Case 2: $c_1\mu _{12}> c_2\mu _2$

In the rest of this chapter, we limit our attention to the case of $c_1\mu _{12}> c_2\mu _2$, where server 2 ``prefers'' to run type 1 jobs in a $c\mu$ sense. Note that condition $c_1\mu _{12}> c_2\mu _2$ is achieved when type 1 jobs are smaller than type 2 jobs, when type 1 jobs are more important than type 2 jobs, and/or in the pathological case when type 1 jobs have good affinity with server 2. (These, in addition, may motivate user of the Beneficiary-Donor model, giving smaller or more important jobs better service.) We will see that, for the T1 policy, the optimal $T_1$ threshold is typically finite when $c_1\mu _{12}> c_2\mu _2$, in contrast to the $c\mu$ rule. For the T2 policy, we will see that the optimal $T_2$ threshold is usually small, but not necessarily $T_2=1$.

Figure 7.5: The mean response time under the T1 policy as a function of $T_1$ (rows 1 and 3), and the mean response time under the T2 policy as a function of $T_2$ (rows 2 and 4), when $c_1\mu _{12}> c_2\mu _2$. Here, $c_1=c_2=1$, $c_1\mu_{12}=1$, and $\rho_2=0.6$ are fixed.

Mean response time of T1 and T2: $c_1\mu _{12}> c_2\mu _2$

$c_2\mu_2=\frac{1}{16}$

T1

T2

$c_2\mu_2=\frac{1}{4}$

T1

T2

(a) $c_1\mu_1=\frac{1}{4}$

(b) $c_1\mu_1=1$

(c) $c_1\mu_1=4$

Figure 7.5 shows the mean response time under the T1 policy as a function of $T_1$ (rows 1 and 3), and the mean response time under the T2 policy as a function of $T_2$ (rows 2 and 4) when $c_1\mu _{12}> c_2\mu _2$ (and $c_1=c_2$). Throughout, $c_1\mu_{12}=1$ is fixed, and we set $c_2\mu_2=\frac{1}{16}$ in the top half, and $c_2\mu_{12}=\frac{1}{4}$ in the bottom half. Again, different columns correspond to different $\mu_1$'s. In each plot, mean response time is evaluated at three loads, $\hat\rho_1 = 0.8, 0.9, 0.95$, by changing $\lambda_1$

In Figure 7.5 (rows 1 and 3), we see that optimal $T_1$ is finite and depends on environmental conditions such as load ($\hat\rho_1$) and job sizes ($\mu_1$). By Theorem 14, a larger value of $T_1$ leads to a larger stability region, and hence there is a tradeoff between good performance at the estimated load, $(\hat\rho_1,\rho_2)$, which is achieved at smaller $T_1$, and stability at higher $\hat\rho_1$ and/or $\rho_2$, which is achieved at larger $T_1$. Note also that the curves have sharper ``V shapes'' in general at higher $\hat\rho_1$ and/or smaller $c_2\mu_2$, which complicates the choice of $T_1$, since mean response time quickly diverges to infinity as $T_1$ becomes smaller. Also, when $\rho_2$ is lower (and thus $\rho_1$ is higher for a fixed $\hat\rho_1$), the optimal $T_1$ tends to become smaller, and hence the tradeoff between low mean response time at the estimated load and stability at higher loads is more significant. This makes intuitive sense, since at lower $\rho_2$, server 2 can help more.

In Figure 7.5 (rows 2 and 4), we see that the mean response time of the T2 policy is minimized at a small $T_2$, as in the case of $c_1\mu _{12}\leq c_2\mu _2$. However, in contrast to the case of $c_1\mu _{12}\leq c_2\mu _2$, the optimal $T_2$ is not necessarily 1 when $c_1\mu _{12}> c_2\mu _2$. Also, observe that the mean response time under the optimized T2 policy can be much higher than that under the optimized T1 policy. This difference becomes larger when $\mu_1$ is smaller (or $\mu_{12}$ is larger), since the type 1 jobs are better served by server 2 and the optimized T1 policy can give more bias toward the type 1 jobs than T2 policies.

Although figures are not shown, we find that the above findings also hold when $c_1\neq c_2$ (and $c_1\mu _{12}> c_2\mu _2$). That is, the value of the $c\mu$ product primarily determines the qualitative behavior of the T1 policy, and individual values of $c$ and $\mu$ have smaller effect.