Transient solutions of Markov processes and generalized continued fractions

Abstract

In 1974 J. A. Murphy and M. R. O'Donohoe numerically approximated the minimal solution of the Kolmogorov forward equation for the generalized birth and death process by use of continued fractions. This paper generalizes this approach by suggesting an algorithm for q-matrices of lower band structure (n, 1). This is achieved by analogy with generalized continued fractions. Applications involving q-matrices of this type include, for example, many types of queueing systems with batch processing or birth-death-catastrophe population processes in biology.