Functional Relationship Between Primary and Secondary Delays on
Functional Relationship Between Primary and Secondary Delays on Railway Lines Trafikdage 2017, Aalborg Dr. Steven Harrod Fabrizio Cerreto, Prof. Otto Anker Nielsen Technical University of Denmark
Polynomial Aggregate Delay Function • Management question • Problem definition • Formulation • Some fun graphical results • Conclusions recommend general “rule of thumb” timetable guidelines 2 DTU Management Engineering, Technical University of Denmark
Strategic Design of Timetable • Goals – Provide service, capacity – Minimize travel time – Promise punctuality • Controls – Frequency of service – Timetable ”slack” • Extra train scheduled time • Extra separation between adjacent trains 3 DTU Management Engineering, Technical University of Denmark
Key Performance Measure: Aggregate Delay • The timetable is a system • Punctuality is a systemwide measure • Measure delayed passenger minutes • Flawed but convenient equivalent –Measure train delays at each station –Accuracy depends on homogenous passenger flow at all stations –Do not measure train delays at non stopping locations 4 DTU Management Engineering, Technical University of Denmark
System Definition, Single Train Accumulated Supplement, 4 a Minimum Safe Separation Headway, h Stations Ideal train path Scheduled Train Path Timetable Supplement, a Time 5 DTU Management Engineering, Technical University of Denmark Performance measurement point
System Definition, Multiple Trains Stations Timetable buffer, b Time 6 DTU Management Engineering, Technical University of Denmark
Impact of a Primary Delay 4 Original schedule 3 Stations Recovery Train Path Primary Delay, p 2 d=p-(s-1)a-(i-1)b d=p-(2 -1)a-(2 -1)b 1 Train 1 7 Train 2 Train 3 Train 4 DTU Management Engineering, Technical University of Denmark Time
Derivation, Bounds of Disruption Solve for: 8 DTU Management Engineering, Technical University of Denmark
Recovery Region Settling Time 1 a/b Stations s* Point of primary delay 1, 1 Trains 9 DTU Management Engineering, Technical University of Denmark i* 1
Symmetric System, c=a=b 10 DTU Management Engineering, Technical University of Denmark
Relaxed Floor Generates Polynomial Cumulative Delay 11 DTU Management Engineering, Technical University of Denmark
Visualizing the Polynomial 12 DTU Management Engineering, Technical University of Denmark
Settling Time Function φ Figure 3‑ 4: Contour of settling time with t=5, h=5, δ=3 and p=5. 13 DTU Management Engineering, Technical University of Denmark
Generic Polynomial, p: {5, 20} 14 DTU Management Engineering, Technical University of Denmark
But Wait, There’s More! If the study area does not extend to end of recovery region, We can calculate e, the delay remaining at any point, as a function of intitial delay p. Stations s* 1 a/b e Γp- Γe Point of primary delay 1, 1 Trains 15 the aggregate delay of the study area is the difference of two polynomials. DTU Management Engineering, Technical University of Denmark i* 1
Flexible Application to Analysis Stations s* 1 Γp- Γe- Γh a/b e Point of primary delay h 1, 1 Trains 16 DTU Management Engineering, Technical University of Denmark i* 1
Conclusions • Cumulative train delay at stations is a key performance measure • Polynomial function is a practical estimate of system delay • Timetable supplement and timetable buffer should be equal • Decreasing marginal benefit of increasing supplement/buffer 17 DTU Management Engineering, Technical University of Denmark
Tak for i dag! 18 DTU Management Engineering, Technical University of Denmark
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