# Simulation-Based Analysis on Operational Control of Batch Processors in Wafer Fabrication
## 3 Operational Control Rules for Simulation Analysis
processing a batch of the wafer lots. In MBS6, an idle BP starts loading only when the number of wafer lots at the queue is equal to the batch capacity (full batch loading). (2) Look-upstream (LKUP) rule: This BP control scheme considers near-future job arrivals from the upstream stations. When a batch processor becomes idle and finds wafer lots of a product type more than the batch capacity, the wafer lots are loaded immediately. When the batch size is less than the batch capacity, the scheduler looks at the upstream stations. If a new wafer lot is expected to arrive at the BP station shortly, waiting for the new wafer lot to arrive to form a larger batch size may lead to less total waiting time at the BP station. The LKUP loading rule is similar to a look-ahead batch loading rule in References [5,13]. The arrival time of the new lot is anticipated, and the trade-off between the expected waiting time savings in the new lot and the waiting time increase in the existing wafer lots in the queue is considered. If the reduction in the waiting time is less than the increase in the waiting time, the batch is immediately loaded. Suppose we are at the decision point, \(t_{0}\), and job type \(i^{*}\) is selected for the next loading with batch size \(q\). There may be jobs from the other job families waiting in the queue. When loading is delayed until the arrival time of the next job of type \(i^{*}\), \(t_{1}\), existing jobs waiting in the queue will experience additional delays while the new upcoming job is loaded without any waiting time. For job type \(i^{*}\), the additional waiting time is \(q(t_{1}-t_{0})\). For the other jobs in the queue, additional waiting takes place only when the jobs are loaded on the machine triggering the control decision. If there are \(m\) parallel batch processors, it is expected that the other jobs in the queue will experience delayed loading with the probability of \(1/m\). Then the loading time delay (increased waiting time) can be calculated by \(\frac{\mathcal{G}^{r}(t_{1}-t_{0})}{m}\) where \(Q^{c}\) is the number of jobs (except job type \(i^{*}\)) in the BP queue. Then, the total increased waiting time, due to delayed loading is \(q(t_{1}-t_{0})+\frac{\mathcal{G}^{r}(t_{1}-t_{0})}{m}\). Now, we need to calculate the time saved by the delayed loading. The saved time occurs only for the incoming job. If the loading is done right away, the loading time of the next incoming job depends on the number of job types, \(n\), and the number of batch processors. When a batch processor is available in the future, the jobs of different job types will compete with each other for the idle BP. Suppose each job type is loaded next with the same probability. Then, on average, the new job is loaded in \(\frac{n+1}{2}\) loading cycles later. The interval of the machine availability (loading cycle) is \(\frac{T}{m}\), where \(T\) is the BP processing time. Then, the expected waiting time is \(\frac{(n+1)T}{2m}\). Now the saved wait time can be calculated by (total saved time - total increased time) as \(\frac{(n+1)T}{2m}-\left[q(t_{1}-t_{0})+\frac{\mathcal{G}^{r}(t_{1}-t_{0})}{m}\right]\). If the saved wait times are negative, the machine starts processing the partial batch immediately. The LKDN control rule is summarized in Figure 4.  
Figure 4: Decision procedure of look-upstream rule (LKUP). MBS, minimum batch size. (3) Look-downstream (LKDN) rule: This rule considers the status of downstream machines in BP control decisions. Unlike existing control rules looking at the immediate downstream [19; 22], this rule looks at a downstream bottleneck (BOT) station possibly a few steps ahead. The bottleneck station is a workstation whose facility utilization is the highest of all workstations. The detailed definition of facility utilization will be given in the subsequent section. Since the bottleneck station determines the capacity of the whole system, any idle time at the bottleneck machine undermines the system capacity. Therefore, it is important for the bottleneck machine to run without a stoppage.