A single-server waiting system is subject to a homogeneous Poisson input with intensity (lambda=30left[h^{-1} ight]). If there

A single-server waiting system is subject to a homogeneous Poisson input with intensity \(\lambda=30\left[h^{-1}\right]\). If there are not more than 3 customers in the system, the service times have an exponential distribution with mean \(1 / \mu=2[\mathrm{~min}]\). If there are more than 3 customers in the system, the service times are exponential with mean \(1 / \mu=1\) [min]. All arrival and service times are independent.

(1) Show that there exists a stationary state distribution and determine it.

(2) Determine the mean length of the waiting queue in the steady state.