Positive recurrence of piecewise Ornstein-Uhlenbeck processes and common quadratic Lyapunov functions

TL;DR

This paper investigates the positive recurrence of piecewise Ornstein-Uhlenbeck processes from queueing systems, employing common quadratic Lyapunov functions to establish the existence of a unique stationary distribution despite regime differences.

Contribution

It extends control theory techniques to prove positive recurrence of piecewise OU processes with regime-dependent behavior, including systems with abandonment.

Findings

Proves positive recurrence for a class of piecewise OU processes.

Constructs Lyapunov functions applicable across different regimes.

Establishes existence of unique stationary distributions.

Abstract

We study the positive recurrence of piecewise Ornstein-Uhlenbeck (OU) diffusion processes, which arise from many-server queueing systems with phase-type service requirements. These diffusion processes exhibit different behavior in two regions of the state space, corresponding to "overload" (service demand exceeds capacity) and "underload" (service capacity exceeds demand). The two regimes cause standard techniques for proving positive recurrence to fail. Using and extending the framework of common quadratic Lyapunov functions from the theory of control, we construct Lyapunov functions for the diffusion approximations corresponding to systems with and without abandonment. With these Lyapunov functions, we prove that piecewise OU processes have a unique stationary distribution.