This paper investigates the characteristics of the MAP/M/1/N queuing system in the transient mode. The matrix method for solving the Kolmogorov equations is proposed. This method makes it possible, in general, to obtain the main characteristics of the considered queuing system in a non-stationary mode: the probability of losses, the time of the transient mode, the throughput, and the number of customers in the system at time t. The developed method is illustrated by numerical calculations of the characteristics of the $MAP/M/1/3$ system in the transient mode.

This paper considers a queuing system with a finite buffer, a constant main $MAP$ flow, and an additional periodic (non-periodic) Poisson piecewise-constant flow of customers. The eigenvalues of the probability translation matrix of a Kolmogorov system with constant input intensities is analyzed. The definition of the transition mode time based on the analysis of the probability translation matrix determinant is introduced for the first time. An analytical solution to the Kolmogorov equation system for a queuing system with piecewise constant arrival and service intensities is found, the solutions for a queuing system with periodic arrival and service intensities are analyzed, and numerical calculations illustrating this approach are presented.

The article deals with an inhomogeneous Markov process with finitely many discrete states, continuous time, and piecewise constant transition intensities. For the first time, analytical expressions are presented that describe both the transient and steady-state modes of the random process. To solve this problem, the fundamental matrix of the Kolmogorov system of differential equations is found in closed form in terms of elementary functions. In addition, an inhomogeneous process with periodically varying transition intensities is considered. For this case, the conditions for the existence of a steady-state mode are presented. Results of numerical calculations are provided for processes without jumps, with jumps, and with periodic jumps in the transition intensities.