The M X /M/1 queue with working breakdown
RAIRO - Operations Research - Recherche Opérationnelle, Tome 48 (2014) no. 3, pp. 399-413.

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.

DOI : 10.1051/ro/2014014
Classification : 68M20, 60K20, 90B22
Mots-clés : MX/M/1 queue, working breakdown, probability generating function (PGF), Laplace-Stieltjes transform (LST), waiting time distribution, stochastic decomposition 
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     title = {The $M^{X}/M/1$ queue with working breakdown},
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     pages = {399--413},
     publisher = {EDP-Sciences},
     volume = {48},
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Liu, Zaiming; Song, Yang. The $M^{X}/M/1$ queue with working breakdown. RAIRO - Operations Research - Recherche Opérationnelle, Tome 48 (2014) no. 3, pp. 399-413. doi : 10.1051/ro/2014014. http://archive.numdam.org/articles/10.1051/ro/2014014/

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