Spectral theory for the differential equations of simple birth and death processes

Abstract

The enumerably infinite system of differential equations describing a temporally homogeneous birth and death process in a population is treated as the limiting case of one or the other of two finite systems of equations. Starting from the expansion of a finite matrix in terms of its associated idempotents, the solutions of the infinite system are displayed in spectral form which, in general, is written as a Stieltjes integral involving a spectral function. This method facilitates the investigation of asymptotic values and of the ergodic property of the system. When the birth- and death-rates satisfy certain conditions of regularity, the spectrum is discrete and the solution can be written down more explicitly. Concrete examples are given, where the system has two distinct solutions for any set of initial conditions. Finally, our method is applied to the known case of linear growth and to a problem in the theory of queues, confirming a result and a conjecture by D. G. Kendall.