The non-preemptive ‘Join the Shortest Queue–Serve the Longest Queue’ service system with or without switch-over times

Abstract

A 2-queue system with a single-server operating according to the combined ‘Join the Shortest Queue–Serve the Longest Queue’ regime is analyzed. Both cases, with or without server’s switch-over times, are investigated under the non-preemptive discipline. Instead of dealing with a state space comprised of two un-bounded dimensions, a non-conventional formulation is constructed, leading to an alternative two-dimensional state space, where only one dimension is infinite. As a result, the system is defined as a quasi birth and death process and is analyzed via both the probability generating functions method and the matrix geometric formulation. Consequently, the system’s two-dimensional probability mass function is derived, from which the system’s performance measures, such as mean queue sizes, mean sojourn times, fraction of time the server resides in each queue, correlation coefficient between the queue sizes, and the probability mass function of the difference between the queue sizes, are obtained. Extensive numerical results for various values of the system’s parameters are presented, as well as a comparison between the current non-preemptive model and its twin system of preemptive service regime. One of the conclusions is that, depending on the variability of the various parameters, the preemptive regime is not necessarily more efficient than the non-preemptive one. Finally, economic issues are discussed and numerical comparisons are presented, showing the advantages and disadvantages of each regime.