The probabilities of extinction for birth-and-death processes that are age-dependent or phase-dependent

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Abstract
In this paper we determine, for an individual of age x existing at some specified point in time, the probability p(x) that his line of descendants will eventually become extinct, for the age-dependent birth-and-death process studied by D. G. Kendall (1949) and Bartlett (1955) in which the time-scale and age-scale are continuous, and also for the analogous age dependent birth-and-death process earlier studied by Bartlett and by Pollard (1966) in which the time-scale and age-scale are discrete. The extinction probabilities are also determined for generalizations of these processes in which, in a unit time-interval in the discrete time, discrete-age process(or in a small time-interval in the continuous-time, continuous-age process), a given individual may produce more than one new individual (multiple births), and his chances of producing new individuals may depend, in ways other than those specified for the original processes, on whether he does or does not live through the time-interval. In addition, the extinction probabilities p(x) are determined for a multiphase birth-and-death process, which is a generalization of the multiphase birth process studied by Kendall (1948 a), in which an individual may move through a number of different phases during his lifetime, and the phase in which he is at a specified point in time may affect his chances of producing new individuals and his chances of dying at that time. For each of these processes, and for any value of x > 0, we present an explicit expression for the extinction probability p(x) in terms of p(0), and we determine p(0) from a single equation.

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