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Mathematics of Parking: Varying Parking Rate
Journal of Statistical Physics ( IF 1.6 ) Pub Date : 2021-01-24 , DOI: 10.1007/s10955-020-02678-x
Pavel B. Dubovski , Michael Tamarov

In the classical parking problem, unit intervals (“car lengths”) are placed uniformly at random without overlapping. The process terminates at saturation, i.e. until no more unit intervals can be stowed. In this paper, we present a generalization of this problem in which the unit intervals are placed with an exponential distribution with rate parameter \(\lambda \). We show that the mathematical expectation of the number of intervals present at saturation satisfies a certain integral equation. Using Laplace transforms and Tauberian theorems, we investigate the asymptotic behavior of this function and describe a way to compute the corresponding limits for large \(\lambda \). Then, we derive another integral equation for the derivative of this function and use it to compute the above limits for small \(\lambda \) with the help of some asymptotic results for integral equations. We also show that the corresponding limits converge to the uniform case as \(\lambda \) vanishes, yielding the well-known Renyi constant. Finally, we reveal the asymptotic behavior of the variance of the intervals at saturation.



中文翻译:

停车数学:变化停车率

在经典的停车问题中,单位间隔(“车长”)均匀地随机放置而不会重叠。该过程在饱和时终止,即直到无法存放更多的单位间隔为止。在本文中,我们对这个问题进行了概括,其中单位间隔以速率参数\(\ lambda \)的指数分布放置。我们表明,饱和状态下存在的区间数的数学期望满足某个积分方程。使用拉普拉斯变换和陶伯定理,我们研究了该函数的渐近行为,并描述了一种计算大\(\ lambda \)的相应极限的方法。然后,我们导出该函数的导数的另一个积分方程,并借助积分方程的一些渐近结果,将其用于计算小\(\ lambda \)的上述极限。我们还表明,随着\(\ lambda \)消失,相应的限制收敛到均匀情况,从而产生众所周知的Renyi常数。最后,我们揭示了饱和时间隔的方差的渐近行为。

更新日期:2021-01-24
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