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Palm–Khintchine theorem

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In probability theory, the Palm–Khintchine theorem, the work of Conny Palm and Aleksandr Khinchin, expresses that a large number of renewal processes, not necessarily Poissonian, when combined ("superimposed") will have Poissonian properties.[1]

It is used to generalise the behaviour of users or clients in queuing theory. It is also used in dependability and reliability modelling of computing and telecommunications.

Theorem

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According to Heyman and Sobel (2003),[1] the theorem states that the superposition of a large number of independent equilibrium renewal processes, each with a finite intensity, behaves asymptotically like a Poisson process:

Let be independent renewal processes and be the superposition of these processes. Denote by the time between the first and the second renewal epochs in process . Define the th counting process, and .

If the following assumptions hold

1) For all sufficiently large :

2) Given , for every and sufficiently large : for all

then the superposition of the counting processes approaches a Poisson process as .

See also

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References

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  1. ^ a b Daniel P. Heyman, Matthew J. Sobel (2003). "5.8 Superposition of Renewal Processes". Stochastic Models in Operations Research: Stochastic Processes and Operating Characteristics.