Classification of States

If state j is accessible from state i, then

• P(i to j in n steps) = 0, for all n

• P(i to j in n steps) > 0, for all n

• P(j to i in n steps) > 0, for all n

• P(i to i in n steps) = 1, for all n

Two states i and j are said to communicate if

• i is accessible from j only

• j is accessible from i only

• i is accessible from j and j is accessible from i

• None of the above

A state is called essential if

• i to j implies j not to i

• i to j implies j to i

• j to i implies i not to j

• j to i implies j is transient

An absorbing state is one where

• The process cannot leave the state once entered

• It always returns to that state

• It transitions to every other state

• The mean recurrence time is finite

A state is called recurrent (persistent) if

• The system might return to the state

• The system never returns to the state

• The system returns to the state with probability 1

• The mean recurrence time is infinite

A state is called transient if

• The process always returns to it

• There is a non-zero chance that the process never returns to it

• It cannot be reached from any other state

• It is periodic

A state is called null if

• It is absorbing

• The process never returns to it

• Its mean recurrence time is infinite

• It is not essential

A persistent non-null state is one that is

• Transient with finite mean return time

• Recurrent with finite mean return time

• Absorbing with infinite mean return time

• Recurrent and periodic

A periodic state is one where

• Return to the state happens only at multiples of an integer > 1

• It has no return paths

• It is also an absorbing state

• It has no fixed recurrence time

A state is aperiodic if

• It is recurrent and has period > 1

• It can only be visited in even steps

• It can be returned to at irregular time steps

• Its greatest common divisor of return steps is more than 1