Class Properties of CTMC

Properties

For a CTMC $\{Y(t):t\ge 0\}$ and its embedded Markov chain $\{X_n\}$, the classification of states is inherited from the embedded chain.

Let $i,j\in S$ and "$\leftrightarrow$" denote the communication property.

Class Properties

Property	Statement
Communication	$i\leftrightarrow j$ in the CTMC $⟺$ $i\leftrightarrow j$ in the embedded MC
State Partition	Since communication partitions the embedded MC into classes, it also partitions the associated CTMC
Irreducibility	The CTMC is irreducible $⟺$ the embedded MC is irreducible
Recurrence	State $i$ is recurrent in the CTMC $⟺$ $i$ is recurrent in the embedded MC
Transience	State $i$ is transient in the CTMC $⟺$ $i$ is transient in the embedded MC
Positive Recurrence	State $i$ is positive recurrent in the CTMC $⟺$ $i$ is positive recurrent in the embedded MC
Class Property	Transient and recurrent are class properties of a CTMC

Interpretation

All structural properties (communication, recurrence, transience, irreducibility) of a CTMC are determined entirely by the behavior of the embedded DTMC. This is because these properties depend only on which states are visited and with what probability — not on how long is spent in each state.

Periodicity

Since $P_{ii}=0$ in the embedded MC (the CTMC always changes state at jump times), periodicity is not possible in CTMCs. All states have period 1.

Related

• Embedded Markov Chain
• Continuous-Time Markov Chain
• Irreducible
• Recurrent vs Transient

Exercises

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Klasifikasi state dari embedded MC. Untuk $Q=\left[\begin{array}{ccc}-2& 2& 0\\ 1& -3& 2\\ 0& 2& -2\end{array}\right]$, tentukan: (a) kelas komunikasi, (b) irreducible atau tidak, (c) recurrent atau transient.

Jawaban: Embedded MC: $P=\left[\begin{array}{ccc}0& 1& 0\\ 1/3& 0& 2/3\\ 0& 1& 0\end{array}\right]$. (a) Satu kelas: $\{0,1,2\}$ (semua state berkomunikasi — 0↔1, 1↔2, jadi 0↔2 via 1). (b) Irreducible. (c) Semua recurrent (state space finite + irreducible → positive recurrent). Tidak ada periodisitas di CTMC.

Periodisitas. Mengapa CTMC tidak memiliki periodisitas?

Jawaban: Karena $P_{ii}=0$ di embedded MC (selalu berubah state saat transisi), dan waktu holding bersifat eksponensial (kontinu). Transisi dapat terjadi kapan saja → periode selalu 1.