Introduction

The present survey deals with stochastic models for planar point patterns which can be applied to investigate the spatial organization of large communication networks. This concerns networks with fixed wired structure as well as wireless and mobile communications. The models discussed in this paper can be used to study questions related to strategic planning of communication networks, like the choice between several potential architectures or the evaluation of network evolution scenarios. In particular, these models are useful for studying the demand for service which arises in cellular networks, e.g. for analyzing the spatial distribution of the locations where wireless calls are initiated.

Besides discussing several classes of stochastic models for planar point patterns, the paper deals with statistical methods interrelating the developed models with empirical spatial data, as well as with Monte Carlo simulation methods which are employed to generate samples of planar point patterns from given models.

A number of stochastic modeling approaches have been proposed for studying these questions. For example, an early paper on stochastic modeling of cellular communication networks is Everitt and Macfadyen (1983). In Kelly (1985, 1986) cell blocking probabilities and related probabilistic problems of optimal routing and dynamical channel assignment are discussed, in particular, the efficient use of a limited number of radio channels by allowing the repeated reuse of each channel in sufficiently separated spatial regions. See also the survey on loss networks given in Kelly (1991). Further results on optimal channel assignment in cellular communication systems can be found e.g. in Everitt and Manfield (1989), Mathar and Mattfeldt (1993), Pallant and Taylor (1994). Blocking probabilities for new and handover calls have been investigated by numerous authors, see e.g. Boucherie and Mandjes (1996), Boucherie and van Dijk (1996), Hellebrandt and Mathar (1997b), Keilson and Ibe (1995), Mandjes (1996), Mandjes and Tutschku (1996), Pallant (1992), Pallant and Taylor (1995), Rezaiifar et al. (1995), Sakamaki et al. (1996), Tran-Gia and Mandjes (1996). Note, however, that in most of these papers a regular (hexagonal) shape of cells is assumed. We also remark that in queueing theory further attempts have been made to build (noncellular) spatial stochastic models where each customer's location in space is taken into account explicitely, see e.g. Cinlar (1995), Coffman and Stolyar (1993), Huang (1996), Kroese and Schmidt (1992, 1994, 1996), Serfozo (1995).

In the present paper we assume that the data available for analysis consist in the locations (coordinates, points) of certain events within a given geographical region. The data may be supplemented by collateral spatial or temporal information (marks) on relevant factors, e.g. the range/scope of an event that has occurred at such a location, the type or the length of a message whose transmission has been initiated at a given location or the direction and speed with which a mobile user moves within a region. We will concentrate on the case in which rather extensive spatial data are available, possibly only at a single time point or at a limited number of time points.

The data are assumed to be generated by some underlying stochastic mechanism which is called a point process if the locations of points alone are considered, or a marked point process if the locations are supplemented by some marks. This kind of spatial stochastic modeling has found an enormous range of applications in areas like biometry, epidemiology, forestry, geosciences, material sciences, medicine, meteorology and others, see e.g. Cox and Isham (1980, 1994), Cox et al. (1997), Cressie (1991), Diggle (1983, 1993, 1996), Kendall et al. (1998), Mattfeldt et al. (1993, 1996), Ripley (1988), Stoyan et al. (1995), Stoyan and Stoyan (1995).

However, for optimal planning, dimensioning and operating a large (wired or wireless) communication system, it is often important to know for example the spatial distribution of those locations where at a given point in time or within a certain (short) time interval, a request for service is sent by a subscriber (fixed or mobile) to a station (e.g. an antenna, concentrator). A closely related question is to know the distribution of the number of subscribers and/or stations in a region of a given size, say a cell, or the joint distribution of these numbers in a whole family of cells, see Lee (1993, 1995). Planar Poisson processes have been used e.g. in Baccelli et al. (1996), Baccelli and Zuyev (1997, 1998), Foss and Zuyev (1996), Gerlich (1997), Hellebrandt and Mathar (1997a), Mathar and Mattfeldt (1995), Tran-Gia and Gerlich (1996), Zuyev et al. (1997) to investigate such problems for large cellular communication systems. Closely related space-time models with special emphasis on mobility of subscribers have been studied in Leung et al. (1994), and Massey and Whitt (1993, 1994), where the mobiles move along a highway according to a deterministic location function from traffic theory.

Clearly, the patterns one observes always consist of finitely many points and lie in a finite (bounded) region. But, for mathematical convenience, it is useful to consider point processes extended for the whole Euclidean space $\RL^2$. Then, a point process X can be regarded either as an infinite (but locally finite, i.e., finite on any bounded set) sequence $X=\{X_1,X_2,\ldots\}$ of random vectors, where X_n denotes the random location of the n-th point, or equivalently, as a family $X=\{X(B),\,B\in \calB\}$ of nonnegative integer-valued random variables, where X(B) denotes the random number of points which lie in the set B. Here B is an arbitrary test set, e.g. a certain part of the whole sampling region, which belongs to the family $\calB$ of all Borel subsets of $\RL^2$.

Although in this paper we will focus on stochastic models for planar point patterns, we remark that some results presented below can be generalized without too much effort to point processes in higher-dimensional Euclidean spaces, for example in $\RL^3$.