Lisa's Master's Thesis Project

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

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Abstract
In this paper I simulate train traffic congestion on Lexington Avenue subway line by
using a numerical model derived from fluid flow. My numerical technique is collocation
method with Chebyshev polynomial interpolation. In addition, I use two separate
collocation implementations. Each highlights different aspects of the situation.


Table of Contents
§ References
§ Theory
§ Traffic Model
§ Findings
§ Collocation and Chebyshev-Gauss quadrature and Computational Results
§ Lag-Time Linearization and Computational Results


References
1. 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.


Theory
Traffic model developed in [1] is based on a fluid dynamic model obtained using the mass conservation equation
and Fick's law.

∑u/∑t + ∑q/∑x = 0

u = u(t, x) is the mass density
q = q(t, x) is the flow

Introducing 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 density

v=1–u

∑u/∑t + ∑u(1-u)/∑x = 0


Traffic Model I
u* = 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 driver's reaction to uncertainty.

Replacing u with u* in the equation for q

q = u*(1 – u*)

= u*– (u*)2

= u [ 1 + h – h u ∑u/∑x ] – ( u [ 1 + h - h u ∑u/∑x] )2


Traffic Model II
Inserting 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)2

ﬂ

∂u/∂t = h u2 ( 1 – u ) (∂2u)/∂x2 + h u ( 2 – 3 u ) (∂u/∂x)2 – ( 1 – 2 u ) ∂u/∂x

By implementing the above differential equation numerically, we model the traffic density, u, for time t and
location x.


Traffic Model III
In 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 center's 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, or
breaking hard to maintain distance. Both drivers and controllers are subject to delayed responses to changing
conditions. 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.


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 35

Express Peak 19 2 3.5 25.85 64.78

Express Off 19 4 5.06 21.05 12.79


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 -


Findings (Local)
Local Peak Local Express
10/21/10 10/27/10
Date/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 18.86
Bleeker St. 5:29 5:39 14.14 15.33 Bleeker St. 12:32 12:45 20.44 13.31
Spring St. 5:30 5:38 14.21 14.76 Spring St. 12:34 12:44 14.26 15.00
Canal St. 5:31 5:36 13.94 15.85 Canal St. 12:35 12:43 14.95 15.41
Brooklyn Brooklyn
5:34 5:35 - - 12:37 12:41 - -
Bridge Bridge
Total Time 21 minutes 19 minutes - - Total Time 21 minutes 19 minutes - -


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.


Findings (Conclusion)
Evidently, train dispatch control center decided to have the trains bypass less crowded stations. When trains run
smoothly during rush hour, the door open-close time is shorter in single line stations. Transfer hubs have a greater
door open-close time.

The findings confirm Bellomo et al's traffic modeling; MTA works hard to prevent the stop and go traffic
phenomenon. They use procedures such as the station skipping as mentioned above. Without that, trains would be
held in tunnels and would move at a slow speed.


Collocation
Assume 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


Chebyshev-Gauss quadrature
Designed 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 error
xi = 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 polynomial
But 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 to
recover u.


Computational Results
i)


Computational Results
ii)


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


Lag-Time Computational Results
The 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 and
areas of relatively low density.

iii)


Lag-Time Computational Results
iv)


Bellomo et al's Computational Results


Lisa's Master's Thesis Project

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