Utilizing phase-type distributions for queueing-based railway junction performance determination

TL;DR

This paper tackles timetable-independent capacity assessment for railway junctions by introducing a Continuous-Time Markov Chain model that uses phase-type distributions to represent independent arrival and service processes (e.g., $L_r$, $\lambda_r$, $\mu_r$). It extends prior exponential-CTMC analyses and evaluates four distribution settings via a capacity-determination algorithm based on Brent's method, comparing queue-length estimates to per-route thresholds $L_{ ext{limit},r}$. The authors validate the PH/PH model against simulations and study approximation quality relative to Hertel and Kingman, showing PH/PH provides the most accurate timetable capacity estimates, albeit with higher computation time. A case study on a mixed freight–passenger double-track junction demonstrates bottleneck identification and practical planning implications, while noting scalability limits and suggesting decomposition or efficient approximations for larger networks.

Abstract

To ensure the effective and objective development of transportation networks, it is crucial to identify performance limitations across various subsystems. A timetable-independent assessment of infrastructure capacity at railway junctions is a fundamental aspect of long-term rail network planning. While recent research introduced queueing-based methods to quantify route-based railway junction performance, modelling arrival and service processes has been limited to exponential distributions. This work utilizes Phase-Type Distributions to propose an extension to a previously described Continuous-Time Markov Chain model. In a comparison between assumed distribution combinations, the effect of a more detailed stochastic process modelling is described. Furthermore, an analysis of the differences to a simulation method is conducted for an exemplary railway junction. The introduced method enables infrastructure managers to accurately model stochastic processes for performance determination in the early stages of the strategic planning phase.

Figures (9)

• Figure 1: Examplary infrastructure of a railway junction
• Figure 2: Examples of the service process with exponential (M) or phase-distributed (Ph) service times
• Figure 3: Mean computation times of all considered models
• Figure 4: Comparison of the queue-length estimations at route $r_3$ for different arrival- and service-process distributions with and without scaling factors
• Figure 5: Comparison of the performance determination by the share of main line traffic $p_{main}$ for the different methods