Comparing DR-A with BB and MK-N

Figure 4.7: Comparison of DR-A, MK-N, and BB approximations for M/PH/2 with 4 priority classes with $\mbox{{\bf\sf E}}\left[ X_4 \right]=1$, $\mbox{{\bf\sf E}}\left[ X_3 \right]=\alpha$, $\mbox{{\bf\sf E}}\left[ X_2 \right]=\alpha^2$, and $\mbox{{\bf\sf E}}\left[ X_1 \right]=\alpha^3$, as a function of $\alpha$. Here, the squared coefficient of variability of the job size distributions are $C^2=1$ (top two rows) or $C^2=8$ (bottom two rows) for all classes. Load is balanced among the classes. Note that MK-N does not appear for $C^2=8$, because the error is beyond the scale of the graphs for most values of $\alpha$.

Comparison of approximations for M/M/2 with 4 priority classes ($C^2=1$)

$\rho=0.3$

$\rho=0.8$

(a) Class 2

(b) Class 3

(c) Class 4

Comparison of approximations for M/PH/2 with 4 priority classes ($C^2=8$)

$\rho=0.3$

$\rho=0.8$

(a) Class 2

(b) Class 3

(c) Class 4

We now evaluate the accuracy of DR-A, BB, and MK-N. In all figures, we assume $k=2$ servers and $m=4$ priority classes. We consider both the case where each priority class has an exponential job size distribution ($C^2=1$; top half of Figure 4.7) and the case of two-phase PH job size distributions with $C^2=8$ (bottom half of Figure 4.7)^4.3. Each class may have a different mean job size, and these are chosen to vary over a large range, determined by parameter $\alpha$. Specifically, the mean job size of class $i$ is set $\mbox{{\bf\sf E}}\left[ X_i \right]=\alpha^{4-i}$, where $\frac{1}{4}\leq\alpha\leq 4$. Thus, $\alpha < 1$ implies small high priority jobs. We equalize the load between the classes, i.e. $\rho_i=\rho/4$, where $\rho_i$ is the load of class $i$. With these settings, the error in mean delay is evaluated for each class of jobs, where the error of an approximation is defined as follows:

$$\mbox{error} = 100 \times \frac{\mbox{(mean delay by approxi... ... by simulation)}}{\mbox{(mean delay by simulation)}}\quad(\%).$$

Thus, positive error means overestimation and negative error means underestimation of the approximation. Simulation is kept running until the simulation error becomes less than 1% with probability 0.95 (see Section 3.9 for more technical details of our simulation).

In evaluating the BB and MK-N approximations, we use accurate methods known to compute their components. For example, BB relies on knowing the mean delay for the M/PH/$k$/FCFS queue. We compute this delay precisely for the PH job size distribution using matrix analytic methods. MK-N relies on being able to analyze the case of two priority classes (since $m$ classes are reduced to two). We analyze the two priority class case in MK-N using DR. We first discuss the accuracy of MK-N and DR-A, and then discuss the accuracy of BB.