Matrix-geometric solutions of M/G/1-type Markov chains: A unifying generalized state-space approach
Date
1998Source Title
IEEE Journal on Selected Areas in Communications
Print ISSN
0733-8716
Volume
16
Issue
5
Pages
626 - 639
Language
English
Type
ArticleItem Usage Stats
197
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274
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Abstract
In this paper, we present an algorithmic approach to find the stationary probability distribution of M/G/1-type Markov chains which arise frequently in performance analysis of computer and communication networ ks. The approach unifies finite- and infinite-level Markov chains of this type through a generalized state-space representation for the probability generating function of the stationary solution. When the underlying probability generating matrices are rational, the solution vector for level k, x k, is shown to be in the matrix-geometric form x k+1 = gF k H, k ≥ 0, for the infinite-level case, whereas it takes the modified form x k+1 = g 1F 1 kH 1 + g 2F 2 K-k-1 H 2, 0 ≤ k < K, for the finite-level case. The matrix parameters in the above two expressions can be obtained by decomposing the generalized system into forward and backward subsystems, or, equivalently, by finding bases for certain generalized invariant subspaces of a regular pencil λE - A. We note that the computation of such bases can efficiently be carried out using advanced numerical linear algebra techniques including matrix-sign function iterations with quadratic convergence rates or ordered generalized Schur decomposition. The simplicity of the matrix-geometric form of the solution allows one to obtain various performance measures of interest easily, e.g., overflow probabilities and the moments of the level distribution, which is a significant advantage over conventional recursive methods.
Keywords
ATM multiplexer analysisGeneralized difference equations
Generalized invariant subspaces
Generalized Schur decomposition
M/G/1-type Markov chains
Matrix-sign function
Polynomial matrix fractional descriptions
Asynchronous transfer mode
Difference equations
Iterative methods
Markov processes
Matrix algebra
Multiplexing equipment
Polynomials
State space methods
Generalized Schur decomposition
Matrix geometric solutions
Matrix sign function
Telecommunication traffic
Permalink
http://hdl.handle.net/11693/25455Published Version (Please cite this version)
http://dx.doi.org/10.1109/49.700901Collections
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