Uncertainty in Trip
Planning in Transportation
Systems
A. P. Sistla
(Joint work with Booth, Wolfson and
Cruz)
Features of Urban Transportation



Multi-modal: Trains, buses, auto,
pedestrians...
Uncertainties in travel times
Provides facilities at different stops
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
Example Network
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.
Trip Example
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
Query Format

SELECT *
FROM ALL-TRIPS (origin,destination)
WITH Stop-vertices
WITH Modes
WITH CERTAINTY prob-value
WHERE condition
OPTIMIZE ttribute
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
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
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.
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.
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.
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
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.
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.
Alternates models of uncertainties


Other possibilities: Using uncertainty intervals
Other query constructs: Definitely and
Possibly.
Questions?