Linear control of a Markov production system

This paper presents a model of a facility that processes many jobs. Each job requires a sequence of activities. The sequence of activities required by a job is random because each activity ends with a test, and the outcome of that test determines the activity that must be performed next. Evolution from activity to activity is Markovian, i.e., is determined by transition probabilities. A few (e.g., six) different types of job can share this facility, each type having its own transition probabilities between the activities. Each activity takes place in a designated sector. Several activities can share a sector. Each sector has a capacity, and buffers of inventory serve to decouple the sectors. In this paper, the authors introduce a family of linear control rules that smooth the flow through each sector and regulate the amount of inventory in each buffer. The operating characteristics of each linear control rule are computed. These include the mean and variance of the flow into each sector, the inventory in each buffer, the throughput of the systems, and the cycle time. These operating characteristics are optimized by a convex nonlinear program.