MATHEMATICAL MODELS THE TIME–DEPENDENT SOLUTION OF THE

Abstract—This work deals with time–dependent analysis of a random process associated with the number of jobs and of their spent service times in the M/G/1 queueing system under the foreground– background (FBPS) processor sharing discipline. This discipline assumes that only the set of jobs with the least amount of attained service share the server in the pure processor sharing fashion. We derive the time–dependent distribution of the number of jobs each of which has an attained service time a ≤ y at time t in terms of the double transforms (with respect to space: the Laplace functional, and with respect to time: the Laplace transform) given the system is empty at time t = 0. In other words, it is obtained the non–stationary distribution of the random counting measure representing the transient state of the FBPS queue in all details. The method of the analysis is an extension and refinement of the principally new approach introduced and developed in the previous authors works (see, for example, [11]). We consider also some important special cases. 1