Queueing systems with long-range dependent input process and subexponential service times

Abstract

We analyze the asymptotic tail distribution of stationary waiting times and stationary virtual waiting times in a single-server queue with long-range dependent arrival process and subexponential service times. We investigate the joint impact of the long range dependency of the arrival process and of the tail distribution of the service times. We consider two traffic models that have been widely used to characterize the long-range dependence structure, namely, the M/G/ 8 input model and the Fractional Gaussian Noise (FGN) model. We focus on the response times of the customers in a First-Come First-Serve (FCFS) queueing system, although the results carry through to the backlog distribution of the system with any arbitrary queueing discipline. When the arrival process is driven by an M/G/ 8 input model we show that if the residual service time tail distribution F e is lighter than the residual session duration G e , then the stationary waiting time is dominated by the long-range dependence structure, which is determined by the residual session duration G e . If the residual service time distribution F e is heavier than the residual session duration G e , then the tail distribution of the stationary waiting time is dominated by that of the residual service time. When the arrival process is modeled by an FGN, we show that the waiting time tail distribution is asymptotically equal to the tail distribution of the residual service time if the latter is asymptotically heavier than Weibull distribution with shape parameter 2-2H, where H is the Hurst parameter of the FGN. If, however, this residual service time is asymptotically lighter than Weibull distribution with shape parameter 2-2H, then the waiting time tail distribution is dominated by the dependence structure of the arrival process so that it is asymptotically equal to Weibull distribution with shape parameter 2-2H.