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Suppose that we have n time slots, and these are divided randomly between multiple users, with each user drawing a different number of time slots as it needs.

How would I measure the distribution of time slots, so that no user starves, or draws all its time slots as one bulk?

I had a method but the problem is that I only get results by simulation, I cannot derive a mathematical formula for it, so I am searching for mathematical form to measure the distribution.

Update: First: I only want a metric that can measure the fairness of the division, as I said before I found a metric but with no mathematical formula.

Second: there are two ways to divide the time slots between users:

  1. Completely random.
  2. Using rounds, example for that if u1 needs 2 time slots, u2 needs 4, and u3 needs 6, then there will be 2 rounds each one has 1+2+3=6 time slots, inside the round the u1 may chose any time slots, u2 may choose any 2 slots and so on.
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For this question to be answerable, you need to be more specific about how the random division occurs: how do the users "draw" their time slots and what is the mechanism whereby their "needs" arise? And please be more precise about the question itself: measuring the distribution of time slots that arise from a process is not the same as optimizing the distribution. You appear to want the latter, but to accomplish that, you need to indicate what variables you can control to achieve better performance and how you would assess that performance. – whuber♦ Dec 4 '12 at 18:23
No whuber, I don't want to optimize anything I just want to measure the fairness of the distribution. – user17470 Dec 4 '12 at 19:13
Do you want a fair allocation strategy or a measurement strategy? Your question isn't clear. Can you also describe what you call your simulation method? In your strategy #2 how do you assign slots within rounds? What if u1 and u2 want the same slots? Who gets the first shot? Are slots scare? More demand than supply? I think you could explain your problem a lot better. – curious_cat Mar 5 at 7:03

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There are many ways to measure the "fairness" of a distribution of resources. One metric is the Gini Coefficient, which is often as an index of income inequality within a country (or other population set).

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thanks Arthur, but that won't do, please read my question again. – user17470 Dec 5 '12 at 10:29

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