# Point Processes

• Philippe Robert
Chapter
Part of the Applications of Mathematics book series (SMAP, volume 52)

## Abstract

This chapter introduces the basic definitions and results concerning point processes. A point process is the mathematical object used to describe the flow of customers in a queueing system: the arrival instants and the value of services they require. It can be an arrival process as well as a departure process. General definitions and properties of point processes are briefly presented. An important subclass is analyzed in detail: Poisson point processes. The Poisson point process is, with Brownian motion, an ubiquitous object in probability theory. It shows up in numerous limit theorems and the explicit form of the distribution of many of its functionals can be easily derived. Poisson point processes are presented in a quite general framework (a locally compact space or a complete metric space) because their main properties are simple and independent of the particular structure of ℝ d where they are generally considered. Moreover, this level of generality is required to study the important marked Poisson point processes. The last part of this chapter is a short summary of the main properties of renewal point processes, including the renewal theorem.