An algorithm for computing the mean response time of a single server queue with generalized on/off traffic arrivals

Abstract

In this paper, an exact solution for the response time distribution of a single server, infinite capacity, discrete-time queue is presented. This queue is fed by a flexible discrete-time arrival process, which follows an on/off evolution. A workload variable is associated with each arrival instant, which may correspond to the service demand generated by a single arrival, or represent the number of simultaneous arrivals (bulk arrivals). Accordingly, the analysis focuses on two types of queues: (On/Off)/G/1 and (Batch-On/Off)/D/1. For both cases, a decomposition approach is carried out, which divides the problem into two contributions: the response time experienced by single bursts in isolation, and the increase on the response time caused by the unfinished work that propagates from burst to burst. Particularly, the solution for the unfinished work is derived from a Wiener-Hopf factorization of random walks, which was already used in the analysis of discrete GI/G/1 queues. Compared to other related works, the procedure proposed in this paper is exact, valid for any traffic intensity and has no constraints on the distributions of the input random variables characterizing the process: duration of on and off periods, and workload. From the general solution, an efficient and robust iterative algorithm for computing the expected response time of both queues is developed, which can provide results at any desired precision. This algorithm is numerically evaluated for different types of input distributions and proved against simulation.