The discrete-time single-server queue (Q809487)

The limiting distribution of the waiting time of a single-server system in discrete time is considered. The service times and the arrival times are arbitrarily distributed and independent. The service time of a customer arriving at an empty system can have a modified distribution. An exact relation between the generating functions of the steady-state waiting-time distribution and the idle-time distribution is given in the general case. Using this relation more explicit results are stated when the generating functions of either the interarrival-time distribution or the service-time distribution is rational. Since the queue-length process can be periodic, a Cesàro limit defines the limiting waiting-time distribution. Some results on the asymptotic behavior of the witing-time distribution are contained, especially a condition for a geometrically fast decreasing tail. The standard technique of complex-valued functions is used to elaborate the analysis of the generating functions. Numerical results for various interarrival-time and service-time distributions illustrate the derived formulas.