The Poisson Process and Renewal Theory

Abstract

In this chapter we analyze a stochastic process termed a renewal process. A stochastic process is a set of random variables {X(t): t ∈ T} defined on a common probability space. It is typical to think of t as time, T as a set of points in time, and X(t)as the value or state of the stochastic process at time t. We classify processes according to time and say that they are discrete time or continuous time depending on whether T is discrete (finite or countably infinite) or continuous. If T is discrete then we typically index the stochastic process with integers, that is, {X ₁, X ₂,...}. In Part I of the book all random variables that we considered either were independent or had a very simple form of dependency (e.g., they were correlated). Stochastic processes are introduced as a way to capture more complex forms of dependency between sets of random variables.