An inverse problem in birth and death processes

An inverse problem of constructing birth and death processes X(t) n finite state space 0, 1, 2, …, N n0 s considered. Given a set of 2N + 1, distinct, nonnegative real numbers one of which is zero, say 0 = s0 < z1 < 2< … < zN < sN, a procedure is established to obtain the birth and death rates of a birth and death process so that and other transient system size probabilities. This technique is illustrated numerically.