1. Metro Scheduling By Philip Anderson & Liza John

2. Metro Scheduling Case Study Real world Practice

3. A simple example Model Station 1 Station 2 Station 3 λ2λ2 λ1λ1 λ3λ3

4. A simple example Arrival Rates Passenger arrival at each station can be modeled as a Poisson process having time variable rate λ. λiλi t

5. A simple example Arrival Rates λ t

6. λiλi λ is t s

7. A simple example Destination Probabilities P ijs Matrix: Probability that a passenger who entered station i will get off at station j. For j ≤ I P = 0. 0P 12 = 1/2 P 13 =1/2 00P 23 = 1 000 1 2 3 123123 j i

8. A simple example Define: Let r be the time interval between trains. From the Central Limit Theorem N is (the number of passengers at M i for period s) is normally distributed having mean r(λis) an variance r(λis). Objective: Create a schedule for period s by specifying r to minimize cost and guarantee capacity constraints

9. A simple example Constraints: Train capacity: C r {4,…,20} Not reaching capacity 95 percent of the time.

10. A simple example Find the smallest r to satisfy all the equations: 95% => z = 1.65 Equation 1: Equation 2:

11. A simple example Results: First select the smallest r from solving equation 1 and 2. If r is > then 20 assign the minimum of the two If r is between 4 and 20 assign that value If r is less then 4 then we cannot guarantee this level of confidence.

12. A simple example Second Look Trains are jobs Stations are machines Flow shop algorithm Fm | prmu | Lmax Because the order of the stations, machines, cannot change, the real problem is figuring out how many trains, jobs, can be completed with the given expressed constraints, and still hold true to the station schedule 12

13. A simple example Second Look Rush Hours 6:30AM - 9:30AM and 3:30PM - 8:00PM Regular Population Density Hours 9:30AM - 3:30PM and 8:00PM - 12:00AM Late Night Hours 12AM - 6:30AM

14. 14 A simple example Second Look Different times in the day allow for different lengths of wait time During rush hours people will be waiting around 4 minutes During regular hours people will be waiting around 7 minutes During late night hours people will be waiting around 20 minutes

15. Simulation TOWARD INCREASED USE OF SIMULATION TRANSPORTATION – Dudley Whitney, Parsons Brinckerhoff Quade & Douglas, Inc. INVESTIGATING THE CAPACITY OF A METRO LINE BY MEANS OF A SIMULATION MODEL – A Ballis*, K Liberis and T Moschovou SIMULATORS USED BY WMATA – Martin Lukes

16. Simulation TOWARD INCREASED USE OF SIMULATION TRANSPORTATION Construction Feasibility: Signal Design: Power Consumption: Traffic Studies: Railroad Capacity Studies: Train Operations Studies:

17. Simulation TOWARD INCREASED USE OF SIMULATION TRANSPORTATION Perceived high cost Tight budgets Tight schedules How to address these issues? http://trainlogic.net/sim_wmata.htm