Forward delay times in multi-phase discrete-time renewal processes

A type of discrete-time Markov renewal process is considered in which the epochs at which the process changes state are not all regeneration points. Such processes arise naturally in breakdown/repair models and in variants of the M/G/1 queue such as occur in modelling road traffic. The key renewal theorem is used to find a relationship between the residual sojourn times (or forward delay times) and sojourn times through the system.