probability terminology for parameter in a Markov process Suppose $$P(\text{feature present at time} \ t \ \text{and} \ t+\Delta t) = \beta^{2}+\beta(1-\beta) \exp(\Delta t/\tau)$$
where $\tau = 1/(\pi_{01}+\pi_{10})$. What is $\tau$?

I don't recognize the formula exactly but $1/\tau$ is proportional to a rate at which something happens and $\tau$ a time scale (as Didier suggests in the comment).
Imagine a light or other two-state system (e.g., feature on / feature off) that is being activated, de-activated, or reversed, at random times according to a random process with some characteristic rate of occurrence of each action per unit of time.  The events ON (1), OFF (0), ON-reversal (10), and OFF-reversal (01) could have rates (or probabilities) $r_0, r_1, \pi_{10}, \pi_{01}$, and the events could happen according to a Poisson process or suitable Markov chain.
If, on average, the light is on for a fraction $\beta$ of the time, then for large $\Delta t$ what happens at the two times is independent and the probability is close to $\beta^2$.   For small $\Delta t$ the probability of a change of state is also small, and the probability of light being on at both times is almost the same as that of the light being on at time $t$, which is $\beta$.
$\exp(-\Delta t/\tau) = \exp (-\Delta t (\pi_{01} + \pi_{10}))$ looks like a probability that no reversals occur.  For example, it is that probability for an interval of length $\Delta t$ if, on average, one person per unit time randomly walks by the light switch and a fraction $\pi_{10}$ of people always will turn the light off (if on) and $\pi_{01}$ will always turn it on (if off).
