Bounded Distributions

I am dealing with a problem for which I need to model the probability distribution of a certain quantity that has both a lower bound and an upper bound. The quantity in question is basically a measure of how likely it will be that a deadline will be met. So, for example, suppose the quantity is "the number of times delivery is achieved in 30 days or less divided by the total number of deliveries", with the result expressed as a percentage. In my application, this percentage should be quite high, say on the order of 90 or 95 %.

For reasons I will not bore you with, I need to assume a certain probabilty distribution for this quantity, with an assumed "peak". I have basically no visibility into the factors that determine whether or not on-time delivery is achieved, so I am inclined to use the gaussian distribution and assume a certain mean (say 90%) and a certain standard distribution (say 10 %). Of course, if I model the distribution this way, and then use Monte Carlo methods, I will get some values greater than 100 %. And I will, albeit rarely, get values less than 0. And, of course, such values are not meaningful

Does anyone have any suggestions as to how I can model this kind of quantity from a probability distribution function perspective?

Thanks.