Mean response time of the ADT policy

We have already seen the benefits of the ADT policy when the load is not exactly known (static robustness). One might also expect that, even when the load is known exactly, the ADT policy might significantly improve upon the T1 policy with respect to mean response time. Earlier work of Ahn et al. and Meyn provides some support for this expectation [4,125]. For example, Meyn shows via numerical examples that, in the case of finite buffers for both queues, the policy that minimizes mean response time is a ``flexible'' T1 policy which allows a continuum of T1 thresholds, $\{T_1^{(i)}\}$, where threshold $T_1^{(i)}$ is used when the length of queue 2 is $i$ [125]. The ADT policy can be seen as an approximation of a ``flexible'' T1 policy, using only two $T_1$ thresholds.

Figure 7.11: Comparison of the ADT policy with the T1 policy with respect to the mean response time. The percentage change (%) in the mean response time of the (locally) optimized ADT policy over the optimized T1 policy at each given load is shown as a function of $\rho_2$ (0% corresponds to the T1 policy optimized at each load). Here, $c_1=c_2=1$, $c_1\mu_1=c_1\mu_{12}=1$, and $\rho_1=1.15$ are fixed.

(a) $c_2\mu_2=\frac{1}{4}$

(b) $c_2\mu_2=\frac{1}{16}$

To evaluate the benefit of the ADT policy, we compare it over a range of $\rho_2$ against the T1 policy optimized for the given $\rho_2$. Since the search space of the threshold values for the ADT policy is large, we find locally optimal threshold values, which are found to be optimal within a search space of $\pm 5$ for each threshold. We measure the percentage change in the mean response time of ADT versus T1:

$$\frac{\mbox{{\bf\sf E}}\left[ R_{ADT} \right] - \mbox{{\bf\s... ...{\mbox{{\bf\sf E}}\left[ R_{T1} \right]} \times 100\quad (\%),$$ (7.2)

where $\mbox{{\bf\sf E}}\left[ R_X \right]$ denotes the mean response time in policy $X\in\{\mbox{T1,ADT}\}$.

Figure 7.11 shows the percentage reduction in the mean response time of the locally optimized ADT policy over the T1 policy optimized at each $\rho_2$, as a function of $\rho_2$. The parameter settings are exactly the same as those in Figure 7.6(b). Figure 7.11 shows that, surprisingly, the benefit of the ADT policy is quite small with respect to mean response time under fixed Poisson arrivals; the improvement of the ADT policy is larger at moderately high $\rho_2$ and at smaller $c_2\mu_2$ value, but overall the improvement is typically within 3%. We conjecture that adding more thresholds (approaching the flexible T1 policy) will not improve mean response time appreciably, given the small improvement from one to two thresholds. Thus, whereas the ADT policy has significant benefits over the simpler T1 policy with respect to static robustness, the two policies are comparable with respect to mean response time.