Classification and unification of the microscopic deterministic traffic models

Abstract

We identify a universal mathematical structure in microscopic deterministic traffic models (with identical drivers), and thus show that all such existing models in the literature, including both the two-phase and three-phase models, can be understood as special cases of a master model by expansion around a set of well-defined ground states. This allows any two traffic models to be properly compared and identified. The three-phase models are characterised by the vanishing of leading orders of expansion within a certain density range, and as an example the popular intelligent driver models (IDM) is shown to be equivalent to a generalized optimal velocity (OV) model. We also explore the diverse solutions of the generalized OV model that can be important both for understanding human driving behaviours and algorithms for autonomous driverless vehicles. It is of great interest, both theoretically and practically, to understand via simple models the emergent behaviours of the complex systems containing a large number of interacting components. Examples like crowd dynamics [1,2], highway traffic systems [3,4], and the more recent urban traffic flows [5] have attracted physicists for decades. Despite the observed complex patterns from these systems, some of the essential characteristics are universally well-defined [6,7]. In contrast to the traditional many-body physical systems (e.g. involving identical particles like electrons), the crowd or traffic systems lack almost any symmetry at the microscopic level: even individual components are different from one another, with intrinsic stochasticity. This poses great challenges in finding the simplest model with enough predictive power to adequately characterise these complex systems.The lack of symmetries at the microscopic level leads to a certain arbitrariness in the model construction. The modelling of the highway traffic system has led to a plethora of traffic models [3,4,6,7]. Analysis of the empirical data has led to profound understanding of the traffic dynamics [8,9]; however due to non-linear interactions between drivers, simple interactions can lead to complex spatiotemporal patterns. It is thus difficult to decide which detailed driving behaviours or specific functional forms should be used in the model. This is the primary concern behind the controversy between the two-phase and three-phase traffic theories [10,12], which leads to two classes of traffic models. While both the two-phase and the three-phase traffic models can simulate the scattering of the data points on the flow-density plane when the traffic is congested, the two classes are distinguished by the presence or absence of a continuous fundamental diagram in the flow-density plot [7]. Moreover, the lack of a fundamental principle in understanding various different traffic models makes it difficult to decide if, in addition to the non-linear interactions between the vehicles, factors like stochasticity or diversity of driving behaviours are also essential to certain observed empirical features.To tack...

Microscopic Statistical Characterisation of the Congested Traffic Flow and Some Salient Empirical Features

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