Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit

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.