# M/M/3 queue - reducing wait time by adding servers

Full question below:

You are the manager of the customer support division in your company. Your division uses 3 telephone lines operated by 3 separate customer service representatives. A customer is put on hold if their call arrives while all 3 customer service representatives are busy serving other customers. You observe that customer calls arrive at a Poisson rate of 5 per hour, and that the length of the customer calls is exponentially distributed. You also observe that 75% of the time, a customer is not put on hold, while the remaining 25% of the time, a customer can expected to be put on hold for an average of 12 minutes. You wish to improve service in the division by making sure that 90% of the time, a customer is not put on hold, while 10% of the time, a customer can expect to be put on hold for an average of only 4 minutes. How many telephone lines will you add to your division to achieve your goal?

So I think the biggest problem here is that I don't know $\mu$. I do know $\rho=\frac{\lambda}{c\mu}=\frac{5}{3\mu}$ for this problem. I understand that "time on hold" refers to to time waiting in the queue. With $W$ is time waiting in the queue, I know: $$E[W]=\frac{\rho}{\lambda(1-\rho)}P(W>0)$$

With 3 operators, I used the fact that "75% of the time, a customer is not put on hold, while the remaining 25% of the time, a customer can expected to be put on hold for an average of 12 minutes" to calculate: $$E[W]=.75(0) + .25(12min)=3min$$ Then using $E[W]$ along with $P(W>0)=.25$, I solved the first equation to find $\mu=\frac{10}{3}$.

Knowing $u$, I used "90% of the time, a customer is not put on hold, while 10% of the time, a customer can expect to be put on hold for an average of only 4 minutes" to find the new $E[W]=1min$ and $P(W>0)=.1$.

To solve for $c$(number of servers) I again plugged these numbers into the original equation for $E[W]$ and found $c=3.3$. You can obviously only have an integer number of servers, so this would be $c=4$, and minus the original 3 would give the addition of just 1 server as the answer.