The problem of sel?sh parking
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 saythat 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. LetXn 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.