# StationaryDistribution

StationaryDistribution[proc]

represents the stationary distribution of the process proc, when it exists.

# Details

• Stationary distribution is also known as limiting distribution, steady-state distribution, and invariant distribution.
• The stationary distribution, if it exists, is a slice distribution that is independent of the time and characterizes the limiting behavior of the process proc after all possible transients have vanished.
• StationaryDistribution[proc] is equivalent to SliceDistribution[proc,].

# Examples

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## Basic Examples(1)

Stationary distribution for an M/M/1 queue:

Probability density function:

Mean and variance:

Compute the probability of an event:

## Scope(3)

Stationary distribution may autoevaluate to known distribution:

Stationary distribution may autoevaluate to a derived distribution:

Compute the stationary distribution for a discrete Markov process:

Some slice distributions:

Stationary distribution:

Visualize the convergence to the stationary distribution using the PDF:

## Properties & Relations(3)

Stationary distribution is the SliceDistribution at infinity:

The stationary distribution may depend on the initial state:

Mean system size is the mean of the stationary distribution for a queue:

## Possible Issues(1)

The stationary distribution may exist only for a certain range of process parameters:

Wolfram Research (2012), StationaryDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/StationaryDistribution.html.

#### Text

Wolfram Research (2012), StationaryDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/StationaryDistribution.html.

#### CMS

Wolfram Language. 2012. "StationaryDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/StationaryDistribution.html.

#### APA

Wolfram Language. (2012). StationaryDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StationaryDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_stationarydistribution, author="Wolfram Research", title="{StationaryDistribution}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/StationaryDistribution.html}", note=[Accessed: 24-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_stationarydistribution, organization={Wolfram Research}, title={StationaryDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/StationaryDistribution.html}, note=[Accessed: 24-July-2024 ]}