Interconnected birth and death processes

SummaryTwo cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f) g of the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution for g. This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends to g as n → ∞ in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λi and μi, i = 1,2, interconnected linearly with transition rates v and δ (see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1 = δ = 0, with the remaining rates time dependent and (ii) λ2 = δ = 0, with the remaining rates constant, explicit solutions for g are obtained and studied.