A Generalization of Rényi’s Parking Problem

Andrii B. Ilienko, Vlad V. Fatenko

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


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

References


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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

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

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