A Generalization of Rényi’s Parking Problem





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


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


increases unboundedly.

Methods. A functional-integral equation satisfied by the function


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




The explicit form of the constant


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.


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).