## 5.8.1 Extensions and Empirical Evidence

Square-root laws for E[D] and E[T] can also be derived for cases other than the one described above. For example, in Chapter 3 it was shown that, when response units are arranged in a symmetric pattern at the centers of equal squares rotated by 45o with respect to the directions of (right-angle) travel, E[D] = 0.47 [cf. (3.107)]. This expression can be used as the starting point for deriving a square-root law for this situation. The same applies when other regular patterns for response unit locations are in efrect (and, as we have seen in Chapter 3, the constant c is likely to be quite insensitive to the precise shape of these patterns).

Another extension would involve the case in which more than one response unit could be placed in some or all of the No stations in the region, as in fire department operations. Then (5.67) must be modified to

 E[D] = c(No - E[number of empty stations])-1/2 (5.67a)

A third extension, also applicable to fire departments, concerns situations in which some requests may be serviced by more than one response unit. It has been shown [CHAI 71] that (5.67) can still be used if the mean service time, -1, is adjusted to reflect the total number of "service time units" spent on requests. For example, if three fire engines spend 20 minutes each at the site of a particular fire alarm, then the service time for that alarm must be set equal to 60 minutes.

The New York City Rand Institute accumulated an impressive amount of data showing that expressions such as (5.64), (5.67), and (5.68) are valid in practice under a considerable variety of conditions [KOLE 75a, KOLE 75b], including dispatching policies that do not always send the nearest available response unit to a call but may instead dispatch the second or third nearest unit (for reasons such as those discussed in Section 5.3). For several different urban regions the constant c has been found to fall in the range 0.55 to 0.61 for fire department operations. This is not surprising in view of the fact that, for right-angle travel, c 0.63 if stations are located completely randomly and c 0.47 if stations are at the corners of a perfectly regular lattice. For most cities the actual pattern of firehouse locations is somewhat between these two extremes.