## 5.4.2 Model Assumptions

We state explicitly below the nine key assumptions underlying the basic hypercube queueing model that we will develop in the subsequent section. Successful generalizations of this basic model have allowed modifications to assumptions 6, 7, and 9 [LARS 74a, 75b, 78, 80; JARV 74; HALP 77; CHEL 80].

1. Geographical atoms. The area in which the system provides service can be broken down into a number NA of "statistical reporting areas" or "geographical atoms." These might correspond, for instance, to census blocks, small collections of city blocks, or police reporting areas. In the model each atom is modeled as a single point, located near the center of the actual atom.

2. Independent Poisson arrivals. Requests for service ("customers") are generated as a Poisson process, independently from each of the atoms. The Poisson arrival rate j, from atom j (j = 1, 2, . . ., NA) is known or can be estimated.

3. Travel times. Data are available to estimate the mean travel time ij from each atom i to each atom j (i, j = 1, 2, . . ., NA). In the absence of such data, plausible approximations for travel times can be made using geometical probability concepts (perhaps employing the Manhattan or Euclidean distance metric or an empirical time-distance relationship).

4. Servers. There are N spatially distributed servers, or response units, each of which can travel to any of the geographical atoms in the serviced region.

5. Server locations. The location of each response unit, while not servicing a customer, is known (at least statistically). For instance, a patrolling police car may allocate 50 percent of its patrol time in atom 12 and 25 percent each in atoms 8 and 11. A fixed position unit, such as an ambulance or delivery truck, would always be located in one particular atom (except when providing service). For non-fixed-position units, the set of atoms in which the unit may be located while available for assignment is called the unit's patrol area. In police applications patrol areas are usually called "beats," "sectors," or "routes." In general, patrol areas may overlap.

6. Server assignment. In response to each request for service, exactly one6 response unit is dispatched to the scene of the request, provided that at least one unit is available within the service region. If no unit is available, the request either enters a queue with other backlogged requests, or it is lost, or it is serviced by some backup system (e.g., police providing backup to an ambulance service or a neighboring community dispatching units into the temporarily saturated community). If the request enters a queue, it is later dispatched according to a queue discipline that does not depend on anticipated service time or geographical location of the request; allowable disciplines are, for instance, first-come, first-served; last-come, first-served; and random. Upon completion of service, a service unit either is dispatched to a request waiting in queue or it immediately returns to its home location.
6The assumption of dispatching exactly one unit to a request indicates that the model.

7. Fixed-preference dispatching. Server assignment takes place according to a fixed-preference procedure. By this we mean the following. Suppose that a request for service arrives from atom 12. Then there is an ordered list of units, say (3, 1, 7, 5, 6, 4, 2) for a seven-unit problem, that specifies the dispatcher's preference for units to assign to atom 12. The possibility of ties in preference is, for convenience, excluded from our discussion here. The dispatcher starts with the first entry in the list, unit 3 in this case, and assigns that unit, if available. Here atom 12 is in unit 3's primary response area.7 If the first preferred unit is not available, then the dispatcher assigns, if available, the first backup unit (the second preferred unit), which is unit 1 in this case. The dispatcher continues down the list until the first available unit is found (if there is one), and assigns that unit. This procedure is called a "fixed-preference" procedure since the ordered list of preferences does not change with the state of the system. However, the list of preferences may change by atoms, as for instance, the list for atom i + 1 may be (5, 6, 3, 1, 7, 2, 4). The model can operate with any fixed-preference dispatch policy. With NA atoms and N response units, there are (N!)NA possible different dispatch policies.8

7A unit's primary response area consists of that set of geographical atoms to which it would be assigned first if it were available. A primary response area may or may not coincide with a unit's patrol area (for the case of non-fixed-position units). For any particular unit, there may be no atoms in its primary response area. However, each atom must be contained in one (and only one) primary response area. Examples of primary response areas are ambulance zones, police beats, and service areas.

8If we include the possibility of ties in dispatch preference (which is excluded from our discussion here), the number of different policies becomes even more enormous.

8. Service times. The service time for a request, including travel time, on-scene time, and possible related follow-up time, has a known average value. In general, each response unit may have its own average service time. Moreover, reflecting the unpredictability of service times in actual systems, there is considerable variation about the average value(s). As one measure of the variability, the standard deviation of the service time is assumed to be approximately equal to the mean. The mathematical analysis assumes negative exponential service times, but reasonable deviations from this assumption have been found not to alter in any practical way the predictive accuracy of the model.

9. Service-time dependence on travel time. Variations in the service time that are due solely to variations in travel time are assumed to be second order compared to variations of on-scene and related followup service time. This assumption, which limits the applicability of the model, is most nearly satisfied by urban police departments, emergency repair services, and some home visit social service agencies, and least nearly satisfied by rural emergency services (especially rural ambulance services). As explained earlier in the chapter, this assumption does not imply that travel time is ignored in computing mean service time; through the process of mean service-time calibration, the mean service time n-1 for server n is equal to the sum of the mean travel time for that server, the mean on-scene time, and the mean off-scene follow-up time.

In practice, no actual service system will ever conform exactly to all the model's assumptions. In applying the model, one must weigh the extent to which the actual system does not fit the rigidities of the model (and the associated loss in predictive accuracy) against alternative methods (with their own limitations) to choose that method which best suits the decision-making goals at hand.

Given the foregoing assumptions, the model computes numerical values for the following performance measures:

1. Region-wide. Mean travel time, average workload and workload imbalance, fraction of assignments outside a unit's primary response area.

2. Response-unit-specific. Workload (measured in fraction of time busy servicing requests), mean travel time, fraction of responses outside the unit's primary response area.

3. Primary-response-area-specific. Internally generated workload (which is an input measure), mean travel time, fraction of requests serviced by other than the primary response unit.

4. Atom-specific. Internally generated workload (which again is an input measure), mean travel time, fraction of requests serviced by each unit n (n = 1, 2, . . ., N).

In addition, if the service being modeled is a police patrol force, one can calculate the frequency of patrol vehicle passings in each of the geographical atoms.