Square-root law of response distance

Kolesar and Blum's square-root law: with available units spread like a random (Poisson) pattern over a region of area area, the mean distance from a call to the nearest available unit is \(c \sqrt{A / (u (1 - b))}\) for units u of which a share busy b is busy; \(c = 1/2\) in straight lines and \(\sqrt{2 \pi} / 4 \approx 0.627\) on a rectangular street grid. Quadrupling the units halves the distance and scaling area and units together leaves it unchanged (proof("sqrtlw")). The busy share can come from hypq()'s workloads.

Usage

sqrtlw(area, units, busy = 0, metric = "euclid", speed = NULL)

Arguments

area: The area served: one positive number (squared distance units).
units: The units deployed: one positive number.
busy: The share of units busy: one number from 0, below 1.
metric: "euclid" (straight lines) or "grid" (right-angle streets).
speed: NULL or the travel speed (distance per unit time): one positive number.

Value

A list: dist, the mean distance; const, the constant c; avail, the units available; time, the mean travel time (NA without speed).

References

Kolesar, P. and Blum, E. H. (1973). Square root laws for fire engine response distances. Management Science, 19(12), 1368-1378. doi:10.1287/mnsc.19.12.1368

Examples

sqrtlw(100, 4)
#> $dist
#> [1] 2.5
#> 
#> $const
#> [1] 0.5
#> 
#> $avail
#> [1] 4
#> 
#> $time
#> [1] NA
#> 
sqrtlw(100, 8, busy = 0.3, metric = "grid", speed = 0.5)
#> $dist
#> [1] 2.648109
#> 
#> $const
#> [1] 0.6266571
#> 
#> $avail
#> [1] 5.6
#> 
#> $time
#> [1] 5.296219
#>