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      Matrix-geometric solutions of M/G/1-type Markov chains: A unifying generalized state-space approach

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      Author(s)
      Akar, N.
      Oǧuz, N.C.
      Sohraby, K.
      Date
      1998
      Source Title
      IEEE Journal on Selected Areas in Communications
      Print ISSN
      0733-8716
      Volume
      16
      Issue
      5
      Pages
      626 - 639
      Language
      English
      Type
      Article
      Item 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 analysis
      Generalized 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/11693/25455
      Published Version (Please cite this version)
      http://dx.doi.org/10.1109/49.700901
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      • Department of Electrical and Electronics Engineering 4011
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