Investigation of the stationary mode existence in a system of conflict service of non-homogeneous demands

This paper studies a nonclassical system which controls several independent conflicting flows and provides service for requests of these flows. It is supposed that there is one high-priority input flow and one high-intensity flow. The input flows can be approximated with a nonordinary Poisson flow. The system includes a service device that provides for each flow a service period and a readjusting period for safe switching between conflicting flows. It is also possible to prolong service for the high-intensity flow until a number of waiting requests in a high-priority flow queue reaches a certain threshold.
The most meaningful characteristics of the system are stated. A mathematical probabilistic model for the system is constructed in the form of a multidimensional homogeneous controllable Markovian chain. The paper determines necessary conditions for the existence of a stationary mode in the system. A sufficient condition for existence of a stationary mode for the high-priority flow is proved as well. All the found conditions can be easily checked in real systems since they deal only with system parameters such as intensities of the input flows, intensities of service, and time periods of the service device states.