## 3 Scheduling Analysis
### Temporal Properties Under THTS  
Similar to TBS, to complete four wafers by a PM for Step \(i\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{2}\) and \(n>2\), by THTS, task sequence \(\sigma_{16}=\langle U_{i}\to M_{i,j+1}\to L_{i+1}\to M_{i+1,j-1} \to A_{(i-1)1}\to U_{i-1}\to M_{i-1,i}\to L_{i}\to M_{i,j-2} \rightarrow\ldots\to M_{3,0}\to U_{0}\to M_{0,1} \to A_{11}\to U_{1}\to U_{1}\to\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\to M_{2,n}\to A_{n2}\to U_{n} \to M_{n,n+1}\to L_{n+1}\to M_{n+1,n-1}\to\ldots\to M_{i+1,j-1} \to A_{(i-1)2}\to U_{i-1}\to M_{i-1,j}\to L_{i}\to L_{i}\to\) the internal chamber of the PM rotates \(\rightarrow\) wafer processing \(\rightarrow\) four wafers stay in the PM after processed (i.e., the delay time)\(\rangle\) should be performed. Figure 6 presents the robot task sequence under the THTS strategy with part of the backward process omitted, in the figure, the meaning of the colored arrows is same as that in Figure 3. The steps with a yellow circular background similarly indicate the internal chamber rotation of the PM before the processing step. Thus, with \(m_{i}\) parallel PMs serving for Step \(i\), according to \(\sigma_{16}\), the time for processing four wafers at Step \(i\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{2}\) and \(n>2\), can be calculated with Equation (34):\[\theta_{i}=[\alpha_{i}+d_{i}+(2n+5)\beta+\beta_{0}+(2n+5)\mu+\upsilon+\sum_{1}^{i-1 }\omega_{i1}+\sum_{i-1}^{n}\omega_{i2}]/m_{i},i\in\mathbf{N}_{n}\backslash \mathbf{N}_{2}\text{ and }n>2 \tag{34}\]  
For Step 2, \(n>2\), to complete four wafers by a PM, task sequence \(\sigma_{17}=\langle U_{2}\to M_{2,3}\rangle\)\(\to L_{3}\to M_{3,0}\to U_{0}\to M_{0,1}\to A_{11}\to U_{1}\)\(\rightarrow\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\to M_{2,n}\)\(\to A_{n2}\to U_{n}\to M_{n,n+1}\to L_{n+1}\to M_{n+1,n-1}\rightarrow\ldots\to M_{3,0}\to U_{0}\to M_{0,1}\to A_{12}\to U_{1}\)\(\rightarrow\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\)\(\rightarrow\) 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. When \(n=2\), for Step 2, to complete four wafers by a PM, the task sequence \(\sigma_{18}\)\(=\langle U_{2}\to M_{2,0}\to U_{0}\)\(\rightarrow\) the robot rotates \(\to L_{0}\to M_{0,1}\to A_{11}\to U_{1}\)\(\rightarrow\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\)\(\rightarrow\) the internal chamber of the PM rotates \(\to U_{2}\to M_{2,0}\to U_{0}\)\(\rightarrow\) the robot rotates \(\to L_{0}\to M_{0,1}\to A_{12}\to U_{1}\)\(\rightarrow\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\)\(\rightarrow\) 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 \(m_{2}\) parallel PMs serving for Step 2, according to \(\sigma_{17}\) and \(\sigma_{18}\), the time for processing four wafers at Step 2 can be calculated using Equations (35) and (36):  
\[\theta_{2}=[\alpha_{2}+d_{2}+(2n+6)\beta+2\beta_{0}+(2n+7)\mu+\upsilon+\omega_ {11}+\sum_{1}^{n}\omega_{i2}]/m_{2},n>2 \tag{35}\]  
\[\theta_{2}=(\alpha_{2}+d_{2}+10\beta+2\beta_{0}+10\mu+2\upsilon+\omega_{11}+ \omega_{12})/m_{2},n=2 \tag{36}\]  
For Step 1, \(n>2\), to complete four wafers by a PM, task sequence \(\sigma_{19}=\langle U_{1}\rightarrow\) the robot rotates \(\to L_{1}\to M_{1,2}\to L_{2}\to M_{2,n}\to A_{n2}\to U_{n} \to M_{n,n+1}\to L_{n+1}\to M_{n+1,n-1}\rightarrow\ldots \to M_{3,0}\to U_{0}\to M_{0,1}\to A_{12}\to U_{1}\) \(\rightarrow\) 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.