# Large Scale and Heavy Traffic Asymptotics for Systems with Unreliable Servers

1 ALGO - Algorithms
Inria Paris-Rocquencourt
Abstract : The asymptotic behaviour of the M/M/n queue, with servers subject to independe- nt breakdowns and repairs, is examined in the limit where the number of servers tends to infinity and the repair rate tends to 0, such that their product remains finite. It is shown that the limiting two-dimensional Markov process corresponds to a queue where the number of servers has the same stationary distribution as the number of jobs in an $M/M/\infty$ queue. Hence, the limiting model is referred to as the M/M/[M/M/\infty ]$queue. Its numerical solution is discussed. Next, the behaviour of the$M/M/[M/M/\infty ]\$ queue is analysed in heavy traffic. When the traffic intensity approaches 1, the distribution of the (suitably normalized) number of jobs in the system is approximately exponential. This result relies on two limiting processes---a diffusion and a normalized heavy traffic limit---being essentially the same.
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https://hal.inria.fr/inria-00072851
Contributor : Rapport de Recherche Inria <>
Submitted on : Wednesday, May 24, 2006 - 11:05:46 AM
Last modification on : Friday, May 25, 2018 - 12:02:02 PM
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### Identifiers

• HAL Id : inria-00072851, version 1

### Citation

Jean-François Dantzer, Isi Mitrani, Philippe Robert. Large Scale and Heavy Traffic Asymptotics for Systems with Unreliable Servers. [Research Report] RR-3807, INRIA. 1999. ⟨inria-00072851⟩

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