Fundamental diagrams and emergent dynamics of mainline rail operations

Emily J Morey, R E Wilson, Kevin Galvin

Research output: Working paper


This paper is concerned with the line capacity of high-speed rail operations. Inspired by highway traffic modelling, we derive analytical ‘fundamental diagrams’ that relate trains’ average speeds, spacings (macroscopically: density), and timetabled frequency (macroscopically: flow rate). Note that the distinction with classical car-following models (CFMs) arises from the block-based signalling rules which provide the only means by which neighbouring trains interact. We then develop simplified simulations in which a large number of identical trains repeatedly circle around a large closed track circuit. Three distinct regimes are observed: (i) trains self-organise to have sufficient space between them to run constantly at their goal speed; (ii) stop-and-go waves, with the circuit divided into regions of maximum speed running, and queues; (iii) jitter, i.e., repeated patterns of acceleration and deceleration when the goal speed is set too high. These regimes are explained in terms of the fundamental diagrams. The analysis provides useful rules-of-thumb for the design of robust mainline operations. Finally, we discuss how the analysis may be adapted to model potential capacity improvements that would result from the roll-out of connected and/or autonomous trains with peer-to-peer communication that is independent of block-based signalling rules.
Original languageEnglish
Publication statusIn preparation - 16 Jun 2022

Structured keywords

  • Engineering Mathematics Research Group


  • Line Capacity
  • Fundamental Diagram
  • Block-Based Signalling


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