Metro Scheduling

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Metro Scheduling
By Philip Anderson Liza John
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Metro Scheduling

• Case Study
• Real world Practice

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A simple example Model
Station 2
Station 1
Station 3
?3
?2
?1
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A simple example Arrival Rates
Passenger arrival at each station can be modeled
as a Poisson process having time variable rate ?.
?i
t
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A simple example Arrival Rates
?
t
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A simple example Arrival Rates
?i
?is
s
t
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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.
j
1 2 3
0 P12 1/2 P13 1/2
0 0 P23 1
0 0 0
1 2 3
i
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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(?is) an variance r(?is)
  .
• Objective
• Create a schedule for period s by specifying r to
  minimize cost and guarantee capacity constraints

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A simple example

• Constraints
• Train capacity C
• r 4,,20
• Not reaching capacity 95 percent of the time.

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A simple example

• Find the smallest r to satisfy all the equations
• 95 gt z 1.65
• Equation 1
• Equation 2

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A simple example
Results First select the smallest r from solving
equation 1 and 2. If r is gt 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.
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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

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A simple exampleSecond Look

• Rush Hours
• 630AM - 930AM and 330PM - 800PM
• Regular Population Density Hours
• 930AM - 330PM and 800PM - 1200AM
• Late Night Hours
• 12AM - 630AM

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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

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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

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Simulation TOWARD INCREASED USE OF SIMULATION
TRANSPORTATION

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

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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