Threshold queueing describes the fundamental diagram of uninterrupted traffic

Greenshields [13] captures the empirical relation between speed, flow and density for uninterrupted traffic in the fundamental diagram, see Figure 1. Mathematical models for uninterrupted traffic have been developed and the fundamental diagram in its basic form is now well-understood, see e.g. Newell [29, 30, 31] for a concise exposition. Traffic jams are a major concern for highway operation and may occur in high density traffic due to variability in driving speed. A wide range of traffic models has been developed over the past decades. These models are mainly from statistical physics and non-linear dynamics, see [9, 18]. Congestion due to variable arrival and/or service processes is the main topic of queueing theory, that, however, has hardly been invoked to analyse the fundamental concepts of uninterrupted traffic flows. Notable exceptions are the models introduced by Heidemann [14] and Jain and Smith [21]. However, these models do not capture the