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

• Published in: Journal of the Korean Statistical Society Vol. 40; no. 3; pp. 325 - 336
• Main Author: Lee, Chihoon
• Format: Journal Article
• Language: English
• Published: Singapore Elsevier B.V 01.09.2011; Springer Singapore; 한국통계학회
• Subjects: Applied Statistics; Bayesian Inference; Birth and death process; Multi-server systems; Skorokhod problem; State-dependent networks; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; Stochastic stability; 통계학; secondary; Multi-server systems; State-dependent networks; Stochastic stability; Skorokhod problem; primary; Birth and death process; primary 60J25; secondary 93D20
• ISSN: 1226-3192; 2005-2863
• DOI: 10.1016/j.jkss.2010.12.003

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.