## 3 Scheduling Analysis
### Temporal Properties Under OHTS
Therefore, \(\omega_{12}\) can be calculated using Equation (17):  
\[\omega_{12}=max(v-(\beta+\beta_{0}+3\mu),0) \tag{17}\]  
The robot-waiting activity \(A_{i3}\), \(i\in\mathbf{N}_{n}\backslash\{0\}\) is caused by the rotation of the internal chamber of a PM for Step \(i\) as well. Similar to OBS, the time taken for \(A_{i3}\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{2}\) is determined by the longest one between the time taken for a rotation operation of a PM for Step \(i\) and \(\langle M_{i,j-1}\to A_{(i-1)2}\to U_{i-1}\to M_{i-1,i}\rangle\). Thus, \(\omega_{i3}\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{2}\), can be obtained using Equation (7). Observing \(\sigma_{12}\), after the robot performs a loading task at Step 2 for the first time, the internal chamber of the PM starts to rotate. At the same time, the robot performs task sequence \(\langle M_{2,0}\to U_{0}\to M_{0,1}\to A_{12}\to U_{1}\to\) the robot rotates \(\to L_{1}\to M_{1,2}\rangle\). Thus, the time taken for \(A_{23}\) is determined by the longest one between the time taken for a rotation operation of the PM and \(\langle M_{2,0}\to U_{0}\to M_{0,1}\to A_{12}\to U_{1}\to\) the robot rotates \(\to L_{1}\to M_{1,2}\rangle\). Therefore, Equation (18) is given to calculate \(\omega_{23}\):  
\[\omega_{23}=max(v-(2\beta+\beta_{0}+4\mu+\omega_{12}),0) \tag{18}\]  
Notice that \(A_{13}\) represents the robot-waiting activity at a PM for Step 1 before the robot loads wafers represented by \(W_{1,34}\) into the PM. Observing \(\sigma_{13}\), since the robot performs a swap operation at the PM such that these wafers are loaded into the PM, it implies that \(A_{13}\) is not necessary. Moreover, similar to OBS, for Step \(i\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{2}\), the wafer sojourn time of \(W_{i,12}\) and \(W_{i,34}\) can be obtained by (9) and (10), respectively. For Step 2, observing \(\sigma_{12}\), the time taken for task sequence \(\langle M_{2,0}\to U_{0}\to M_{0,1}\to A_{12}\to U_{1}\to\) the robot rotates \(\to L_{1}\to M_{1,2}\to A_{23}\to L_{2}\to\) the internal chamber of the PM rotates \(\to\) wafer processing \(\to\) four wafers stay in the PM after processed (i.e., the delay time)\(\rangle\) equals the wafer sojourn time of \(W_{2,12}\). Also, the time taken for task sequence \(\langle\)the internal chamber of the PM rotates \(\to\) wafer processing \(\to\) four wafers stay in the PM after processed (i.e., the delay time) \(\to U_{2}\to M_{2,3}\to L_{3}\to M_{3,2}\to A_{22}\rangle\) equals the wafer sojourn time of \(W_{2,34}\). Then, Equation (19) is given to obtain \(\tau_{2,12}\), while \(\tau_{2,34}\) can be obtained by (10). \[\tau_{2,12}=\alpha_{2}+d_{2}+v+\beta_{0}+3\beta+4\mu+\omega_{12}+\omega_{23} \tag{19}\]  
For Step 1, observing \(\sigma_{13}\), the time taken for task sequence \(\langle M_{1,2}\to L_{2}\to M_{2,0}\to U_{0}\to M_{0,1}\to A_{12}\to U_{1}\to\) the robot rotates \(\to L_{1}\to\) the internal chamber of the PM rotates \(\to\) wafer processing \(\to\) four wafers stay in the PM after processed (i.e., the delay time)\(\rangle\) equals the wafer sojourn time of \(W_{1,12}\), and the time taken for task sequence \(\langle\)the internal chamber of the PM rotates \(\to\) wafer processing \(\to\) four wafers stay in the PM after processed (i.e., the delay time)\(\rangle\) equals the wafer sojourn time of \(W_{1,12}\), and the time taken for task sequence \(\langle\)the internal chamber of the PM rotates \(\to\) wafer processing \(\to\) four wafers stay in the PM after processed (i.e., the delay time)\(\rangle\) equals the wafer sojourn time of \(W_{1,12}\) and the time taken for task sequence \(\langle\)the internal chamber of the PM rotates \(\to\) wafer processing \(\to\) four wafers stay in the PM after processed (i. \(L_{2}\to M_{2,n}\)), respectively.