EX 4.5 time headway

“Time headway” in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let $X=$ the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow. The following pdf of $X$ is essentially the one suggested in “The Statistical Properties of Freeway Traffic” (Transp. Res., vol. 11: 221-228):

$$f\left(x\right)=\{\begin{array}{cc}.15e^{-.15\left(x-.5\right)}& x\ge .5\\ 0& \text{ otherwise }\end{array}$$

The graph of $f\left(x\right)$ is given in Figure 4.4;

Figure 4.4 The density curve for time headway in Example 4.5

There is no density associated with headway times less than $.5$, and headway density decreases rapidly (exponentially fast) as $x$ increases from .5. Clearly, $f\left(x\right)\ge 0$ ; to show that ${\int }_{-\infty }^{\infty }f\left(x\right)dx=1$, we use the calculus result ${\int }_a^{\infty }{e}^{-kx}dx=\left(1/k\right)e^{-k\cdot a}.$ Then

$$\begin{array}{rl}{\int }_{-\infty }^{\infty }f\left(x\right)dx& ={\int }_{.5}^{\infty }.15e^{-.15\left(x-.5\right)}dx\\ & =.15e^{.075}{\int }_{.5}^{\infty }{e}^{-.15x}dx\\ & =.15e^{.075}\cdot \frac{1}{.15}{e}^{-\left(.15\right)\left(.5\right)}=1\end{array}$$

The probability that headway time is at most 5 sec is

$$\begin{array}{rl}P\left(X\le 5\right)& ={\int }_{-\infty }^{5}f\left(x\right)dx\\ & ={\int }_{.5}^5.15e^{-.15\left(x-.5\right)}dx\\ & =.15e^{.075}{\int }_{.5}^5{e}^{-.15x}dx\\ & =.15e^{.075}\cdot \left(-{\frac{1}{.15}{e}^{-.15x}|}_{x=.5}^{x=5}\right)\\ & =e^{.075}\left(-e^{-.75}+e^{-.075}\right)\\ & =1.078\left(-.472+.928\right)\\ & =.491\\ & =P\left(\text{less than }5\sec \right)\\ & =P\left(X<5\right)\end{array}$$