Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Hyperbolic Transport Model for Passenger Flow on Tram Networks

Thomas Schillinger

Abstract

We introduce a modeling framework for an urban tram network based on a hyperbolic partial differential equation describing the transport of passengers along the network, coupled with a family of stochastic processes representing passenger boarding. Solutions are considered in a measure-valued sense. The system is further extended and subjected to uncertainties such as delays and service interruptions through a numerical study. Its robustness is assessed using appropriate risk measures.

A Hyperbolic Transport Model for Passenger Flow on Tram Networks

Abstract

We introduce a modeling framework for an urban tram network based on a hyperbolic partial differential equation describing the transport of passengers along the network, coupled with a family of stochastic processes representing passenger boarding. Solutions are considered in a measure-valued sense. The system is further extended and subjected to uncertainties such as delays and service interruptions through a numerical study. Its robustness is assessed using appropriate risk measures.
Paper Structure (27 sections, 4 theorems, 72 equations, 14 figures, 3 tables, 1 algorithm)

This paper contains 27 sections, 4 theorems, 72 equations, 14 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Modeling a tram network
  3. Admissible tram schedule
  4. Conditions at the vertex for $m-n$ junctions
  5. 1-1 junction.
  6. 1-2 junction.
  7. 2-1 junction.
  8. m-n junction.
  9. A system of hyperbolic balance laws for tram passengers
  10. Measure-valued formulation
  11. Modeling of passengers at the vertex
  12. Boarding and alighting passengers
  13. Weak measure-valued solution on one edge
  14. Construction of solutions on the network
  15. 1--1 junction.
  16. ...and 12 more sections

Key Result

Lemma 3.2

Let the arrival rates $\lambda^e(t)$ of the Poisson process be uniformly bounded, and assume that the capacities prescribed by the admissible tram schedule $\tau^e(t)$ are uniformly bounded as well. Then the queue $q^e(t)$ defined in Equation eq:queue is well-defined and nonnegative for all $t \ge 0

Figures (14)

  • Figure 1: Tram line routes at the junction $v$. Line 1 (red), line 2 (green), line 3 (blue), line 4 (orange).
  • Figure 2: Exemplary evolution of the passenger queue at a tram station during the morning and afternoon periods, together with the corresponding tram departure times.
  • Figure 3: Evolution of trams in a small 1-1-1-1-1 tram network. The lines show the movement of the tram in space and time and the color represents the number of traveling passengers. Tram stops are marked with black vertical lines.
  • Figure 4: Line 1 from Schönau to Rheinau, with all stops up to the central station.
  • Figure 5: The passenger waiting times (in hours) per tram station (left) and hourly interval (right) for different tram frequencies.
  • ...and 9 more figures

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 3.1: Borel measure
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • ...and 4 more