Busy period analysis of the level dependent PH/PH/1/K queue

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12 Scopus citations


In this paper, we study the transient behavior of a level dependent single server queuing system with a waiting room of finite size during the busy period. The focus is on the level dependent PH/PH/1/K queue. We derive in closed form the joint transform of the length of the busy period, the number of customers served during the busy period, and the number of losses during the busy period. We differentiate between two types of losses: the overflow losses that are due to a full queue and the losses due to an admission controller. For the M/PH/1/K, M/PH/1/K under a threshold policy, and PH/M/1/K queues, we determine simple expressions for their joint transforms.

Original languageEnglish
Pages (from-to)221-249
Number of pages29
JournalQueueing Systems
Issue number3
StatePublished - Mar 2011
Externally publishedYes

Bibliographical note

Funding Information:
Acknowledgements We first thank the anonymous reviewer for the valuable comments which helped to improve the presentation of the paper. The author would like to thank Prof. Onno Boxma for stimulating discussions during the paper’s preparation when the author held a postdoctoral position at EURANDOM Institute, Eindhoven. The author thanks De Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) for their financial support. In the Netherlands, the three universities of technology have formed the 3TU.Federation. This article is the result of joint research in the 3TU.Centre of Competence NIRICT (Netherlands Institute for Research on ICT).


  • Absorbing Markov chains
  • Busy period
  • Level dependent queues
  • Matrix analytical approach
  • PH/PH/1/K queue
  • Phase-type distributions
  • Transient analysis

ASJC Scopus subject areas

  • Statistics and Probability
  • Computer Science Applications
  • Management Science and Operations Research
  • Computational Theory and Mathematics


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