Poisson process
2020 Mathematics Subject Classification: Primary: 60G55 Secondary: 60K25 [MSN][ZBL]
A stochastic process with independent increments
,
, having a Poisson distribution. In the homogeneous Poisson process
![]() | (1) |
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for any . The coefficient
is called the intensity of the Poisson process
. The trajectories of the Poisson process
are step-functions with jumps of height 1. The jump points
form an elementary flow describing the demand flow in many queueing systems. The distributions of the random variables
are independent for
and have exponential density
,
.
One of the properties of a Poisson process is that the conditional distribution of the jump points when
is the same as the distribution of the variational series of
independent samples with uniform distribution on
. On the other hand, if
is the variational series described above, then as
,
and
one obtains in the limit the distribution of the jumps of the Poisson process.
In an inhomogeneous process the intensity depends on the time
and the distribution of
is defined by the formula
![]() |
Under certain conditions a Poisson process can be shown to be the limit of the sum of a number of independent "sparse" flows of fairly general form as this number increases to infinity. For certain paradoxes which have been obtained in connection with Poisson processes see [3].
References
[1] | A.A. Borovkov, "Wahrscheinlichkeitstheorie" , Birkhäuser (1976) (Translated from Russian) |
[2] | I.I. Gikhman, A.V. Skorokhod, M.I. Yadrenko, "Probability theory and mathematical statistics" , Kiev (1979) (In Russian) |
[3] | W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1971) pp. Chapt. 1 |
Comments
References
[a1] | J.W. Cohen, "The single server queue" , North-Holland (1982) |
[a2] | G.G. Székely, "Paradoxes in probability theory and mathematical statistics" , Reidel (1986) |
Poisson process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poisson_process&oldid=23643