Approximate Solutions for Some Two-Stage Tandem Queues, Part 1: Individual Arrivals at the Second Stage

The analysis of tandem queues in which the output of each stage immediately becomes the input to the next is, in general, quite difficult. Although the analytic solution for the steady-state departure distribution from an M/M/N queue is well-known and that of an M/G/1 queue can be obtained if the analyst is willing to view it through a “Laplacian curtain,” the list of existing analytic solutions is not long. Some analytic “tricks” exist, but their utility is usually limited to the single-server queue. The treatment of tandem queues not fitting into these categories has been largely left to simulation. Yet this approach has its drawbacks. It is expensive and involves difficulties in designing and analyzing the simulation experiment. Approximation techniques are beginning to emerge as a wiser, faster, cheaper, and less troublesome alternative to simulation. In this paper we develop approximate solutions for average steady-state queue length in four different but related tandem queues that arise in connection with the service sys-of an air terminal complex. The approximate expressions obtained are validated by simulation.