MapK/GK/1 queue with the generalized foreground-background processor sharing discipline

Queueing systems with Markov arrival flow, customers of several types, generalized foreground-background processor-sharing discipline and either separated buffers of finite capacity, or common buffer of finite capacity for customers of all types, or common buffer of infinite capacity for customers of all types are under consideration. The mathematical relationships among the steady-state joint distributions of the number of customers of all types in these systems are obtained. The Laplace-Stieltjes transform of the steady-state distribution of the sojourn time for a customer of each type for the system with common buffer of infinite capacity is derived too.