Static robustness of the ADT policy

Figure 7.10 illustrates static robustness of the ADT policy, showing the mean response time under the ADT policy (a) as a function of $\rho_1$ and (b) as a function of $\rho_2$. For comparison, the mean response time under two T1 policies are plotted with dotted lines. The parameter settings in the middle row are exactly the same as those in Figure 7.6. In the top and bottom rows, only the value of $c_1\mu_1$ is changed, as labeled.

Figure 7.10: Static robustness of the ADT policy is illustrated by plotting the mean response time as functions of (a) $\rho_1$ ($\rho_2=0.6$ is fixed) and (b) $\rho_2$ ($\rho_1=1.15$ is fixed). Here, $c_1=c_2=1$, $c_1\mu_1=c_1\mu_{12}=1$, and $c_2\mu_2=\frac{1}{16}$ are fixed. For comparison, the mean response time under two T1 policies are also shown.

Static robustness: ADT vs. T1

$c_1\mu_1=\frac{1}{2}$

$c_1\mu_1=1$

$c_1\mu_1=2$

(a)

(b)

Figure 7.10 shows that the ADT policy with parameters ( $T_1^{(1)},T_1^{(2)},T_2$) achieves at least as low mean response time as the better of the two T1 policies, one with parameter $T_1^{(1)}$ and the other with parameter $T_1^{(2)}$, throughout the range of $\rho_1$ and $\rho_2$, if $T_2$ is chosen appropriately. In plotting Figure 7.10, we found ``good'' $T_2$ values manually by trying a few different values so that the ADT policy provides low mean response time throughout the range of load, which took only a few minutes. Observer that if $T_2$ is set too low, the ADT policy behaves like the T1 policy with parameter $T_1^{(2)}$, degrading the mean response time at lower loads, since $T_1^{(2)}$ is larger than the optimal $T_1$ in the T1 policy at lower loads. If $T_2$ is set too high, the ADT policy behaves like the T1 policy with parameter $T_1^{(1)}$. This worsens the mean response time at higher loads.

Static robustness of the ADT policy can be attributed to the following. The dual thresholds on queue 1 make the ADT policy adaptive to misestimation of load, in that the ADT policy with parameters ( $T_1^{(1)},T_1^{(2)},T_2$) operates like the T1 policy with parameter $T_1^{(1)}$ at the estimated load and like the T1 policy with parameter $T_1^{(2)}$ at a higher load, where $T_1^{(2)}>T_1^{(1)}$. Thus, server 2 can help queue 1 less when there are more type 2 jobs, preventing server 2 from becoming overloaded. This leads to the increased stability region and improved performance.

Figure 7.10 makes an additional point about the effect of $c_1\mu_1$. When $c_1\mu_1$ is smaller (top row), the difference in the stability regions of the two T1 policies becomes smaller, and the difference in the mean response times of the two T1 policies at low loads becomes larger. Thus, the benefit of the ADT policy over the T1 policy becomes smaller at smaller $c_1\mu_1$, since the T1 policy with smaller $T_1$ can provide low mean response time with only a slightly reduced stability region. This makes intuitive sense, since in the limit as $c_1\mu_1\rightarrow 0$, the Beneficiary-Donor model reduces to a single (donor) server with two queues, where the policy following the $c\mu$ rule is provably optimal [45]. Also, when $c_1\mu_1$ is larger (bottom row), the difference in the stability region of the two T1 policies becomes larger, and the difference in the mean response times of the two T1 policies at low loads becomes smaller. Again, the benefit of the ADT policy over the T1 policy becomes smaller at larger $c_1\mu_1$, since the T1 policy with larger $T_1$ can provide a wide stability region with only a slightly high mean response time at lower loads. This makes intuitive sense, since the jobs in queue 1 are much better served by server 1 and the help from server 2 becomes negligible when $c_1\mu_1$ is much larger than $c_1\mu_{12}$. Thus, the ADT policy is most effective when $c_1\mu_1$ and $c_1\mu_{12}$ are comparable and $c_1\mu _{12}> c_2\mu _2$, where the T1 policy with $1<T_1<\infty$ provides lower mean response time than T1(1) and T1($\infty$) but suffers from static robustness.