We record measurements of an appartus every day. If apparatus doesn't break (it has probability equal to $1-p_2$), it will measure zero with probability $p_1$. If apparatus breaks (probability $p2$), that and every consecutive measurement will be zero, until the apparatus is fixed.
I already know equations for conditional probability of apparatus breaking n days ago, given m consecutive zeros $P(n, m)$; conditional probability of apparatus being broken today given m consecutive zeros $P'(m)$ and some other probabilities. These are the functions of $m$, $n$, $p_1$ and $p_2$, but are rather complex, so I'll paste them if someone need them.
Let's assume that the cost of checking apparatus (and fixing if it's broken) is $C$, value of a correct measurement is $1$ and if we find that apparus is broken, we must discard all the consecutive zeros before it, because we don't know if any of them is a correct measurement.
What is the optimal strategy? At what length of run of zeros ($m$) should we check the apparatus? Checking the apparatus after a small $m$ would result in many unnecessary checks of apparatus, on the other hand checking the apparatus only after very big $m$ would mean we keep discarding big number of measurements, also lowering our score.
For what $m$ we get best scores? Sorry, that I don't know how to formalize this problem better.