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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.
His two-volume textbook on probability theory and its applications was called "the most successful treatise on probability ever written" by Gian-Carlo Rota. [8] By stimulating his colleagues and students in Sweden and then in the United States, Feller helped establish research groups studying the analytic theory of probability.
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.
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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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Markov renewal processes are a class of random processes in probability and statistics that generalize the class of Markov jump processes.Other classes of random processes, such as Markov chains and Poisson processes, can be derived as special cases among the class of Markov renewal processes, while Markov renewal processes are special cases among the more general class of renewal processes.