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 R\'enyi's parking problem [Magyar Tud. Akad. Mat. Kutat\'{o} Int. K\"{o}zl. 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.

翻译：本文提出了一种高精度且高效的方法，用于计算随机最大规模非重叠单位直径圆盘集合（该集合与一个固定单位直径圆盘接触）的中心向量的计数、总和及其平方范数的期望值。该方法扩展了Rényi停车问题（[Magyar Tud. Akad. Mat. Kutat. Int. Közl. 3 (1-2), 1958, pp. 109-127]）的早期研究。其核心在于对第一个圆盘状态进行条件拆分。我们证明了该拆分方法，并由此推导出期望的积分方程。这些方程具有下块三角形式，通过代入法并结合块内多项式近似对积分进行极高精度逼近来求解。