# Joint queue length distribution of multi-class, single server queues with preemptive priorities

A.V. Sleptchenko, J. Selen, I.J.B.F. Adan, G.J.J.A.N. Houtum, van

### Abstract

In this paper we analyze an \$M/M/1\$ queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an \$M/M/1\$ queueing system with non-preemptive priorities between customer classes.
Original language English s.n. 15 Published - 2014

### Publication series

Name arXiv.org 1411.3176 [math.PR]

Markov processes
Logistics
Repair
Servers

### Cite this

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title = "Joint queue length distribution of multi-class, single server queues with preemptive priorities",
abstract = "In this paper we analyze an \$M/M/1\$ queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an \$M/M/1\$ queueing system with non-preemptive priorities between customer classes.",
author = "A.V. Sleptchenko and J. Selen and I.J.B.F. Adan and {Houtum, van}, G.J.J.A.N.",
year = "2014",
language = "English",
series = "arXiv.org",
publisher = "s.n.",

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s.n., 2014. 15 p. (arXiv.org; Vol. 1411.3176 [math.PR]).

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AU - Selen, J.

AU - Houtum, van, G.J.J.A.N.

PY - 2014

Y1 - 2014

N2 - In this paper we analyze an \$M/M/1\$ queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an \$M/M/1\$ queueing system with non-preemptive priorities between customer classes.

AB - In this paper we analyze an \$M/M/1\$ queueing system with an arbitrary number of customer classes, with class-dependent exponential service rates and preemptive priorities between classes. The queuing system can be described by a multi-dimensional Markov process, where the coordinates keep track of the number of customers of each class in the system. Based on matrix-analytic techniques and probabilistic arguments we develop a recursive method for the exact determination of the equilibrium joint queue length distribution. The method is applied to a spare parts logistics problem to illustrate the effect of setting repair priorities on the performance of the system. We conclude by briefly indicating how the method can be extended to an \$M/M/1\$ queueing system with non-preemptive priorities between customer classes.

M3 - Report

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BT - Joint queue length distribution of multi-class, single server queues with preemptive priorities

PB - s.n.

ER -