Palm–Khintchine Theorem
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probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, the Palm–Khintchine theorem, the work of
Conny Palm Conrad "Conny" Rudolf Agaton Palm (May 31, 1907 – December 27, 1951) was a Swedish electrical engineer and statistician, known for several contributions to teletraffic engineering and queueing theory. Rolf B. HaugenThe life and work of Conny Pa ...
and
Aleksandr Khinchin Aleksandr Yakovlevich Khinchin (, ), July 19, 1894 – November 18, 1959, was a Soviet mathematician and one of the most significant contributors to the Soviet school of probability theory. Due to romanization conventions, his name is sometim ...
, expresses that a large number of
renewal process Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponential distribution, exponentially distributed holding times, a renewal process may have any IID, independent and ...
es, not necessarily Poissonian, when combined ("superimposed") will have Poissonian properties. It is used to generalise the behaviour of users or clients in
queuing theory Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because th ...
. It is also used in dependability and reliability modelling of computing and
telecommunications Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically through cables, radio waves, or other communication technologies. These means of ...
.


Theorem

According to Heyman and Sobel (2003), 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 \, i=1,2,\ldots, m be independent renewal processes and \ be the superposition of these processes. Denote by X_ the time between the first and the second renewal epochs in process j. Define N_(t) the jth counting process, F_(t)=P(X_\leq t) and \lambda_=1/(E((X_)). If the following assumptions hold 1) For all sufficiently large m: \lambda_+\lambda_+\cdots+\lambda_=\lambda<\infty 2) Given \varepsilon>0, for every t>0 and sufficiently large m: F_(t)<\varepsilon for all j then the superposition N_(t)=N_(t)+N_(t)+\cdots+N_(t) of the counting processes approaches a Poisson process as m \to \infty.


See also

*
Law of large numbers In probability theory, the law of large numbers is a mathematical law that states that the average of the results obtained from a large number of independent random samples converges to the true value, if it exists. More formally, the law o ...


References

{{DEFAULTSORT:Palm-Khintchine theorem Queueing theory Network performance Point processes Theorems in probability theory