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.

This paper considers the reliability approximation of the queueing system PH/PH/1 with server breakdown and repair, where the life time of the server has the exponential distribution, and its repair time has the phase type distribution. First, we obtain the steady availability and steady failure frequency of the server. Then, we get the reliability function of the server, its probability density function and its Laplace-Stiltjes transform, which also gives an effective approximation of the reliability function of the server, where the error of the approximation can be easily bounded and is uniform in t. Finally, we discuss a special case and we can show that the reliability approximation has a wider application to various Markovian queueing systems with a repairable server.

this paper, we first introduce a symmetric phase type (PH) distribution, and then propose a symmetric exponential-polynomial type (EP) distribution based on the symmetric PH distribution. The class of symmetric EP distributions is illustrated to be so large that an arbitrary symmetric probability density function in L2 (-∞, +∞), which is the space of square integrable functions on the real line, can be approximated in (-∞, +∞) by a sequence of symmetric EP probability density functions. We prove this result by means of a constructive approach based on two orthogonal decompositions: Laguerre spectrum decomposition and Hermite spectrum decomposition. We also provide a moment-based approach for simply determining a symmetric EP probability density function to approximatively express a symmetric random variable under which some moments of the symmetric random variable are given. We further propose multidimensional symmetric PH and EP distributions and provide their useful structures and properties. Finally, we apply the symmetric PH and EP distributions to study the probability Hough transform.

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.