A short route to Rényi's parking constant

A classical parking problem due to A. Rényi \cite{Slavik:2} asks for the expected number of cars of unit length that can be parked randomly in a street of a given length. The problem leads to a delay differential equation, and when the length goes to infinity, the mean density of the cars approaches Rényi's parking constant \[ C=\int_0^\infty \exp\left(-2\int_0^u\frac{1-{\mathrm e}^{-t}}{t}\,{\mathrm d}t\right)\,{\mathrm d} u \approx 0.7475979. \] We propose an alternative derivation of this constant, which is inspired by N. G. de Bruijn's analysis of the Buchstab function in number theory \cite{Slavik:1}. It is shorter and more elementary than Rényi's original approach, and relies on the duality between differential equations with delayed and advanced arguments.
\begin{thebibliography}{2} \bibitem{Slavik:1} \textsc{N. G. de Bruijn}, On the number of uncancelled elements in the sieve of Eratosthenes, \textit{Indag. Math.}, \textbf{12}(1950), 247-256. \bibitem{Slavik:2} \textsc{A. R\'enyi}, On a one-dimensional problem concerning random space filling, \textit{Publ. Math. Inst. Hung. Acad. Sci.}, \textbf{3}(1958), 109--127. \end{thebibliography}