APPROXIMATIONS OF STATE TRANSITION-PROBABILITIES IN FINITE BIRTH-DEATH PROCESSES

This paper studies approximations to the transient probabilities P(ij)(t) (i, j = 0, 1, 2, ..., n) for a transition from state i at t = 0 to state j at time t in the n-channel birth-death processes. First, P(0n)(t) is considered as an extension of Gnedenko's approximate expressions P(0n)(t) when state n is regarded as an absorbing state for the models M/M/1/n/infinity, M/M/1/n/N, M/M/n/n/infinity, and M/M/n/n/N. That is to say, if P(n) is the steady-state probability of state n, the approximations P(0n)(t)/P(n) when state n is not an absorbing state can be obtained from the function 1 - exp{-Q(t)} (Q(t) greater-than-or-equal-to 0 is an analytic function). Based on these considerations, transition diagrams are derived to obtain P(0n)(t) for the models M/M/S/ n/infinity and M/M/S/n/N. Finally, P(ij)(t) can be expressed with this P(0n)(t). Several examples show that the approximations of the transient probabilities are nearly equal to the exact values calculated numerically using the Runge-Kutta method on a personal computer. As the approximations in this paper are very precise and calculable instantaneously on a personal computer, they may be applicable for time-dependent traffic theory which will be useful, for instance, in real-time network management technology.