Queueing Systems

, Volume 91, Issue 3–4, pp 297–318 | Cite as

Discrete-time modified number- and time-limited vacation queues

  • Dieter ClaeysEmail author
  • Stijn De Vuyst


A vast amount of literature has appeared on vacation queues. In the well-known number- and time-limited vacation policies, the server goes on vacation if the number of customers, respectively, work (time slots) served since the previous vacation reaches a specified value, or if the system becomes empty, whichever occurs first. However, in practice, the server does not always go on vacation when the system is empty if the number of customers/work to be served has not yet reached the specified amount. Therefore, we study modified number- and time-limited vacation policies, where we account for this feature. We complement our recent work on these vacation policies by considering a discrete time, instead of a continuous-time, setting. We therefore adopt a different analysis approach, which enables us to obtain similar as well as new results as compared to our previous work. The results in this paper are valid for a memoryless distribution, but also for distributions with finite support, and a mixture of geometric distributions.


Modified vacation policies Queueing Discrete time Matrix generating functions Decomposition 

Mathematics Subject Classification

60K25 68M20 90B22 



The authors would like to thank Ivo Adan, Onno Boxma, and Offer Kella, as was well as the reviewers, for their valuable comments on this paper. We believe that these comments have considerably improved the quality of this paper.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Industrial Systems Engineering and Product DesignGhent UniversityZwijnaardeBelgium
  2. 2.Industrial Systems Engineering (ISyE)Flanders MakeLommelBelgium

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