Generalizations of the parking problem

Abstract

The well-known parking problem of the Hungarian mathematician Rényi is about the asymptotic behavior of the mathematical expectation of the number of open unit intervals randomly filling a long interval. The length of the interval being filled increases without bound.

The paper studies generalizations of the parking problem in two directions. The first is the case where the length of the placed intervals is a random variable. Unlike in the original setting of the problem, the behavior of the expectations of both the number of placed intervals and the measure of the occupied part of the long interval are studied. The second direction is the case where the distribution of the position of a placed unit interval is nonuniform, unlike in the classical parking problem.