On some generalization of Lorden's inequality for renewal processes and its applications

In queueing theory, the Lorden's inequality can be used to finding the bounds of the moments of backward renewal time at the fixed (non-random) time. The Lorden's inequality is true for the renewal process, so the expectation of backward renewal time bounded by the ratio of two first moments of renewal times. This fact was proved for a "classic" renewal process where the renewal times are i.i.d. We found the conditions when a "generalized" renewal process satisfies the analogue Lorden's inequality. In "generalized" renewal process the renewal times can be dependent and can have different distributions. Applications of Lorden's inequality generalization for ergodicity analysis will be considered. The bounds estimation of convergence rate for some queueing and reliability system with dependent parameters will be presented.