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 be-tween 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).

Automated summary

This paper introduces the Laplace-Stieltjes transform (LST) and the two-dimensional joint LST for the inter-departure time distribution in a PH/PH/c queue. These LSTs are constructed using the properties of phase-type (PH) random variables and the steady-state distribution of the continuous-time Markov chain. Numerical examples are provided to validate the approach, and the correlation between consecutive inter-departure times in various queues is analyzed.