Queue methods for variability in congested traffic

Time-dependent queue methods are extended to calculate variances of stochastic queues along with their means, and thereby provide a tool for evaluation and better understanding of travel time variability and reliability in congested traffic networks and other systems, including through probability distributions estimated from moments. Objectives include developing computationally efficient analytical methods, and achieving robustness by reflecting the underlying structure of queuing systems rather than relying on statistical fitting, New deterministic and equilibrium formulae for queue variance are developed, acting also as constraints on estimating time-dependent queues generated by a range of processes, enabling improved accuracy and reliability estimates. New methods for approximating equilibrium and dynamic probability distributions use respectively doubly-nested geometric distributions and exponentially-weighted combinations of exponential and Normal functions, avoiding the need to rely on empirical functions, costly simulation, or equilibrium distributions inappropriate in dynamic cases. For growing queues, corrections are made to the popular sheared approximation, that combines deterministic and Pollaczek-Khinchin equilibrium mean formulae in one time-dependent function. For decaying queues, a new exponential approximation is found to give better results, possibly through avoiding implicit quasi-static assumption in shearing. Predictions for M/M/1 (yield) and M/D/1 (signal) processes applied to 34 oversaturated peaks show good agreement when tested against Markov simulations based on recurrence relations. Looking to widen the range of queues amenable to time-dependent methods, dependence of stochastic signal queues on green period capacity is confirmed by an extended M/D/1 process, for which new formulae for equilibrium moments are obtained and compared with earlier approximations. A simple formulation of queuing on multiple lanes with shared service is developed, two-lane examples with turning movements showing fair match to simulation. The main new methods are implemented in a spreadsheet demonstrator program, incorporating a database of time-sliced peak cases together with a procedure for estimating dynamic probability distributions from moments