Transition probability ($P_{ij}$)

The transition probability $P_{ij}$ is the conditional probability that a Markov chain moves to state $j$ on its next step given that its current state is $i$.

At state $x$, the holding probability of a discrete-time Markov chain is the self-transition probability $P(x,x)$. It is the conditional probability that the next state remains $x$. In a nearest-neighbour birth-death chain, it is $1-p_x-q_x$, where $p_x$ and $q_x$ are the probabilities of stepping right and left. Holding changes the elapsed time but not the embedded chain of actual moves or its boundary exit probabilities.