Averaging in Markov models with fast Markov switches and applications to queueing models

An approximation of Markov type queueing models with fast Markov switches by Markov models with averaged transition rates is studied. First, an averaging principle for two-component Markov process (xn(t),ζn(t)) is proved in the following form: if a component xn(·) has fast switches, then under some asymptotic mixing conditions the component ζn(·) weakly converges in Skorokhod space to a Markov process with transition rates averaged by some stationary measures constructed by xn(·). The convergence of a stationary distribution of (xn(·),ζn(·)) is studied as well. The approximation of state-dependent queueing systems of the type MM,Q/MM,Q/m/N with fast Markov switches is considered.