Multi-level reflecting Brownian motion on the half line and its stationary distribution
Abstract: 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 author (2024). 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, Castiel and Reiman (2022,2023). 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.
- The BAR-approach for multiclass queueing networks with SBP service policies. Stochastic Systems. Published online in Articles in Advance, URL https://doi.org/10.1287/stsy.2023.0011.
- Introduction to Stochastic Integration. 2nd ed. Birkhäuser, Boston.
- Stochastic Calculus and Applications. 2nd ed. Birkhäuser.
- Harrison, J. M. (2013). Brownian Models of Performance and Control. Cambridge University Press, New York.
- Kallenberg, O. (2001). Foundations of Modern Probability. 2nd ed. Springer Series in Statistics, Probability and its applications, Springer, New York.
- Brownian motion and stochastic calculus, vol. 113 of Graduate text in mathematics. 2nd ed. Springer, New York.
- On Harris recurrence in continuous time. Mathematics of Operations Research, 19 211–222.
- An explicit formula for the Skorokhod map on (0,a]. The Annals of Probability, 35 1740–1768.
- Generalized resolvents and Harris recurrence of Markov processes. In 50 years after Doeblin: Developments on the Theory of Markov Chains, Markov Processes and Sums of Random Variables (H. Cohn, ed.). Contemporary Mathematics, Amer. Math. Soc., Providence, RI.
- Miyazawa, M. (2024). Diffusion approximation of the stationary distribution of a two-level single server queue. Tech. rep. Submitted for publication, URL https://arxiv.org/abs/2312.11284.
- Tanaka, H. (1963). Note on continuous additive functionals of the 1-dimensional Brownian path. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 1 251–257.
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