A tandem queue with server slow-down and blocking

N.D. van Foreest, J.C.W. van Ommeren, M.R.H. Mandjes, W.R.W. Scheinhardt*

*Corresponding author for this work

Research output: Contribution to journalConference articleAcademicpeer-review

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Abstract

We consider two variants of a two-station tandem network with blocking. In both variants the first server ceases to work when the queue length at the second station hits a 'blocking threshold.' In addition, in variant 2 the first server decreases its service rate when the second queue exceeds a 'slow-down threshold, ' which is smaller than the blocking level. In both variants the arrival process is Poisson and the service times at both stations are exponentially distributed. Note, however, that in case of slow-downs, server 1 works at a high rate, a slow rate, or not at all, depending on whether the second queue is below or above the slow-down threshold or at the blocking threshold, respectively. For variant 1, i.e., only blocking, we concentrate on the geometric decay rate of the number of jobs in the first buffer and prove that for increasing blocking thresholds the sequence of decay rates decreases monotonically and at least geometrically fast to max�?1, �?2, where �?i is the load at server i. The methods used in the proof also allow us to clarify the asymptotic queue length distribution at the second station. Then we generalize the analysis to variant 2, i.e., slow-down and blocking, and establish analogous results.
Original languageEnglish
Pages (from-to)695-724
Number of pages30
JournalStochastic models
Volume21
Issue number2-3
DOIs
Publication statusPublished - 2005
Event5th International Conference on Matrix Analytic Methods in Stochastic Models, MAM 2005 - Pisa, Italy
Duration: 21 Jun 200524 Jun 2005
Conference number: 5
http://www.dm.unipi.it/~mam5/

Keywords

  • Tandem queue
  • Slowdown
  • Feedback
  • Geometric decay rate
  • Blocking

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