Heavy-traffic analysis of the $M^X/\text{semi-Markov}/1$ queue

Abhishek; Rudesindo Núñez Queija; Marko Boon

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arXiv · 1812.02404

Heavy-traffic analysis of the $M^X/\text{semi-Markov}/1$ queue

Abhishek·Rudesindo Núñez Queija·Marko Boon

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Abstract

In this paper we analyze a single server queue with batch arrivals and semi-Markovian service times. We also include the feature that the first service of each busy period might have a different distribution than subsequent service times. Our generating function based approach allows us to determine the heavy traffic limit of the scaled queue-length distribution. It turns out that this distribution converges to an exponential distribution. Nonsurprisingly, the exceptional first service does not influence this limiting distribution. We identify a sufficient and necessary condition under which the dependence between successive service times disappears in the limit, which we illustrate in a numerical example.

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BibTeX RIS

Abhishek, Rudesindo Núñez Queija, Marko Boon. 2018-12-06. Heavy-traffic analysis of the $M^X/\text{semi-Markov}/1$ queue. https://arxiv.org/abs/1812.02404

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