The adaptive dual threshold (ADT) policy

The key idea in the design of the ADT policy is that, in contrast to the T1T2 policy, the ADT policy is always operating as a T1 policy, but unlike the standard T1 policy, the value of $T_1$ adapts, depending on the length of queue 2, to provide static robustness. Specifically, the ADT policy behaves like the T1 policy with parameter $T_1^{(1)}$ if the length of queue 2 is less than $T_2$ and otherwise like the T1 policy with parameter $T_1^{(2)}$, where $T_1^{(2)}>T_1^{(1)}$.

We will see that the ADT policy, unlike the T1T2 policy, is far superior to the T1 policy with respect to static robustness, due to the dual thresholds on queue 1. In addition, one might also expect that the mean response time of the optimized ADT policy will significantly improve upon that of the optimized T1 policy, since the ADT policy generalizes the T1 policy (the ADT policy is reduced to the T1 policy by setting $T_1^{(1)}=T_1^{(2)}$). However, this turns out to be largely false, as we see below. Formally, the ADT policy is characterized by the following rule.

Definition 19   The ADT policy with parameters ( $T_1^{(1)},T_1^{(2)},T_2$) operates as the T1 policy with parameter $T_1^{(1)}$ if $N_2\leq T_2$; otherwise, it operates as the T1 policy with parameter $T_1^{(2)}$.

Below, the ADT policy with parameters ( $t_1^{(1)},t_1^{(2)},t_2$) is also denoted by the ADT( $t_1^{(1)},t_1^{(2)},t_2$) policy.

Figure 7.9: Figure shows whether server 2 works on jobs from queue 1 or queue 2 as a function of $N_1$ and $N_2$ under the ADT policy with parameters $(T_1^{(1)}, T_1^{(2)}, T_2)$.

Figure 7.9 shows the jobs processed by server 2 under the ADT policy as a function of $N_1$ and $N_2$. At high enough $\hat\rho_1$ and $\rho_2$, $N_2$ usually exceeds $T_2$, and the policy behaves similar to the T1 policy with parameter $T_1^{(2)}$. Thus, the stability condition for the ADT policy is the same as that for the T1 policy with parameter $T_1^{(2)}$. The following theorem can be proved in a similar way as Theorem 14.

Theorem 19   The stability condition for the ADT policy with parameters ( $T_1^{(1)},T_1^{(2)},T_2$) is given by the stability condition for the T1 policy with parameter $T_1^{(2)}$ (Theorem 14).