TY - JOUR
T1 - On moment stability properties for a class of state-dependent stochastic networks
AU - Lee, Chihoon
PY - 2011/9
Y1 - 2011/9
N2 - 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.
AB - 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.
KW - Birth and death process
KW - Multi-server systems
KW - Primary
KW - Secondary
KW - Skorokhod problem
KW - State-dependent networks
KW - Stochastic stability
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U2 - 10.1016/j.jkss.2010.12.003
DO - 10.1016/j.jkss.2010.12.003
M3 - Article
AN - SCOPUS:79960306221
SN - 1226-3192
VL - 40
SP - 325
EP - 336
JO - Journal of the Korean Statistical Society
JF - Journal of the Korean Statistical Society
IS - 3
ER -