Resumen
Birth-multiple catastrophe processes are analyzed where the birth transition rates are assumed to be constant while catastrophes are distinguished by having possibly different destinations and possibly different transition rates. The transient probability functions of such birth-multiple catastrophe systems are determined. The solution method uses dual processes, randomization, and sample path counting. Solutions are explicit in terms of being a finite linear combination of products of exponential functions of time, t, and nonnegative integer powers of t. The coefficients within this expansion follow a pattern of rational functions of the transition rates.

Resumen
A representation of the transient probability functions of finite birth–death processes (with or without catastrophes) as a linear combination of exponential functions is derived using a recursive, Cayley–Hamilton approach. This method of solution allows practitioners to solve for these transient probability functions by reducing the problem to three calculations: determining eigenvalues of the Q-matrix, raising the Q-matrix to an integer power and solving a system of linear equations. The approach avoids Laplace transforms and permits solution of a particular transition probability function from state i to j without determining all such functions.