A Generalization of Rényi’s Parking Problem

Abstract

Background . A generalization of the A. Renyi’s stochastic parking model is considered. In our model, the probability distribution of the left end of a parked car is a mixture of a uniform distribution and a degenerated one. This allows distinguishing drivers with different skills. Objective. The aim of the paper is an asymptotic study of the mean number of parked cars \[m_{p}(x) \] when the length of parking space \[x\] increases unboundedly. Methods. A functional-integral equation satisfied by the function \[m_{p}\] is derived. This equation admits an explicit solution in terms of the Laplace transform which allows for applying Tauberian theorems to study the required asy­mp­totics. Results. For the above model it is shown, that \[m_{p}(x)=\lambda_{p}x+o(x) \] as \[x\rightarrow\infty.\] The explicit form of the constant \[\lambda_{p}\] is given by \[\frac{1}{1-p}\int_{0}^{\infty} \mathrm{exp}\left(-2\int_{0}^{S}\frac{e^{\tau}-1}{\tau({e^{\tau}-p})}d\tau\right)ds.\] A similar asymptotic bound is obtained for a wider class of models in which the uniform component of the mixture of distributions is replaced by a more general one. Conclusions. As a generalization of the classical A. Renyi’s random parking model, a new parking model is proposed. In this model the uniform distribution of the parking point is replaced by a mixture of a uniform distribution and a degenerated one. In the new model, a counterpart of the Renyi’s theorem on the asymptotic behavior of the mean number of parked cars is deduced.