Motivation

It is interesting to compare the results of our analysis with the approximation proposed by Buzen and Bondi (the BB approximation) [23], which is the only prior work to investigate the multiserver system with multiple priority classes under general job size distributions. The BB approximation uses the following intuitive framework to approximate the mean delay in a priority system, where $\mbox{{\bf\sf E}}\left[ D^{\mbox{\scriptsize M/GI/}k\mbox{\scriptsize /prio}} \right]$ is the overall expected delay (response time minus service time) under the M/GI/$k$ queue with priority classes.

$$\mbox{{\bf\sf E}}\left[ D^{\mbox{\scriptsize M/GI/}k\mbox{\script... ...t]}{\mbox{{\bf\sf E}}\left[ D^{\mbox{\scriptsize M/GI/1/FCFS}} \right]} \right)$$ (4.1)

Figure 4.6 shows the optimal number of servers given by our analysis as compared with that predicted by the BB approximation. Here, the size of the high priority jobs has a two phase PH distribution with mean 10, and the low priority jobs have an exponential job size distribution with mean 1, as before. To compute the mean delay via BB, the mean delay in an M/GI/$k$/FCFS, $\mbox{{\bf\sf E}}\left[ D^{\mbox{\scriptsize M/GI/}k\mbox{\scriptsize /FCFS}} \right]$, is computed exactly via matrix analytic methods as before, as job sizes have a PH distribution; the mean delay in an M/GI/1/prio, $\mbox{{\bf\sf E}}\left[ D^{\mbox{\scriptsize M/GI/1/prio}} \right]$, is calculated via a known closed form expression [101].

Figure 4.6 shows that the BB approximation gives qualitatively misleading conclusion regarding the optimal number of servers. As load increases, so should the optimal number of servers; an effect not true under the BB approximation.

Figure 4.6: Comparison of (a) DR with (b) BB with respect to the question of ``how many servers are best?'' Case shown assumes the high priority jobs have a smaller mean job size ( $\mbox{{\bf\sf E}}\left[ X_H \right]=1$ and $\mbox{{\bf\sf E}}\left[ X_L \right] = 10$), and the load made up by high priority jobs equals that comprised by low priority jobs.

Comparison: Optimal number of servers

(a) DR

(b) BB

Since the only existing approximation, BB, leads to misleading conclusions, and computational complexity of DR becomes high when the number of classes increases, as we have discussed in Section 3.9, we propose a new approximation based on DR, which we refer to as DR-A (dimensionality reduction with class aggregation). DR-A allows us to efficiently compute the mean response time (per class and overall) of an M/PH/$k$ queue with many priority classes. We will see that the error in DR-A in predicting mean delay is within 5% against simulation for a range of parameters, which is much smaller than the error in BB. We will evaluate the error in DR-A in predicting mean delay rather than mean response time, since the error in mean delay is larger.