Investigation of a Queueing System with Two Classes of Jobs, Bernoulli Feedback, and a Threshold Switching Algorithm

Abstract

A server with jobs of two classes, Bernoulli feedback, and setup times is considered. Jobs arrive in batches according to a Poisson process. Different job classes have different arrival intensities and different batch size distributions. Service times and setup times are random with exponential probability distributions. A control algorithm is parametrized by a threshold L: first-class jobs are taken for service only if the number of the second-class jobs in the system doesn’t exceed L. This kind of queueing models is often used to model computer systems and other information-processing systems. On the other hand, threshold-type controls for multiclass jobs have not been widely studied yet. A time-homogeneous Markov process describing the server state and numbers in the queues is introduced, its infinitesimal intensities are identified, a necessary and sufficient condition for the existence of the stationary probability distribution is found. Steady-state probabilities for the server states are found explicitly, and independence of the threshold parameter L is established. An algorithm for the numerical solution of a system of functional equations for the steady-state probability generating functions, the main objective of the talk, is presented for several particular values of the threshold L. This algorithm is a result of the problem investigation by means of computer algebra software since the size of intermediate formulas exceeds human capabilities to manage them by hand.