Matrixgeometric solutions of M/G/1type Markov chains: A unifying generalized statespace approach
Author
Akar, N.
Oǧuz, N.C.
Sohraby, K.
Date
1998Source Title
IEEE Journal on Selected Areas in Communications
Print ISSN
07338716
Volume
16
Issue
5
Pages
626  639
Language
English
Type
ArticleItem Usage Stats
32
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Show full item recordAbstract
In this paper, we present an algorithmic approach to find the stationary probability distribution of M/G/1type Markov chains which arise frequently in performance analysis of computer and communication networ ks. The approach unifies finite and infinitelevel Markov chains of this type through a generalized statespace 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 matrixgeometric form x k+1 = gF k H, k ≥ 0, for the infinitelevel case, whereas it takes the modified form x k+1 = g 1F 1 kH 1 + g 2F 2 Kk1 H 2, 0 ≤ k < K, for the finitelevel 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 matrixsign function iterations with quadratic convergence rates or ordered generalized Schur decomposition. The simplicity of the matrixgeometric 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/1type Markov chains
Matrixsign 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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