Balls into Bins with Variable Sized Balls

Hello,

Sorry if this is not the right group to send, I tried to seek for
statistics group but I didn't find. If this isn't the right group, I'd
hope someone would guide me to the right one. Thanks.

The classical balls into bins problem discusses the probability of
certain events occurring on throwing n balls at random on m bins. Like
the event that no bin is empty and so on. It doesn't care about the
size of the balls, however.

My problem requires that we distribute n balls among m bins ( usually
n > m, but n <= m is possible too ). Given only the mean and variance
of the balls set ( where the random variable is the size of the
ball ), I need to calculate the mean and variance of the random
variable which is the total amount of size per bin ( sum of sizes of
all the balls in some bin ), such that the variance is the minimum
possible variance.

That is I need some way to get the minimum possible variance of that
variable from some set of balls, so I'd compare it to the output of
some load-balancing algorithm.

For the input set (ball sizes set) 1 1 2 2 2 3 3 3, and n = 3, they
would be optimally distributed as 5 5 4, which has the mean 4.666667
and the (biased) variance of 0.222222. In contrast to 6 6 2 which has
the same mean and the vairance 3.555555.

At least I need some keywords to guide me where to look. I am seeking
a concrete answer, I at least want to know how to approach such
problem and solve it.

Any help is greatly appreciated.

.

Relevant Pages

  • Re: I function F(x) that tells me,how do i put the balls in the beans
    ... but the sequence should be ... The number of ways of putting 128 indistinguishable balls into 8 ... distinguishable bins is not the same as the number of ways of choosing ... I'm not familiar enough with the literature to point OP to a source. ...
    (sci.math)
  • Re: Balls in Bins problem
    ... Given m bins of infinite capacity and p balls. ... ball is thrown ...
    (sci.math)
  • Balls in Bins problem
    ... can be restated in the familiar balls & bins form. ... Given m bins of infinite capacity and p balls. ...
    (sci.math)
  • Re: I function F(x) that tells me,how do i put the balls in the beans
    ... but the sequence should be ... The number of ways of putting 128 indistinguishable balls into 8 ... distinguishable bins is not the same as the number of ways of choosing ...
    (sci.math)
  • Re: Please test this encryption
    ... > that it's probably not a straight monoalphabetic substitution cipher. ... For comparison, here's Emacs's expected-frequency table, using the ... if q balls are thrown uniformly ... at random into n bins then the expected number of bins containing k ...
    (sci.crypt)