Marginal and joint distribution of inter-departure times for a PH/PH/c queue

In this paper, we introduce the Laplace–Stieltjes transform (LST) of the inter-departure time distribution in a PH/PH/c queue, and the two-dimensional joint LST of two consecutive inter-departure times to construct their correlation structure. We exploit the properties of phase-type (PH) random variables, as well as the steady-state distribution of the underlying continuous-time Markov chain in a PH/PH/c queue to construct these LSTs. We demonstrate our approach through numerical examples, while validating the results. Later, we analyze the correlation between two consecutive inter-departure times for various PH/PH/c queues. We observe that, if the fundamental elements of the queue have high (low) variability, then the correlation is positive (negative).