Abstract
We consider state-dependent stochastic networks in the heavy-traffic diffusion limit represented by reflected jump-diffusions in the orthant ℝ n+ with state-dependent reflection directions upon hitting boundary faces. Jumps are allowed in each coordinate by means of independent Poisson random measures with jump amplitudes depending on the state of the process immediately before each jump. For this class of reflected jump-diffusion processes sufficient conditions for the existence of a product-form stationary density and an ergodic characterization of the stationary distribution are provided. Moreover, such stationary density is characterized in terms of semi-martingale local times at the boundaries and it is shown to be continuous and bounded. A central role is played by a previously established semi-martingale local time representation of the regulator processes.
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F.J. Piera’s research supported in part by CONICYT, Chile, FONDECYT Project 1070797.
R.R. Mazumdar’s research supported in part by NSF, USA, Grant 0087404 through Networking Research Program, and a Discovery Grant from NSERC, Canada.
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Piera, F.J., Mazumdar, R.R. & Guillemin, F.M. Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit. Queueing Syst 58, 3–27 (2008). https://doi.org/10.1007/s11134-007-9056-3
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DOI: https://doi.org/10.1007/s11134-007-9056-3
Keywords
- Stochastic networks
- Heavy-traffic limits
- Jump-diffusions
- State-dependent oblique reflections
- Product-form stationary distributions
- Local times
- Skorokhod maps
- Reflection maps
- Regulator processes