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

In this paper, we provide a novel approach for studying heavy-tailed asymptotics for the stationary probability vector of a Markov chain of GI/G/1 type, whose transition matrix is constructed by two matrix sequences referred to as a boundary matrix sequence and a repeating matrix sequence. We first provide a necessary and sufficient condition under which the stationary probability vector is heavy-tailed. Then we derive long-tailed asymptotics for the R-measure in terms of the RG-factorization for the repeating matrix sequence and a Wiener-Hopf equation for the boundary matrix sequence. Based on this, we are able to provide a detailed analysis for subexponentail asymptotics of the stationary probability vector.

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 this paper, the censoring technique is used to deal with block-structured Markov renewal processes. Two probabilistic measures the R-and G-measures are defined and a censoring invariant property for the R- and G-measures is obtained The RG-factorization for the ransition probability matrix is dcrivcd based on the Wiener-Hopf type the transition probability matrix consists of two sequences of matrices one is refered to as the probability matrix consists of two sequences of matrices one is referred to as the boundary sequence and the other the repeating sequence Thw, RG-facteiation of the double transformations for the reqeating matrix sequence and four matrix inequalities of the double transformations for the boundary matrix sequence are given.

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.