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It is more important to estimate the rate of convergence to a stationarydistribution rather than only to prove the existence one in many appliedproblems of reliability and queuing theory. This can be done via standardmethods, but only under assumptions about an exponential distribution ofservice time, independent intervals between recovery times, etc. Results forsuch simplest cases are well-known. Rejection of these assumptions results torather complex stochastic processes that cannot be studied using standardalgorithms. A more sophisticated approach is needed for such processes. Thatrequires generalizations and proofs of some classical results for a moregeneral case. One of them is the generalized Lorden's inequality proved in thispaper. We propose the generalized version of this inequality for the case ofdependent and arbitrarily distributed intervals between recovery times. Thisgeneralization allows to find upper bounds for the rate of convergence for awide class of complicated processes arising in the theory of reliability. Therate of convergence for a two-component process has been obtained via thegeneralized Lorden's inequality in this paper.

Lorden's inequality and the polynomial rate of convergence of some extended Erlang-Sevastyanov queuing system