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{\bf Sergey Agievich}
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{\bf Two-Stage Allocations and the Double $Q$-Function}
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Let $m+n$ particles be thrown randomly, independently of each other
into $N$ cells, using the following two-stage procedure.


\item{1.}
The first $m$ particles are allocated equiprobably, that is,
the probability of a particle falling into any particular cell is $1/N$.
Let the $i$th cell contain $m_i$ particles on completion.
Then associate with this cell the probability $a_i=m_i/m$
and withdraw the particles.

\item{2.}
The other $n$ particles are then allocated polynomially, that is,
the probability of a particle falling into the $i$th cell is $a_i$.


Let $\nu=\nu(m,N)$ be the number of the first particle that falls into
a non-empty cell during the second stage.  We give exact and
asymptotic expressions for the expectation ${\bf E}\nu$.

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