Space—Time Poisson Models

Abstract

This chapter covers space-time Poisson models for queueing networks, spatial service or storage systems, and particle systems. Such a model describes the collective movement of units or customers in space and time, where the units enter the system according to a Poisson space-time process and then move about independently of each other. Because of these properties, the evolution of the system can be formulated by certain "random transformations" of Poisson point processes in space and time. We characterize these transformations and then use them in a variety of models. An important example is a network with time-dependent Poisson arrival process and infinite-server nodes with general service times. We also consider models for systems in which the input process is not Poisson, but the system is sparsely populated. The sparseness leads to Poisson space-time models that are justified by convergence theorems. An example is a network of infinite-server nodes with a non-Poisson arrival process and general service times. 9.1 Introductory Examples The following are two classic examples of space-time Poisson models that give a glimpse of what lies ahead. Example 9.1. Treelike Network of M/G/co Service Stations. Consider an open network of m service stations (or nodes), where the service times at node j are independent and identically distributed with mean JL j I. There is no queueing for service, since only a finite number of the servers are busy at any time. For simplicity , assume the network forms a tree with a single root, and each customer enters R. Serfozo, Introduction to Stochastic Networks