Study of two-dimensional marked MMPP under the high rate limit condition

This paper considers a mathematical model of a heterogeneous flow in the form of a two-dimensional marked MMPP. The study of such models is necessary to analyze the load on multimodal systems. Multimodal interfaces are capable of processing multiple natural human input methods, each of which requires specific resources for recognition, processing and transmission. To design such systems, it is necessary to estimate the required resources. These estimates can be based on the joint probability distribution of the number of calls of each type over a certain period of time. The paper proposes an asymptotic method to estimating the two-dimensional probability distribution of the number of arrivals in a high-intensity marked Markov Modulated Process. The limiting condition of high intensity is determined by the parameter of the rate of arrivals in the process over a certain time. The asymptotic analysis method is carried out in two stages. At the first stage, the parameters are obtained that determine the asymptotic mean numbers of arrivals of the first and second types that occurred in the high-intensity flow. At the second stage, the parameters are found that determine the asymptotic variances and the covariance of the number of events of the first and second types. It is shown that the limiting distribution of the number of events that occurred in a high-intensity marked MMPP is a two-dimensional Gaussian. The resulting formulas for finding the distribution and its characteristics have fairly simple expressions, the unknowns in which are found by solving systems of linear equations.