Module 3 : Microscopic Traffic Flow Modeling

Lecture 12 : Vehicle Arrival Models: Headway

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Numerical Example

An observation from 2434 samples is given table below. Mean headway and the standard deviation observed is 3.5 and 2.6 seconds respectively. Fit a negative exponential distribution.

Table 1: Observed headway distribution

$h$	$h+dh$	$p_i^o$
0.0	1.0	0.012
1.0	2.0	0.178
2.0	3.0	0.316
3.0	4.0	0.218
4.0	5.0	0.108
5.0	6.0	0.055
6.0	7.0	0.033
7.0	8.0	0.022
8.0	9.0	0.013
9.0	$>$	0.045
Total		1.00

Solution:

The solution is shown in Table 2. The headway range and the observed probability (or proportion) is given in column (2), (3) and (4). The observed frequency for the first interval (0 to 1) can be computed as the product of observed frequency $p_i$ and the number of observation (N). That is, $f_i^o = p_i\times N = 0.012\times2434 = 29.21$ and is shown in column (5). The probability that the headway greater than $t=0$ is computed as $p(t\ge0)=e^{-0}=1$ (refer equation ) and is given in column (6). These steps are repeated for the second interval, that is $f_i^o=0.178\times2434=433.25$, and $p(t\ge 1)=e^{-1}=0.751$. Now, the probability of headway lies between 0 and 1 for the first interval is given by the probability that headway greater than zero from the first interval minus probability that headway greater than one from second interval. That is $p_i(0\le t\le 1)=p_i(t>0)-p_i(t> 1)=1.00-0.751=0.249$ and is given in column (7). Now the computed frequency $f_i^c$ is $p_i\times N = 0.249\times2434=604.904$ and is given in column (8). This procedure is repeated for all the subsequent items. It may be noted that probability of headway $>$ 9.0 is computed by 1-probability of headway less than 9.0 $= 1-(0.249+0.187+\dots)=0.076.$

Table 2: Illustration of fitting a negative exponential distribution

$No$	$h$	$h+dh$	$p_i^o$	$f_i^o$	$p(t>=h)$	$p_i^c$	$f_i^c$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
1	0.0	1.0	0.012	29.21	1.000	0.249	604.904
2	1.0	2.0	0.178	433.25	0.751	0.187	454.572
3	2.0	3.0	0.316	769.14	0.565	0.140	341.600
4	3.0	4.0	0.218	530.61	0.424	0.105	256.705
5	4.0	5.0	0.108	262.87	0.319	0.079	192.908
6	5.0	6.0	0.055	133.87	0.240	0.060	144.966
7	6.0	7.0	0.033	80.32	0.180	0.045	108.939
8	7.0	8.0	0.022	53.55	0.135	0.034	81.865
9	8.0	9.0	0.013	31.64	0.102	0.025	61.520
10	9.0	$>$	0.045	109.53	0.076	0.076	186.022
	Total		2434			1.000	2434