Regenerative process

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Regenerative processes have been used to model problems in inventory control. The inventory in a warehouse such as this one decreases via a stochastic process due to sales until it gets replenished by a new order.[1]

In applied probability, a regenerative process is a class of stochastic process with the property that certain portions of the process can be treated as being statistically independent of each other.[2] This property can be used in the derivation of theoretical properties of such processes.


Regenerative processes were first defined by Walter L. Smith in Proceedings of the Royal Society A in 1955.[3][4]


A regenerative process is a stochastic process with time points at which, from a probabilistic point of view, the process restarts itself.[2] These time point may themselves be determined by the evolution of the process. That is to say, the process {X(t), t ≥ 0} is a regenerative process if there exist time points 0 ≤ T0 < T1 < T2 < ... such that the post-Tk process {X(Tk + t) : t ≥ 0}

  • has the same distribution as the post-T0 process {X(T0 + t) : t ≥ 0}
  • is independent of the pre-Tk process {X(t) : 0 ≤ t < Tk}

for k ≥ 1.[5] Intuitively this means a regenerative process can be split into i.i.d. cycles.[6]

When T0 = 0, X(t) is called a nondelayed regenerative process. Else, the process is called a delayed regenerative process.[5]



\lim_{t \to \infty} \frac{1}{t}\int_0^t X(s) ds= \frac{\mathbb{E}[R]}{\mathbb{E}[\tau]}.
where \tau is the length of the first cycle and R=\int_0^\tau X(s) ds is the value over the first cycle.
  • A measurable function of a regenerative process is a regenerative process with the same regeneration time[7]


  1. Hurter, A. P.; Kaminsky, F. C. (1967). "An Application of Regenerative Stochastic Processes to a Problem in Inventory Control". Operations Research. 15 (3): 467. JSTOR 168455. doi:10.1287/opre.15.3.467. 
  2. 2.0 2.1 2.2 2.3 2.4 Ross, S. M. (2010). "Renewal Theory and Its Applications". Introduction to Probability Models. pp. 421–641. ISBN 9780123756862. doi:10.1016/B978-0-12-375686-2.00003-0.  Cite error: Invalid <ref> tag; name "Ross" defined multiple times with different content
  3. Schellhaas, Helmut (1979). "Semi-Regenerative Processes with Unbounded Rewards". Mathematics of Operations Research. 4: 70–78. JSTOR 3689240. doi:10.1287/moor.4.1.70. 
  4. Smith, W. L. (1955). "Regenerative Stochastic Processes". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 232 (1188): 6–4. Bibcode:1955RSPSA.232....6S. doi:10.1098/rspa.1955.0198. 
  5. 5.0 5.1 Haas, Peter J. (2002). "Regenerative Simulation". Stochastic Petri Nets. Springer Series in Operations Research and Financial Engineering. pp. 189–273. ISBN 0-387-95445-7. doi:10.1007/0-387-21552-2_6. 
  6. 6.0 6.1 Asmussen, Søren (2003). "Regenerative Processes". Applied Probability and Queues. Stochastic Modelling and Applied Probability. 51. pp. 168–185. ISBN 978-0-387-00211-8. doi:10.1007/0-387-21525-5_6. 
  7. 7.0 7.1 Sigman, Karl (2009) Regenerative Processes, lecture notes