Non-Stationary Characteristics of Queueing systems with the Foreground-Background Processor Sharing Discipline

Two queueing systems with foreground-background processor sharing discipline are considered. Non-stationary joint distribution of the number of customers served until the moment t and being in the system at time t taking into account their lengths (for the customers being in the system at the moment t their served lengths are considered) is derived in terms of a triple transform (the Laplace transform on time and generation functions on number of customers)