Single-threshold allocation policies: T1 and T2

The T1 and T2 policies are formally defined as follows:

Definition 16   Let $N_1$ (respectively, $N_2$) denote the number of jobs at queue 1 (respectively, queue 2). The T1 policy with parameter $T_1$ is characterized by the following set of rules, all of which are enforced preemptively (preemptive-resume):

• Server 1 serves only its own jobs.
• Server 2 serves jobs from queue 1 if either (i) $N_1 \geq T_1$ or (ii) $N_2 = 0$ & $N_1 \geq 2$. Otherwise, server 2 serves jobs from queue 2.

To achieve maximal efficiency, we assume the following exceptions. When $N_1=1$ and $N_2 = 0$, the job is processed by server 2 if and only if $\mu_1<\mu_{12}$. Also, when $T_1=1$ and $N_1=1$, the job in queue 1 is processed by server 2 if and only if $\mu_1<\mu_{12}$ regardless of the number of type 2 jobs.

Definition 17   The T2 policy with parameter $T_2$ is characterized by the following set of rules, all of which are enforced preemptively (preemptive-resume):

• Server 1 serves only its own jobs.
• Server 2 serves jobs from queue 1 if $N_2 < T_2$. Otherwise server 2 serves jobs from queue 2.

When $N_1=1$ and $N_2 = 0$, we allow the same exception as in the T1 policy.

Below, the T1 policy with parameter $t_1$ is also denoted by the T1($t_1$) policy, and the T2 policy with parameter $t_2$ is also denoted by the T2($t_2$) policy.

Figure 7.2: Figure shows whether server 2 works on jobs from queue 1 or queue 2 as a function of $N_1$ and $N_2$ under (a) the T1 policy with parameter $T_1$ and (b) the T2 policy with parameter $T_2$.

(a) T1 policy

(b) T2 policy

Figure 7.2 shows the jobs processed by server 2 as a function of $N_1$ and $N_2$ (a) under the T1 policy and (b) under the T2 policy. Note that the T1(1) policy is the c$\mu$ rule when $c_1\mu _{12}> c_2\mu _2$, and the T1($\infty$) policy is the c$\mu$ rule when $c_1\mu_{12}<c_2\mu_2$; thus the $c\mu$ rule falls within the broader class of T1 policies.