β-invariant measures for transition matrices of GI/M/1 type

β-invariant measures for transition matrices of GI/M/1 type

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Article ID: iaor2004677
Country: United States
Volume: 19
Issue: 2
Start Page Number: 201
End Page Number: 233
Publication Date: Jan 2003
Journal: Communications in Statistics - Stochastic Models
Authors: ,
Keywords: queues: theory
Abstract:

In this paper, we study transition matrices of GI/M/1 type by using the approach proposed in Li and Zhao. We obtain conditions on the α-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/1 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 β-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 β-invariant measure provided in this paper and in the literature.

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