5.8 EXPECTED TRAVEL DISTANCES AND EXPECTED TRAVEL TIMES, REVISITEDIn Chapter 3 we derived several expressions for the expected travel distance and the expected travel time of urban service units responding to calls. In this section we shall extend our results to account, under certain conditions, for the effects of congestion (i.e., for the fact that some response units may be busy and unavailable at the time when a call for service occurs). Consider a large region of a city of area A_{o} and assume that response unit stations have been located in this region according to a spatial Poisson process with intensity of stations per square mile. Assume that each of the N_{o} stations holds exactly one response unit. In the following, we shall assume that:
Note that assumption 2 implies that E[time spent on the scene of a call] >> E[travel time to a call] a condition that is true for many urban service systems. When A_{o} is sufficiently large so that the effects of the boundaries of the region can be ignored and when all N_{o} response units are available, it was shown in Chapter 3 [cf. (3.104a) and (3.105)] that, for the system described, the expected travel distance to a call is given by
Here, c is a constant that depends on the travel metric in use and possibly other geographical characteristics of the region in question. In general, the number of available response units, N, will fluctuate with workload and, over a long period of time, will take values ranging from 0 to N_{o}. It would clearly be very useful if we could find a simple relationship beeen E[D] and the average number E[N] of available response units in the area. Such a relationship would be helpful for planning purposes since it would take into account the effects of system workload--as reflected in E[N]. To develop such a relationship, we shall assume that rarely, if ever, are N_{o} units in the region busy. We have already shown in the previous section that under these circumstances one can use the M / G / system results, and we proved that the busy response units at any instant in time are distributed as a spatial Poisson process with parameter / busy units per unit area. Since the N_{o} unit stations are distributed as a spatial Poisson process as well, it follows that, at any instant, the available response units are approximately distributed as a spatial Poisson process.^{9} Hence, given any value of N, we haveve, as in (5.59), that ^{9}This assumption is an approximation, since the dispatch policy that assigns the closest available server creates "holes in coverage"; these holes are somewhat larger than those ordinarily found in a spatial Poisson process. The net efFect is that assignment of servers to busy status does not constitute "random erasures" of an existing spatial Poisson process, but rather "correlated erasures," yielding a residual (unerased) process that is not strictly a Poisson process.
For the case N = 0, which is very unlikely anyway, we can assume that a response unit from outside the region responds to calls and that the expected travel distance is then given by a constant D_{o}. In the steady state, we therefore have
where
are the steady-state probabilities of having k units available, or (N_{o} - k) units busy, as given by an M / G / N_{o} queueing model with no waiting space [cf. (4.85)]. This approach was first utilized by Larson in analyzing overlapping police beats [LARS 72a, Eqs. (7.40), (7.42)]. Since P_{o} is very small by assumption, we can safely ignore the first term in (5.61) and write
where the expectation is taken over the state probabilities P_{k}. It is much more convenient, however, to use a further simplifying approximation, by writing
In truth (5.64) provides only a lower bound for (5.63) since E[()^{-1}] < (E[N])^{-1/2} . This follows from the fact that if h(X) is a convex function of a nonnegative random variable X [such as h(X) = ()^{-1}], then^{10}
^{10}Inequality (5.65) is also known as Jensen's inequality (cf. Problem 3.5). However, we show in Problem 5.5 that the substitution of 1 / for E[1 / is quite reasonable for E[N] sufficiently large and N "compactly distributed" about its mean. Then, using the fact that P_{o} is very small and that, consequently, the M / G / N_{o} model is virtually indistinguishable from an M / G / model, we finally have
from which it follows by substitution into (5.64) that
These results can now be extended in a straightforward way to travel times. Using, for instance, the approximate acceleration/cruising speed model of Chapter 3 [cf. (3.93)],
we have
Equations (5.64), (5.67), and (5.69) are often referred to as square-root laws, since they relate E[D] and E[T] to the square root of the density of available response units in an urban area [KOLE 75a]. |