Application of these "limiting probabilities"

Question

I have been taking a look at some basic properties of discrete markov chains. Let $S$ be a Markov Chain with a finite state space and with transition matrix $P$. It is well know that under some regularities properties there is an invariant distribution such as $\Pi=P^\prime \Pi$. My question is about the probability $\Pi_i P_{ij}$ related to the transition from $i$ to $j$, i.e., $\Pi_i P_{ij}$ is the probability that the transition $i$ to $j$ happens. Are these probabilities important somewhere? I mean "Do they arise in any important context?