# A Generalization of Rényi’s Parking Problem

## Authors

• Andrii B. Ilienko Igor Sikorsky Kyiv Polytechnic Institute, Ukraine
• Vlad V. Fatenko Igor Sikorsky Kyiv Polytechnic Institute, Ukraine

## Keywords:

Rényi’s parking problem, Mixture of distributions, Asymptotic behavior, Functional-integral equation, Laplace transform, Tauberian theorems

## Abstract

Background. A generalization of the A. Rényi’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. Rényi’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 Rényi’s theorem on the asymptotic behavior of the mean number of parked cars is deduced.

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