Consider a system in which individuals at any time are classified as being in one of r possible states, and assume that an individual changes states in accordance with a Markov chain having transition probabilities Pij, i, j = 1, . . . , r.

That is, if an individual is in state i during a time period then, independently of its previous states, it will be in state j during the next time period with probability Pij. The individuals are assumed to move through the system independently of each other. Suppose that the numbers of people initially in states 1, 2, . . . , r are independent Poisson random variables with respective means λ1, λ2, . . . , λr. We are interested in determining the joint distribution of the numbers of individuals in states 1, 2, . . . , r at some time n.