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April 2019 The critical greedy server on the integers is recurrent
James R. Cruise, Andrew R. Wade
Ann. Appl. Probab. 29(2): 1233-1261 (April 2019). DOI: 10.1214/18-AAP1434


Each site of $\mathbb{Z}$ hosts a queue with arrival rate $\lambda $. A single server, starting at the origin, serves its current queue at rate $\mu $ until that queue is empty, and then moves to the longest neighbouring queue. In the critical case $\lambda =\mu $, 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 $1/4$.


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James R. Cruise. Andrew R. Wade. "The critical greedy server on the integers is recurrent." Ann. Appl. Probab. 29 (2) 1233 - 1261, April 2019.


Received: 1 December 2017; Revised: 1 August 2018; Published: April 2019
First available in Project Euclid: 24 January 2019

zbMATH: 07047448
MathSciNet: MR3910027
Digital Object Identifier: 10.1214/18-AAP1434

Primary: 60J27
Secondary: 60K25 , 68M20 , 90B22

Keywords: Greedy server , iterated logarithm law , queueing system , recurrence

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.29 • No. 2 • April 2019
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