Which practicable solutions exist to fair distribution of a finite resource over a finite time horizon?

Question

Imagine we have $X$ amount of some resource available over a fixed, known, and finite time period $T$.
An unknown but finite number $N$ of users can each at arbitrary times request some share $Y = y/X$ of the resource $X$. We do not know in advance when future requests arrive and each user may request several times during $T$.
On one hand we wish to have spent all of $X$ by the end of $T$. On the other hand, we do not wish to run out of $X$ before the end of $T$.
How can we find a fair strategy for granting $Y$ to each user, i.e. determining the fraction $y$? I would like to adapt $y$ over the time period $T$ to adapt to observed demand. For example: if the demand turns out to be lower than expected early in the period, we will want to increase $y$ to increase the probability that we spend all of $X$. Or, if the demand turns out to be higher, we will want to decrease $y$ to decrease the probability that we run out of our resource way before the end of $T$.
I am interested in identifying in some sense optimal or close to optimal approaches for solving this problem in practice. I am convinced this is such a common problem that someone must have treated it many times over the years and I am guessing an answer may be some kind of dynamic programming?

Answer 1

The number of requests in a time seems a Poisson($\lambda$) distribution ($\lambda$ expected events in 1 unity of time), the optimal strategy would be $Y=\frac{X}{\lambda T}$. If you don't know the real parameter of the population, you could estimate it using the sample mean (which is the maximum likelihood estimation of the real parameter). This value will be getting closer to the real parameter over time, so you could use it to adapt your $y$.