A single-server waiting system is subject to a homogeneous Poisson input with intensity If there are not more than 3 customers in the system, the service times have an exponential distribution with...

A single-server waiting system is subject to a homogeneous Poisson input with intensity

If there are not more than 3 customers in the system, the service times have an exponential distribution with mean 1/μ = 2 [min]. If there are more than 3 customers in the system, the service times are exponential with mean 1/μ = 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.