A queueing model with independent arrivals, and its fluid and diffusion limits

摘要

We introduce the {Delta}(i)/GI/1 queue, a new queueing model. In this model,customers from a given population independently sample a time to arrive fromsome 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 ageneral distribution G, with independent service times. The exact model isanalytically intractable. Thus, we develop fluid and diffusion limits for thevarious stochastic processes, and performance metrics. The fluid limit of thequeue length is observed to be a reflected process, while the diffusion limitis observed to be a function of a Brownian motion and a Brownian bridgeprocess, and is given by a 'netput' process and a directional derivative of theSkorokhod reflected fluid netput in the direction of a diffusion refinement ofthe netput process. We also observe what may be interpreted as a transientLittle's law. Sample path analysis reveals various operating regimes where thediffusion limit switches between a free diffusion, a reflected diffusionprocess and the zero process, with possible discontinuities during regimeswitches. The weak convergence is established in the M1 topology, and it isalso shown that this is not possible in the J1 topology.