DES - Classification of the States of a CTHMC

Classification of the States of a CTHMC:

Recall the definition of transition functions matrix:

$$p_{i,j}(t)\triangleq P(X(s+t)=j\mid X(s)=i)$$

Which is independent of $s$

Then a state $x\in \mathcal{X}$ can be defined:

• Reachable
• In a Closed Subset
• In an Irreducible Subset
• Transient or Recurrent

Irreducible CTHMC:

The CTHMC is said irreducible if the state set $\mathcal{X}$ is irreducible

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~Example: