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.

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 provide UL-type and LU-type RG-factorizations for an irreducible continuous-time level-dependent quasi-birth-and-death (QBD) process with either finitely-many levels or infinitely-many levels and then apply the RG-factorizations to solve a class of linear QBD-equations which is always crucial for analyzing a stochastic model described as a QBD process Based on the results obtained for the linear QBD-equations we analyze up-, down-and return-integral functinals We explicityly express the Laplace transforms of the conditional distributions of the three types of stochastic integral functionals and their conditional moments.

We consider the reliability proble of the SM/PH/1 queue with repairable server of PH life time and PH repai time. The generalizedservices are proved to be dependent on each other and not to have the same distribution if the system is at transient stata For this reason the transient behavior of the system is so complex that we can not give any available results On the contrary the repairable queueing system is shown to be equivalent to the classical SMSM(PH)/1 queue when the system is at steady state, where SM(PH) denoted Markovian semi-Markov process which was discussed in Latouche [10] and Sengupta [23] By means of the equivalent model SM/SM(PH)/1 the explicit expressions of the reliability quantities are derived, for instance, the steady state availability and the steady state failure frequency Copyright1999IFAC