Introduction to Stochastic Processes and Simulation

7Various Applications

CONCEPTS DISCUSSED IN THIS CHAPTER.– This section discusses various applications of random processes.

First, we look at a classic application: reliability and availability of equipment.

Then, we focus on the large research field of genetics.

This is followed by population dynamics, with the predator-prey model.

Finally, the chapter analyzes Brownian motion, which takes us from physics to finance.

Recommended reading: [DUP 05, FAU 14, LEB 12, LES 14, PEL 97, RUE 89].

7.1. Reliability and availability of equipment

7.1.1. Reliability and instantaneous failure rate

In many processes, a technical device is expected to perform a required operation for a given time period. This quality is called “reliability”. Knowing that the device operates at time t = 0, the reliability R(t) is the probability that it still operates at a later time t. The operating time T of the device is a random variable, because generally we cannot predict failures that may occur. T is a continuous variable that takes positive values. Therefore, we put:

Since R(t) is a probability, it is between 0 and 1. It is assumed that at moment 0, the device operates, so R(0) = 1. Subsequently, the device may fail and therefore R(t) < 1 for t ∈]0, + ∞[. The more time passes, the higher the probability of failure and the reliability decreases.

T is the lifetime or service time of the device. If it is ...