A Queueing Model with Independent Arrivals, and its Fluid and Di↵usion Limits

Abstract We study a queueing model with ordered arrivals, which can be called the (i)/GI/1 queue. Here, customers from a fixed, finite, population independently sample a time to arrive from some given distribution F , and enter the queue in order of the sampled arrival times. Thus, the arrival times are order statistics, and the inter-arrival times are di↵erences of consecutive ordered statistics. They are served by a single server with independent and identically distributed service times, with general service distribution G. The discrete event model is analytically intractable. Thus, we develop fluid and di↵usion limits for the performance metrics of the queue. The fluid limit of the queue length is observed to be a reflection of a ‘fluid netput’ process, while the di↵usion limit is observed to be a function of a Brownian motion and a Brownian bridge process or ‘di↵usion netput’ process. The di↵usion limit can be seen as being reflected through the directional derivative of the Skorokhod regulator of the fluid netput process in the direction of the di↵usion netput process. We also observe what may be interpreted as a sample path Little’s law. Sample path analysis reveals various operating regimes where the di↵usion limit switches between a free di↵usion, a reflected di↵usion process and the zero process, with possible discontinuities during regime switches. The weak convergence results are established in the M1 topology.