Equal mean sizes

In the topmost plot of Figure 4.2(a), we see that the high priority jobs often prefer multiple servers. This is to be expected based on our results in Section 4.3, since the high priority jobs are served in FCFS order and are not affected by the low priority jobs.

Surprisingly, however, the second plot in column (a) shows that the number of servers preferred by low priority jobs is much greater than that preferred by high priority jobs. Low priority jobs prefer more servers because low priority jobs are preempted by high priority jobs and thus their mean response time improves with more servers, which allows them to escape from the dominance of high priority jobs. That is, multiple servers reduce the impact of prioritization on the mean response time of low priority jobs.

The preferred number of servers with respect to the overall mean response time (the average of all jobs, including both low and high priority jobs) is shown in the third plot in column (a), where we see that the number of servers preferred by the overall mean, as expected, is a hybrid of that preferred by low and high priority jobs. Note though that this hybrid is more weighted toward the preference of low priority jobs because adding extra servers only hurts high priority jobs a small amount, whereas adding extra servers helps low priority jobs enormously. Interestingly, the number of servers preferred with respect to the overall mean is nearly identical to that shown for a single aggregate class of high and low priority jobs, shown in the bottom most plot in column (a). To understand why, observe that all jobs in this case have the same mean, and thus prioritizing in favor of some of them over others does not affect the mean response time greatly. Even though the classes have different variabilities, that is a smaller-order effect. This will not remain true in general.

Figure 4.2(b) shows that when the high priority jobs make up a smaller fraction of the load, the same trends are evident, but the specific numbers are quite different. For example, the topmost plot in column (b) shows that the number of servers preferred by high priority jobs is much fewer, since there are fewer high priority jobs in the systems, and they benefit from a fewer but faster servers (as many slow servers lead to low utilization). Less obvious is the fact that the number of servers preferred by low priority jobs in column (b) is also fewer than that in column (a). This follows from the same reasoning: the low priority jobs are strongly affected by prioritization (interruptions by high priority jobs), and with fewer high priority jobs, there are fewer interruptions and thus fewer servers are needed to avoid queueing behind high priority jobs.

Since both the high and low priority jobs in column (b) prefer fewer servers than in column (a), it makes sense that their overall mean also indicates that fewer servers are desired (the third plot of column (b)). This third plot also matches the bottom most plot in column (b) consisting of a single aggregate class, as in column (a).

Not shown in Figure 4.2 is the case where high priority jobs comprise more of the load. In this case, both classes prefer more servers and, therefore, the average of the two classes also prefers more servers. The reason for this is the converse of the above situation: since the load made up by the high priority jobs is high, large high priority jobs are likely to block other high priority jobs in a single server system. Further, the low priority jobs are preempted more frequently by high priority jobs and therefore also want more servers to alleviate the effect. Again the single aggregate class is very similar to the two priority class with respect to the optimal number of servers.

Figure 4.3: How many servers are best when the high priority jobs have a smaller mean job size?

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

High Priority

Low Priority

Overall mean

1 Aggregate Class

(a) $\rho_H = \rho_L$

(b) $2 \rho_H = \rho_L$