On Lévy-driven vacation models with correlated busy periods and service interruptions

Offer Kella*, Onno Boxma, Michel Mandjes

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


This paper considers queues with server vacations, but departs from the traditional setting in two ways: (i) the queueing model is driven by Lévy processes rather than just compound Poisson processes; (ii) the vacation lengths depend on the length of the server's preceding busy period. Regarding the former point: the Lévy process active during the busy period is assumed to have no negative jumps, whereas the Lévy process active during the vacation is a subordinator. Regarding the latter point: where in a previous study (Boxma et al. in Probab. Eng. Inf. Sci. 22:537-555, 2008) the durations of the vacations were positively correlated with the length of the preceding busy period, we now introduce a dependence structure that may give rise to both positive and negative correlations. We analyze the steady-state workload of the resulting queueing (or: storage) system, by first considering the queue at embedded epochs (viz. the beginnings of busy periods). We show that this embedded process does not always have a proper stationary distribution, due to the fact that there may occur an infinite number of busy-vacation cycles in a finite time interval; we specify conditions under which the embedded process is recurrent. Fortunately, irrespective of whether the embedded process has a stationary distribution, the steady-state workload of the continuous-time storage process can be determined. In addition, a number of ramifications are presented. The theory is illustrated by several examples.

Original languageAmerican English
Pages (from-to)359-382
Number of pages24
JournalQueueing Systems
Issue number4
StatePublished - Apr 2010

Bibliographical note

Funding Information:
The research of O. Boxma and M. Mandjes has been partly funded by the Dutch BSIK/BRICKS (Basic Research in Informatics for Creating the Knowledge Society) project. O. Kella’s research is partially supported by Grant No. 964/06 from the Israel Science Foundation and the Vigevani Chair in Statistics.


  • Busy period
  • Lévy process
  • Queues with server vacations
  • Storage process


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