General solution scheme for the static link transmission model

Most static traffic assignment models, both in the literature and in practice, are neither capacity constrained nor storage constrained. They allow flows to exceed the link capacity and/or queues to exceed the link length. Recent studies in this area have resulted in novel approaches that do consider capacity constraints, resulting in residual queues, and sometimes even storage constraints, resulting in possible queue spillback. We build upon the results of these works and in particular on the model formulated in Bliemer and Raadsen (2020) that introduced a static network loading model formulation that is both capacity constrained as well as storage constrained. Their static network loading model formulation is derived from – and consistent with - the link transmission model, a well-established dynamic network loading model. It is referred to as the static link transmission model (sLTM) in this paper. This model considers a general concave fundamental diagram for each link and a general first order node model. It is well known that ignoring queue spillback can result in significant underestimation of path travel times. This is especially true for paths that do not traverse any of the bottleneck(s) directly, but that are affected by space occupied by queues that are spilling back. The prospect of being able to capture queuing and spillback effects in static assignment provides new opportunities for improving the modelling capabilities of this paradigm. In this paper, we propose a solution scheme to sLTM capable of finding a solution on large scale networks. This is the first time that an algorithm is proposed for solving an analytical static model with queue spillback. The inclusion of a node model in a static context - while enhancing the model's capabilities – generally results in the absence of a guaranteed convergent algorithm, and introducing spillback exacerbates the issue. Given the challenges of finding a stable solution, we discuss a base solution scheme and three (configurable) extensions. Further, we investigate algorithmic settings with respect to convergence and its impact on computational cost. A large-scale case study demonstrates the feasibility of the proposed scheme by finding solutions under the most challenging of conditions in a real-world setting. We show that improving convergence capabilities negatively affects computational efficiency. To this end, several potential improvements based on our findings are discussed. Lastly, we discuss the potential of this line of research in more general terms, highlighting strengths and weaknesses following our experiences.