Question

The probability that $k$ number of vehicles arrive (i.e., cross a predefined line) in time $t$ is given as $(\lambda t{)}^k{e}^{-\lambda t}/k!$ where $\lambda$ is the average vehicle arrival rate. What is the probability that the time headway is greater than or equal to time $t_1$ ?

• A. $\lambda e^{\lambda t_1}$
• B. $\lambda e^{-t_1}$
• C. ${\lambda }^{t_1}$
• D. $e^{-\lambda t_1}$

Answer 1

The correct option is D $e^{-\lambda t_1}$

Time headway-The time interval between the passes of successive vehicles moving in the same lane and measured from head to head as they pass a point on the road is known as the time headway.
Probability of arriving $k$ number of vehicles in time (\t\)
$P\left(k\right)=\frac{(\lambda t{)}^k{e}^{-\lambda t}}{k!}$
If time headway is greater than or equal to time $t_1$ then no. of vehicles arrive in time $t_1=0$.
$P\left(0\right)=\frac{(\lambda t_1{)}^0{e}^{-\lambda t}}{0!}=e^{-\lambda t_1}$