  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.
