In Section 6.1 we introduce Sakasegawa's formula, which approximates the expected waiting time in a \(G/G/c\) queue. With some license, one could say that the philosophy behind lean manufacturing and the Toyota production system rests on principles derived from this formula. We then illustrate how to use Sakasegawa's formula to estimate waiting times in three queueing settings where the service process is interrupted. In the first case, Section 6.2, the server produces jobs from different families with a change-over time required to switch from one family to another. Such setups reduce the server's available time, thereby increasing the load. To counter this, the server produces in batches of fixed size. In the second case, Section 6.3, the server occasionally requires small adjustments, for example to prevent production quality from degrading below a certain level. Such adjustments typically do not occur during a job's service, but can take place between any two jobs. Consequently, the number of jobs served between adjustments (or setups) varies, unlike the situation in which batch sizes are fixed. In the third example, Section 6.4, quality issues or breakdowns may occur during a job's service. These increase service-time variability, leading to longer expected queueing times.

Along the way, we use some interesting results from probability theory and the Poisson process.