## 3 Scheduling Analysis
### Temporal Properties Under THTS
Hence, the time taken for robot-waiting activity \(A_{12}\), \(n>2\), is determined by the longest one between the time taken for a rotation operation of the PM and \(\langle M_{1,2}\to L_{2}\)\(\rightarrow\)\(M_{2,n}\)\(\rightarrow\)\(A_{n2}\)\(\rightarrow\)\(U_{n}\)\(\rightarrow\)\(M_{n,n+1}\)\(\rightarrow\)\(L_{n+1}\)\(\rightarrow\)\(M_{n+1,n-1}\)\(\rightarrow\)\(\ldots\)\(\rightarrow\)\(M_{5,3}\)\(\rightarrow\)\(A_{32}\)\(\rightarrow\)\(U_{3}\)\(\rightarrow\)\(M_{3,4}\)\(\rightarrow\)\(L_{4}\)\(\rightarrow\)\(M_{4,2}\)\(\rightarrow\)\(A_{22}\)\(\rightarrow\)\(U_{2}\)\(\rightarrow\)\(M_{2,3}\)\(\rightarrow\)\(L_{3}\)\(\rightarrow\)\(M_{3,0}\)\(\rightarrow\)\(U_{0}\)\(\rightarrow\)\(M_{0,1}\)). When \(n=2\), the time taken  
\[\theta_{i}=[\alpha_{i}+d_{i}+(2n+5)\beta+\beta_{0}+(2n+5)\mu+\upsilon+\sum_{1}^{i-1 }\omega_{i1}+\sum_{i-1}^{n}\omega_{i2}]/m_{i},i\in\mathbf{N}_{n}\backslash \mathbf{N}_{2}\text{ and }n>2 \tag{34}\]0 Description of robot movements under the THTS strategy. for robot waiting activity \(A_{12}\) is determined by the longest one between the time taken for a rotation operation of the PM and \(\langle M_{1,2}\to L_{2}\to A_{22}\to U_{2}\to M_{2,0}\to U_{0}\to\) the robot rotates \(\to L_{0}\to M_{0,1}\rangle\). Thus, \(\omega_{12}\) should be calculated with Equations (43) and (44):  
\[\theta_{i}=[\alpha_{i}+d_{i}+(2n+5)\beta+\beta_{0}+(2n+5)\mu+\upsilon+\sum_{1}^{i-1 }\omega_{i1}+\sum_{i-1}^{n}\omega_{i2}]/m_{i},i\in\mathbf{N}_{n}\backslash \mathbf{N}_{2}\text{ and }n>2 \tag{34}\]1  
\[\theta_{i}=[\alpha_{i}+d_{i}+(2n+5)\beta+\beta_{0}+(2n+5)\mu+\upsilon+\sum_{1}^{i-1 }\omega_{i1}+\sum_{i-1}^{n}\omega_{i2}]/m_{i},i\in\mathbf{N}_{n}\backslash \mathbf{N}_{2}\text{ and }n>2 \tag{34}\]2  
Similar to TBS, 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 with (29) and (30), respectively. For Step 2, \(n>2\), the wafer sojourn time of \(W_{2,12}\) and \(W_{2,34}\) can be obtained using (31) and (32), respectively. For Step 2, \(n=2\), the time taken for task sequence \(\langle\)the internal chamber of the PM rotates \(\to U_{2}\to M_{2,0}\to U_{0}\to\) the robot rotates \(\to L_{0}\to M_{0,1}\to A_{12}\to U_{1}\to\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\to\) 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 \(W_{2,12}\). Furthermore, 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) \(\to\)\(U_{2}\to M_{2,0}\to U_{0}\to\) the robot rotates \(\to L_{0}\to M_{0,1}\to A_{11}\to U_{1}\to\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\to\) the internal chamber of the PM rotates \(\rangle\) equals the wafer sojourn time of \(W_{2,34}\). Therefore, the wafer sojourn time of \(W_{2,12}\) and \(W_{2,34}\) can be calculated using Equations (45) and (46):  
\[\theta_{i}=[\alpha_{i}+d_{i}+(2n+5)\beta+\beta_{0}+(2n+5)\mu+\upsilon+\sum_{1}^{i-1 }\omega_{i1}+\sum_{i-1}^{n}\omega_{i2}]/m_{i},i\in\mathbf{N}_{n}\backslash \mathbf{N}_{2}\text{ and }n>2 \tag{34}\]3  
\[\theta_{i}=[\alpha_{i}+d_{i}+(2n+5)\beta+\beta_{0}+(2n+5)\mu+\upsilon+\sum_{1}^{i-1 }\omega_{i1}+\sum_{i-1}^{n}\omega_{i2}]/m_{i},i\in\mathbf{N}_{n}\backslash \mathbf{N}_{2}\text{ and }n>2 \tag{34}\]4  
For Step 1, \(n>2\), the time taken for the task sequence \(\langle M_{1,2}\to L_{2}\to M_{2,n}\to A_{n2}\to U_{n}\to M_{n,n+1}\to L_{n+1} \to M_{n+1,n-1}\to\ldots\to M_{3,0}\to U_{0}\to M_{0,1}\to A_{12}\to U_{1}\to\) 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 \(W_{1,12}\).