# A Generalization of Rényi’s Parking Problem

## DOI:

https://doi.org/10.20535/1810-0546.2017.4.105391## 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.

## References

A. Rényi, “On a one-dimensional problem concerning random space-filling”, *Publ. Math. Inst. Hung. Acad. Sci.*, vol. 3, pp. 109–127, 1958.

A. Dvoretzky and H. Robbins, “On the parking problem”, *Publ. Math. Inst. Hung. Acad. Sci.*, vol. 9, pp. 209–224, 1964.

P.E. Ney, “A random interval filling problem”, *Ann. Math. Stat.*, vol. 33, pp. 702–718, 1962.

D. Mannion, “Random space-filling in one dimension”, *Publ. Math. Inst. Hung. Acad. Sci.*, vol. 9, pp. 143–154, 1964.

H. Solomon and H.J. Weiner, “A review of the packing problem”, *Comm. Statist. Th. Meth.*, vol. 15, pp. 2571–2607, 1986.

M.D. Penrose and J.E. Yukich, “Limit theory for random sequential packing and deposition”, *Ann. Appl. Probab.*, vol. 12, no. 1, pp. 272–301, 2002. doi: 10.1214/aoap/1015961164

S.M. Ananjevskii, “Some generalizations of “parking” problem”, *Vestnik Sankt-Peterburgskogo Universiteta. Ser. 1. Matematika.** Mehanika*. *Astronomiya*, vol. 3 (61), no. 4, pp. 525–532, 2016 (in Russian). doi: 10.21638/11701/spbu01.2016.401

A.M. Samoilenko *et al.*, *Differential Equations. Textbook*, 2nd ed.Kyiv,Ukraine: Lybid’, 2003 (in Ukrainian).

W. Feller, *An Introduction to Probability Theory and its Applications*, vol. 2 (Transl. from the 2nd English ed). Moscow, SU: Mir, 1984 (in Russian).

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2017 Igor Sikorsky Kyiv Polytechnic Institute

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work