On Transitory Queueing

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-offices, 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 ∆(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 Donsker’s Theorems to triangular arrays. Our analysis uses a technique we call population acceleration, which we discuss in some detail.