Queues with delays in two-state strategies and Lévy input

R. Bekker*, O. J. Boxma, O. Kella

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.

Original languageAmerican English
Pages (from-to)314-332
Number of pages19
JournalJournal of Applied Probability
Volume45
Issue number2
DOIs
StatePublished - Jun 2008

Keywords

  • Delayed feedback control
  • Lévy exponent
  • M/G/1 queue
  • Reflected Lévy process
  • Storage process
  • Two-state strategy
  • Workload process

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