A generalized queuing model and its solution properties

Modeling queuing behavior is central to the analysis of transportation and other service systems. To date, several queuing models been developed, but analytical insights on their global properties are hard to obtain. This is because in most cases, queuing dynamics are formulated as differential or difference equations, with possible discontinuities in their solutions, making most conventional analytical tools inadequate. As a result, simulations are often used to study these models, and if not properly treated, negative flows could arise from the simulation near certain discontinuities. In this paper, we propose a continuous-time queuing model that captures generalized queuing dynamics, where bottleneck discharging capacity and demand can vary simultaneously. We provide insights on the global properties of this model, upon deriving its closed-form variational solutions. Rather than resorting to the usual Hamilton-Jacobi theory, our derivations are built on an intrinsic periodicity property of the general queuing dynamics combined with measure-theoretic analysis. This treatment allows us to obtain results with more complex boundary conditions and make further extensions. We demonstrate its applications and show its solution properties in queuing simulation and performance bounding. In particular, we provide graphical, iterative and linearized solution schemes, which are all devoid of the well-known negative flow issue associated with numerical solutions to the point queue model.