Consider a queuing system that follows the model M / M / 1 wherein the average time between consecutive arrivals is 12 secons and the average service time is 3 seconds.
$\lambda=\displaystyle\frac{1}{12}$ and $\mu=\displaystyle\frac{1}{3}$
What is the probability that a customer has to wait in queue? My options are: $$\displaystyle\frac{1}{2}$$$$\displaystyle\frac{2}{3}$$ $$\displaystyle\frac{3}{4}$$$$\displaystyle\frac{1}{4}$$$$\displaystyle\frac{1}{3}$$
but i obtained that $$w_q=\rho*w=\displaystyle\frac{1}{4}*4=1$$ what's wrong?
What is the probability that a customer has to wait more than six seconds in queue? How do i calculate this?
You calculated the average waiting time in the queue instead of the desired probability. The probability that the server is busy upon a customer arrival is simply $\rho$.