The asymptotic analysis of the income in closed HM-structure with priority messages

In the article the asymptotic analysis of closed exponential queueing HM-structure with priority messages is carried out with a large total number of messages, depending on time. The number of service lines in systems, the intensity of service messages in them, the probabilities of message transitions between systems also depends on time. It is proved that the density of the income distribution in the network systems in asymptotic satisfies differential equations in partial derivatives. This provided the inhomogeneous differential equation for the expected incomes system structure. An example of transport logistics shows how to solve such equations.