Waiting time at competing M/M/n queues


Customers arrive following a Poisson process with rate $ \lambda$ . Each customer can decide between three different queues which may satisfy their demand. Two of these queues have one server and one of the queues has n servers. Each of these servers has service time that are exponentially distributed with rate $ \mu$ . In essence, two M/M/1 queues and one M/M/n queue are competing for customers.

Customers are waiting time sensitive and they know the expected waiting time $ E[t_i]$ at each queue $ i$ . Therefore, in equilibrium holds that $ E[t1]=E[t_2]=E[t_3]$ .

What is the expected waiting time $ E[t_i]$ in this system?

I already looked up the QueuingProcess[] function, but I do not see the solution.

Any hint into the right direction is highly appreciated.