DES - Definition of 'Recurrent State or Transient State'

Transient or Recurrent State:

Consider the following problem: The system is in state $i\in \mathcal{X}$, and a transition to another state $j\ne i$ occurs.

Will the system surely return to state $i$ in the future?

• Consider the “future” as infinity time from the state change.

If the answer is yes, the state $i$ is said Recurrent Otherwise, it is said Transient

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~Example:

State $1$ is Recurrent. State $2$ is Recurrent. State $3$ is Transient.

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~Example:

State $1$ is Transient. State $2$ is Transient. State $3$ is Transient. State $4$ is Recurrent. State $5$ is Recurrent. State $6$ is Recurrent.