Starting at time 0, unit-length intervals arrive and are placed on
the positive real line by a unit-intensity Poisson process in two dimensions;
the left endpoints of intervals appear at the rate of 1 per unit time per unit
distance. An arrival is accepted if and only if, for some given x , the
interval is contained in $[o, x]$ and overlaps no interval already accepted.
This stochastic, on-line interval packing problem generalizes the classical
parking problem, the latter corresponding only to the absorbing states of the
interval packing process, where successive packed intervals are separated by
gaps less than or equal to 1 in length. In earlier work,the authors studied the
distribution of the number of intervals accepted during 0 t .This paper is
concerned with the vacant intervals (gaps)between consecutive packed
intervals.Let p t y be the limit x 8of the fraction of the gaps at time t which
are at most y in length.We prove that
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where $\alpha(t) = \int_0^t
\beta(v)dv, \beta(t) = \exp[-2 \int_0^t((1-e^{-v})/v)dv]$. We briefly discuss
the recent importance acquired by interval packing models in connection with
resource-reservation systems. In these applications, our vacant intervals
correspond to times between consecutive reser- vation intervals. The results of
this paper improve our understanding of the fragmentation of time that occurs
in reservation systems.