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Lisa's Master's Thesis Project

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It is a research and programming (mathematica) project, about transportation (traffic) modeling.

I spent about 3 months (September - early December 2010) taking the trains, relying on my Baby-G's stop watch function to keep record of door open and close time for each stop and duration between stops.

After that the concentration went to testing the differential equation, and then expanded the original coding method with a different numerical approach, that's where the lag-time process came in.

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Lisa's Master's Thesis Project

  1. 1. Numerical Modeling of Train Traffic: Lexington 4, 5, 6 Trains, Using Collocation Method with Chebyshev Interpolation by Kuoi “Lisa” Ueda Department of Mathematics and Statistics Hunter College (CUNY) Adviser: Professor John Loustau March 2011 This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  2. 2. AbstractIn this paper I simulate train traffic congestion on Lexington Avenue subway line byusing a numerical model derived from fluid flow. My numerical technique is collocationmethod with Chebyshev polynomial interpolation. In addition, I use two separatecollocation implementations. Each highlights different aspects of the situation. This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  3. 3. Table of Contents § References § Theory § Traffic Model § Findings § Collocation and Chebyshev-Gauss quadrature and Computational Results § Lag-Time Linearization and Computational Results This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  4. 4. References1. Bellomo, Nicola, Bertrand Lods, Roberto Revelli, Luca Ridolfi, "Generalized Collocation Methods, Solutions to Nonlinear Problems", Birkhauser, 2002.2. Hildebrand, F. B., "Introduction to Numerical Analysis", McGraw - Hill, 1956. This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  5. 5. TheoryTraffic model developed in [1] is based on a fluid dynamic model obtained using the mass conservation equationand Ficks law.∑u/∑t + ∑q/∑x = 0u = u(t, x) is the mass densityq = q(t, x) is the flowIntroducing a new variable, velocity v = v(t, x), writing q as a product of u and v,and expressing velocity so that it varies inversely to densityv=1–u∑u/∑t + ∑u(1-u)/∑x = 0 This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  6. 6. Traffic Model Iu* = u [1 + h (1 – u) ∑u/∑x]is the "apparent local density" or "local fictitious density" proposed by Bellomo et al.The coefficient h is intended to represent the drivers reaction to uncertainty.Replacing u with u* in the equation for qq = u*(1 – u*) = u*– (u*)2 = u [ 1 + h – h u ∑u/∑x ] – ( u [ 1 + h - h u ∑u/∑x] )2 This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  7. 7. Traffic Model IIInserting the last expression into the basic differential equation for the proposed variables u and v yields∑u/∑t + ∑/∑x( u [ 1 + h – h u ∑u/∑x ] – ( u [ 1 + h - h u ∑u/∑x] )2 ) = 0∑u/∑t + ∑/∑x( u [ 1 + h – h u ∑u/∑x ] ) = ∑ /∑x (u [ 1 + h - h u ∑u/∑x] )2)∑u/∑t + ( 1 – 2 u ) ∑u/∑x = h u2 ( 1 – u ) (∑2u)/∑x2 + h u ( 2 – 3 u ) (∑u/∑x)2fl∂u/∂t = h u2 ( 1 – u ) (∂2u)/∂x2 + h u ( 2 – 3 u ) (∂u/∂x)2 – ( 1 – 2 u ) ∂u/∂xBy implementing the above differential equation numerically, we model the traffic density, u, for time t andlocation x. This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  8. 