## 3 Scheduling Analysis
### Temporal Properties Under TBS  
Under the steady state, to complete four wafers at a PM for Step \(i\), \(i\in\mathbf{N}_{n}\backslash\mathbf{N}_{1}\), by TBS, with the robot-waiting activities being considered, task sequence \(\sigma_{14}=\langle U_{i}\to M_{i,i+1}\to L_{i+1}\to M_{i+1}\to M_{i+1,i-1}\to A_{ (i-1)1}\to U_{i-1}\to M_{i-1,i}\to L_{i}\to M_{i,i-2}\to\ldots\to M_{3,1}\to A_{ 11}\to U_{1}\to M_{1,2}\)\(\to L_{2}\to M_{2,0}\to U_{0}\to M_{0,1}\to L_{1}\to M_{1,n}\to A_{n2}\to U_{ 1}\to M_{n,n+1}\to L_{n+1}\to M_{n+1,n-1}\to\ldots\to M_{i+1,i-1}\to A_{(i-1)2 }\to U_{i-1}\to M_{i-1,i}\to L_{i}\to\) the internal chamber of the PM rotates \(\to\) wafer processing \(\to\) four wafers stay in the PM after processed (i.e., the delay time)\(\rangle\) should be performed. Figure 5 shows the robot task sequence under the TBS 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. With \(m_{i}\) parallel PMs serving for Step \(i\), according to \(\sigma_{14}\), the time for processing four wafers can be calculated with Equation (22):  
\[\theta_{i}=\Big{[}\alpha_{i}+d_{i}+(2n+5)\beta+\beta_{0}+(2n+5)\mu+v+\sum_{1} ^{i-1}\omega_{i1}+\sum_{i-1}^{n}\omega_{i2}\Big{]}/m_{i},i\in\mathbf{N}_{n} \backslash\mathbf{N}_{1} \tag{22}\]  
Similarly, to complete four wafers by a PM for Step \(1\), by TBS, with the robot-waiting activities being considered, task sequence \(\sigma_{15}=\langle U_{1}\to M_{1,2}\to L_{2}\to M_{2,0}\to U_{0}\to M_{0,1} \to L_{1}\to M_{1,n}\to A_{n2}\to U_{n}\to M_{n,n+1}\to L_{n+1}\to M_{n+1,n-1} \to\ldots\to M_{2,0}\to U_{0}\to M_{0,1}\to L_{1}\to\) the internal chamber of the PM rotates \(\to\) wafer processing \(\to\) four wafers stay in the PM after processed (i.e., the delay time)\(\rangle\) should be performed. With \(m_{1}\) parallel PMs serving for Step \(1\), according to \(\sigma_{15}\), the time for processing four wafers can be calculated using Equation (23):  
\[\theta_{1}=\Big{[}\alpha_{1}+d_{1}+(2n+4)\beta+2\beta_{0}+(2n+5)\mu+v+\sum_{1 }^{n}\omega_{i2}\Big{]}/m_{1} \tag{23}\]  
In \(\sigma_{14}\), \(A_{i2}\) is also caused by the rotation of the internal chamber of a PM for Step \(i\). After the robot performs a loading task at the PM in \(\sigma_{14}\) for the first time, the internal chamber of the PM starts to rotate. At the same time, the robot performs task sequence \(\langle M_{i,i-2}\to A_{(i-2)1}\to U_{i-2}\to M_{i-2,i-1}\to L_{i-1}\to M_{i-1,i- 3}\to\ldots\to M_{2,0}\to U_{0}\to M_{0,1}\to L_{1}\to M_{1,n}\to A_{n2}\to U_{ n}\to M_{n,n+1}\to M_{n+1,n-1}\to\ldots\to M_{i+2,i}\rangle\). It follows from this sequence that \(A_{12}\) and \(A_{22}\) are different, implying that \(A_{i2}\) is dependent on \(n\). Thus, we divide it into two cases: \(n=2\) and \(n>2\). If \(n\geq 2\), \(A_{i2}\), \(i\in\{3,4,\ldots,n-1\}\) should be performed. And the time taken for \(A_{i2}\) is determined by the longest one between the time taken for a rotation operation of the PM and \(\langle M_{i,i-2}\to A_{(i-2)1}\to U_{i-2}\to M_{i-2,i-1}\to L_{i-1}\to M_{i-1,i- 3}\to\ldots\to M_{2,0}\to U_{0}\to M_{0,1}\to L_{1}\to M_{1,n}\to M_{n,n+1}\to L_{n+1} \to M_{n+1,n-1}\to\ldots\to M_{i+2,i}\rangle\). When \(i=2\) and \(n>2\), the time taken for \(A_{22}\) is determined by the longest one between the time taken for a rotation operation of the PM and \(\langle M_{2,0}\to U_{0}\to M_{0,1}\to L_{1}\to M_{1,n}\to A_{n2}\to U_{n}\to M_{n,n+1} \to L_{n+1}\to M_{n+1,n-1}\to\ldots\to M_{i+2,i}\to A_{i2}\to U_{i}\to M_{i,i+1} \to L_{i+1}\to M_{i+1,j-1}\to\ldots\to M_{5,3}\to A_{32}\to U_{3}\to M_{3,4} \to L_{4}\to M_{4,2}\rangle\). When \(n\) = \(2\), \(A_{22}\) should be performed. And the time taken for \(A_{22}\) is determined by the longest one between the time taken for a rotation operation of the PM and \(\langle M_{2,0}\to U_{0}\to M_{0,1}\to L_{1}\to M_{1,2}\rangle\).