Introduction to Stochastic Models

Chapter 4

Renewal Models

Renewal theory was developed during the studies carried out on practical problems that arise from the degradation and replacement of components of complex systems. The fundamental theory can be found in [FEL 66], while [COX 62] is a monograph on the topic.

4.1. Fundamental concepts and examples

Note that, very often, to an arbitrary stochastic process there can naturally be associated sequences of i.i.d. r.v.

Let X₁ ,X₂,... be a sequence of real i.i.d. r.v., with the same d.f. F, F(−0) = 0, F(0) < 1, defined on a probability space . The sequence (S_n), where

[4.1]

is called a renewal chain (process).

In the applications that led to the renewal theory, the r.v. S₁, S₂, … represent the replacement (renewal) times of worn out components, while X₁, X₂, … are the working times (the times between successive arrivals). The expected value μ = (X_k), that exists and can be possibly +00, is called the mean lifetime of the components.

The counting function, i.e. the number of renewals during a time interval [0, t], is an r.v. defined by

[4.2]

and, for n ≥ 1, we have

where ...