8. Traffic Model IIIIn a subway system – § Density can be measured via train arrival frequency at any station. The maximal density is governed by work rules designating the minimum distance between trains. § Locations of trains in front or behind are unknown to train drivers. Drivers rarely, if ever, see another train. § Drivers rely on the control centers instruction to stop, go, slowdown, and speedup. Information is transmitted by signal lights and radio communication.A train driver at a fixed point in a tunnel feels the "apparent local density" or the "local fictitious density, when – § Told by the control center that train congestion is ahead. § Seeing large number of passengers on the platform when train is pulling into the station.Drivers and controllers can over-react to the condition in the subway system by over-accelerating the train, orbreaking hard to maintain distance. Both drivers and controllers are subject to delayed responses to changingconditions. A stopped train needs time to regain speed causing further delays in the trains that are behind.The result of over-reaction would create a "stop and go" traffic pattern. This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  9. 9. Findings (Summary) Number of Arrivals Mean Arrival Arvl. Frequency Mean Station Time Station Time Uptown & Frequency (in Variance (in seconds) Variance Downtown minutes) Local Peak 63 2 2.55 22.82 448.59 Local Off 26 5 11.36 21.09 35Express Peak 19 2 3.5 25.85 64.78 Express Off 19 4 5.06 21.05 12.79 This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  10. 10. Findings (Express) Express Peak Date/Time Station 59th Street 42nd Street 14th Street Brooklyn Bridge Total Time To 8:54 8:59 9:04 9:09 15 minutes Back 9:26 9:23 9:17 9:11 15 minutes 11/19/10 8:54 am 25.02 23.81 17.54 30.65 - Station Time (in seconds) 18.26 32.92 20.65 19.50 - Express Off Peak Date/Time To 6:51 6:55 6:58 7:03 13 minutes Back 7:16 7:13 7:08 7:04 13 minutes 11/17/10 17.98 25.14 23.37 14.36 - 6:51 pm Station Time (in seconds) 20.99 24.63 21.42 20.54 - This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  11. 11. Findings (Local) Local Peak Local Express 10/21/10 10/27/10Date/Time Date/Time 5:15 pm 12:16 pm Station Time Station Time Station To Back Station To Back (in seconds) (in seconds) 68th St. 5:13 5:54 20.89 13.95 68th St. 12:16 13:00 17.62 19.94 59th St. 5:15 5:52 24.80 17.55 59th St. 12:17 12:58 23.25 20.10 st st 51 St. 5:16 5:51 24.88 15.37 51 St. 12:18 12:56 20.78 19.31 nd nd 42 St. 5:18 5:48 20.97 21.73 42 St. 12:21 12:55 22.21 20.76 33rd St. 5:20 5:46 14.39 15.16 33rd St. 12:23 12:53 17.18 14.69 th th 28 St. 5:22 5:45 20.27 13.18 28 St. 12:25 12:51 16.60 16.43 rd rd 23 St. 5:23 5:43 15.02 13.64 23 St. 12:26 12:50 17.75 17.85 th th 14 St. 5:26 5:42 21.03 19.76 14 St. 12:28 12:48 23.92 17.50 Astor Pl. 5:27 5:40 13.38 13.75 Astor Pl. 12:30 12:47 15.33 18.86Bleeker St. 5:29 5:39 14.14 15.33 Bleeker St. 12:32 12:45 20.44 13.31Spring St. 5:30 5:38 14.21 14.76 Spring St. 12:34 12:44 14.26 15.00Canal St. 5:31 5:36 13.94 15.85 Canal St. 12:35 12:43 14.95 15.41Brooklyn Brooklyn 5:34 5:35 - - 12:37 12:41 - - Bridge BridgeTotal Time 21 minutes 19 minutes - - Total Time 21 minutes 19 minutes - - This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  12. 12. Findings (Exceptional) 11/05/10 11/08/10 Date/Time 5:53 pm 6:20 pm Station To Back Station Time To Back Station Time *5:53 34.36 68th St. 6:40 25.28 6:20 7:02 22.22 25.70 5:55 25.28 59th St. 5:58 6:39 33.21 28.65 6:22 7:00 30.39 24.57 51st St. 6:00 6:36 35.68 27.19 6:24 6:58 36.26 26.49 42nd St. *6:03 6:33 50.16 227.66 6:27 *6:56 29.63 37.14 33rd St. 6:05 6:31 36.52 18.39 6:30 6:54 21.65 16.97 28th St. 6:07 6:30 42.17 19.51 6:31 6:53 20.83 14.76 23rd St. 6:09 6:28 40.12 17.41 6:33 6:51 23.04 17.57 14th St. 6:10 6:26 39.95 22.73 6:35 6:50 36.99 16.29 Astor Pl. Skipped 6:24 - 18.25 Skipped 6:48 - 17.68 Bleeker St. 6:13 6:23 30.96 17.17 6:38 6:47 21.53 15.66 Spring St. Skipped 6:22 - 15.20 Skipped 6:45 - 14.46 Canal St. Skipped 6:21 - 17.75 Skipped 6:44 - 14.32 Brooklyn 6:16 6:19 - - 6:42 6:43 - - Bridge Total Time 21 minutes 21 minutes 22 minutes 19 minutes* On Nov. 8 2010: I documented two arrivals for 68th Street station, because I had to get on the 2nd train arrived at 5:55 PM after I failed to squeeze myself into the first train. This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  13. 