Steady-state queue size distribution of discrete-time \(PH/Geo/1\) retrial queues.

*(English)*Zbl 1042.60543Summary: This paper studies discrete-time single server retrial queues with phase-type interarrival times and geometric service times. A matrix analytical method is applied to derive the analytical solution for the joint steady-state distribution of arrival phases and queue sizes of the system. The necessary and sufficient condition for system stability is also determined. Based on the special form of the matrix analytic solution of the system, upper and lower bounds for the joint steady-state distribution of arrival phases and queue sizes are developed. It is shown that the errors between these upper and lower bounds can be made as small as desired. Finally, an efficient and numerically stable algorithm for computing the joint steady-state probabilities of arrival phases and queue sizes is presented.

##### MSC:

60K25 | Queueing theory (aspects of probability theory) |

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\textit{H. Li} and \textit{T. Yang}, Math. Comput. Modelling 30, No. 3--4, 51--63 (1999; Zbl 1042.60543)

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##### References:

[1] | Falin, G.I., A survey of retrial queues, Queueing systems, 7, 127-168, (1990) · Zbl 0709.60097 |

[2] | Yang, T.; Templeton, J.G.C., A survey on retrial queues, Queueing systems, 2, (1987) · Zbl 0658.60124 |

[3] | Yang, T.; Li, H., On the steady-state queue size distribution of the discrete-time geo/G/1 queue with repeated customers, Queueing systems, 21, 199-215, (1995) · Zbl 0840.60085 |

[4] | Neuts, M.F., Matrix-geometric solutions in stochastic models, (1981), The John Hopkins University Press Baltimore, MD · Zbl 0469.60002 |

[5] | Diamond, J.E.; Alfa, A.S., Matrix analytic methods for M/PH/1 retrial queues, Commun. statist.â€”stochastic models, 11, 3, 447-470, (1995) · Zbl 0829.60091 |

[6] | Berman, A.; Plemmons, R.J., Nonnegative matrices in the mathematical science, (1979), Academic Press New York |

[7] | Horn, R.; Johnson, C.A., Matrix analysis, (1985), Cambridge University Press New York · Zbl 0576.15001 |

[8] | Neuts, M.F.; Rao, B.M., Numerical investigation of a multiserver retrial model, Queueing systems, 7, (1990) · Zbl 0711.60094 |

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