Generalised Lorden's Inequality and Convergence Rate Estimates for Non-regenerative Stochastic Models

Abstract: We present a generalisation of Lorden's inequality tailored for stochastic systems with dependent and non-identically distributed renewal times modelled as generalised renewal processes. This framework introduces the new concept of a generalised intensity, enabling the analysis of systems where classical renewal theory is inapplicable. The proposed approach is applied to a variety of complex systems, including reliability models, queueing networks and Markov-modulated processes, to establish ergodicity and derive convergence rate estimates. Unlike traditional methods relying on transition matrices or generators, our technique provides an alternative with some rigorous upper bounds, but computationally much more efficient than existent ones. This work offers a unified framework for analysing systems with dependencies and heterogeneities, bridging gaps in classical stochastic modelling.