Modeling Traffic Characteristics

Lecture notes in Transportation Systems Engineering

4 August 2009

Modeling time-headway's

Modeling inter arrival time or time headway or simply headway is he time interval between the successive arrival of two vehicles at a given point. This is a continuous variable and can be treated as a random variable.

Classification of headway distribution

One can observe three types of flow in the field:
  1. Low volume flow
    1. Headway follow a random process as there is no interaction between the arrival of two vehicles.
    2. The arrival of one vehicle is independent of the arrival of other vehicle.
    3. The minimum headway is governed by the safety criteria.
    4. A negative exponential distribution can be used to model such flow
  2. High volume flow
    1. This is characterized by 'near' constant headway
    2. The flow is very high and is near to the capacity
    3. The mean is very low and so is the variance
    4. A normal distribution can used to model such flow
  3. Intermediate flow
    1. Some vehicle travel independently and some vehicle has interaction
    2. More difficult to analyzed and has more application in the field.
    3. Pearson Type III Distribution can be used which is a very general case of negative exponential distribution and normal distribution.

Negative exponential distribution

Normal distribution

Pearson type III distribution

Example

An obseravtion from 3424 samples is given table below. Mean headway observed was 3.5 seconds and the standard deviation 2.6 seconds. Fit a (i) negatie expoetial distribution, (ii) normal distrbution and (iii) Person Type III Distribution.
Table 1: Obsered headway distribution
t t+dt p (obs)
0.0 0.5 0.012
0.5 1.0 0.064
1.0 1.5 0.114
1.5 2.0 0.159
2.0 2.5 0.157
2.5 3.0 0.130
3.0 3.5 0.088
3.5 4.0 0.065
4.0 4.5 0.043
4.5 5.0 0.033
5.0 5.5 0.022
5.5 6.0 0.019
6.0 6.5 0.014
6.5 7.0 0.010
7.0 7.5 0.012
7.5 8.0 0.008
8.0 8.5 0.005
8.5 9.0 0.007
9.0 9.5 0.005
9.5 $ >$ 0.033
Total   1.00

Solutions

Table 2: Solution using negative exponential distribution
t t+dt p (obs) Pobs=p*N $ p(h>=t)$ N-exp (p) P=p*N
0.0 0.5 0.012  41.1 1.000 0.133 455.8
0.5 1.0 0.064 219.1 0.867 0.115 395.1
1.0 1.5 0.114 390.3 0.751 0.100 342.5
1.5 2.0 0.159 544.4 0.651 0.087 296.9
2.0 2.5 0.157 537.6 0.565 0.075 257.4
2.5 3.0 0.130 445.1 0.490 0.065 223.1
3.0 3.5 0.088 301.3 0.424 0.056 193.4
3.5 4.0 0.065 222.6 0.368 0.049 167.7
4.0 4.5 0.043 147.2 0.319 0.042 145.4
4.5 5.0 0.033 113.0 0.276 0.037 126.0
5.0 5.5 0.022  75.3 0.240 0.032 109.2
5.5 6.0 0.019  65.1 0.208 0.028  94.7
6.0 6.5 0.014  47.9 0.180 0.024  82.1
6.5 7.0 0.010  34.2 0.156 0.021  71.2
7.0 7.5 0.012  41.1 0.135 0.018  61.7
7.5 8.0 0.008  27.4 0.117 0.016  53.5
8.0 8.5 0.005  17.1 0.102 0.014  46.4
8.5 9.0 0.007  24.0 0.088 0.012  40.2
9.0 9.5 0.005  17.1 0.076 0.010  34.8
9.5 $ >$ 0.033 113.0 0.066 0.066 226.8
Total   1.000  3424   1.000  3424
Table 3: Solution using normal distribution
h t+dt p (obs) P=p*N $ p(h<=t)$ $ p(t<h<t+0.5)$ P=p*N
0.0 0.5 0.012 41.09 0.010 0.013 44.289
0.5 1.0 0.064 219.14 0.023 0.025 85.738
1.0 1.5 0.114 390.34 0.048 0.043 148.673
1.5 2.0 0.159 544.42 0.091 0.067 230.928
2.0 2.5 0.157 537.57 0.159 0.094 321.299
2.5 3.0 0.13 445.12 0.252 0.117 400.433
3.0 3.5 0.088 301.31 0.369 0.131 447.033
3.5 4.0 0.065 222.56 0.500 0.131 447.033
4.0 4.5 0.043 147.23 0.631 0.117 400.433
4.5 5.0 0.033 112.99 0.748 0.094 321.299
5.0 5.5 0.022 75.33 0.841 0.067 230.928
5.5 6.0 0.019 65.06 0.909 0.043 148.673
6.0 6.5 0.014 47.94 0.952 0.025 85.738
6.5 7.0 0.01 34.24 0.977 0.013 44.289
7.0 7.5 0.012 41.09 0.990 0.006 20.492
7.5 8.0 0.008 27.39 0.996 0.002 8.493
8.0 8.5 0.005 17.12 0.999 0.001 3.153
8.5 9.0 0.007 23.97 1.000 0.000 1.048
9.0 9.5 0.005 17.12 1.000 0.000 0.312
9.5 $ >$ 0.033 112.99 1.000 0.010 33.716
    3424        

Table 4: Solution using Pearson type III distribution
h t+dt p (obs) P=p*N f(t) $ p(t<h<t+0.5)$ P=p*N
0.0 0.5 0.012 41.09   0.000 0.00
0.5 1.0 0.064 219.14 0.000 0.066 225.91
1.0 1.5 0.114 390.34 0.264 0.127 433.26
1.5 2.0 0.159 544.42 0.242 0.114 389.45
2.0 2.5 0.157 537.57 0.213 0.099 339.13
2.5 3.0 0.13 445.12 0.183 0.085 291.12
3.0 3.5 0.088 301.31 0.157 0.072 247.86
3.5 4.0 0.065 222.56 0.133 0.061 209.90
4.0 4.5 0.043 147.23 0.112 0.052 177.08
4.5 5.0 0.033 112.99 0.095 0.044 148.97
5.0 5.5 0.022 75.33 0.079 0.037 125.04
5.5 6.0 0.019 65.06 0.067 0.031 104.78
6.0 6.5 0.014 47.94 0.056 0.026 87.67
6.5 7.0 0.01 34.24 0.047 0.021 73.28
7.0 7.5 0.012 41.09 0.039 0.018 61.18
7.5 8.0 0.008 27.39 0.033 0.015 51.04
8.0 8.5 0.005 17.12 0.027 0.012 42.54
8.5 9.0 0.007 23.97 0.023 0.010 35.44
9.0 9.5 0.005 17.12 0.019 0.009 29.51
9.5 $ >$ 0.033 112.99 0.016 0.102 350.83
  x 3424        

Comparison of distributions

Examples compared

Evaluating the selected distribution

Modeling vehicle arrival

Modeling speed

Estimation of population and sample means

Bibliography

1 Adolf D. May. Fundamentals of Traffic Flow. Prentice - Hall, Inc. Englewood Cliff New Jersey 07632, second edition, 1990.

Prof. Tom V. Mathew 2009-08-04