Some further results on the birth-and-death process and its integral

Abstract

AbstractIn a simple homogeneous birth-and-death process with λ and μ as the constant birth and death rates respectively, let X(t) denote the population size at time t, Z(t) the number of deaths and N(t) the number of events (births and deaths combined) occurring during (0, t). Also let . The results obtained include the following:(a) An explicit formula for the characteristic quasi-probability generating function of the joint distribution of X(t), Y(t) and Z(t).(b) Let X(0) = 1. It is shown that, if t → ∞ while λ ≤ μ, N(t) ↑ N a.s., where N takes only positive odd integral values. If λ > μ, then P[N(t) ↑ ∞] = 1 − μ/λ. Given that N(t)∞, the limiting distribution of N(t) is similar to that of N. It was reported earlier (Puri (11)), that the limiting distribution of Y(t) is a weighted average of certain chi-square distributions. It is now found that these weights are nothing but the probabilities P[N = 2k + 1] (k = 0, 1,…).(c) Let λ = μ, and MXω), MYω and MZω be defined as in (36), then aswhere the c.f. of (X*; Y*; Z*) is given by (38).(d) Exact expressions for the p.d.f. of Y(t) are derived for the cases (i) λ = 0, μ > 0, (ii) λ > 0, μ = 0. For the case (iii) λ gt; 0, μ > 0, since the complete expression is complicated, only the procedure of derivation is indicated.(e) Finally, it is shown that the regressions of Y(t) and of Z(t) on X(t) are linear for X(t) ≥ 1.