Asymptotic analysis of a multi-flow heterogeneous QS under conditions of extremely rare state changes manager of input flows Markov chains

Currently, multimodal systems are gaining popularity with the development of multimodal interfaces. Multimodal streams are integrated streams of different types, including the transmission of voice, text data and video, so it is logical to use non-Poisson models to describe them. As a mathematical model of a multimodal servicing system, a multi-threaded heterogeneous queuing system with flows changing their intensity depending on the states of the Markov random environment is considered. Incoming requests from various flows are serviced during an exponentially distributed random time with parameters determined by the type of flow. Expressions are obtained for finding the maximum values of the main probabilistic characteristics of the number of occupied devices of each type. Asymptotic research is carried out under the condition of extremely rare changes in the states of the environment. The form of the multidimensional asymptotic characteristic function is obtained. It is proven that one-dimensional (marginal) stationary probability distributions of the number of occupied devices of each type are weighted sums of Poisson distributions. A numerical analysis of the range of applicability of the obtained approximation was carried out.