## 3 Scheduling Analysis
### Temporal Properties Under OHTS  
By OHTS, with the robot-waiting activities being considered, similar to OBS, task sequence \(\sigma_{9}\) or \(\sigma_{10}\) is executed so as to complete four wafers by a PM for Step \(i\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{2}\). Thus, with \(m_{i}\) parallel PMs serving for Step \(i\), according to \(\sigma_{9}\) and \(\sigma_{10}\), the time for processing four wafers at Step \(i\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{2}\), can be calculated using Equations (3) and (4). We can see the task sequence of the robot among PM\({}_{1}\), PM\({}_{2}\), and LL as presented in Figure 4, where the meaning of colored arrows is same as that in Figure 3. In the steps with a yellow circular background, the internal chamber of the PM rotates just before starting to process wafers. For Step 2, \(n>2\), to complete four wafers at a PM, task sequence \(\sigma_{12}=\langle U_{2}\to M_{2,3}\to L_{3}\to M_{3,2}\to A_{22}\to U_{2} \to M_{2,3}\to A_{33}\to L_{3}\to M_{3,0}\to U_{0} \to M_{0,1}\to A_{11}\to U_{1}\to\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\to M_{2,0}\to U_{0} \to M_{0,1}\to A_{12}\to U_{1}\to\) the robot rotates \(\rightarrow\)\(L_{1}\to M_{1,2}\to A_{23}\to L_{2}\to\) the internal chamber of the PM rotates \(\rightarrow\) wafer processing \(\rightarrow\) four wafers stay in the PM after processed (i.e., the delay time)) should be performed. With \(m_{2}\) parallel PMs serving for Step 2, according to \(\sigma_{12}\), the time for processing four wafers at Step 2 can be calculated using Equation (14):  
\[\theta_{2}=(\alpha_{2}+d_{2}+2\beta_{0}+10\beta+11\mu+v+\omega_{33}+\omega_{22}+ \omega_{23}+\omega_{11}+\omega_{12})/m_{2},n>2 \tag{14}\]For Step 2, \(n=2\), to complete four wafers at a PM, the robot activity time is equivalent to that of Step 2 with \(n>2\) in the task sequence. However, the waiting time is not the same due to the absence of Step 3, which is \(A_{33}\). With \(m_{2}\) parallel PMs serving for Step 2, \(n=2\), the time for processing four wafers at Step 2 can be calculated using Equation (15). \[\theta_{2}=(\alpha_{2}+d_{2}+2\beta_{0}+10\beta+11\mu+v+\omega_{22}+\omega_{23 }+\omega_{11}+\omega_{12})/m_{2},\,n=2 \tag{15}\]  
For Step 1, to complete four wafers by a PM, task sequence \(\sigma_{13}=\langle U_{1}\rightarrow\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\to M_{2,0}\to U_{0}\to M_{0,1} \to A_{12}\to U_{1}\rightarrow\) the robot rotates \(\to L_{1}\rightarrow\) the internal chamber of the PM rotates \(\rightarrow\) wafer processing \(\rightarrow\) four wafers stay in the PM after processed (i.e., the delay time)\(\rangle\) should be performed. With \(m_{1}\) parallel PMs serving for Step 1, according to \(\sigma_{13}\), the time for processing four wafers at Step 1 can be calculated using Equation (16):  
\[\theta_{1}=(\alpha_{1}+d_{1}+\beta_{0}+5\beta+5\mu+v+\omega_{12})/m_{1} \tag{16}\]  
The robot-waiting activity \(A_{i2}\), \(i\in\mathbf{N}_{n}\backslash\{0\}\) is caused by the rotation of the internal chamber of a PM for Step \(i\). Similar to OBS, the time taken for \(A_{i2}\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\), is determined by the longest one between the time taken for a rotation operation of the PM and \(\langle M_{i,i+1}\to L_{i+1}\to M_{i+1,i}\rangle\). Thus, \(\omega_{2}\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\) can be obtained using Equation (6). Figure 4: Description of robot movements under the OHTS strategy. Observing \(\sigma_{13}\), after the robot performs a loading task at Step 1 for the first time so as to complete a robot rotation operation, the internal chamber of the PM starts to rotate. At the same time, the robot performs the task sequence \(\langle M_{1,2}\to L_{2}\to M_{2,0}\to U_{0}\to M_{0,1}\rangle\). Thus, the time taken for \(A_{12}\) is determined by the longest one between the time taken for a rotation operation of the PM and \(\langle M_{1,2}\to L_{2}\to M_{2,0}\to U_{0}\to M_{0,1}\rangle\).