The critical greedy server on the integers is recurrent

Cruise, James R.; Wade, Andrew R.

Abstract

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.

Citation

Cruise, J. R., & Wade, A. R. (2019). The critical greedy server on the integers is recurrent. Annals of Applied Probability, 29(2), 1233-1261. https://doi.org/10.1214/18-aap1434

• Journal Article Type: Article
• Acceptance Date: Sep 20, 2018
• Online Publication Date: Jan 24, 2019
• Publication Date: Apr 30, 2019
• Deposit Date: Jan 25, 2018
• Publicly Available Date: Oct 16, 2018
• Journal: Annals of Applied Probability
• Print ISSN: 1050-5164
• Publisher: Institute of Mathematical Statistics
• Peer Reviewed: Peer Reviewed
• Volume: 29
• Issue: 2
• Pages: 1233-1261
• DOI: https://doi.org/10.1214/18-aap1434
• Public URL: https://durham-repository.worktribe.com/output/1340962