Module 3 : Microscopic Traffic Flow Modeling
Lecture 16 : Microscopic Traffic Simulation
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Building Traffic Simulator

The basic steps involved in the development are same irrespective of the type of model. The different activities involved are the following.
  1. Define the problem and the model objectives
  2. Define the system to be studied - Roadway, Vehicle and Driver characteristics
  3. Model development
  4. Model calibration
  5. Model verification
  6. Model validation
  7. Documentation
The most significant steps among the above are described with the help of stating the procedure for developing a microscopic model.

Model development

The framework of a model consists of mainly three processes as mentioned below.
  1. Vehicle generation
  2. Vehicle position updation
  3. Analysis
The flow diagram of a microscopic traffic simulation model is given in Figure 16.2. The basic structure of a model includes various component models like car following models like car following models, lane changing models etc. which come under the vehicle position updation part. In this chapter, the vehicle generation stage is explained in detail. The vehicles can be generated either according to the distributions of vehicular headways or vehicular arrivals. Headways generally follow one of the following distributions.
  1. Negative Exponential Distribution (Low flow rate)
  2. Normal Distribution (High flow rate)
  3. Erlang Distribution (Intermediate flow rate)
The generation of vehicles using negative exponential distribution is demonstrated here. The probability distribution function is given as follows.

$\displaystyle f(x) = \lambda e^{-\lambda x}$ (1)

From the above equation, the expression for exponential variate headway X can be derived as:

$\displaystyle X = \mu(-\log_e~R)$ (2)

where, $ \mu$ is the mean headway, R is the random number between 0 and 1
Figure 1: Flow diagram of a Microscopic traffic simulation model
\includegraphics[width = 14cm]{qfflowdiagram}
Random number generation is an essential part in any stochastic simulation model, especially in vehicle generation module. Numerous methods in terms of computer programs have been devised to generate random numbers which appear to be random. This is the reason why some call them pseudo-random numbers. Therefore headways can be generated using the above expression by giving a random number and the mean headway as the input variables.

In a similar way, the vehicular arrival pattern can be modeled using Poisson's distribution. The probability mass function is given as:

$\displaystyle p(x) = \frac{\lambda^x e^{-\lambda}}{x!}$ (3)

where, p(x) is the probability of x vehicle arrivals in an interval t, $ \lambda$ is the mean arrival rate of vehicles If the probability of no vehicle in the interval t is given as p(0), then this probability is same as the probability that the headway greater than or equal to t.

Numerical example

Given flow rate is 900 veh/hr. Simulate the vehicle arrivals for 1 min using negative exponential distribution.

Solution

Step 1: Calculate the mean headway $ \mu =\frac{1}{(900/3600)}=4 sec$. Step 2: Generate the random numbers between 0 and 1. Step 3: Calculate the headways and then estimate the cumulative headways. The calculations are given in Table 1

$\displaystyle X = \mu(-log_eR)$    


Table 1: Vehicle arrivals using Negative exponential distribution
Veh. No. R X Arrival time (sec)
1 0.73 1.23 1.23
2 0.97 0.14 1.37
3 0.27 5.26 6.63
4 0.44 3.25 9.88
5 0.52 2.63 12.51
6 0.77 1.05 13.55
7 0.43 3.39 16.94
8 0.81 0.84 17.79
9 0.08 9.96 27.75
10 0.74 1.18 28.93
11 0.53 2.58 31.51
12 0.81 0.83 32.34
13 0.15 7.46 39.80
14 0.44 3.26 43.06
15 0.29 5.02 48.08
16 0.68 1.56 49.63
17 0.05 12.09 61.72


Numerical example

The hourly flow rate in a road section is 900 veh/hr. Use Poisson distribution to model this vehicle arrival for 10 min.

Solution

Step 1: Calculate the no. of vehicles arriving per min. $ \lambda$ = 900/60 = 15 veh/min. Step 2: Calculate the probability of 0, 1, 2, ... vehicles per minute using Poisson distribution formula. Also calculate the cumulative probability as shown below.

$\displaystyle p(x) = \frac{\lambda^x e^{-\lambda}}{x!}$    

Step 3: Generate random numbers from 0 to 1. Using the calculated cumulative probability values, estimate the no. of vehicles arriving in that interval as shown in Table below.

Table 2: Calculation of probabilities using Poisson distribution
n p(x=n) p(x<=n)
0 0.000 0
1 0.000 0.000
2 0.000 0.000
3 0.000 0.000
4 0.001 0.001
5 0.002 0.003
6 0.005 0.008
7 0.010 0.018
8 0.019 0.037
9 0.032 0.070
10 0.049 0.118
11 0.066 0.185
12 0.083 0.268
13 0.096 0.363
14 0.102 0.466
15 0.102 0.568
16 0.096 0.664
17 0.085 0.749
18 0.071 0.819
19 0.056 0.875
20 0.042 0.917



Table 3: Vehicle arrivals using Poisson distribution
t (min) R n
1 0.231 11
2 0.162 10
3 0.909 19
4 0.871 18
5 0.307 12
6 0.008 6
7 0.654 15
8 0.775 17
9 0.632 15
10 0.901 20
    143


Here the total number of vehicles arrived in 10 min is 143 which is almost same as the vehicle arrival rate obtained using negative exponential distribution.