The A(i)/GI/l Queue: A new model of transitory queueing

We introduce the Δ_(i)/GI/1 queue, a new model of transitory queueing, where demand for service exists only in a fixed interval of time and a large, but finite, number of customers enter the system. Customers independently arrive according to some given distribution F. Thus, the arrival times are an ordered statistics, and the inter-arrival times are differences of consecutive ordered statistics. They are served by a single server which provides service according to a general distribution G, with independent service times. The exact model is analytically intractable. Thus, we develop fluid and diffusion limits for the various stochastic processes as the population size is increased. The fluid limit of the queue length is observed to be a reflected process, while the diffusion limit is observed to be a function of a Brownian motion and a Brownian bridge, and given by a netput process and a directional derivative of the Skorokhod reflected fluid netput in the direction of a diffusion refinement of the netput process.