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570:{\displaystyle Q={\begin{pmatrix}B_{00}&B_{01}\\B_{10}&A_{1}&A_{2}\\&A_{0}&A_{1}&A_{2}\\&&A_{0}&A_{1}&A_{2}\\&&&A_{0}&A_{1}&A_{2}\\&&&&\ddots &\ddots &\ddots \end{pmatrix}}}
259:{\displaystyle P={\begin{pmatrix}A_{1}^{\ast }&A_{2}^{\ast }\\A_{0}^{\ast }&A_{1}&A_{2}\\&A_{0}&A_{1}&A_{2}\\&&A_{0}&A_{1}&A_{2}\\&&&\ddots &\ddots &\ddots \end{pmatrix}}}
32:. As with the birth-death process it moves up and down between levels one at a time, but the time between these transitions can have a more complicated distribution encoded in the blocks.
954:
837:; Scheinhardt, W. R. W.; Taylor, P. G. (2004). "Spectral properties of the tandem Jackson network, seen as a quasi-birth-and-death process".
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Palugya, S. N.; Csorba, M. T. J. (2005). "Modeling Access
Control Lists with Discrete-Time Quasi Birth-Death Processes".
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978:
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Latouche, G.; Pearce, C. E. M.; Taylor, P. G. (1998). "Invariant measures for quasi-birth-and-death processes".
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876:"Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators"
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are matrices. The process can be viewed as a two dimensional chain where the block structure are called
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The stationary distribution of a quasi-birth-death process can be computed using the
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802:. Stochastic Modelling and Applied Probability. Vol. 51. pp. 302–339.
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Latouche, G. (2011). "Level-Independent Quasi-Birth-and-Death
Processes".
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can be considered as quasi-birth-death processes with infinitely (but
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Usually the blocks have finitely many phases, but models like the
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Encyclopedia of Operations Research and Management Science
640:(as transition times are then not exponentially distributed).
632:. When describing the process by both level and phase it is a
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are irregular matrices for the first and second levels.
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798:Asmussen, S. R. (2003). "Markov Additive Models".
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28:describes a generalisation of the
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628:and the intra-block structure
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