We are in a supermarket and the arrival rate of customers to a cashier is $\lambda=240$ people per hour. There are always at least 5 desks open and there is a single waiting line. If a customer arrives and finds more than 10 people in the waiting line (excluding those in the process of paying) then the manager opens another desk, unless the maximum of 10 desks is already open at that point. When a customer leaves and the number of customers waiting in line drops below 10, the manager closes the desk where that customer just paid, unless the minimum of 5 desks is open at that point. The mean service time is 2 minutes, so we have $\mu=30$ people per hour.
Now I have to determine the steady state distribution of the total number of customers and the steady state distribution of the number of customers in the process of paying. We have an $M/M/10$ queue, but I am confused how to draw the state diagram. At first I thought the equilibrium equations would be \begin{align*}0=-\lambda p_0+\mu p_1 & \\ 0=\lambda p_{n-1}-(\lambda+n\mu)p_n+(n+1)\mu p_{n+1} & \quad 0\leq n<5 \\ 0=\lambda p_{n-1}-(\lambda+5\mu)p_n+5\mu p_{n+1} & \quad 5\leq n<15 \\ 0=\lambda p_{n-1}-(\lambda+(n-10)\mu)p_n+(n+1-10)\mu p_{n+1} & \quad 15\leq n<20 \\ 0=\lambda p_{n-1}-(\lambda+10\mu)p_n+10\mu p_{n+1} & \quad n\geq 20. \end{align*} But I got stuck working this out. So maybe because there are always at least 5 desks open, we have for $n=0,\dots,5$ already a service rate of $5\mu$?