Static robustness of single-threshold allocation policies

We next study static robustness of the T1 and T2 policies by evaluating the mean response time of these policies under a range of loads. In particular, we examine how a policy (T1 or T2) tuned for a certain load behaves at other loads.

Figure 7.6: Static robustness of single-threshold allocation policies is illustrated by plotting their 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. Bottom row compares the mean response time under the ``optimized'' T2 policy and two T1 policies.

Static robustness: T1 policy

Static robustness: T2 policy

Static robustness: T1 vs. T2

(a)

(b)

Figure 7.6 (top row) highlights static robustness of the T1 policy, plotting the mean response time of two T1 policies as a function of $\rho_1$ (only $\lambda_1$ is changed) in column (a) and as a function of $\rho_2$ (only $\lambda_2$ is changed) in column (b). For example, in column (b), $T_1=3$ is the optimal threshold when $\rho_2=0.4$. However, if it turns out that $\rho_2=0.8$ is the actual load, then the T1(3) policy leads to instability (infinite mean response time), while the T1(19) policy minimizes mean response time. Thus, choosing a higher $T_1$ (=19) guarantees stability against misestimation of $\rho_2$, but results in worse performance at the estimated load. Similar observation holds for stability against misestimation of $\rho_1$ (see column (a)), although the T1 policy appears to have more stability against misestimation of $\rho_1$ than against misestimation of $\rho_2$. Overall, we conclude that the T1 policy optimized at a lower load is poor with respect to static robustness. While it is possible to tune the T1 policy for higher loads (by choosing a larger $T_1$), this would degrade mean response time at lower loads.

Figure 7.6 (middle row) illustrates the static robustness of the T2 policy, plotting the mean response time of three T2 policies as functions of $\rho_1$ and $\rho_2$, as in the top row. In both columns (a) and (b), the T2(2) policy minimizes the mean response time in the class of T2 policies for a range of loads. That is, $T_2=2$ is the optimal choice for these parameter settings ( $c_1,c_2,\mu_1,\mu_2,\mu_{12}$) regardless of the loads ( $\lambda_1,\lambda_2$). Recall that the T2 policy is optimized at small $T_2$ values ($T_1=1$ or 2) for a wide range of parameter settings (see Figures 7.4-7.5). Since the T2 policy has the widest possible stability region regardless of its $T_2$ value, its mean response time is not very sensitive to the changes (or misprediction) in loads for a wider range of load, as compared to the T1 policy with parameter $T_1<\infty$. In this sense, the T2 policy has more static robustness than the T1 policy.

However, Figure 7.6 (bottom) shows that the mean response time under the optimized T2 policy, T2(2), can be much higher than that under the T1 policy optimized at each load, which is in parallel to our previous findings (see Figure 7.5). For example, in column (b), the mean response time under T2(2) is twice as high as that under T1(3) when $\rho_2=0.4$, and the mean response time under T2(2) is eight times as high as that under T1(19) when $\rho_2=0.8$. Only at very high $\rho_2$, the mean response time of T2(2) becomes lower than that of T1(19). Similar observation holds for column (a) as well. We conclude that the T2 policy performs worse than the T1 policy with respect to mean response time.