The problem of selfish parking

Abstract

In this paper, we propose to investigate one of the models of the discrete analogue of the Renyi problem known as the “parking problem”. Let n, i be integers, n ≥ 0 and 0 ≤ i ≤ n−1. On the segment [0, n] we will place an open interval (i, i+1), where i is a random variable with equal probability taking the values 0, 1, 2 . . . , n−1 for all n ≥ 2. If n < 2, then we say that the interval does not fit. After placing the first interval, two free segments [0, i] and [i + 1, n], are formed, which are filled with intervals of unit length according to the same rule, independently of each other, etc. At the end of the process of filling the segment [0, n] with single intervals, between any two adjacent intervals the distance will not exceed 1. Let Xn denote the number of placed intervals. We study the asymptotic behavior of moments of a random variable Xn. Unlike the classical case, for the first moments it is possible to establish exact expressions for the moments.