A lot sizing model with queueing delays: The issue of safety time

The objective of this paper is to introduce the concept of safety time in a make-to-order production environment. The production facility is represented as a queueing model, explicitly including a non-zero setup time. A methodology is presented to quantify the safety time and to compute the associated service level based on the queueing delay. The main result is a convex relationship of the expected waiting time, the variance of the waiting time and the quoted lead time as a function of the lot size and a concave relationship of the service level as a function of the lot size. Most models in the literature assume batch arrivals. We relax that assumption so that an individual customer arrival process is allowed. We therefore have to derive a new closed form analytical expression for the expected waiting time. Both the deterministic and a stochastic case are studied.