Diffusion models and steady-state approximations for exponentially ergodic Markovian queues
| Autor: | Gurvich, Itai |
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| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | Annals of Applied Probability 2014, Vol. 24, No. 6, 2527-2559 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1214/13-AAP984 |
| Popis: | 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. Comment: Published in at http://dx.doi.org/10.1214/13-AAP984 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org) |
| Databáze: | arXiv |
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