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

By BY A. B. Dieker, Xuefeng Gao, X. Gao. 2011

50 citations recorded by the source corpus.

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.

View Paper: https://www.semanticscholar.org/paper/58bab0cf7b79c4dca1e3b88bf6484b6fe49d8054

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Citations in the corpus

• Diffusion models and steady-state approximations for exponentially ergodic Markovian queues. (2014, background): “For the second example—a queue with many servers and phase-type service time distribution—this task is substantially more difficult, but recent work [9] provides us with the required function.”
• Stein's method for steady-state diffusion approximations of $M/Ph/n+M$ systems (2015, background): “firms that it is positive definite. Thus Σ exists. Observe that Y depends only on β,α,p,P, and ν, all of which are held constant throughout this paper. The diffusion process in (2.6) has been studied by [17]. They prove that Y is positive recurrent by finding an appropriate Lyapunov function. In particu”