Regenerative process

A regenerative process is a special stochastic process that occurs among others in queuing theory and renewal theory .

definition

Be , with or , a stochastic process with values ​​in a state space . We call the (un- / delayed) process regenerative (in a broad sense) if there is an (un- / delayed) renewal process , so that for all the post- process is both independent of (or ) and its distribution is not of dependent. We name the embedded renewal process and the regeneration times . ${\displaystyle \{X_{t}\}_{t\in I}}$ ${\displaystyle I=\mathbb {N} }$ ${\displaystyle I={\mathopen {[}}0,\infty {\mathclose {[}}}$ ${\displaystyle (E,{\mathcal {E}})}$ ${\displaystyle \{X_{t}\}_{t\in I}}$ ${\displaystyle \{S_{n}\}_{n\in \mathbb {N} }=\{Y_{0}+\ldots +Y_{n}\}_{n\in \mathbb {N} }}$ ${\displaystyle n\geq 0}$ ${\textstyle S_{n}}$ ${\displaystyle \left(Y_{n+1},Y_{n+2},\ldots ,\{X_{S_{n}+t}\}_{t\in I}\right)}$ ${\displaystyle S_{0},\ldots ,S_{n}}$ ${\displaystyle Y_{0},\ldots ,Y_{n}}$ ${\displaystyle n}$ ${\displaystyle \{S_{n}\}_{n\in \mathbb {N} }}$ ${\displaystyle S_{n}}$

This definition allows a certain dependency between the cycles, in contrast to the classical definition, which requires that the post- process is independent of and . ${\textstyle S_{n}}$ ${\displaystyle S_{0},\ldots ,S_{n}}$ ${\displaystyle \{X_{t}\}_{t<S_{n}}}$

Examples

Classic examples of regenerative processes are:

• positively recurrent, irreducible Markov chains (in continuous and discrete time)
• G / G / 1 queues
• Aging process and remaining life process of a renewal process

Boundary behavior

Regenerative processes have a limit distribution under comparatively mild conditions, which are automatically fulfilled in many applications.

Individual evidence

1. ↑ Gerold Alsmeyer: Renewal Theory : Analysis of stochastic regeneration schemes . BG Teubner, Stuttgart 1991, ISBN 978-3-663-09977-2 .
2. ^ Søren Asmussen: Applied probability and queues . 2nd ed. Springer, New York 2003, ISBN 0-387-00211-1 .