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

TL;DR

This paper generalizes Lorden's inequality to dependent, arbitrarily distributed recovery intervals, enabling the estimation of convergence rates in complex reliability and queuing systems beyond classical exponential assumptions.

Contribution

The paper introduces a generalized Lorden's inequality applicable to dependent and arbitrarily distributed recovery intervals, extending classical results to more complex stochastic processes.

Findings

Derived an upper bound for convergence rate in a two-component process

Extended Lorden's inequality to dependent, non-exponential recovery times

Provided tools for analyzing convergence in complex reliability models

Abstract

It is more important to estimate the rate of convergence to a stationary distribution rather than only to prove the existence one in many applied problems of reliability and queuing theory. This can be done via standard methods, but only under assumptions about an exponential distribution of service time, independent intervals between recovery times, etc. Results for such simplest cases are well-known. Rejection of these assumptions results to rather complex stochastic processes that cannot be studied using standard algorithms. A more sophisticated approach is needed for such processes. That requires generalizations and proofs of some classical results for a more general case. One of them is the generalized Lorden's inequality proved in this paper. We propose the generalized version of this inequality for the case of dependent and arbitrarily distributed intervals between recovery times. This generalization allows to find upper bounds for the rate of convergence for a wide class of complicated processes arising in the theory of reliability. The rate of convergence for a two-component process has been obtained via the generalized Lorden's inequality in this paper.