High priority class has smaller mean

In Figure 4.3, we continue to hold the mean high priority job size at 1 and increase the low priority job size to 10. Note that giving high priority jobs (smaller jobs) preference lower the overall mean response time, in this case.

Notice that the preferred number of servers for the high priority jobs is identical to that in Figure 4.2 because the high priority job size distribution is unchanged. However, the number of servers preferred by low priority jobs is now very different: they almost always prefer only one server. This is explained by observing that the effect of multiple servers in reducing the impact of prioritization on low priority jobs is small when the high priority jobs are smaller, and that a few fast servers has an advantage over many slow servers with respect to utilization.

The overall preferred number of servers, averaged over the two priority classes, is again a hybrid of the preferences of the two classes, but this time is biased toward the preferences of the high priority jobs because they are in the majority, implying a preference for fewer servers than the corresponding graph in Figure 4.2. Recall that adding servers is a way to help small jobs avoid queuing behind larger jobs. Since we are in the case where small jobs have priority already, we do not need the effect of multiple servers. Thus, in this case, priority classes can be viewed as a substitute for adding more servers.

Comparing the overall preferred number of servers for the case of dual priorities with that preferred under a single aggregate class, we see that there is a significant difference in preferences. The single aggregate class prefers many more servers. This is in contrast to the case where the high priority jobs and the low priority jobs have the same mean size (Figure 4.2). This again is a consequence of the fact that in this case prioritization is a substitute for increasing the number of servers.

Figure 4.3(b) illustrates the same graphs for the case where the high priority jobs comprise less of the total load. The trends are the same as in column (a); however the preferred number of servers is significantly smaller in all figures. This follows from the same argument as that given for Figure 4.2(b). In the case where high priority jobs make up a greater proportion of the total load (not shown), the number of servers preferred is, as before, always higher than in column (a).

Figure 4.4: How many servers are best when the high priority class has a larger mean job size?

(iii) $\mbox{{\bf\sf E}}\left[ X_H \right]=1$ and $\mbox{{\bf\sf E}}\left[ X_L \right] = 1/10$

High Priority

Low Priority

Overall mean

1 Aggregate Class

(a) $\rho_H = \rho_L$

(b) $2 \rho_H = \rho_L$