How many servers are best in priority system?

In this section, we study the question of ``how many servers are best?'' in an M/PH/$k$ queue with two priority classes with respect to minimizing mean response time. The M/PH/$k$ queue with two priority classes can be modeled as a foreground-background (FB) process, as discussed in Section 3.4, and the analysis of the FB process via DR (see Chapter 3) provides mean response time of high priority jobs and low priority jobs, respectively.

Throughout our evaluations, we will consider a range of load typically shown on the horizontal axis, and a range of variability in the high priority job sizes, typically shown on the vertical axis. When we vary the load, we vary only the arrival rates and keep the job size distributions. In all the results shown, the size of the high priority jobs has a two phase Coxian$^+$ PH distribution (as defined in Section 2.2). The mean job size for the high priority jobs is held fixed at $\mbox{{\bf\sf E}}\left[ X_H \right]=1$, and its squared coefficient of variation is varied ($C^2 \geq 1$)^4.2. On the other hand, throughout, we assume that the size of the low priority job has an exponential distribution ($C^2=1$), since the effect of its variability is less interesting to study, as it affects only the performance of low priority jobs. We consider three different mean sizes, $\mbox{{\bf\sf E}}\left[ X_L \right]$, for the low priority jobs: (i) $\mbox{{\bf\sf E}}\left[ X_L \right] = 1$, (ii) $\mbox{{\bf\sf E}}\left[ X_L \right] = 10$, and (iii) $\mbox{{\bf\sf E}}\left[ X_L \right] = 1/10$. Note that the mean service time changes depending on how many servers are in the system (1 fast server or $k$ slow servers) so that the systems are comparable. The values specified above for $\mbox{{\bf\sf E}}\left[ X_H \right]$ and $\mbox{{\bf\sf E}}\left[ X_L \right]$ are the values for the maximum number of servers used in each plot and the mean sizes for each of the other number of servers is scaled appropriately. Finally, note that when we compare a dual priority system with its single priority counterpart (high priority jobs and low priority jobs are aggregated into a single class and served in FCFS order), the job size of the single aggregate class has a PH distribution that is a mixture of the two-phase PH distribution (for the high priority job size) and the exponential distribution (for the low priority job size).

Figure 4.2: How many servers are best when the two priority classes have the same mean job size?

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

High Priority

Low Priority

Overall mean

1 Aggregate Class

(a) $\rho_H = \rho_L$

(b) $2 \rho_H = \rho_L$

Figures 4.2-4.4 illustrate the results of our analysis for the three cases: (i) $\mbox{{\bf\sf E}}\left[ X_H \right]=1$ and $\mbox{{\bf\sf E}}\left[ X_L \right] = 1$, (ii) $\mbox{{\bf\sf E}}\left[ X_H \right]=1$ and $\mbox{{\bf\sf E}}\left[ X_L \right] = 10$, and (iii) $\mbox{{\bf\sf E}}\left[ X_H \right]=1$ and $\mbox{{\bf\sf E}}\left[ X_L \right] = 1/10$, respectively. For each figure, column (a) shows the case where the load made up by high and low priority jobs is equal ( $\rho_H = \rho_L$), and column (b) shows the case where $\rho_{H} < \rho_{L}$. We also discuss, but omit showing, the case where $\rho_{H} > \rho_{L}$. For each figure we consider both the case of dual priority classes and the case of a single aggregate class.