ORF 510: Directed Research II

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• ORF 510 Directed Research II
• Analyzing travel time distributions using GPS
  data
• Santiago Arroyo
• Advisor Prof. Alain L. Kornhauser
• January 13th, 2004

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Objectives

• Obtain travel time distributions between
  monuments on the US road network in order to
• Find more realistic shortest paths (according to
  travel time, not distance)
• Obtain travel time patterns according to
  categories
• Day of week (week/weekend)
• Time of day
• Road type
• Forecast travel times in real time

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Concepts

• Road network US (Copilot)
• Different levels
• Level 0 30 x 106 arcs
• Level 1 500.000 arcs
• Level 2 10.000 arcs
• Level 3 1.000 arcs
• Nodes ??
• Monuments
• Midpoints of all/some arcs on level 1 (used
  originally to build network)
• ??

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Data

• Copilot
• Generated on Nov 14th, 2003
• Every 3 seconds
• Matched to link using position and heading
• Biased geographically and by users
• More precise than Qualcomm
• Identified by vehicle, position, heading, speed,
  date and time

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Data

• 1.500.000 monument to monument (m2m) times
• Max of observations for m2m pair 1202
• Total of m2m pairs 34879
• Only 1730 (4.96) pairs have 100 observations or
  more
• Biased around Princeton area

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Time vs. Speed

• Time
• Absolute (dont care about what happens in
  between measures)
• Readily extractable from data

• Speed
• Calculate estimated time using speed measurements
• More difficult to take into account changes in
  speed between points (intersections, left turns,
  stops)
• Significant errors around intersections due to
  matching

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Case Study
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Case Study
Mean 191.63 Median 115 St Dev 508.81
Mean 308.39 Median 197 St Dev 647.00
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Case Study
Mean 122.18 Median 114 St Dev 17.87
Mean 197.58 Median 197 St Dev 20.10
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Case Study
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Problem

• Stochastic Shortest Path
• Travel time is a function of departure time
• Discrete time of day intervals?
• Continuous function relating travel time and
  departure time
• Edge weights are random variables
• What are their distributions?
• Are they independent?
• How do we add distributions on a path?
• Memory limitations
• Cant store all possible paths
• Which paths should we store?

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Travel time as a function of departure time

• Schrader Kornhauser (2003)
• Milwaukee Highway System (Weekday travel time)

where TT travel time t departure time
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Travel time as a function of departure time

• Schrader Kornhauser (2003)
• Ten-parameter function fit to data from the
  Milwaukee Highway System

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Edge weights are random variables

• Extension of deterministic algorithms
• Dijkstras, Bellman-Ford, A and variations
• Use expected value as edge weight
• Need to know travel time distribution
• Stochastic Shortest Path algorithms
• Priority First Search (PFS) with dominance
  pruning (Wellman et al., 1995)
• Adaptive Path Planning (Wellman et al., 1995)
• Vertex-Potential Model (Cooper et al., 1997)
• Path Optimality Indexes (Sigal et al., 1980)

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Edge weights are random variables

• PFS with dominance pruning
• Stochastically consistent network
• cij(x) time dependent travel time from i to j
• For all i, j, sltt, and z Prs
  cij(s)ltzgtPrt cij(t)ltz
• i.e. the probability of arriving by any given
  time z cannot be increased by leaving later

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Edge weights are random variables

• PFS with dominance pruning
• Stochastic dominance
• Arrival time distribution at a node dominates
  another iff cumulative probability function is
  uniformly greater or equal to that of the other

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Edge weights are random variables

• PFS with dominance pruning
• Utility is nonincreasing with respect to arrival
  time
• How do we find arrival time distributions from a
  path with two edges or more?

• Priority queue only keep stochastically
  undominated paths
• Apply variations like A

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Further Research

• Establish m2m time travel distribution
  approximations (normal, lognormal, exponential,
  etc.)
• Addition of distributions on a path (stochastic
  model)
• Investigate on independence of distributions in a
  path (maybe as a function of edges in the path)
• Categorize by time of day, day of week, road type
• Develop travel time as a function of departure
  time using Copilot data
• Construct network with expected travel times
  instead of distances (PTNM)
• Use travel times from Copilot data to forecast
  travel times, incorporating real time data

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Bibliography

• C. Cooper, A. Frieze, K. Mehlhorn and V. Priebe,
  Average-case of shortest-paths problems in the
  vertex-potential model, International Workshop
  RANDOM97, Bologna, Italy
• A.M. Frieze and G.R. Grimmett, The Shortest-Path
  Problem for Graphs with Random Arc-Lengths,
  Discrete Applied Mathematics, 10 (1985) 57-77.
• S. Pallottino and M. Scutella, Shortest Path
  Algorithms in Transportation Models Classical
  and Innovative Aspects. In Marcotte, P., Nguyen,
  S., eds. Equilibrium and Advanced Transportation
  Modelling. Kluwer, Amsterdam (1998) 245-281.
• C. Schrader and Alain L. Kornhauser, Using
  Historical Information in Forecasting in Travel
  Times, BSE Thesis, Princeton University, 2003
• C. Elliot Sigal, A. Alan B. Pritsker and James J.
  Solberg, The Stochastic Shortest Route Problem,
  Operations Research, (1980) 1122-1129.
• Michael P. Wellman, Kenneth Larson, Matthew Ford
  and Peter R. Wurman, Path Planning Under
  Time-Dependent Uncertainty, Eleventh Conference
  on Uncertainty in Artificial Intelligence, 28-5
  (1995) 532-539.
• Fastest Path Problems in Dynamic Transportation
  Networks, in www.husdal.com

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