# Computing expected moments of the Rényi parking problem on the circle

## DOI:

https://doi.org/10.21914/anziamj.v64.17955## Keywords:

piecewise polyomial approximation, point processes, one dimensional random packing, lower triangular linear systems## Abstract

A highly accurate and efficient method to compute the expected values of the count, sum, and squared norm of the sum of the centre vectors of a random maximal sized collection of non-overlapping unit diameter disks touching a fixed unit-diameter disk is presented. This extends earlier work on Renyi's parking problem [Magyar Tud. Akad. Mat. Kutato Int. Kozl. 3 (1–2), 1958, pp. 109–127]. Underlying the method is a splitting of the the problem conditional on the value of the first disk. This splitting is proven and then used to derive integral equations for the expectations. These equations take a lower block triangular form. They are solved using substitution and approximation of the integrals to very high accuracy using a polynomial approximation within the blocks.

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