Level product form QSF processes and an analysis of queues with Coxian interarrival distribution

Abstract

Abstract:In this article, 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 Cox(k)/M Y /1 -queue that can be modeled 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.

A successive censoring algorithm for a system of connected LDQBD-processes

“…We will briefly discuss the relation between our model and the models discussed in Gaver et al (1984), Smit (2012), andErtiningsih et al (2019).…”

“…Finally, it is worth noting that in this paper there is no restriction that the up transition matrices should have a rank of one, e.g., equal to the product of a column and a row vector, as assumed in Ertiningsih et al (2019).…”

“…The single entrance states in Katehakis and Smit (2012), Katehakis et al (2015) are called mandatory states in Kim and Smith (1989) and input states in Feinberg and Chiu (1987) in which a parallel lumping procedure was introduced. Most recently, Ertiningsih et al (2019) studied a class of Quasi-Skip Free processes where the transition rate submatrices in the Skip Free direction, either to the right or the left, have a column times row structure. For this class of Markov chains they derived the stationary distributions and its properties.…”

“…(3) In Ertiningsih et al (2019) a two dimensional ski-free process is considered with up transition matrices having a rank of one, e.g., equal to the product of a column and a row vector. In our case, we consider a three-dimensional process with the rank of matrices larger than one.…”

Matrix geometric approach for random walks: Stability condition and equilibrium distribution

“…Regarding QBDs with a simple structure the literature is mainly devoted to stochastic processes for which the rate matrix has a simple property, see, e.g., [11] and the references therein, or to stochastic process with a special structure in the allowed transitions, see, e.g., [19,20] and the references therein. In [11], the authors consider an infinite rate matrix R that can be written as the product of a vector column times a row vector. This simple structure permits, under the assumptions of homogeneity and irreducibility, to show that the stationary distributions of these processes have a product form structure as a function of the level.…”