Minute is the chosen time scale. A highway ramp has only one tollbooth. Cars introduce themselves according to a Poisson process of $\lambda = 0.3$. The at the toolboth follows an exponential law with a mean of 1 minute.
- How many cars introduce themselves in an hour at the toolboth? What is the queueing networks corresponding to this toolboth?
I said that E(L), the client average number in the system, was $E(L)=\frac{\rho}{1-\rho}=\frac{0.3}{0.7}$ and that the system was an $M/M/1$ because there were only one network in the system which was the toolboth.
- What is the utilization rate? What is the mean number of cars waiting to pay? What is the mean time for cars to pay? What is the mean time at the toll (waiting and paying)?
- Here I said that the utilization rates was $\rho=\frac{\lambda}{\mu}=0.3$, $\mu$ being the service rate, the number of clients who are being served, but I'm not sure about it...
- Then I said that the number of clients waiting to pay was $E(L^q)=\bar \lambda E(W^q)$, with $E(W^q)$ the mean time passed by clients in the queue without being served. Yet, $$E(L)=\lambda E(W)$$ and $$E(W)=E(W^q)+\frac{1}{\mu}$$ then $$E(L^q)=\lambda(\frac{E(L)}{\lambda} -\frac{1}{\mu})$$ Which leads to $$E(L^q)=\frac{0,3}{0,7}-0,3$$
- What is odd is that I'm looking for $E(W^q)$ which I already calculated...
Thus did I misunderstood something with $E(L^q)$?