{o}(‖*y*‖) d*y* for *s*, *∞*) in (Eq. 20) by (0, *∞*)*.*

**Proof. **Since *C*_{θ}(*y*, *x*) in (Eq. 13) and in *θ* > 1 since

### 4.4 Numerical Experiments

We present the results of numerical experiments for the analytical results obtained in Sections 4.2, 4.3. We set *d* = 2 and the distribution *Q* for the location of the offspring points as *Q*(d*y*) = *f*_{o}(‖*y*‖) d*y* and *s* ≥ 0; that is, *Q* is the isotropic normal distribution with variance *σ*^{2}, so that the resulting PPCP Ψ is the Thomas point process. Furthermore, the path-loss function is set as *ℓ*(*r*) = *r*^{−β}, *r* > 0, with *β* > 2.

The numerical results for the coverage probability are given in Figure 2, where the values of *θ* and *σ*^{2} are plotted. The other parameters are fixed at *λ*_{Φ} = *π*^{−1}, *μ* = 10, *p* = 0.5, *β* = 4 and *N* = 0. For comparison, the values when the devices are located according to a homogeneous PPP are also displayed in the figure with the label “*σ*^{2} → *∞*.” From Figure 2, we can see that, as the value of *σ*^{2} increases, the coverage probability decreases and is closer to that for the homogeneous PPP. This is contrary to the case of cellular networks, where the coverage probability increases and is closer to that for the homogeneous PPP from below as the variance of the locations of offspring points increases [see (Miyoshi, 2019)]. This difference is thought to be due to the fact that the locations of a receiving device and its communication partner are near to each other in the PPCP-deployed D2D network since they are both points of the same PPCP, whereas the location of a receiver is likely far from that of the associated BS in the PPCP-deployed cellular network since their locations are independent of each other.

**FIGURE 2**. Coverage probability as a function of SINR threshold (*λ*_{Φ} = *π*^{−1}, *μ* = 10, *p* = 0.5, *β* = 4 and *N* = 0).

The results of the device discovery is given in Figure 3, where we know that the closed form expression of the expected number of discovered devices is obtained as *N* ≡ 0 [see, e.g., (Hamida et al., 2008)]. The figure shows similar features to the coverage probability.

**FIGURE 3**. Expected number of discovered devices as a function of SINR threshold (*λ*_{Φ} = *π*^{−1}, *μ* = 10, *p* = 0.5, *β* = 4 and *N* = 0).

## 5 Conclusion

In this paper, we have presented an alternative form of Neveu’s exchange formula for jointly stationary point processes on

## Data Availability Statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

## Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

## Funding

This work was supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (C) 19K11838.

## Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s Note

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Keywords: stationary point processes, Palm calculus, Neveu’s exchange formula, cluster point processes, device-to-device networks, hotspot clusters, coverage probability, device discovery

Citation: Miyoshi N (2022) Neveu’s Exchange Formula for Analysis of Wireless Networks With Hotspot Clusters. *Front. Comms. Net* 3:885749. doi: 10.3389/frcmn.2022.885749

Received: 28 February 2022; Accepted: 17 May 2022;

Published: 28 June 2022.

Edited by:

Harpreet S. Dhillon, Virginia Tech, United StatesReviewed by:

Mehrnaz Afshang, Other, United StatesPraful Mankar, International Institute of Information Technology, India

Copyright © 2022 Miyoshi. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Naoto Miyoshi , miyoshi@is.titech.ac.jp