Analysis of make-to-stock queues with general processing times and start-up and lost sales costs

We consider a make-to-stock environment with a single production unit that corresponds to a single machine or a line. Production and hence inventory are controlled by the two-critical-number policy. Production times are independent and identically distributed general random variables and demands are generated according to a stationary Poisson process. We model this production-inventory system as an M/G/1 make-to-stock queue. The main contribution of the study is to extend the control of make-to-stock literature by considering general production times, lost sales and fixed production costs at the same time. We characterize the long-run behaviour of the system and also propose a simple but very effective approximation to calculate the control parameters of the two-critical-number policy. An extensive numerical study exhibits the effects of the production time distribution and the system parameters on the policy control levels and average system cost.