Metro Scheduling

By Philip Anderson & Liza John

Metro Scheduling

Case Study Real world Practice

A simple example Model

Station 1 Station 2 Station 3

λ2λ1 λ3

A simple example Arrival Rates

Passenger arrival at each station can be modeled as a Poisson process having time variable rate λ.

λi

t

A simple example Arrival Rates

λ

t

A simple example Arrival Rates

λi

λis

t s

A simple example Destination Probabilities

Pijs Matrix: Probability that a passenger who entered station i will get off at station j. For j ≤ I P = 0.

0 P12 = 1/2 P13 =1/2

0 0 P23 = 1

0 0 0

1 2 3

1

2

3

j

i

A simple example

Define:

•Let r be the time interval between trains.

•From the Central Limit Theorem Nis (the number of passengers at Mi 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

A simple example

Constraints:

Train capacity: C

r {4,…,20}

Not reaching capacity 95 percent of the time.

A simple example

Find the smallest r to satisfy all the equations:

95% => z = 1.65

Equation 1:

Equation 2:

65.1)(

)(

1

1

s

s

r

rC

65.1)()1)((

)()1)((

2121

2121

ss

ss

rPr

rPrC

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.

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

A simple exampleSecond 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

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

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

Simulation TOWARD INCREASED USE OF SIMULATION TRANSPORTATION

Construction Feasibility: Signal Design: Power Consumption: Traffic Studies: Railroad Capacity Studies: Train Operations Studies:

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