5.8 EXPECTED TRAVEL DISTANCES AND EXPECTED TRAVEL TIMES, REVISITED

In 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:

1. Calls for service are independently and uniformly distributed in the city region and are generated in a Poisson manner at a rate of calls per hour per unit area.

2. Each call is handled by a single response unit and service times are independent and approximately identically distributed random variables with mean ^-1 (this includes travel time to the call, time spent on the scene, and travel time back to the response unit's station).

3. A nearest-response-unit dispatching policy is used [i.e., the response unit dispatched to a call is always the available (nonbusy) unit which is closest to the location of that call at the time when the call is received].

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

E[D] = c

(5.59)

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

⁹This 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.

E[D | N = k] c

k = 1, 2, 3, . . . , No

(5.60)

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

E[D] = PoDo + Pk c

(5.61)

where

Pk =

k = 0, 1, 2, . . ., No

(5.62)

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

E[D] c E

(5.63)

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

E[D] c

(5.64)

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], then¹⁰

E[g(X)] g(E[X])

(5.65)

¹⁰Inequality (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

E[N] = No - E[number of busy response units] No - No

(5.66)

from which it follows by substitution into (5.64) that

(5.67)

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)],

(5.68)

we have

(5.69)

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].