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 existence of a moment generating function in a neighborhood of zero, with respect to the unique invariant measure of the state process;

• (ii) 

  the convergence of the expected value of unbounded functionals of the state process 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.