## 5.6 FRACTION OF DISPATCHES THAT ARE INTERRESPONSE AREA DISPATCHES
Often an agency administrator will want to know the number of
dispatches that take units outside of their primary response areas. Such
responses increase travel time and they may result in degraded service
due to unfamiliarity with the neighborhood (its geography, people, and
traditions). While we could compute this quantity exactly with the
hypercube model, it is instructive to obtain some approximate
"back-of-the-envelope" results for situations with many servers
For a given region with many response units
Given a request that arrives from primary response area - Dispatch unit
*n*, if available. - Otherwise, dispatch some unit
*m, m n*, where the particular choice depends on the state of the system (and, perhaps, other factors).
Invoking a large- Suppose that a service request arrives from response area Thus, the likelihood that a random request arrives in _{n}(t)
Summing over all
If the system is non-time-varying, with _{n}, _{n}(t) = _{n}, (5.55) reduces to
In Problem 5.14 we explore some special cases of (5.56). We will
find, in general, that f to equal (or slightly exceed) the
average fraction of time units are busy, averaged over all units,
provided that the system is non-time-varying._{I}What if the system is time-varying? We can show that this usually
"makes things worse," assuming that interresponse area dispatches are
undesirable. That is, given one further assumption, we can obtain a
bound for and _{n}(t) with their time averages,_{n}(t) = _{n}_{n}(t)dt = _{n}_{n}(t)dt
The additional assumption we require is that _{n}(t) and that _{n}_{n}(t) whenever _{n}_{n}(t). This says that a unit's
workload should be above (or below) average whenever the service request
rate from its primary response area is above (or below) average. Is this
reasonable? Can you think of counterexamples?_{n}We wish to prove, given the foregoing assumptions, that
= _{n}(t) + _{n}(t) = _{n}(t) + _{n}(t)
Clearly, the perturbation terms on the right-hand side integrate to zero; that is, (t) dt = 0(t) dt = 0
From (5.55) we have f = [_{I} + _{n}_{n} + _{n}(t) + _{n}(t)(t)(t)]
dt
Since the second and third terms in the integrand integrate to zero, f = _{I} + _{n}(t)(t) dt
Now, since (t)] = sgn [(t)]
then (t)(t)
dt 0
and thus (5.57) must be true. Problem 5.14 asks you to reexamine this analysis for systems that do not always give first preference to the primary response area's unit (e.g., a system incorporating an automatic vehicle locator system, which would allow the dispatcher to assign the vehicle closest to the scene of the service request). |