Occupancy distribution for batched allocations?
I've searched here and in my limited collection of combinatorics books for a solution to this with no luck.
Say I have $b$ bins labeled $1$...$b$.
A process selects $k$<=$b$ integers from $[1,b]$ without replacement, and a single ball is placed into each of the correspondingly labeled boxes. (n.b.: the "without replacement" applies only to each iteration of the process, not throughout the iterations, that is, at each iteration, all of the integers $[1,b]$ are available for sampling).
If I repeat the process (with $k$ fixed for each iteration) $z$ times, what is the distribution of fillings for the bins - that is, I'd like to be able to get the probability that some bin has $0$<=$n$<=$z$ balls at the completion of the iterations of the process.
I naively thought this would be a simple extension of the same problem where one ball is allocated at each iteration, but it's clearly more complex and beyond my limited combinatorics knowledge.
Help, even if only in a literature reference, will be most appreciated.
combinatorics balls-in-bins
asked Nov 28 at 22:36
Tom G
184
add a comment |
I've searched here and in my limited collection of combinatorics books for a solution to this with no luck.
Say I have $b$ bins labeled $1$...$b$.
A process selects $k$<=$b$ integers from $[1,b]$ without replacement, and a single ball is placed into each of the correspondingly labeled boxes. (n.b.: the "without replacement" applies only to each iteration of the process, not throughout the iterations, that is, at each iteration, all of the integers $[1,b]$ are available for sampling).
If I repeat the process (with $k$ fixed for each iteration) $z$ times, what is the distribution of fillings for the bins - that is, I'd like to be able to get the probability that some bin has $0$<=$n$<=$z$ balls at the completion of the iterations of the process.
I naively thought this would be a simple extension of the same problem where one ball is allocated at each iteration, but it's clearly more complex and beyond my limited combinatorics knowledge.
Help, even if only in a literature reference, will be most appreciated.
combinatorics balls-in-bins
asked Nov 28 at 22:36
Tom G
184
add a comment |
I've searched here and in my limited collection of combinatorics books for a solution to this with no luck.
Say I have $b$ bins labeled $1$...$b$.
A process selects $k$<=$b$ integers from $[1,b]$ without replacement, and a single ball is placed into each of the correspondingly labeled boxes. (n.b.: the "without replacement" applies only to each iteration of the process, not throughout the iterations, that is, at each iteration, all of the integers $[1,b]$ are available for sampling).
If I repeat the process (with $k$ fixed for each iteration) $z$ times, what is the distribution of fillings for the bins - that is, I'd like to be able to get the probability that some bin has $0$<=$n$<=$z$ balls at the completion of the iterations of the process.
I naively thought this would be a simple extension of the same problem where one ball is allocated at each iteration, but it's clearly more complex and beyond my limited combinatorics knowledge.
Help, even if only in a literature reference, will be most appreciated.
combinatorics balls-in-bins
asked Nov 28 at 22:36
Tom G
184
I've searched here and in my limited collection of combinatorics books for a solution to this with no luck.
Say I have $b$ bins labeled $1$...$b$.
A process selects $k$<=$b$ integers from $[1,b]$ without replacement, and a single ball is placed into each of the correspondingly labeled boxes. (n.b.: the "without replacement" applies only to each iteration of the process, not throughout the iterations, that is, at each iteration, all of the integers $[1,b]$ are available for sampling).
If I repeat the process (with $k$ fixed for each iteration) $z$ times, what is the distribution of fillings for the bins - that is, I'd like to be able to get the probability that some bin has $0$<=$n$<=$z$ balls at the completion of the iterations of the process.
I naively thought this would be a simple extension of the same problem where one ball is allocated at each iteration, but it's clearly more complex and beyond my limited combinatorics knowledge.
Help, even if only in a literature reference, will be most appreciated.
combinatorics balls-in-bins
combinatorics balls-in-bins
asked Nov 28 at 22:36
Tom G
184
asked Nov 28 at 22:36
Tom G
184
asked Nov 28 at 22:36
Tom G
184
asked Nov 28 at 22:36
Tom G
184
asked Nov 28 at 22:36
Tom G
184
184
add a comment |
add a comment |
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