Suppose $(X_1,X_2,..X_i,..,X_b)$ as multinomial vector of random variables with $N=\sum_{i=1}^b X_i$ and probabilities $p_i$ to parametrize the $X_i$.

Let us take the following imagination to describe my problem: Assume there is an array of $b$ tubes that can hold maximal $N$ balls. All $N$ balls are randomly thrown towards the tubes and with probability $p_i$ a ball enters the tube labeled with index $i$. Ideally, we want to have the balls equally distributed among the tubes. Therefore let us define a non-negative penalty function (a truncated proxy random variable) for each tube as follows: $$ y(X_i) = \begin{cases} X-1 &,~for~ X_i \geq 2\\ 0 &,~else\\ \end{cases} $$ Finally, I am interested in the summation of the tubes overhang $\hat N=\sum_{i}~y(X_i)$ and the distribution of $\hat N$.

As the number of tubes $b$ and the number of balls $N$ are in the order of $10^5..10^6$ I see no way to calculate the distribution of $\hat N$ analytically. $$$$ In fact I am interested in showing a probabilistic bound on the maximal overhang like ${\rm Pr}(\hat N \leq k) \geq \mathcal F(k,b,N,I(\mathbf p))$, as are functioncal relation $\mathcal F$, which incorporate an information theoretic measure $\mathcal I(\mathbf p)$, e.g. the entropy of the distribution $\mathbf p$, the total variation distance of $\mathbf p$ to uniform distribution or maybe some useful f-divergence. At least a integrating $\max_i p_i$ and $\min_i p_i$ would be helpful.

If you have any idea how to tackle my problem, I am glad to hear your advice or answer. Thanks in advance.