I Introduction
Ia Motivation
In wireless system level analysis, most models analyze propagation using distancebased path loss functions
[rappaport1996wireless, goldsmith2005wireless]. Such a modeling is justified in the free space case but does not capture realworld environments with obstacles. By introducing a shadowing term, it is possible to model the effect of obstacle blockage. This term accounts for the fact that the received signal power is strongly attenuated by obstacles on the propagation path between transmitter and receiver. For a single link, this attenuation is typically modeled by a lognormal distribution
[goldsmith2005wireless], which is justified by the multiplicative blockage loss and the central limit theorem
[coulson1998statistical]. However, this does neither capture the fact that nearby links are often blocked by common obstacles nor the fact that the shadowing statistics highly depend on the spatial geometry of obstacles.Stochastic geometry has been widely studied to analyze the performance of both infrastructure (e.g., cellular) and infrastructureless (e.g., D2D) networks. The papers in this research field provide highly tractable performance evaluation results in several scenarios. Since shadowing is a significant part of wireless communication, it is important to incorporate this feature in stochastic geometric models. However, as explained above, even though the shadowing effect is spatially correlated in real networks [gudmundson1991correlation], most previous stochastic geometric models assume that shadowing is spatially independent over links.
The main purpose of this paper is to question this independence assumption and to analyze the effect of correlated shadowing fields when using stochastic geometry. For this, we provide the Laplace transforms of the interference associated with Poisson networks under spatially correlated and independent shadowing assumptions, and prove general ordering relations between them. Using the Laplace stochastic ordering [shaked1995stochastic], we also give the ordering of some important performance metrics of the two shadowing models. Especially when the metric is coverage probability, Shannon throughput or local delay, we show that the performance metric under the independent shadowing is in fact always evaluated in a pessimistic way compared to the correlated case.
IB Related Works
IB1 Correlated Shadowing
In real networks, shadowing fields are spatially correlated [goldsmith2005wireless]. However, few generative or tractable models have been proposed to represent this correlation. Gudmundson proposed the first model of correlation [gudmundson1991correlation]
to model the lognormal shadowing random process between a fixed base station and a moving user by an autoregressive process with an exponentially decaying autocorrelation. As a result, the spatial dependence of shadowing can be formulated by joint Gaussian distributions. The multibase station
[graziosi2002general] and multihop network [agrawal2009correlated] cases were also considered based on similar ideas. This approach also forms the basis of the models suggested by the 3GPP[access2010further] and the 802.11 standardization groups[erceg2004tgn].These models have shortcomings. It is hard to give a clear physical interpretation to the joint Gaussian distribution used to model spatially correlated shadowing. These models give limited intuition on large and dense wireless networks. Also, complex simulation platforms are required.
IB2 Stochastic Geometry and Shadowing Models
Over the past decades, stochastic geometric models, and most notably the planar Poisson point process (PPP) model, have become popular for the analysis of network performance in wireless communications, in both the D2D [weber2010overview, baccelli2006aloha, zhang2012random, zhang2012delay] and the cellular contexts [andrews2011tractable, dhillon2012modeling, dhillon2012coverage]. While an independent shadowing field can easily be incorporated into the basic models [ilow1998analytic, blaszczyszyn2015wireless, bai2014analysis], there is no known approach to combine general stochastic geometry models with correlated shadowing where links at nearby locations can be blocked by the same physical obstacles. Recently, by using a Poisson line process, correlated shadowing fields of urban networks [baccelli2015correlated, zhang2015indoor] and inbuilding networks[lee20163] have been analyzed in a way taking this correlation into account. However, these models use blockagebased path loss functions and fail taking the distance basedterm into account.
IB3 Comparison of Point Processes
Stochastic comparison tools have been used to investigate the clustering properties among point processes by evaluation of the Ripley K function, the paircorrelation function or the empty space function [chiu2013stochastic]. To quantify the impact of clustering properties among point processes, the directionally convex order on point processes [blaszczyszyn2009directionally] and the properties of positive and negative association [burton1985scaling, pemantle2000towards] have been proposed. This was for instance used to compare certain point processes with the Poisson point process [blaszczyszyn2014comparison].
In this paper, we consider a new type of comparison which is that of interference fields when a shadowing random field is introduced to model the blockage effects. We compare the cases where this random field is spatially correlated or not. Other propagation effects such as reflection are not considered.
IC Problem Statement and Main Contributions
As already explained, most of the previous research papers assigned shadowing variables to links independently by using an empirical distribution [goldsmith2005wireless] or based on link length when topology is incorporated [ilow1998analytic, blaszczyszyn2015wireless, bai2014analysis]. We will call these models (spatially) independent shadowing models.
A typical instance of the independent shadowing model is provided in [bai2014analysis]
. Under this model, both the centers of blockages and the base stations are deployed as Poisson point processes. The shadowing random variable of a given link is determined by an independent Poisson random variable with mean proportional to the length of the link.
In contrast, in the present paper, in order to represent the spatial correlation property, we assign some shadowing value based on the obstacle topology. For example, in Fig. LABEL:fig:network_ppp, obstacles are random segments and the plane is divided into cells in which all base stations are blocked by the same number of obstacles, when seen from the origin. Such cells are not necessarily convex but connected. Another example is depicted in Fig. LABEL:fig:network_pcp, when the base stations form a cluster process. In this network, we assign the same shadowing random variable to the base stations which share the same mother point. In contrast to the situation of Fig. LABEL:fig:network_ppp, even very closeby points can have different shadowing random variables. This is meant to model the situation where each cluster is located at a different altitude and has different shadowing properties. From these observations, we introduce the concept of Shadowing cell where base stations in the same Shadowing cell have the same shadowing random variable. Each base station should not belong to more than one Shadowing cell.
The main question of this paper is the comparison of the interference distribution under the correlated and the independent shadowing models in the stochastic ordering sense. To provide a fair comparison, we use the same marginal shadowing laws in both cases. We compute the Laplace transforms of the interference observed by the typical user which is located at the origin under these two models, and we then provide the ordering relation of the three metrics for the typical user, i.e., 1) coverage probability, 2) Shannon throughput, and 3) local delay. We show that these three metrics are completely monotone functions of the interference. From well known results on the relation between the Laplace transform ordering and completely monotone functions, we obtain the ordering relations under the two shadowing assumptions.
For the case where base stations form either a homogeneous Poisson point process or a Matrn cluster process on , we provide exact expressions for the Laplace transform of interference. These expressions are provided conditioned on the Shadowing cells, but provide a general ordering relation of the above metrics by deconditioning. Especially, if the Shadowing cells are Matrn disks [haenggi2012stochastic], we can obtain further closed form expressions by deconditioning with respect to the Shadowing cells.
Our key findings can be summarized as follows:

We provide closedform expressions for the Laplace transform of the interference measured at the origin under the two shadowing assumptions for some generic network examples.

We investigate the Laplace transforms of interference and their ordering relationship for point processes with the same point configuration but different joint shadowing distributions.

By using the Laplace stochastic ordering and the formalism of completely monotone functions, we also give the ordering relation of the three key performance metrics under the two different shadowing assumptions.
Ii Laplace Stochastic Ordering and Completely Monotone Functions
We first introduce some mathematical preliminaries. The following results and definitions are borrowed from [alzaid1991laplace]. They will be used to investigate the ordering of the network performance metrics in the next sections.
Definition 1 (Laplace stochastic ordering)
Let and be random variables in . is said to be less than in the Laplace stochastic ordering (written ), if the Laplace transforms and satisfy
(1) 