## 5.5 HYPERCUBE APPROXIMATION PROCEDURE (INFINITE LINE CAPACITY)

As just discussed, the storage and execution time required to solve the hypercube model equations roughly doubles with each additional server. Thus, one is motivated to find an efficient approximation procedure that computes to reasonable accuracy all the hypercube performance measures. Such a procedure has been developed that requires solution to only N simultaneous equations rather than 2N, as in the exact model [LARS 75a].

The approximation has been found to compute results to within 1 or 2 percent of the exact results. In most practical situations such approximate solutions should suffice. For instance, data inaccuracies may not justify use of a highly precise model. Or the system planner may not have access to a sophisticated computer system necessary to perform calculations with the exact model. Or certain nonquantifiable concerns, perhaps involving political, legal, spatial, or administrative constraints, may play an important role in system design, thereby making precise estimates of quantifiable performance measures unnecessary. Finally, the approximation procedure can be carried out on a computer for any reasonable number of servers N, whereas the exact model is impractical for N greater than 15.

In this section we will simply sketch out the key ideas of the approximation procedure, as applied to the basic infinite queue capacity hypercube model of the previous section. Full details are found in [LARS 75a]. In complex urban service systems, we expect such approximations to play an increasingly important role in bringing computationally practical models to the aid of the urban decision maker.

The following are the main features of the approximation procedure:

1. As with the exact model, one assumes that the dispatcher has a rank-ordered list of preferred units to dispatch to service requests from each geographical atom and that he always dispatches the most preferred available (free) unit (i.e., fixed-preference dispatching).

2. In addition, if n is the fraction of time that unit n is busy, we use (1 - n) as the probability that unit n will be available to be dispatched to a service request when all units more preferred than unit n are busy. A correction factor is used as a multiplier to make this approximation as exact as possible. For instance, if unit 5 is the third preferred for a particular request, and units 7 and 2 are preferred to unit 5, the probability that unit 5 will be dispatched is equal to 72(1 - 5)(correction factor).

3. The correction factor, which can deviate significantly from 1, is derived to account for the fact that the statuses of servers are not independent (as would be assumed if the correction factor were always 1).

4. Given features 2 and 3, we can write N simultaneous equations relating the N unknowns (the workloads) to the dispatch policy and the arrival rates from the various geographical atoms.

5. The N simultaneous equations are solved iteratively, thereby yielding estimates of the workloads of the units.

6. If we desire other performance measures of the system (e.g., the mean travel time to each geographical atom or the fraction of dispatches that are interresponse area), then the values of the utilization factors found in feature 5 may be used to estimate the fraction of dispatches that send unit n to atom j, for all n and j. These fractions are then entered into simple equations to obtain estimates of the values of the desired performance measures.

The derivation of the approximation procedure is carried out assuming the M / M / N infinite-capacity model with indistinguishable (homogeneous) servers. As usual, the arrival rate is , the mean service time (for each server) is -1, and N < 1. The approximation arises from the fact that in the context of an urban service system, the servers are distinguishable and thus have differing performance characteristics.