Reliability-Based Timepoint Schedules
for Long Headway Transit Routes
Peter G. Furth, Northeastern University
with Theo H.J. Muller, Delft University of Technology
1
Outline
1. Translating Reliability into User Cost
2. Operations model
– Segment running time
– Timepoint holding
– Layover and dispatch holding
3. Example results
2
Reliability Affects Passenger Travel
Time
• Passengers arrive so that P[miss the bus] < 2%
• Passengers budget for 95-percentile [wait + ride] time
• Potential Travel Time =
Budgeted Travel Time – mean [Wait + Ride Time]
3
4
Passenger Travel Cost Components
Waiting Time:
Riding Time:
Potential Travel Time:
$9/hr
$6/hr
$4.5/hr
Reliability has been captured:
Cost = f(Tails of departure and arrival time distributions)
Note: Estimating tails requires archived AVL data.
5
Operating Cost for a Route with Holding
= Cycle Time
cSchedule = Design Parameter
• Can be fixed or optimized
cactual = an outcome, the sum of 3 components:
– Mean uncontrolled running time
– Mean holding time (running time supplement)
– Mean layover time (layover slack)
Inconsistent unless cactual ≈ cSchedule
– Steady state: f(StartDeviationcycle n) ≈ f(StartDeviationcycle n+1)
6
Operations Model
Segments (includes necessary dwell time)
•
Random, independent running times
•
Ideally, get distribution from AVL data
Timepoints
•
Hold early arrivals
•
Add random holding supplement
End of Line
•
Add a random needed layover to become “Ready”
•
Hold early “Ready”
•
Add random dispatch supplement
7
Timepoints: Random Holding
Supplement
-1
0
1
min
Holding Supplement (min)
8
End of Line: Needed Layover and
Dispatch Supplement
1
6
Needed Layover (min)
-1
0
1
2
Dispatch Supplement (min)
9
Layover Model: Planning View
Labor policies on minimum layover constrain
cSchedule
Finding:
• Unreliable service: reliability governs optimal cSchedule
• Highly reliable service: labor policy governs
10
Analysis
• Track discretized probability distributions using MatLab
• 2 Warm-up cycles to achieve quasi-steady state
• Example route
– 17 stops, 16 segments
– mean running time = 40 min, s = 5 min for base case
– mean boardings = 74, max load = 36 pax
• Optimize w.r.t. two overlapping schedule parameters
– Cycle supplement = CycleTime – MeanUncontrolledRunningTime
– Running Time supplement = ScheduledRunningTime –
MeanUncontrolledRunningTime
11
Cost vs. Running Time Supplement
Optimized cycle length
$40
riding time
$20
operating cost
$0
-15.0
-10.0
-5.0
0.0
5.0
10.0
15.0
potential travel time
-$20
total cost
-$40
waiting time
-$60
Running Time Supplement (min)
12
Slack Distributions vs. Running Time
Supplement
cSchedule optimized; dashed line = only 1 timepoint
0.40
0.30
Total holding
0.20
0.10
Layover
holding
Timepoint holding
-15.0
-10.0
-5.0
0.0
5.0
10.0
15.0
Running Time Supplement (min)
13
(Bars)
Change in
Cost
(Base = No
Timepoints)
($/trip)
σroute = 3 min
σroute = 5
σroute = 7
Cost vs. Number of Timepoints
$0
3.5
-$10
Cycle suppl, σroute=3
3
-$20
Cycle suppl, σroute=5
Cycle suppl, σroute=7
2.5
-$30
2
-$40
RT suppl, σroute=7
1.5
-$50
RT suppl, σroute=5
1
-$60
RT suppl, σroute=3
0.5
-$70
(Lines)
Schedule
Supplement
as multiple
of σroute
0
0
1
3
7
14
Number of Timepoints
14
Optimal Schedule Supplements vs sroute
dashed line for a single timepoint
3.00
Cycle time supplement
2.00
1.00
Running time supplement
0.00
2.0
3.0
4.0
5.0
6.0
7.0
8.0
s route (min)
15
Conclusion and Remarks
1.
2.
Archived AVL data makes reliability analysis possible
Capturing reliability in the cost function facilitates tradeoff against riding time
and operating cost; contrast rules of thumb
3.
Schedules should probably have more en-route slack
4.
To a large degree, en-route slack and recovery slack simply substitute for
one another, meaning en-route slack can be added without increasing cycle
time
5.
Amount of en-route & layover slack are not easily calculated
6.
Optimal departure time depends on boardings & other factors
7.
On more reliable routes, layover is governed by operator rest needs
8.
What about operator behavior?
9.
What about short headway service?
10. What about passengers who make transfers?
16