Set of palm khinchin

The set of Palm-Khinchin the stochastics tells himself that the superposition (superposition) of a sufficiently large number of not necessarily Poisson renewal processes asymptotically a Poisson process approach when the events in the individual processes are relatively rare. The set is based on works by Conny Palm from 1943 and Aleksander Chintschin from 1955. It is used in queuing theory and reliability analysis , for example when modeling customer arrival processes or rare events in actuarial mathematics.

statement

Be for , independent renewal processes and ${\displaystyle \{N_{mi}(t),t\geq 0\}}$ ${\displaystyle m\in \mathbb {N} }$ ${\displaystyle i=1,2,\dotsc ,m}$

${\displaystyle N_{m}(t)=N_{m1}(t)+N_{m2}(t)+\ldots +N_{mm}(t)}$

the superposition of these processes. Next denote the time between the first and second renewals in process as well . Under the assumptions ${\displaystyle X_{mi}}$ ${\displaystyle i}$ ${\displaystyle \lambda _{mi}=1/E(X_{mi})}$

1. For all sufficiently large applies: . ${\displaystyle m}$ ${\displaystyle \lambda _{m1}+\lambda _{m2}+\ldots +\lambda _{mm}=\lambda <\infty }$
2. Given , for everyone and sufficiently large applies: for everyone . ${\displaystyle \varepsilon >0}$ ${\displaystyle t>0}$ ${\displaystyle m}$ ${\displaystyle P(X_{mi}\leq t)<\varepsilon }$ ${\displaystyle i}$

then strives to superimpose the counting processes for against against a Poisson process with rate . ${\displaystyle N_{m}(t)}$ ${\displaystyle m}$ ${\displaystyle \infty }$ ${\displaystyle \lambda }$

Extensions

There are numerous extensions such as: B. Grigelionis' theorem, which generalizes the assumptions and derives a non-homogeneous Poisson process as a boundary process. In software reliability, there are numerous extensions for software reliability growth models, classic z. B. Littlewood's theorem, in which the failure process for complex software systems, the internal structure of which can be described by Markov chains , also tends towards a Poisson process.

Individual evidence

1. ↑ Conny Palm: Intensity fluctuations in telephone traffic, Ericsson Techniks 44, 1–189 (1943)
2. ↑ Aleksander Chintschin: Matematicheskie metody teorii massovogo obsluzhivaniia , Trudy Matematicheskogo Instituta Steklov, Akad. Nauk, USSR, Vol. 49 (1955)
3. ^ Daniel P. Heyman, Matthew J. Sobel: Stochastic Models in Operations Research: Stochastic Processes and Operating Characteristics , Courier Corporation, 2003, ISBN 978-0-48643-259-5 , pp. 156-161
4. ↑ Alessandro Birolini : Reliability Theory , 7th edition, Springer, Heidelberg, 2013, Chapter A7.8.3
5. ↑ Littlewood, B .: A reliability model for systems with Markov structure , Applied Statistics 24 (1975), 172-177.