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Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) holding times that have finite mean.
Renewal processes are regenerative processes, with T 1 being the first renewal. [5] Alternating renewal processes, where a system alternates between an 'on' state and an 'off' state. [5] A recurrent Markov chain is a regenerative process, with T 1 being the time of first recurrence. [5] This includes Harris chains.
On the elementary renewal theorem for non-identically distributed variables, in Pacific Journal of Mathematics, 14(2):673–699, 1964 Congestion Theory , Proceedings of the Symposium on Congestion Theory, The University of North Carolina Monograph Series in Probability and Statistics., 1965.
Renewal theory texts usually also define the spent time or the backward recurrence time (or the current lifetime) as () = (). Its distribution can be calculated in a similar way to that of the residual time. Likewise, the total life time is the sum of backward recurrence time and forward recurrence time.
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In the mathematical theory of probability, a generalized renewal process (GRP) or G-renewal process is a stochastic point process used to model failure/repair behavior of repairable systems in reliability engineering. Poisson point process is a particular case of GRP.