One of the purposes of Markov analysis is to predict the future. Given the vector of state probabilities and the matrix of transition probabilities, it is not very difficult to determine the state probabilities at a future date. With this type of analysis, we are able to compute the probability that a person will be shopping at one of the grocery stores in the future. Because this probability is equivalent to market share, it is possible to determine future market shares for American Food Store, Food Mart, and Atlas Foods. When the current period is 0, calculating the state probabilities for the next period (period 1) can be accomplished as follows:

Furthermore, if we are in any period n, we can compute the state probabilities for period $n+1$ as follows:

Equation 14-3 can be used to determine what the next period’s market shares will be for the grocery stores. The computations are

As you can see, the market shares for American Food Store and Food Mart have increased, while the market share for Atlas Foods has decreased. Will this trend continue in the next period and the one after that? From Equation 14-4, we can derive a model that will tell us what the state probabilities will be in any time period in the future. Consider two time periods from now:

Since we know that

we have

In general,

Thus, the state probabilities for n periods in the future can be obtained from the current state probabilities and the matrix of transition probabilities.

In the grocery store example, we saw that American Food Store and Food Mart had increased market shares in the next period, while Atlas Food had lost market share. Will Atlas eventually lose its entire market share? Or will all three groceries reach a stable condition? ­Although Equation 14-5 provides some help in determining this, it is better to discuss this in terms of equilibrium or steady state conditions. To help introduce the concept of equilibrium, we present a second application of Markov analysis: machine breakdowns.