# MarkovProcessProperties

MarkovProcessProperties[mproc]

gives a summary of properties for the finite state Markov process mproc.

MarkovProcessProperties[mproc,"property"]

gives the specified "property" for the process mproc.

# Details

• MarkovProcessProperties can be used for finite state Markov processes such as DiscreteMarkovProcess and ContinuousMarkovProcess.
• MarkovProcessProperties[mproc,"Properties"] gives a list of available properties.
• MarkovProcessProperties[mproc,"property","Description"] gives a description of the property as a string.
• Basic properties include:
•  "InitialProbabilities" initial state probability vector "TransitionMatrix" conditional transition probabilities m "TransitionRateMatrix" conditional transition rates q "TransitionRateVector" state transition rates μ "HoldingTimeMean" mean holding time for a state "HoldingTimeVariance" variance of holding time for a state "SummaryTable" summary of properties
• For a continuous-time Markov process "TransitionMatrix" gives the transition matrix of the embedded discrete-time Markov process.
• The holding time is the time spent in each state before transitioning to a different state. This takes into account self-loops which may cause the process to transition to the same state several times.
• Structural properties include:
•  "CommunicatingClasses" sets of states accessible from each other "RecurrentClasses" communicating classes that cannot be left "TransientClasses" communicating classes that can be left "AbsorbingClasses" recurrent classes with a single element "PeriodicClasses" communicating classes with finite period greater than 1 "Periods" period for each of the periodic classes "Irreducible" whether the process has a single recurrent class "Aperiodic" whether all classes are aperiodic "Primitive" whether the process is irreducible and aperiodic
• The states of a finite Markov process can be grouped into communicating classes where from each state in a class there is a path to every other state in the class.
• A communicating class can be transient when there is a path from the class to another class or recurrent when there is not. A special type of recurrent class, called absorbing, consist of a single element.
• A state is periodic is if there is a non-zero probability that you return to the state after two or more steps. All the states in a class have the same period.
• Transient properties before the process enters a recurrent class:
•  "TransientVisitMean" mean number of visits to each transient state "TransientVisitVariance" variance of number of visits to each transient state "TransientTotalVisitMean" mean total number of transient states visited
• A Markov process will eventually enter a recurrent class. The transient properties characterize how many times each transient state is visited or how many different transient states are visited.
• Limiting properties include:
•  "ReachabilityProbability" probability of ever reaching a state "LimitTransitionMatrix" Cesaro limit of the transition matrix "Reversible" whether the process is reversible
• If a property is not available, this is indicated by Missing["reason"].

# Examples

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## Basic Examples(2)

Summary table of properties:

Find the values of a specific property:

Description of the property:

## Scope(5)

Find the communicating classes, highlighted in the graph through different colors:

The process is not irreducible:

Find the recurrent classes, represented by square and circular vertices in the graph:

Find the transient classes, represented by diamond vertices in the graph:

Find the absorbing classes, represented by square vertices in the graph:

Define a Markov process with self-loops:

The self-loops make the class aperiodic:

Markov process with no self-loops:

Here both classes are periodic:

Summary table of properties for a continuous-time Markov process:

Obtain the value for a specific property for a continuous Markov chain:

Find the conditional mean number of total transitions starting in state 1 and ending in state 4:

Compare with the results from simulation:

Find the conditional mean number of transitions from state 2 to state 3:

Compare with the results from simulation:

## Applications(2)

A gambler starts with \$3 and bets \$1 at each step. He wins \$1 with a probability of 0.4:

Find the expected number of times the gambler has units:

Find the expected time until the gambler wins \$7 or goes broke:

Total states visited before the gambler wins \$7 or goes broke:

In a game of tennis between two players, suppose the probability of the server winning a point is . There are 17 possible states:

Visualize the random walk graph for :

Find the probability of the server winning the game if :

Find the mean time to absorption, that is, the number of points played:

Find the mean number of states visited:

Find the average number of times the score will be tied at deuce:

## Properties & Relations(1)

The transition matrix of this Markov process is not irreducible:

Hence the stationary distribution depends on the initial state probabilities:

## Possible Issues(1)

Some property values may not be available:

This property is available only for continuous Markov processes:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_markovprocessproperties, author="Wolfram Research", title="{MarkovProcessProperties}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/MarkovProcessProperties.html}", note=[Accessed: 29-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_markovprocessproperties, organization={Wolfram Research}, title={MarkovProcessProperties}, year={2012}, url={https://reference.wolfram.com/language/ref/MarkovProcessProperties.html}, note=[Accessed: 29-May-2024 ]}