### Presentation (PPT)

```Reliability-Based Timepoint Schedules
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
•
End of Line
•
•
•
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.