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

### A Generalization of Rényi’s Parking Problem

#### 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.

#### Keywords

#### Full Text:

PDF (Українська)#### 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).

#### GOST Style Citations

Copyright (c) 2017 Igor Sikorsky Kyiv Polytechnic Institute

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