Consider throwing $n$ balls uniformly into $k$ bins, each of capacity $c$.
Denoting by $X_i$ the number of balls thrown to bin $i$, I'm interested in understanding the quantity $$S_{n,k,c}=\sum_{i=1}^k\max(0,X_i-c)\qquad,$$ which is the number of balls that spill from the bins.
What is known about $S_{n,k,c}$? Can we upper bound its expectation? What about the variance?
It seems easy to write an explicit formula for the expectation: $$ \mathbb E[S_{n,k,c}]=k\sum_{j=c+1}^n\Pr[X_i=j]\cdot(j-c)=k\sum_{j=c+1}^n{n\choose j}k^{-j}(1-1/k)^{n-j}(j-c)\quad{}, $$ but I'm hoping for a bound that's easier to work with.