Rényi's Parking Constants

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Given the Closed Interval with , let 1-D cars'' of unit length be parked randomly on the interval. The Mean number of cars which can fit (without overlapping!) satisfies

 (1)

The mean density of the cars for large is
 (2)

Furthermore,
 (3)

for all (Rényi 1958), which was strengthened by Dvoretzky and Robbins (1964) to
 (4)

Dvoretzky and Robbins (1964) also proved that
 (5)

Let be the variance of the number of cars, then Dvoretzky and Robbins (1964) and Mannion (1964) showed that

 (6)

where

 (7) (8)

and the numerical value is due to Blaisdell and Solomon (1970). Dvoretzky and Robbins (1964) also proved that
 (9)

and that
 (10)

Palasti (1960) conjectured that in 2-D,

 (11)

but this has not yet been proven or disproven (Finch).

References

Blaisdell, B. E. and Solomon, H. On Random Sequential Packing in the Plane and a Conjecture of Palasti.'' J. Appl. Prob. 7, 667-698, 1970.

Dvoretzky, A. and Robbins, H. On the Parking Problem.'' Publ. Math. Inst. Hung. Acad. Sci. 9, 209-224, 1964.

Finch, S. Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/renyi/renyi.html

Mannion, D. Random Space-Filling in One Dimension.'' Publ. Math. Inst. Hung. Acad. Sci. 9, 143-154, 1964.

Palasti, I. On Some Random Space Filling Problems.'' Publ. Math. Inst. Hung. Acad. Sci. 5, 353-359, 1960.

Rényi, A. On a One-Dimensional Problem Concerning Random Space-Filling.'' Publ. Math. Inst. Hung. Acad. Sci. 3, 109-127, 1958.

Solomon, H. and Weiner, H. J. A Review of the Packing Problem.'' Comm. Statist. Th. Meth. 15, 2571-2607, 1986.