Brian H. Fralix | Searcharxiv

arXiv2013

First passage times to congested states of many-server systems in the Halfin-Whitt regime

We consider the heavy-traffic approximation to the $GI/M/s$ queueing system in the Halfin-Whitt regime, where both the number of servers $s$ and the arrival rate $λ$ grow large (taking the service rate as unity), with $λ=s-β\sqrt{s}$ and $β$ some constant. In this asymptotic regime, the queue length process can be approximated by a diffusion process that behaves like a Brownian motion with drift above zero and like an Ornstein-Uhlenbeck process below zero. We analyze the first passage times of this hybrid diffusion process to levels in the state space that represent congested states in the original queueing system.

math.PR↗

arXiv2013

Factorization identities for reflected processes, with applications

We derive factorization identities for a class of preemptive-resume queueing systems, with batch arrivals and catastrophes that, whenever they occur, eliminate multiple customers present in the system. These processes are quite general, as they can be used to approximate Levy processes, diffusion processes, and certain types of growth-collapse processes; thus, all of the processes mentioned above also satisfy similar factorization identities. In the Levy case, our identities simplify to both the well-known Wiener-Hopf factorization, and another interesting factorization of reflected Levy processes starting at an arbitrary initial state. We also show how the ideas can be used to derive transforms for some well-known state-dependent/inhomogeneous birth-death processes and diffusion processes.

math.PR↗