## 5.5.3 Return to the Three-Server ExampleWe 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
Q(3, ½, 0) = 1Q(3, ½, 1) = 0.73684Q(3, ½, 2) = 0.63158We 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 Once we have the f is the
fraction of assignments that send unit _{nj}n to atom j. The
details of this are worked out in Problem 5.11. Once we have
f for all _{nj}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 |