This paper refers to some closed two-stage queueing systems with state-dependent vacation policy. By means of the markovian renewal process and Ihe stochastic decomposition formulas of the quantities in equilibrium for M/G/I vacation models. we derive both stationary distributions of tbe queue length and the cyclic time for the closed stale-dependent vacation model We try to solve the problems of closed sustems by using the known results of the related opened Models, and for the purpose give an example in

In theis paper, we study a BMAP/M/1 generalized processor-sharing queue, We propose a RG-factorization approach which can be applied to a wider class ofMarkovian block-structured processor-sharing queues We obtain the expressions for bothe the distribution of the stationary queue length and the Laplace transform of the sojourn time distribution From these two expressions we develop an algorithm to compute the mean and variance of the sojourn time approximately.

We consider the MIMic queue with server multiple vacations, where each of the c servers leaves for a vacation of an exponentially distributed duration when it finds no waiting units in line. If the server returns to an empty waiting line, it immediately takes another vacation. This paper is concerned with the upper triangular structure of Lhe rate matrix R and the determinaLion of the stationary distnbuliolls of queue length and waiting time. The emphasis is on obtaining the conditional stochastic decompositions of the stationary queue length and waiting time conditioned by the event [B=c], where [B=c) denotes the event that all of the c servers are busy.

In this paper, we present an algorithmic approach for sensitivity analysis of stationary and transient performance measures of a perturbed continuous-time level-dependent quasi-birth-and-death (QBD) process with infinitely-many levels. By developing a new LU-type RG-factorization using the censoring technique, we obtain the maximal negative inverse of the infinitesimal generator of the QBD process. The derivatives of the stationary performance measures of the QBD process can then be expressed and computed in terms of the maximal negative inverse, overcoming the computational difficulty arising from the use of group inverses of infinite size in the current literature (see Cao and Chen [11]). We also use a stochastic integral functional to study the transient performance measure of the QBD process and show how to use the algorithmic approach for its sensitivity analysis. As an example, a perturbed MAP/PH/1 queue is also analyzed.

In this paper, we study transition matrices of GI/M/1 type by using the approach proposed in Li and Zhao.[13] We obtain conditions on the a-classification of states for the transition matrix of GI/M/1 type. Unlike for matrices of M/G/1 type where association of the matrix multiplication can be easily justified, for matrices of GI/M/type, we first construct formal expressions for the β-invariant measure based on a representation of factorization of the transition matrix, and then show that it is a β-invariant measure directly. We also prove some spectral properties for the matrix of GI/M/1 type, which are not only used in constructing a formal expression for the b-invariant measure, but also of their own interest. We point out that the spectral analysis required for studying matrices of GI/M/1 type is much more sophisticated than that for matrices of M/G/1 type. Finally, we discuss connections of expressions for the b-invariant measure provided in this paper and in the literature.