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 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 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.

In this paper, we study the transition matrix of M/G/1 type, The radius of conver-gence is discussed, conditions onf the a-classificaion of the states are obtained and expressions of the β-invariant measure are constructed. The censoring technique is generalized to deal with nonnegative matices which may be neither stochastic nor substochastic. This allows us to prove a factorization result for the discounted transition matrix. This factorization provides a unifiedalgorithmic approach for expressing theβ-invariant measure for transition matrices with a block-sturcture, including the matrix of M/G/1 type.

In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period.