We consider a storage allocation model with a finite number of storage spaces. There are m primary spaces that are ranked {1,2,. . .,m} and R secondary spaces ranked {m + 1, m + 2,. . .,m + R}. Items arrive according to a Poisson process, occupy a space for a random exponentially distributed time, and an arriving item takes the lowest ranked available space. Letting N1 and N2 denote the numbers of occupied primary and secondary spaces, we study the joint distribution Prob[N1 = k, N2 = r] in the steady state. The joint process (N1, N2) behaves as a random walk in a lattice rectangle. We shall obtain explicit expressions for the distribution of (N1, N2), as well as the marginal distribution of N2. We also give some numerical studies to illustrate the qualitative behaviors of the distribution(s). The main contribution is to study the effects of a finite secondary capacity R, whereas previous studies had R = ∞.

Consider n random intervals I1, …, IN chosen by selecting endpoints independently from the uniform distribution. A packing of I1, …, IN is a disjoint sub-collection of these intervals: its wasted space is the measure of the set of points not covered by the packing. We investigate the random variable WN equal to the smallest wasted space among all packings. Coffman, Poonen and Winkler proved that EWN is of order (log N)2/N. We provide a sharp estimate of log P(WN ≥ t (log N)2 / N) and log P(WN ≤ t (log N)2 / N) for all values of t.