1 Reliability-Based Timepoint Schedules for Long Headway Transit Routes Peter G. Furth, Northeastern University with Theo H.J. Muller, Delft University of Technology

2 Outline 1.Translating Reliability into User Cost 2.Operations model –Segment running time –Timepoint holding –Layover and dispatch holding 3.Example results

3 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]

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5 Passenger Travel Cost Components Waiting Time: $9/hr Riding Time:$6/hr Potential Travel Time:$4.5/hr Reliability has been captured: Cost = f(Tails of departure and arrival time distributions) Note: Estimating tails requires archived AVL data.

6 Operating Cost for a Route with Holding = Cycle Time c Schedule = Design Parameter Can be fixed or optimized c actual = an outcome, the sum of 3 components: –Mean uncontrolled running time –Mean holding time (running time supplement) –Mean layover time (layover slack) Inconsistent unless c actual ≈ c Schedule –Steady state: f(StartDeviation cycle n ) ≈ f(StartDeviation cycle n+1 )

7 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

8 Timepoints: Random Holding Supplement Holding Supplement (min)

9 End of Line: Needed Layover and Dispatch Supplement Needed Layover (min) Dispatch Supplement (min) 16

10 Layover Model: Planning View Labor policies on minimum layover constrain c Schedule Finding: Unreliable service: reliability governs optimal c Schedule Highly reliable service: labor policy governs

11 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,  = 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

12 Cost vs. Running Time Supplement Optimized cycle length -$60 -$40 -$20 $0 $20 $ Running Time Supplement (min) riding time operating cost potential travel time total cost waiting time

13 Slack Distributions vs. Running Time Supplement c Schedule optimized; dashed line = only 1 timepoint Running Time Supplement (min) Layover holding Timepoint holding Total holding

14 Cost vs. Number of Timepoints -$70 -$60 -$50 -$40 -$30 -$20 -$10 $ Number of Timepoints (Lines) Schedule Supplement as multiple of σ route (Bars) Change in Cost (Base = No Timepoints) ($/trip) RT suppl, σ route =3 RT suppl, σ route =7 RT suppl, σ route =5 Cycle suppl, σ route =3 Cycle suppl, σ route =5 Cycle suppl, σ route =7 σ route = 3 min σ route = 5 σ route = 7

Optimal Schedule Supplements vs  route dashed line for a single timepoint 15

16 Conclusion and Remarks 1.Archived AVL data makes reliability analysis possible 2.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?