A building has $N$ floors. Each floor has a single trash bin of capacity $C$. On every day only once the trash is put into each basket at every floor until the basket is full (at which time no trash can be put to basket). The amount of trash that is put on $k$-th day in $i$-th floor is $T_{i,k}$. All $T_{i,k}$ for $i \in \{1,2..N\}, k \in \{1,2,3...\}$ are i.i.d random variables. The amount of trash present in $n$-th floor basket at the end of $d$-th day can be written as follows $$B_{n,d}=\min\left(\sum_{k=1}^{d}T_{i,k},C\right)$$ I want to answer following question. "After how many days (on average) $F\leq N$ baskets will be full?" Unfortunately, I do not know where to start from. I will be thankful if somebody could help me in solving this problem or could provide me some reference or hint.
Many thanks in advance.
My attempt:
Let $\overline{D}$ denotes the average number of days that $F$ baskets are full then we can write $$\overline{D}=1P(1)+2P(2)+3P(3)+...$$ where $P(x)$ denotes the probability that in $x$ days $F$ baskets are full. So far I can only find $P(1)$ (which is actually in the form of CDF of $\min\left(T_{i,1},C\right)$ ) but the summation above in the expectation goes to infinity and I have no idea about how to find $P(2)$. Maybe if I could find $P(2)$ then I can develop some pattern and later on use some approximation to take only few terms in the expectation term.