Stochastic models of simple controlled systems just-in-time

We propose a new and simple approach for the mathematical description of a stochastic system that implements the well-known just-in-time principle. This principle (abbreviated JIT) is also known as a just-in-time manufacturing or Toyota Production System.
The models of simple JIT systems are studied in this article in terms of point processes in the reverse time. This approach allows some assumptions about the processes inherent in real systems. Thus, we formulate and solve some, very simple, optimal control problems for a multi-stage just-in-time system and for a system with the bounded intensity. Results are obtained for the objective functions calculated as expected linear or quadratic forms of the deviations of the trajectories from the planned values. The proofs of the statements utilize the martingale technique. Often, just-in-time systems are considered in logistics tasks, and only (or predominantly) deterministic methods are used to describe them. However, it is obvious that stochastic events in such systems and corresponding processes are observed quite often. And it is in such stochastic cases that it is very important to find methods for the optimal management of processes just-in-time. For this description, we propose using martingale methods in this paper. Here, simple approaches for optimal control of stochastic JIT processes are demonstrated. As examples, we consider an extremely simple model of rescheduling and a method of controlling the intensity of the production process, when the probability of implementing a plan is not necessarily equal to one (with the corresponding quadratic loss functional).