Open AccessDiscretization of the parking problem
Author(s) -
Nikolai А. Kryukov
Publication year - 2020
Publication title -
vestnik sankt-peterburgskogo universiteta. matematika. mehanika. astronomiâ/vestnik sankt-peterburgskogo universiteta. seriâ 1, matematika, mehanika, astronomiâ
Language(s) - English
Resource type - Journals
eISSN - 2587-5884
pISSN - 1025-3106
DOI - 10.21638/spbu01.2020.408
Subject(s) - interval (graph theory) , mathematics , combinatorics , sequence (biology) , random variable , discretization , unit interval , discrete mathematics , statistics , mathematical analysis , genetics , biology
The present work consider a natural discretization of R´enyi’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.