A new transitory queueing model and its process limits

We introduce the Δ
_(i)
/GI/1 queue, a new queueing model. In this model, customers from a given population 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. 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, reflected through a directional derivative of the usual Skorokhod reflection map. 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.