Palm–Khintchine theorem

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.