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

TL;DR

This paper introduces a new queueing model with independent arrivals and general service times, analyzing its fluid and diffusion limits, and revealing complex operating regimes and transient behaviors.

Contribution

It develops fluid and diffusion limit results for the {}(i)/GI/1 queue, a novel model with independent sampling arrivals, providing new insights into its stochastic behavior.

Findings

Fluid limit is a reflected process.

Diffusion limit involves Brownian motion and Brownian bridge.

Multiple operating regimes with regime switches.

Abstract

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 from 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, and performance metrics. 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 process, and is 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. We also observe what may be interpreted as a transient 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 is established in the M1 topology, and it is also shown that this is not possible in the J1 topology.