Analytical Modeling of the Dynamics of Random Processes During Combat Use of a Military Tetrasystem

A military system consisting of four autonomous subsystems (a tetrasystem) is considered: air, land, sea, and drone. During combat, each subsystem is subject to a stream of random events involving losses and restorations. The dynamics of random processes is studied using a continuous-time Markov chain with sixteen asymmetric possible states. The corresponding mathematical model of the random processes is constructed in the form of sixteenth-order Kolmogorov differential equations. Formulas are found for the sixteen roots of the characteristic Kolmogorov equation, expressed in terms of the intensities of the tetrasystem's loss and restoration flows. The analytical solution to the Kolmogorov differential equations for the tetrasystem is represented in the form of ordered matrices and sixteenth-order determinants, which allows for a compact description of a large volume of initial data, overcomes limitations associated with the problem's dimensionality, and ensures adaptability to computer technologies, including the problem of verification.