Abstract: The paper presents a description of a mathematical model of a stochastic dynamic productive system with discrete states. The model is given in terms of point counting processes, their compensators and intensities of jumps. The article provides a generalizing definition of “just-in-time” processes. The criterion formulated in the article represents the necessary and sufficient “just-in-time” conditions for the processes of reproduction and death with jump intensities, which are random functions, determined by initial conditions. The mathematical description of the “just-in-time” processes is given for a fairly general case that goes beyond the model in the well-known case of a Poisson bridge.The paper discusses two types of models of stochastic systems - with finite and infinite carriers of the probability density of the execution time of the production process.A criterion is formulated and proved for the process of performing “just-in-time” operations in a random environment. Random environment is random functions - the intensity coefficients of the process jumps. The descriptions are given in trajectory martingale terms, allowing to analyze the behavior of the process passes, and solve simulation problems.
Index terms: mathematical modelling, productive system, just-in-time, martingale, birth and death processes, random environment, intensity, compensator.

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