On moment stability properties for a class of state-dependent stochastic networks

We consider a class of stochastic networks with state-dependent arrival and service rates. The state-dependency is described via multi-dimensional birth/death processes, where the birth/death rates are dependent upon the current population size in the system. Under the uniform (in state) stability condition, we establish several moment stability properties of the system: (i) the unique invariant measure of the state process admits a finite moment generating function in a neighborhood of zero; (ii) the expected value of unbounded functionals of the state process converges to the expectation under the invariant measure at an exponential rate; (iii) uniform (in time and initial condition) estimates on exponential moments of the process; (iv) growth estimates of polynomial moments of the process as a function of the initial conditions. Our approach provides elementary proofs of these stability properties without resorting to the convergence of the scaled process to a stable fluid limit model.