On a dual hybrid queueing system

Abstract

This article deals with a hybrid system, in which a single server processes two different queues of units, one called primary and the other one — secondary. The queueing process in the primary system is formed by a Poisson flow of groups of units, while the secondary system is closed. The server’s primary appointment (in hybrid mode I) is to process units in batches until the buffer content drops significantly. In this case, the server takes over a queue in the secondary system (activating hybrid mode II), and he is to complete some minimum amount of jobs (rendered in groups of random sizes during random times). When he is done with this work, he returns to the primary system. If the queue there is not long enough, he waits, thereby activating hybrid mode III. The authors first apply and embellish some techniques from fluctuation theory to find the exit times from respective hybrid modes and queue levels in both systems in terms of their joint functionals. The results are then utilized for the subsequent (semi-regenerative) analysis of the evolution of queueing processes. The authors obtain explicit formulas for the limiting distribution of the queueing process and the mean number of units processed in the secondary system.