Level product form QSF processes and an analysis of queues with Coxian inter-arrival distribution

TL;DR

This paper introduces a class of QSF processes with a specific structure, deriving their stationary distributions and applying the results to analyze queues with Coxian inter-arrival times, revealing monotonicity properties.

Contribution

It characterizes the stationary distribution of a new class of QSF processes and applies it to Cox(k)/M^Y/1 queues, analyzing key queue metrics.

Findings

Stationary distributions have a product form depending on the level.

Monotonicity of mean queue length and sojourn time with respect to k.

Analytical insights into queues with Coxian inter-arrival distributions.

Abstract

In this paper we study a class of Quasi-Skipfree (QSF) processes where the transition rate submatrices in the skipfree direction have a column times row structure. Under homogeneity and irreducibility assumptions we show that the stationary distributions of these processes have a product form as a function of the level. For an application, we will discuss the ${\it Cox(k)}/M^Y/1$-queue, that can be modelled as a QSF process on a two-dimensional state space. In addition we study the properties of the stationary distribution and derive monotonicity of the mean number of the customers in the queue, their mean sojourn time and the variance as a function of $k$ for fixed mean arrival rate.