# Matrix-analytic stochastic models

Reference work entry

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**DOI:**https://doi.org/10.1007/1-4020-0611-X_598

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A rich class of models for queues, dams, inventories, and other stochastic processes has arisen out of matrix/vector generalizations of classical approaches. We present three specific examples in the following, namely, matrix-analytic solutions for M/G/1-type queueing problems, matrix-geometric solutions to GI/ M/1-type queueing problems, and finally, the Markov arrival process (MAP) generalization of the renewal point process.

## MATRIX-ANALYTIC M/G/1-TYPE QUEUES

The unifying structure that underlies these models is an imbedded Markov renewal process whose transition probability matrix is of the form:

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© Kluwer Academic Publishers 2001