Transient solutions of Markov processes and generalized continued fractions

In 1974 J. A. Murphy and M. R. O'Donohoe numerically approximatedthe minimal solution of the Kolmogorov forward equation forthe generalized birth and death process by use of continuedfractions. This paper generalizes this approach by suggestingan 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, forexample, many types of queueing systems with batch processingor birth–death–catastrophe population processesin biology.