This is merely a concept question. I have an inclination, unfortunately with no proof, that the stationary distribution of a Continuous Time Markov Chain and its embedded Discrete Time Markov Chain should be if not the same very similar. Discrete Time Markov chains operate under the unit steps whereas CTMC operate with rates of time. Still, in the long run I see the proportions spent on each state to be the same.
Anyway, can anything be said about the stationary distribution of a CTMC and its embedded DTMC stationary distribution?
[Math] CTMC stationary distribution vs. embedded DTMC stationary distribution
markov chains
Best Answer
Here is a formal proof of my comment above. Let $S$ be the state space. Let $q_{ij}$ be the transition rates of the CTMC. For each $i \in S$ define $v_i = \sum_{j \in S} q_{ij}$. It is assumed that $v_i >0$ for all $i \in S$. Let $P_{ij}$ be the transition probabilities for the embedded DTMC. Thus, $P_{ij} = \frac{q_{ij}}{v_i}$ for all $i, j \in S$.
Claim:
Let $\{\pi_i\}_{i\in S}$ be any collection of real numbers. Fix $C \in \mathbb{R}$ and define $p_i = C\pi_i/v_i$ for all $i \in S$. If $\{\pi_i\}_{i\in S}$ satisfies the following global balance equations for the DTMC: $$ \pi_j = \sum_{i \in S} \pi_i P_{ij} \quad , \forall j \in S $$ Then $\{p_i\}_{i\in S}$ satisfies the following global balance equations for the CTMC: $$ p_jv_j = \sum_{i \in S} p_i q_{ij} \quad, \forall j \in S $$
Proof:
Assume $\{\pi_i\}_{i \in S}$ satisfies the global balance equations for the DTMC. Fix $j \in S$. Since $\sum_{i \in S} P_{ji} = 1$, the DTMC global balance equation for state $j$ is:
$$ \pi_j\underbrace{\sum_{i \in S} P_{ji}}_{1} = \sum_{i \in S} \pi_i P_{ij} $$ Substituting $P_{ij} = q_{ij}/v_i$ and $P_{ji} = q_{ji}/v_j$ and rearranging terms gives the result. $\Box$
Note: Of course, to represent a steady state distribution, we require the $\pi_i$ values to be nonnegative and sum to 1. Then, by defining $p_i = C\pi_i/v_i$, the constant $C$ must be chosen so that the $p_i$ values sum to 1. This is always possible for finite state DTMCs. In general, existence of steady state follows standard steady state theory and requires some additional assumptions, including a regularity assumption for the CTMC that ensures (with prob 1) that there are a finite number of transitions over any finite interval of time.