Consider a renewal process whose interrenewal-time distribution is phase type with representation (α,T). The authors show that the (time-dependent) excess-life distribution is phase type with representation (α',T), where α' is an appropriately modified initial probability vector. Using this result, they derive the (time-dependent) distributions for the current life and the total life of the phase-type renewal process. They in turn enable the authors to obtain the equilibrium distributions for the three random variables. These results simplify the computation of the respective distribution functions and consequently enhance the potential use of renewal theory in stochastic modeling-particularly in inventory, queueing, and reliability applications.
