Customers arrive at a store according to a Poisson process with a fixed rate $\lambda$ per hour. Now we only know that the store have served $m$ people during $T$ hours. What is the probability that the store is closed during this time period? (Assume the store has unlimited service capability and cannot serve customers if it is closed.)
I think, if the expected number of customers $\lambda T > m$, then the store might be closed during the time period $T$. Specifically, if $m=0$ for a very large time period $T$, then the probability that the store is closed during $T$ would be close to $1$.
But I’m getting stuck into the problem to formulize the probability that the store is closed during $T$. Is there any other information that has to be provided?
Would you please help me?