An Iterative Solution of Two-Dimensional Birth and Death Processes

This paper presents an iterative, seminumerical method for solving the balance equations of finite two-dimensional birth and death processes. The method is seminumerical in that it uses the formal knowledge of the stationary probability distribution of one variable, and the iteration is applied to the conditional probabilities of the second variable given the first one. Sufficient convergence conditions for this approach are discussed. An always convergent entirely numerical alternative solution is also presented. Empirical results indicate that both methods perform, in many cases, several times better (in terms of time required) than the commonly used Gauss-Seidel method. Possible generalizations to processes of more than two dimensions are also indicated.