# 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:

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Find the values of a specific property:

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Description of the property:

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