The critical greedy server on the integers is recurrent

Cruise, James R.; Wade, Andrew R.

doi:10.1214/18-aap1434

The critical greedy server on the integers is recurrent

Cruise, James R.; Wade, Andrew R.

Authors

James R. Cruise

Professor Andrew Wade andrew.wade@durham.ac.uk
Professor

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