Answer to Question #295003 in Statistics and Probability for Rio

Question

In a railway yard goods trains arrive that at the rate of 30 trains per day. Assuming

that the inter-arrival time follows an exponential distribution and the service time

distribution is also exponential with an average of 36 minutes, calculate the probability the yard is empty 2) the average length assuming that the line capacity of the yard is 9 trains

Answer 1

Solution;

Here;

$\frac{\lambda}{\mu}=\rho=0.75$

(a)Probability that the yard is empty;

$P_o=\frac{1-\rho}{1-\rho^{N+1}}$

Where N=9

$P_o=\frac{1-0.75}{1-0.75^{9+1}}=0.2649$

(b)the average length

$L=\frac{1-\rho}{1-\rho^{N+1}}\displaystyle{\sum}_{n=0}^Nn\rho^n$

$L=\frac{1-0.75}{1-0.75^{9+1}}\displaystyle{\sum}_{n=0}^9n(0.75)^n$

$L=0.28×9.58=3trains$