# CHAPTER 4

# MODELING WITH SYSTEMS OF LINEAR DIFFERENCE EQUATIONS

# 4.1. MODELING WITH MARKOV CHAINS

In this section we discuss a special class of finite stochastic processes. We consider a sequence of finite experiments in which the outcomes and associated probabilities at each stage depend only on the outcomes and associated probabilities of the immediately preceding stage. Such a process is called a **Markov process** or a **Markov chain**. The Markov chains (processes) are named after the Russian mathematician Andrey Andreyevich Markov (1856–1922). Let’s investigate the following population movement model.

**4.1.1. A Population Movement Model**

We consider a simple model of population movement between a certain city and its surrounding suburbs. For simplicity, we assume the following:

i. The people who move from the city go to the suburbs, and the people who move from the suburbs go to the city,

ii. We assume that during a year the total population in the city and its surrounding suburbs is fixed—that is we ignore other factors such as births and deaths.

Under these two assumptions the total population in the city and the suburbs is the same every year. Assume that the demographic studies showed that during 2000, 6% of the city population move to suburbs and 2% of the suburb population move to the city. This means that 94% of the city population stays in the city and 98% of the suburb population stays in the suburb.