The MX/M/1 queue with working breakdown

Zaiming Liu; Yang Song

RAIRO - Operations Research - Recherche Opérationnelle (2014)

  • Volume: 48, Issue: 3, page 399-413
  • ISSN: 0399-0559

Abstract

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In this paper, we consider a batch arrival MX/M/1 queue model with working breakdown. The server may be subject to a service breakdown when it is busy, rather than completely stoping service, it will decrease its service rate. For this model, we analyze a two-dimensional Markov chain and give its quasi upper triangle transition probability matrix. Under the system stability condition, we derive the probability generating function (PGF) of the stationary queue length, and then obtain its stochastic decomposition, which shows the relationship with that of the classical MX/M/ 1 queue without vacation or breakdown. Besides, we also give the Laplace-Stieltjes transform (LST) of the stationary waiting time distribution of an arbitrary customer in a batch. Finally, some numerical examples are given to illustrate the effect of the parameters on the system performance measures.

How to cite

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Liu, Zaiming, and Song, Yang. "The MX/M/1 queue with working breakdown." RAIRO - Operations Research - Recherche Opérationnelle 48.3 (2014): 399-413. <http://eudml.org/doc/275051>.

@article{Liu2014,
abstract = {In this paper, we consider a batch arrival MX/M/1 queue model with working breakdown. The server may be subject to a service breakdown when it is busy, rather than completely stoping service, it will decrease its service rate. For this model, we analyze a two-dimensional Markov chain and give its quasi upper triangle transition probability matrix. Under the system stability condition, we derive the probability generating function (PGF) of the stationary queue length, and then obtain its stochastic decomposition, which shows the relationship with that of the classical MX/M/ 1 queue without vacation or breakdown. Besides, we also give the Laplace-Stieltjes transform (LST) of the stationary waiting time distribution of an arbitrary customer in a batch. Finally, some numerical examples are given to illustrate the effect of the parameters on the system performance measures.},
author = {Liu, Zaiming, Song, Yang},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {MX/M/1 queue; working breakdown; probability generating function (PGF); Laplace–Stieltjes transform (LST); waiting time distribution; stochastic decomposition; queue; Laplace-Stieltjes transform (LST)},
language = {eng},
number = {3},
pages = {399-413},
publisher = {EDP-Sciences},
title = {The MX/M/1 queue with working breakdown},
url = {http://eudml.org/doc/275051},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Liu, Zaiming
AU - Song, Yang
TI - The MX/M/1 queue with working breakdown
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 3
SP - 399
EP - 413
AB - In this paper, we consider a batch arrival MX/M/1 queue model with working breakdown. The server may be subject to a service breakdown when it is busy, rather than completely stoping service, it will decrease its service rate. For this model, we analyze a two-dimensional Markov chain and give its quasi upper triangle transition probability matrix. Under the system stability condition, we derive the probability generating function (PGF) of the stationary queue length, and then obtain its stochastic decomposition, which shows the relationship with that of the classical MX/M/ 1 queue without vacation or breakdown. Besides, we also give the Laplace-Stieltjes transform (LST) of the stationary waiting time distribution of an arbitrary customer in a batch. Finally, some numerical examples are given to illustrate the effect of the parameters on the system performance measures.
LA - eng
KW - MX/M/1 queue; working breakdown; probability generating function (PGF); Laplace–Stieltjes transform (LST); waiting time distribution; stochastic decomposition; queue; Laplace-Stieltjes transform (LST)
UR - http://eudml.org/doc/275051
ER -

References

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