Queueing Systems

, Volume 83, Issue 1–2, pp 29–55 | Cite as

Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines

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Abstract

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue-length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse.

Keywords

Markov-modulation Heavy traffic Discriminatory processor sharing Single-server queue 

Mathematics Subject Classification

60K25 60K37 

Notes

Acknowledgments

The authors would like to thank Urtzi Ayesta, Joke Blom and Michel Mandjes for helpful discussions. This research was partially supported by the SMI Program of INP Toulouse.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.CWIAmsterdamThe Netherlands
  2. 2.CNRS, IRITToulouseFrance
  3. 3.Universitè de Toulouse, INPToulouseFrance

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