Uncertainty in Trip Planning in Transportation Systems

1. Uncertainty in Trip Planning in Transportation Systems A. P. Sistla (Joint work with Booth, Wolfson and Cruz)

2. Features of Urban Transportation • Multi-modal: Trains, buses, auto, pedestrians... • Uncertainties in travel times • Provides facilities at different stops

3. Graph Model of the network • Use labeled Multi-graph • Nodes --- Represent stops and stations • Edges --- Connections by different modes • Node Labels --- Name, Geometry and Facilities • Facilities : restaurants, supermarkets, etc. • Edge Labels--- Mode, Route#, Run#, Departure time, Duration

4. Example Network

5. Querying Trips • Find a trip from work to home leaving after 5PM and reaching before 6PM. • Trip may involve mutiple modes, require some stopping points. • Some Definitions: LEG: A path with all edges having same mode, route#, run#. TRIP: A sequence of legs (L1,L2,...,Ln) end point of each leg is the start if the next leg.

6. Trip Example

7. Query Language • Should be easy to use. • Should be able to specify intermediate stops, • Certainty as probability • Conditions involving on travel times • Use an operator ALL-TRIPS(Origin, Destination) • Employ SQL-like syntax

8. Query Format • SELECT * FROM ALL-TRIPS (origin,destination) WITH Stop-vertices WITH Modes WITH CERTAINTY prob-value WHERE condition OPTIMIZE ttribute

9. Example • SELECT * FROM ALL-TRIPS(work, home) as t WITH MODES pedestrian, bus WITH CERTAINTY 0.8 WHERE finish-time(t) <= 5PM • CERTAINTY clause specifies the probability that the trip satisfies the where condition

10. Another Example • SELECT * FROM ALL-TRIPS(home, theater) as t WITH STOP POINT v WHERE “ATM” in v.facilities AND start-time(t) >= 2PM AND finish-time(t) <=3PM WITH CERTAINTY 0.95

11. Semantics and Processing of Queries • Need precise semantics of queries including the certainty clause. • Consider the “home-to-theater” query • Let F be the set of possible trips • Consider a trip t = (L1,L2) from F • Let v be the intermediate stop. • Evaluate and simplify the where condition on trip t. Assume it satisfies non-temporal conditions. Let C be the resulting condition.

12. Query Processing Continued • In the example C is --- start-time(t)>= 2PM AND finish-time(t)<=3PM. • Let Y1, Y2 be random variables denoting departure times of legs L1 and L2. • Let Z1, Z2 be the random variables representing durations of L1 and L2. • We want to compute the probability that trip t satisfies condition C.

13. Computation of Probabilities • We need joint density function of Y1,Y2,Z1,Z2. Let f(y1,y2,z1,z2) be this function. • Departure time of L1 is Y1. • Arrival time at destination is (Y2+Z2). • Transform the condition C using y1,y2,z1,z2: y1 >= 2PM AND y2+z2 <= 3PM. • Also need to make sure transfer at v is successful.

14. Successful transfer probability • What is the probability that the transfer at v is successful? • Required stop time for successful transfer: RT • Arrival time of L1 --- Y1+Z1 • Duration at v: Y2 – (Y1+Z1). • Transfer condition: (y2- (y1+z1)) >= RT

15. Final Probability • Combined condition: y1 >=2PM AND y2+z2 <= 3PM AND (y2-y1-z1) >=RT. • The combined condition defines a region X in the 4-dimensional space. • Probability t satisfies the where condition is the definite integral f(y1,y2,z1,z2) dy1 dy2 dz1 dz2 over the region X.

16. Practical Approach • Assume that travel times on different routes are independent. • The joint density function can be written as product of independent density functions. • Density functions can be maintained as tables. • Further research needed for implementation.

17. Alternates models of uncertainties • Other possibilities: Using uncertainty intervals • Other query constructs: Definitely and Possibly.

18. Questions?