Analysis of the queueing systems with mixed priorities

A one-line queuing system with an infinite number of waiting places, an arbitrary distribution of service time, and Poisson incoming flows of customers is studied. Two models of mixed priorities are considered. In the first model, there is either preemptive repeat priority discipline between the customers of different flows or a head of the line priority discipline. In the second model — preemptive priority discipline either with loss or with repeat of a newly interrupted customer. By the method of additional components, a multidimensional random process is investigated, the components of which are the number of customers of each priority in the system and the time elapsed since the start of servicing the customer located on the device at time $t$. The distribution of the specified process in the nonstationary mode of the system is found.