A formula for tail probabilities of Cox distributions

We derive a simple recursive scheme for calculating tail probabilities of Cox distributions. This is particularly useful for the computation of certain performance measures in queueing systems. An example of a call center model is provided. A Cox distribution with n > 0 phases can be defined as the time until absortion into state 0, starting from state n, of the Markov process depicted in Figure 1. The process remains in state 1 ≤ k ≤ n an exponentially distributed amount of time with parameter μk. Upon departure from state k the process moves to state 0 with probability αk and moves to state k − 1 with probability ᾱk = 1 − αk. To avoid trivial situations we assume that αk < 1 for all k.