Module 3 : Microscopic Traffic Flow Modeling

Lecture 13 : Vehicle Arrival Models: Count

1

2

3

4

5

	Introduction As already noted in the previous chapter that vehicle arrivals can be modelled in two inter-related ways; namely modelling how many vehicle arrive in a given interval of time, or modelling what is the time interval between the successive arrival of vehicles. Having discussed in detail the former approach in the previous chapter, the first part of this chapter discuss how a discrete distribution can be used to model the vehicle arrival. Traditionally, Poisson distribution is used to model the random process, the number of vehicles arriving a given time period. The second part will discuss methodologies to generate random vehicle arrivals, be it the generation of random headways or random number of vehicles in a given duration. The third part will elaborate various ways of evaluating the performance of a distribution. Poisson Distribution Suppose, if we plot the arrival of vehicles at a section as dot in a time axis, it may look like Figure 1. Figure 1: Illustration of vehicle arrival modeling Let $h_1$, $h_2$, ... etc indicate the headways, then as mentioned earlier, they take some real values. Hence, these headways or inter arrival time can be modelled using some continuous distribution. Also, let $t_1$, $t_2$, $t_3$ and $t_4$ are four equal time intervals, then the number of vehicles arrived in each of these interval is an integer value. For example, in Fig. 1, 3, 2, 3 and 1 vehicles arrived in time interval $t_1$, $t_2$, $t_3$ and $t_4$ respectively. Any discrete distribution that best fit the observed number of vehicle arrival in a given time interval can be used. Similarly, any continuous distribution that best fit the observed headways (or inter-arrival time) can be used in modelling. However, since these process are inter-related, the distributions that describe these relations should also be inter-related for better explanation of the phenomenon. Interestingly, there exist distributions that meet the above requirements. First, we will see the distribution to model the number of vehicles arrived in a given duration of time. Poisson distribution is commonly used to describe such a random process. The probability density function of the Poisson distribution is given as: $$p(x)= \frac{\mu^x e^{-\mu}}{x!}$$ (1) where $p(x)$ is the probability for $x$ events will occur in the time interval, and $\mu$ is the expected rate of occurrence of that event in that interval. Some special cases of this distribution is given below. $$p(0) = e^{-\mu}$$ $$p(1) = \frac{\mu e^{-\mu}}{1} = \mu~p(0)$$ $$p(2) = \frac{\mu^2 e^{-\mu}}{2!} = \frac{\mu}{2}~p(1)$$ $$\therefore p(n) = \frac{\mu}{n}~p(n-1).$$ Since the events are discrete, the probability that certain number of vehicles ($n$) arriving in an interval can be computed as: $$p(x\le n) = \sum_{i=0}^{n} p(i),~i\in~I.$$ Similarly, the probability that the number of vehicles arriving in the interval is exactly in a range (between $a$ and $b$, both inclusive and $a<b$) is given as: $$p(a\le x\le b) = \sum_{i=a}^{b} p(i),~i\in~I.$$