Cruise, James R. and Wade, Andrew R. (2019) 'The critical greedy server on the integers is recurrent.', Annals of applied probability., 29 (2). pp. 1233-1261.
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.
|Full text:||(AM) Accepted Manuscript|
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|Full text:||(VoR) Version of Record|
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|Publisher Web site:||https://doi.org/10.1214/18-AAP1434|
|Date accepted:||20 September 2018|
|Date deposited:||21 September 2018|
|Date of first online publication:||24 January 2019|
|Date first made open access:||16 October 2018|
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