A finite capacity $BMAPK/GK/1$ queue with the generalized foreground-background processor-sharing discipline

A queueing system with a batch Markov arrival process, several types of customers, generalized foreground-background processor-sharing discipline with minimal served length, and separate finite buffers for customers of different types or a common finite buffer for customers of all types is studied. Mathematical relations for computing the stationary joint distributions of the number of customers of all types in the system are derived.