Analysis of a production system with early set-up and post-operation

We consider a production system with post-operation and early set-up. As soon as there are no more units (raw materials) to process and the machine becomes idle, it undergoes a post-operation. After the post-operation, if the number of waiting units is greater than or equal to m, it begins a set-up. After the set-up, if the number of units is greater than or equal to N (≥m), it begins to process the units. We first analyze the system by employing the decomposition property of the vacation queue. We, then, propose an algorithm that finds the optimal threshold values (m*, N*). Finally we present some numerical examples and interpret the system behavior.