  We introduce a framework and develop a theory of transitory queueing models.
These are models that are not only non-stationary and time-varying but also
have other features such as the queueing system operates over finite time, or
only a finite population arrives. Such models are relevant in many real-world
settings, from queues at post-offces, DMV, concert halls and stadia to
out-patient departments at hospitals. We develop fluid and diffusion limits for
a large class of transitory queueing models. We then introduce three specific
models that fit within this framework, namely, the Delta(i)/GI/1 model, the
conditioned G/GI/1 model, and an arrival model of scheduled traffic with epoch
uncertainty. We show that asymptotically these models are distributionally
equivalent, i.e., they have the same fluid and diffusion limits. We note that
our framework provides the first ever way of analyzing the standard G/GI/1
model when we condition on the number of arrivals. In obtaining these results,
we provide generalizations and extensions of the Glivenko-Cantelli and Donskers
Theorem for empirical processes with triangular arrays. Our analysis uses the
population acceleration technique that we introduce and develop. This may be
useful in analysis of other non-stationary and non-ergodic queuing models.
