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      A general theory on spectral properties of state-homogeneous finite-state quasi-birth-death processes

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      Author(s)
      Fadıloğlu, M. M.
      Yeralan, S.
      Date
      2001
      Source Title
      European Journal of Operational Research
      Print ISSN
      0377-2217
      Publisher
      Elsevier
      Volume
      128
      Issue
      2
      Pages
      402 - 417
      Language
      English
      Type
      Article
      Item Usage Stats
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      173
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      Abstract
      In this paper a spectral theory pertaining to Quasi-Birth–Death Processes (QBDs) is presented. The QBD, which is a generalization of the birth–death process, is a powerful tool that can be utilized in modeling many stochastic phenomena. Our theory is based on the application of a matrix polynomial method to obtain the steady-state probabilities in state-homogeneous finite-state QBDs. The method is based on finding the eigenvalue–eigenvector pairs that solve a matrix polynomial equation. Since the computational effort in the solution procedure is independent of the cardinality of the counting set, it has an immediate advantage over other solution procedures. We present and prove different properties relating the quantities that arise in the solution procedure. By also compiling and formalizing the previously known properties, we present a formal unified theory on the spectral properties of QBDs, which furnishes a formal framework to embody much of the previous work. This framework carries the prospect of furthering our understanding of the behavior the modeled systems manifest.
      Keywords
      Eigenvalues and eigenfunctions
      Markov processes
      Mathematical models
      Polynomials
      Probability distributions
      Queueing theory
      Theorem proving
      Jordan canonical forms
      Quasi-birth-death processes
      Operations research
      Permalink
      http://hdl.handle.net/11693/24900
      Published Version (Please cite this version)
      http://dx.doi.org/10.1016/S0377-2217(99)00367-7
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