Transitory Queueing Systems

“Transitory” queueing systems, namely those that exist only for finite time are all around us. And yet, current queueing theory provides with very few models and even fewer tools to understand them. In this talk, I’ll introduce a model that may be more appropriate in “transitory” queueing situations. I’ll start by first talking about the “concert arrival game” that we introduced. This model captures the tradeoff in decision-making when users must decide whether to arrive early and wait in a queue, or arrive late but miss out on opportunities (e.g., best seats at a concert). With such decision-making by the users, it is a folly to assume that the arrival process is exogenous, and a well-behaved renewal process. Thus, under the fluid model, we characterize the equilibrium arrival profile and characterize the “price of anarchy” (PoA). We also give some simple mechanisms to reduce it. We then present various extensions including to general feedforward networks when routing itself is strategic. The concert queueing game led us to the Delta(i)/GI/1 queue – a new model for transitory queueing systems, in which a each user picks their time of arrival from a common distribution. We derive the fluid and diffusion process limits for such a model. We find that the limiting queue length process is a reflection of a combination of a Brownian motion and a Brownian bridge. The convergence is established in the M1 topology, and we also show that it is not possible to obtain it in the stronger J1 topology. Such models allow us to study transitory queueing systems which are not very amenable to extant techniques in queueing theory.