Discretization of the parking problem

Abstract

The present work consider a natural discretization of Rényi’s so-called “parking problem”. Let l, n, i be integers satisfying l ≥ 2, n ≥ 0 and 0 ≤ i ≤ n − l. We place an open interval (i, i + l) in the segment [0, n] with i being a random variable taking values 0, 1, 2,...,n − l with equal probability for all n ≥ l. If n<l we say that the interval does not fit. After placing the first interval two free segments [0, i] and [i+ l, n] are formed and independently filled with the intervals of length l according to the same rule, etc. At the end of the filling process the distance between any two adjacent unit intervals is at most l−1. Let ξn,l denote the cumulative length of the intervals placed. The asymptotics behavior of expectations of the aforementioned random sequence have already been studied. This contribution has an aim to continue this investigation and establish the behavior of variances of the same sequence.