This is really an extended comment, to large for a comment. The overhang is simply $\hat{N} = N - b + g$, where $g$ is the number of multinomial components equal to zero. So the only random part of that is $G$, the random variable defined as the number of $X_i=0$.

Consider the following game. You have $b$ boxes, throwing balls at them, independently. Box $i$ have hit probability $p_i$. Then we can think about varios scenarios, as example:

1. How large must $N$ (number throws) be to hit all the boxes, that is , to get $G=0$? That is the coupons collector problem.

2. How large must $N$ be to have al least one box with two (or more)balls? That is the birthday problem.

You want the complete distribution of $G$, or some approximation, so can be considered a generalization of the coupon collector problem.

This is really an extended comment, to large for a comment. The overhang is simply $\hat{N} = N - b + g$, where $g$ is the number of multinomial components equal to zero. So the only random part of that is $G$, the random variable defined as the number of $X_i=0$.

Consider the following game. You have $b$ boxes, throwing balls at them, independently. Box $i$ have hit probability $p_i$. Then we can think about varios scenarios, as example:

1. How large must $N$ (number throws) be to hit all the boxes, that is , to get $G=0$? That is the coupons collector problem.

2. How large must $N$ be to have al least one box with two (or more)balls? That is the birthday problem.

You want the complete distribution of $G$, or some approximation. That is a question about the number of occupied boxes, which is known as the occupancy problem, closely related to coupons collector problem. See for example https://probabilityandstats.wordpress.com/2010/03/27/the-occupancy-problem/