The critical greedy server on the integers is recurrent

James R. Cruise, Andrew R. Wade

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
32 Downloads (Pure)


Each site of Z hosts a queue with arrival rate λ. A single server, starting at the origin, serves its current queue at rate μ until that queue is empty, and then moves to the longest neighbouring queue. In the critical case λ=μ, we show that the server returns to every site infinitely often. We also give a sharp iterated logarithm result for the server’s position. Important ingredients in the proofs are that the times between successive queues being emptied exhibit doubly exponential growth, and that the probability that the server changes its direction is asymptotically equal to 1/4.
Original languageEnglish
Pages (from-to)1233-1261
Number of pages29
JournalAnnals of Applied Probability
Issue number2
Early online date24 Jan 2019
Publication statusPublished - Apr 2019


  • Greedy server
  • Iterated logarithm law
  • Queueing system
  • Recurrence

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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