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1 Origin, Destination, and Travel Time Distribution Without Surveys Copyright, 1996 © Dale Carnegie & Associates, Inc. INFORMS SLC May 2000 Larry George Problem Solving Tools

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4/28/00PST2 Sample or population? zAir travel surveys ~10%, on the day surveyors choose zCommuter surveys have non-response, bias. “Why are they surveying me?” zWant O-D population proportions and travel time distributions “It’s human nature to doubt statistically significant conclusions based on a sample that is a small fraction of the population”

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4/28/00PST3 M/G/ G(t) is estimable, without customer id Input Poisson, service time distribution is G(t), infinite servers Output is Poisson at rate G(t) zArrival and departure times or counts, without id, are sufficient zDitto M t /G/ under mild restrictions

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4/28/00PST4 Tandem M t /G/ Input Poisson G 1 (t) to a second M t /G 2 / zOutput is Poisson at rate G 1 (u)G 2 (t-u)dG 1 (u) zEstimate G 1 and G 2 from input, intermediate, and output observations

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4/28/00PST5 M/G 2 / 2 /p zTwo M/G/ systems, one N-S and the other E-W, customers switch with pr. p zObserve 2 inputs and 2 outputs Estimate 1 and 2 from inputs yEstimate p and G(t) from output rates r 1 = ((1-p) 1 +p 2 )G(t) r 2 = ((1-p) 2 +p 1 )G(t)

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4/28/00PST6 M/G 2 / 2 /p with one obs. zObserve output rate r j at time t p = ( 2 *r 1 - 1 *r 2 ) / ((- 1 + 2 )*(r 1 + r 2 )) G(t) = (r 1 + r 2 )/( 1 + 2 ) zExample r 1 = 4, r 2 = 3, 1 = 10, 2 = 20 y=> p = 5/7 and G(t) = 7/30

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4/28/00PST7 Solutions for several obs. zTwo obs. gives multiple solutions. Use lse. r 1 (1) = 3, r 2 (1)= 4, r 1 (2) = 6, r 2 (2)= 8, 1 = 10, 2 = 20 y=> p = 0.285714, G(1) = 0.233333, G(2) = 0.466667 zMultiple obs. yp 1, p 2, G 11 (t), G 12 (t), G 21 (t), G 22 (t) yDepends on whether you want t or all (i,j)

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4/28/00PST8 M t /G k / k net with switching zObserve node I/O; inputs can have nonstationary rates zAssume stationary switching and travel time distributions zRoute frequency is product of switch probabilities (assume independence) zTravel time is convolution of intra-node travel time distributions

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4/28/00PST9 Summary zO-D proportions and travel time distributions don’t require customer id zPopulation data beats sample data zWorks for transient processes: e.g. commutes, BART, other daily cycles and traffic light cycles

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