Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical Sciences1991-86152310-7081Samara State Technical University2060310.14498/vsgtu1633Original ArticleStochastic models of simple controlled systems just-in-timeButovAlexander ADr. Phys. & Math. Sci., Professor; Head of Dept.; Dept. of Applied Mathematicsbutov.a.a@gmail.comKovalenkoAnatoly AM. Sc.; Postgraduste Student; Dept. of Applied Mathematicsanako09@mail.ruUlyanovsk State University1509201822351853114022020Copyright © 2018, Samara State Technical University2018We 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).modelingmartingaleintensityoptimizationreschedulingjust-in-timeмоделированиемартингалинтенсивностьоптимизацияперепланированиеточно-в-срокIntroduction. In this paper, we consider some stochastic models of simple just-in-time systems. The well-known principle of just-in-time system (abbreviated as JIT system) is used in many areas. Examples include just-in-time production systems (see [1, 2] and references therein), pedagogical strategies of just-in-time teaching (often abbreviated to JiTT ; see, e.g., [3,4]), and just-in-time compilation methods in computer programming (see [5, 6]). It should be noted that at present mathematical, especially stochastic, models for JIT systems are not sufficiently developed. Such models are necessary for solving optimal control problems, which could allow optimizing the allocation of system resources and implementing optimal planning of a stochastic JIT system. The purpose of this article is to present an approach to the stochastic description of JIT systems, which would be suitable for both analytical methods and computer simulation. Mathematical models of such systems should allow assuming that the trajectories of processes must take the given values at a fixed time. Such behavior of processes is known in stochastic bridges and stochastic processes in the reverse time. Thus, we should consider models of systems with the requirement of JIT in terms of processes with the behavior of trajectories close to stochastic bridges. Models should also allow investigating possible violations of this requirement that are unavoidable for real systems. The time reversal of stochastic processes has been studied for many years. For example, see [7-10] and references therein. We note that a number of works related to stochastic bridges (for example, the Brownian bridge, the Poisson bridge, also known as the Poisson bridge), is devoted to the investigation of these processes. In addition, some works on reversible Markov processes adjoin process descriptions in reverse time (see, e.g., [11]). In this article, we study models of simple JIT systems in semimartingale terms for point processes close to the Poisson bridge mentioned above. Here we allow some assumptions about the processes inherent in real systems. Thus, simple cases of multi-stage JIT systems and a system with bounded intensity are investigated. As shown, for these cases simple optimal control problems can be formulated and solved. The proofs of the results utilize the semimartingale technique. 1. Time reversal method for a simple JIT system. Consider a JIT system that can be described in terms of point (counting) processes. We assume that in the system some integer number[Sugimori Y., Kusunoki K., Cho F., Uchikawa S. Toyota production system and kanban system materialization of just-in-time and respect-for-human system, Int. J. Prod. Res., 1977, vol. 15, no. 6, pp. 553-564. doi: 10.1080/00207547708943149.][Yavuz M., Akçali E. Production smoothing in just-in-time manufacturing systems: a review of the models and solution approaches, Int. J. Prod. Res., 2007, vol. 45, no. 16, pp. 3579-3597. doi: 10.1080/00207540701223410.][Killi S., Morrison A. Just-in-Time Teaching, Just-in-Need Learning: Designing towards Optimized Pedagogical Outcomes, Universal Journal of Educational Research, 2015, vol. 3, no. 10, pp. 742-750. doi: 10.13189/ujer.2015.031013.][McGee M., Stokes L., Nadolsky P. Just-in-Time Teaching in Statistics Classrooms, Journal of Statistics Education, 2016, vol. 24, no. 1, pp. 16-26. doi: 10.1080/10691898.2016.1158023.][Aycock J. A brief history of just-in-time, ACM Computing Surveys, 2003, vol. 35, no. 2, pp. 97-113. doi: 10.1145/857076.857077.][Pape T., Bolz C. F., Hirschfeld R. Adaptive just-in-time value class optimization for lowering memory consumption and improving execution time performance, Science of Computer Programming, 2017, vol. 140, pp. 17-29. doi: 10.1016/j.scico.2016.08.003.][Elliott R. J., Tsoi A. H. Time reversal of non-Markov point processes, Ann. Inst. Henri Poincaré, 1990, vol. 26, no. 2, pp. 357-373, https://eudml.org/doc/77383.][Jacod J., Protter P. Time Reversal on Levy Processes, Ann. Probab., 1988, vol. 16, no. 2, pp. 620-641. doi: 10.1214/aop/1176991776.][Főllmer H. Random fields and diffusion processes, In: École d’Été de Probabilités de Saint-Flour XV-XVII, 1985-87, Lecture Notes in Mathematics, 1362; eds. PL. Hennequin. Berlin, Heidelberg, Springer, 1988, pp. 101-203. doi: 10.1007/BFb0086180.][Privault N., Zambrini J.-C. Markovian bridges and reversible diffusion processes with jumps, Annales de l’I.H.P. Probabilités et statistiques, 2004, vol. 40, no. 5, pp. 599-633. doi: 10.1016/j.anihpb.2003.08.001.][Longla M. Remarks on limit theorems for reversible Markov processes and their applications, J. Stat. Plan. Inf., 2017, vol. 187, pp. 28-43. doi: 10.1016/j.jspi.2017.02.009.][Conforti G., Léonard C., Murr R., Roelly S. Bridges of Markov counting processes. Reciprocal classes and duality formulas, Electron. Commun. Probab., 2015, vol. 20, no. 18, pp. 1-12. doi: 10.1214/ECP.v20-3697.][Dellacherie C. Capacités et processus stochastiques. Berlin, Springer-Verlag, 1972, ix+155 pp.][Butov A. A. Some estimates for a one-dimensional birth and death process in a random environment, Theory Probab. Appl., 1991, vol. 36, no. 3, pp. 578-583. doi: 10.1137/1136067.][Butov A. A. Martingale methods for random walks in a one-dimensional random environment, Theory Probab. Appl., 1994, vol. 39, no. 4, pp. 558-572. doi: 10.1137/1139043.][Butov A. A. Random walks in random environments of a general type, Stochastics and Stochastics Reports, 1994, vol. 48, pp. 145-160. doi: 10.1080/17442509408833904.][Butov A. A. On the problem of optimal instant observations of the linear birth and death processes, Statistics and Probability Letters, 2015, vol. 101, pp. 49-53. doi: 10.1016/j.spl.2015.02.021.]