Holding probability of a Markov chain ($P(x,x)$)

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.