5.5.3 Return to the Three-Server Example
We apply the workload approximation procedure to the three-server, 10-atom example of the previous section. Recall that = 1.5, = 1, implying that = /(3 x 1) = 0.5. To initiate the procedure we need to specify the dispatch preference sets Gnj, and these are shown in Table 5-8. From (5.48) we compute the numerical values for the three required correction factors:
Q(3, ½, 1) = 0.73684
Q(3, ½, 2) = 0.63158
We are now ready to proceed through the steps of the algorithm. One complete iteration of the algorithm is shown in each of Figures 5.20-5.22. As can be seen, the procedure has converged for a very stringent convergence criterion ( = 0.0011 or greater) on the third iteration. The resulting workloads are estimated to be 1 = 0.5627, 2 = 0.4716, and 3 = 0.4657. The exact workloads, as computed from the exact hypercube model, are 1 = 0.5574, 2 = 0.4734, and 3 = 0.4693. The maximum error is |1 - 1| 0.0053, or about 1 percent error. The average error is about 0.7 percent. These error magnitudes are typical of those achieved with the approximation procedure.
Once we have the n's, as derived above, it is relatively straightforward to obtain an estimate for the fnj's, where fnj is the fraction of assignments that send unit n to atom j. The details of this are worked out in Problem 5.11. Once we have fnj for all n, j, then all other hypercube performance measures (including travel times) can be computed simply by substituting into the simple algebraic equations derived earlier for the exact hypercube model. Approximation errors for these measures, too, rarely exceed 2 percent and often are near 1 percent.
There are extensions to the basic approximation procedure described above, paralleling (but not as extensive as) those of the basic hypercube model. In particular, one can derive an approximation procedure for the zero-line-capacity queue and for the case of unequal service times. However, there is no known extension to dispatch policies other than fixed preference. Details can be found in [LARS 75a] and [JARV 75].
We have now completed our tour of N-server spatial queueing models for finite N. We conclude the chapter with two applications of many-server queues to derive analytically several important (rule-of-thurnb) performance characteristics of server-to-customer queueing systems.