Interval packing: the vacant interval distribution

Abstract

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 $$p(t,y)= \begin{cases} \frac{2\int^t_0(1−e^{−vy})\beta(v)dv}{\alpha(t)}, \qquad\qquad\qquad y ≤ 1,\\ p(t,1)+\frac{(1−\exp(−t(y−1)))t\beta(t)}{\alpha(t)}, \quad y > 1,\end{cases} $$ 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.