Bounding Queuing System Performance with Variational Theory

Queuing models are often used for traffic analysis, but analytical results concerning a system of queues are rare, thanks to the interdependence between queues. In this paper, we present an analysis of queuing systems to obtain bounds of their performance without studying the details of individual queues. Queuing dynamics is formulated in continuous-time, subject to variations of demands and bottleneck capacities. Our analysis develops new techniques built on the closed-form solution to a generalized queuing model for a single bottleneck. Taking advantage of its variational structure, we derive the upper and lower bounds for the total queue length in a tandem bottleneck system and discuss its implication for the kinematic wave counterpart. Numerical experiments are conducted to demonstrate the appropriateness of the derived upper and lower bounds as approximations in a stochastic setting.