The Annals of Applied Probability

The critical greedy server on the integers is recurrent

James R. Cruise and Andrew R. Wade

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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 $1/4$.

Article information

Ann. Appl. Probab., Volume 29, Number 2 (2019), 1233-1261.

Received: December 2017
Revised: August 2018
First available in Project Euclid: 24 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B22: Queues and service [See also 60K25, 68M20]

Greedy server queueing system recurrence iterated logarithm law


Cruise, James R.; Wade, Andrew R. The critical greedy server on the integers is recurrent. Ann. Appl. Probab. 29 (2019), no. 2, 1233--1261. doi:10.1214/18-AAP1434.

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