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

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Abstract

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.

Original languageEnglish
Pages (from-to)325-336
Number of pages12
JournalJournal of the Korean Statistical Society
Volume40
Issue number3
DOIs
StatePublished - Sep 2011

Keywords

  • Birth and death process
  • Multi-server systems
  • Primary
  • Secondary
  • Skorokhod problem
  • State-dependent networks
  • Stochastic stability

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