# Probability of short sojourns in an alternating renewal process?

Let $X_t$ be an alternating renewal process on the state space {'off', 'on'}. We can think of the process as sojourning in those states. Let $[0,\tau]$ be an interval of time and $s < \tau$.

What is the probability that during $[0,\tau]$, the process has at least one sojourn of duration less than $s$ ? (eg It flicks from 'on' to 'off' and then back to 'on' within duration $s$, at least once during the time interval.)

(I'm attempting to address an earlier question that I posed, with specialization to alternating renewal processes.)