Multi-level Reflecting Brownian Motion on the Half Line and Its Stationary Distribution

A semi-martingale reflecting Brownian motion is a popular process for diffusion approximations of queueing models including their networks. In this paper, we are concerned with the case that it lives on the nonnegative half-line, but the drift and variance of its Brownian component discontinuously change at its finitely many states. This reflecting diffusion process naturally arises from a state-dependent single server queue, studied by the Miyazawa (Diffusion approximation of the stationary distribution of a two-level single server queue, 2024. https://arxiv.org/abs/2312.11284). Our main interest is in its stationary distribution, which is important for application. We define this reflecting diffusion process as the solution of a stochastic integral equation, and show that it uniquely exists in the weak sense. This result is also proved in a different way by Atar et al. (Parallel server systems under an extended heavy traffic condition: A lower bound, 2022. https://arxiv.org/pdf/2201.07855). In this paper, we consider its Harris irreducibility and stability, that is, positive recurrence, and derive its stationary distribution under this stability condition. The stationary distribution has a simple analytic expression, likely extendable to a more general state-dependent SRBM. Our proofs rely on the generalized Ito formula for a convex function and local time.