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

Functional Relationship Between

Primary and Secondary Delays on

Railway Lines

Polynomial Aggregate Delay Function

• Management question

• Problem definition

• Formulation

• Some fun graphical results

• Conclusions recommend general

“rule of thumb” timetable

guidelines

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

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

System Definition, Single Train

Time

St at io ns

Ideal train path

Performance measurement

point

Scheduled Train Path

Minimum Safe Separation Headway, h

Timetable Supplement, a

Accumulated

Supplement, 4a

System Definition, Multiple Trains

Time

St at io ns

Timetable

buffer, b

Impact of a Primary Delay

Time

St at io ns

Primary Delay, p Original schedule Recovery Train Path

1 2 3 4

Train 1

Train

2 Train

3

Train 4

d=p-(s-1)a-(i-1)b

d=p-(2-1)a-(2-1)b

Derivation, Bounds of Disruption

Γ = ∑

¿ � ∈� ∨ � _{� , �} ≥ �

¿ � ∈�

� _{� , �}

� _� ^∗ = ⌊ � + � − �

� − � �

� ⌋ + 1∨ � ≥ � + �

� _� ^∗ = ⌊ � + � − �

� − � �

� ⌋ + 1 ∨� ≥ � + �

� _{� , �} ≥ �

Solve for:  

Recovery Region

s ^* ₁

i ^* ₁

1,1

a/b

St at io ns

Trains

Point of primary delay

� + �

�

=��� { ^{( ¿ �}

∗

, ( h + � ) �

}

Settling Time

Symmetric System, c=a=b

� _� ^∗ = ⌊ � − �

� ⌋ − � + 2 ∨� ≥ � + �

� _� ^∗ = ⌊ � − �

� ⌋ − � + 2 ∨� ≥ � + �

� + �

� + 2 � − � ( ¿ )

¿

Γ _s = ∑

¿ � ∈ { ^{1,2 , … , ⌊ � −} � ^� ⌋ − � + 2 }

¿ � ∈ { ^{1,2 , … ⌊ � −} � ^� ⌋+ 1 }

¿

Relaxed Floor Generates

Polynomial Cumulative Delay

Γ _� = � ³

6 � ² + � ²

2 � + ( 2 � ² + 3 � � − 3 � ² ) _�

6 � ² + 4 � ² � − 6 � � ² + 2 � ³ 6 � ²

D e l a y P 1 0 0 0

2 0 0 0 3 0 0 0 4 0 0 0



Visualizing the Polynomial

Settling Time Function φ

Figure 3 4: Contour of settling time with t=5, h=5, δ=3 and p=5. ‑

Generic Polynomial, p: {5, 20}

But Wait, There’s More!

s ^* ₁

i ^* ₁

1,1

a/b

St at io ns

Trains

Point of primary delay

If the study area does not extend to end of recovery region,

the aggregate delay of the study area is the

difference of two polynomials.

e We can calculate e,

the delay remaining at any point, as a

function of intitial delay p.

Γ _p - Γ _e

Flexible Application to Analysis

s ^* ₁

i ^* ₁

1,1

a/b

St at io ns

Trains

Point of primary delay

e

h

Γ _p - Γ _e - Γ _h

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

Tak for i dag!