In
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
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
*
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