the holding (sojourn) time

The holding time (or sojourn time) is the random length of time a continuous-time Markov chain spends in a state before its next jump. It is the basic timing ingredient of a CTMC: the path sits at state i for a holding time, jumps, sits at the next state for another holding time, and so on. Understanding why these times are exponential — and what their rate means — is the key to building and reasoning about every CTMC.

The Markov property forces the holding time to be memoryless: given that the chain is in i and has been there for time s, the chance it stays a further t cannot depend on s, because conditioning on the present state erases the past. On the half-line, the only distribution with this lack-of-memory property is the exponential. So the holding time in state i is exponential with some rate q_i >= 0; equivalently its mean is 1/q_i. A state with q_i = 0 is absorbing (the chain never leaves). The construction is most transparent through competing exponential clocks: assign to each possible target j an independent exponential clock of rate q_ij; the holding time is the minimum of these clocks, which is exponential with rate q_i = sum_{j != i} q_ij, and the destination is the clock that rang first, with probability q_ij/q_i — and crucially, the minimum and the argmin are independent, which is exactly why the next state and the holding time are independent.

Holding times are why CTMC dynamics are clean: the process is determined by, and only by, the rates q_ij. A common confusion is to think the holding time depends on where the chain will go next; it does not — the time spent in i is exponential(q_i) regardless of the destination, a consequence of the independence of the min and the argmin of independent exponentials. A second subtlety: if many destinations are open with large total rate q_i, the mean holding time 1/q_i is small. If the q_i grow without bound along a path of states, the holding times can sum to a finite value and the chain explodes.