Diffusion models and steady-state approximations for exponentially ergodic Markovian queues.
By Itai Gurvich. 2014
84 citations recorded by the source corpus.
Motivated by queues with many servers, we study Brownian steady-state approximations for continuous time Markov chains (CTMCs). Our approximations are based on diffusion models (rather than a diffusion limit) whose steady-state, we prove, approximates that of the Markov chain with notable precision. Strong approximations provide such "limitless" approximations for process dynamics. Our focus here is on steady-state distributions, and the diffusion model that we propose is tractable relative to strong approximations. Within an asymptotic framework, in which a scale parameter $n$ is taken large, a uniform (in the scale parameter) Lyapunov condition imposed on the sequence of diffusion models guarantees that the gap between the steady-state moments of the diffusion and those of the properly centered and scaled CTMCs shrinks at a rate of $\sqrt{n}$. Our proofs build on gradient estimates for solutions of the Poisson equations associated with the (sequence of) diffusion models and on elementary martingale arguments. As a by-product of our analysis, we explore connections between Lyapunov functions for the fluid model, the diffusion model and the CTMC.
Subfields: Management Information Systems, Management Science and Operations Research, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/c39dd6b03e69b3fdda637c8ea11f8163a43c66de
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In Management Information Systems
In Management Science and Operations Research
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Citations in the corpus
- Stein's method for steady-state diffusion approximations: an introduction through the Erlang-A and Erlang-C models (2015, methodology): “The framework developed in [10] was inspired largely by the work of Gurvich in [28] who developed methodologies to prove statements similar to Theorem 1 for a broad class of multidimensional CTMCs.”
- Heavy traffic approximation for the stationary distribution of a generalized Jackson network: The BAR approach (2015, methodology): “Our use of MGFs in this paper, instead of CFs, is not accidental.”