Module 3 : Microscopic Traffic Flow Modeling

Lecture 12 : Vehicle Arrival Models: Headway

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	Negative exponential distribution The low flow traffic can be modeled using the negative exponential distribution. First, some basics of negative exponential distribution is presented. The probability density function $f(t)$ of any distribution has the following two important properties: First, $$p[-\infty < t < +\infty] = \int_{-\infty}^{+\infty}f(t)~dt=1$$ (1) where $t$ is the random variable. This means that the total probability defined by the probability density function is one. Second: $$p[a \le t\le b]=\int_{a}^{b}f(t)~dt$$ (2) This gives an expression for the probability that the random variable $t$ takes a value with in an interval, which is essentially the area under the probability density function curve. The probability density function of negative exponential distribution is given as: $$f(t)=\lambda e^{-\lambda t},~~~t\ge 0$$ (3) where $\lambda$ is a parameter that determines the shape of the distribution often called as the shape parameter. The shape of the negative exponential distribution for various values of $\lambda$ (0.5, 1, 1.5) is shown in figure 1. Figure 1: Shape of the Negative exponential distribution for various values of $\lambda$ The probability that the random variable $t$ is greater than or equal to zero can be derived as follow, $$p(t\ge0) = \int_{0}^{\infty}\lambda~e^{-\lambda t}~dt$$ (4) $$= \lambda~\int_{0}^{\infty} e^{-\lambda t}~dt$$ $$= \lambda \left\vert\frac{e^{-\lambda t}}{-\lambda}\right\vert^{\infty}_{0}$$ $$= \left\vert-e^{-\lambda t}\right\vert^{\infty}_{0}$$ $$= -e^{-\lambda \infty}+e^{-\lambda 0}$$ $$= 0+1 = 1$$ The probability that the random variable $t$ is greater than a specific value $h$ is given as $$p(t\ge h) = 1-p(t<h)$$ (5) $$= 1-\int_{0}^{h}\lambda.e^{-\lambda t}~dt$$ $$= 1-\lambda \left[\frac{e^{-\lambda t}}{-\lambda}\right]^h_0$$ $$= 1+\left\vert e^{-\lambda t}\right\vert^h_0$$ $$= 1+\left[e^{-\lambda h}-e^{-\lambda 0}\right]$$ $$= 1+e^{-\lambda h}-1$$ $$= e^{-\lambda h}$$ Unlike many other distributions, one of the key advantages of the negative exponential distribution is the existence of a closed form solution to the probability density function as seen above. The probability that the random variable $t$ lies between an interval is given as: $$p[h\le t \le~(h+\delta h)] = p[t\ge h]-p[t\ge (h+\delta h)]$$ (6) $$= e^{-\lambda~h}-e^{\lambda~(h+\delta h)}$$ This is illustrated in figure 2. Figure 2: Evaluation of negative exponential distribution for an interval The negative exponential distribution is closely related to the Poisson distribution which is a discrete distribution. The probability density function of Poisson distribution is given as: $$p(x)=\frac{\lambda^x~e^{-\lambda}}{x!}$$ (7) where, $p(x)$ is the probability of $x$ events (vehicle arrivals) in some time interval ($t$), and $\lambda$ is the expected (mean) arrival rate in that interval. If the mean flow rate is $q$ vehicles per hour, then $\lambda=\frac{q}{3600}$ vehicles per second. Now, the probability that zero vehicle arrive in an interval $t$, denoted as $p(0)$, will be same as the probability that the headway (inter arrival time) greater than or equal to $t$. Therefore, $$p(x=0) = \frac{\lambda^0~e^{-\lambda}}{0!}$$ $$= e^{-\lambda}$$ $$= p(h\ge t)$$ $$= e^{-\lambda~t}$$ Here, $\lambda$ is defined as average number of vehicles arriving in time $t$. If the flow rate is $q$ vehicles per hour, then, $$\lambda=\frac{q\times~t}{3600}=\frac{t}{\mu}$$ (8) Since mean flow rate is inverse of mean headway, an alternate way of representing the probability density function of negative exponential distribution is given as $$f(t)=\frac{1}{\mu}~e^\frac{-t}{\mu}$$ (9) where $\mu=\frac{1}{\lambda}$ or $\lambda=\frac{1}{\mu}$. Here, $\mu$ is the mean headway in seconds which is again the inverse of flow rate. Using equation 6 and equation 5 the probability that headway between any interval and flow rate can be computed. The next example illustrates how a negative exponential distribution can be fitted to an observed headway frequency distribution.