13. Findings (Conclusion)Evidently, train dispatch control center decided to have the trains bypass less crowded stations. When trains runsmoothly during rush hour, the door open-close time is shorter in single line stations. Transfer hubs have a greaterdoor open-close time.The findings confirm Bellomo et als traffic modeling; MTA works hard to prevent the stop and go trafficphenomenon. They use procedures such as the station skipping as mentioned above. Without that, trains would beheld in tunnels and would move at a slow speed. This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  14. 14. CollocationAssume track length = 1. Hence our spatial variable x lies in [0, 1] § Set of collocation points Z = {xi}, i = 1, . . ., 20. § pi[xj] = dij. § For any continuous function s, polynomial interpolation satisfies PZ(s) = ∑i s(xi)pi in 19. § 0 = h u2[(1 – u)∑2u/∑x2 + h u(2 – 3u)(∑u/∑x)2 – (1 – 2u)∑u/∑x. To simplify, we rewrite this as 0 = L(u). Then the collocation solution to the equation is an element uP in 19 with the property that 0 = PZ(L(uP)). PZ is zero on a function if and only if L(uP)(xi) = 0 for all i. This yields a system of equations. § ajn+1 = t [ L( ∑j ajn pj )](xi) + ajn This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  15. 15. Chebyshev-Gauss quadratureDesigned for functions that have vertical tangents at the end points of the domain.s(x) = u(x)/Sqrt[x(1-x)]Select interpolation points (collocation points) that will minimize the errorxi = 1/2 – 1/2 Cos ((i - 1)/(n - 1))§ Steep tangents at the boundary of the congested region§ It is possible that s behaves like a polynomial function and may be approximated by a polynomialBut then up is an approximation of s, not u. Hence at the end we must multiply the result by Sqrt[x(1-x)] in order torecover u. This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  16. 16. Computational Resultsi) This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  17. 17. Computational Resultsii) This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  18. 18. Lag-Time Linearization § Alternative implementation § A time step procedure where each time state is a linear function of the prior one. Our differential equation given by ∑u/∑t = h u2[(1 – u)∑2u/∑x2 + h u(2 – 3u)(∑u/∑x)2 – (1 – 2u)∑u/∑x is non-linear. With the time lag process, we can linearize the differential equation by asserting prior state ajn-1 ,and then calculate the next state ajn+1 as a linear transformation of the current state ajn. § ajn+1 = t [ N( ∑j ajn-1 pj ) ( ∑j ajn pj )](xi) + ajn This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  19. 19. Lag-Time Computational ResultsThe following graphs show the output of the lag-time process.Graph iii) shows a folding effect that is actually rubber-banding – between areas of relatively high density andareas of relatively low density.iii) This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  20. 20. Lag-Time Computational Resultsiv) This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.
  21. 21. Bellomo et als Computational Results This PDF file is Created by trial version of Quick PDF Converter Suite. Please use purchased version to remove this message.

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