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

We study a queueing model with ordered arrivals, which can be called the \(\varDelta _{(i)}/GI/1\) 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 differences of consecutive order 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 diffusion 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 diffusion limit is observed to be a function of a Brownian motion and a Brownian bridge process or ‘diffusion netput’ process. The diffusion 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 diffusion 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 diffusion limit switches between a free diffusion, a reflected diffusion process, and the zero process, with possible discontinuities during regime switches. The weak convergence results are established in the \(M_1\) topology.