Transition Rate Matrix

Definition

For a homogeneous CTMC on discrete state space $S$, the transition rate from state $i$ to state $j$ ($i\ne j$) is defined as:

$q_{ij}={\nu }_i{P}_{ij}$

where:

• ${\nu }_i$ is the rate of the exponential sojourn time in state $i$
• $P_{ij}$ is the probability that the process enters state $j$ when leaving $i$

The transition rate matrix (or Q-matrix, generator matrix) is:

$Q=(q_{ij}{)}_{i,j\in S}$

with the diagonal entries:

$q_{ii}=-{\sum }_{j\ne i}{q}_{ij}=-{\nu }_i$

so that each row sums to $0$.

Interpretation

The Q-matrix is one of two equivalent ways to specify a CTMC (the other being the transition probability function). The off-diagonal entry $q_{ij}$ represents the instantaneous rate at which the process transitions from $i$ to $j$.

From Q-matrix to Parameters

Given $Q$, we can recover ${\nu }_i$ and $P_{ij}$:

${\nu }_i={\sum }_{j\ne i}{q}_{ij}$

$P_{ij}=\frac{q_{ij}}{{\nu }_i}=\frac{q_{ij}}{{\sum }_{j\ne i}{q}_{ij}}$

CTMC via Transition Rates

A CTMC can be defined directly through its transition rates. For small $h>0$:

$\mathrm{Pr}\{X(t+h)=j\mid X(t)=i\}=\{\begin{array}{ll}{q}_{ij}h+o(h),& i\ne j\\ 1-{\nu }_{i}h+o(h),& i=j\end{array}$

Related

• Continuous-Time Markov Chain
• CTMC Transition Probability Function
• Transition Probability Matrix

Exercises

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Kuis 2 2025 No. 3. Reaksi kimia: $N$ molekul A berubah menjadi B secara irreversibel. Jika ada $j$ molekul, laju transisi per molekul adalah $q$. Tentukan entri matriks generator $Q$ untuk transisi $j\to j-1$.

Jawaban: $q_{j,j-1}=qj$. Karena ada $j$ molekul independen, total laju transisi $j\to j-1$ adalah $qj$. Diagonal: $q_{jj}=-qj$. Semua entri lain nol (hanya transisi $j\to j-1$ yang mungkin).

Kuis 2 2025 No. 8. Diberikan $Q=\left[\begin{array}{ccc}-2& 2& 0\\ 1& -3& 2\\ 0& 2& -2\end{array}\right]$ pada $S=\{0,1,2\}$. Tentukan ${\nu }_0,{\nu }_1,{\nu }_2$ dan matriks embedded MC $P$.

Jawaban: ${\nu }_0=2$, ${\nu }_1=3$, ${\nu }_2=2$. $P=\left[\begin{array}{ccc}0& 1& 0\\ 1/3& 0& 2/3\\ 0& 1& 0\end{array}\right]$. Diperoleh dari $P_{ij}=q_{ij}/{\nu }_i$ untuk $i\ne j$.