Michel Mandjes (Leiden)
Title:听L茅vy driven queues: the workload correlation function is positive, decreasing and convex
Abstract:In this talk I will consider L茅vy-driven queues, i.e., reflected L茅vy processes, with a focus on structural properties of the workload correlation function. After having introduced the objects studied, I'll proceed by stating the conjecture that has been around for quite a while, namely that the workload correlation is a positive, decreasing and convex function of time. As a historic account, I'll briefly discuss the seminal contribution by Ott on the special case of the M/G/1 queue, based on exploiting properties of complete monotone functions. The same methodology has been used in the extension (by Es-Saghouani and me) to queues with spectrally positive L茅vy input, whereas later (in a paper by Glynn and me) the spectrally negative case was dealt with. For a long time, there was little hope to prove the conjecture for general L茅vy input (and, for that matter, for reflected random walks in discrete-time). In a recent paper (that I wrote with Berkelmans and Cichocka), we provide an elementary proof, only relying on basic properties of L茅vy processes and their reflected version. Importantly, the argumentation extends to double reflection, and also covers reflected random walks. Time permitting, I also discuss various ramifications due to Kella and me, and I comment on the question whether the structural properties carry over to the Markov modulated case.
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