I have an undesired event that follows a Poisson distribution, it happens at a constant rate, every m seconds. If two or more of these events occurs, my system will crash. I can periodically (every n seconds, n ≤ m) perform a procedure (her called 'controlled reset') to revert the effects of one event, to avoid two events to accumulate.
Is there a way to calculate the probability of an event to occur, after a 'reset'? And to calculate the probability of two events to occur in a time span less than n?
Thank you!
If the number of events in a fixed time period is Poisson with parameter $\lambda$, then the time it will take for the next event to occur has an exponential distribution, also with parameter $\lambda$. If you expect one event to happen on average every $m$ seconds, then $\lambda = \frac{1}{m}$ (i.e. the expected number of events to happen each second).
The exponential distribution is "memoryless", meaning that the distribution is the same regardless of your starting point. So if you have just reset the system, the probability that the next event will happen in $T$ seconds is $\lambda e^{-\lambda T} = \frac{1}{m} e^{-\frac{T}{m}}$.
This also means that the Poisson distribution scales - the number of events that will occur in $T$ seconds will be distributed as $Poisson(\frac{T}{m})$, so the probability that two events will happen within $T$ seconds is $(\frac{T}{m})^2 \frac{e^{-\frac{T}{m}}}{2!}$.