We consider a closed two-stage cyclic queueing model for a multiprogrammed computer system. Assuming one general server and one server whose service time distribution is a mixture of Erlang distributions, we compute both the steady-state distribution at departure points and the Laplace-Stieltjes transform of the cycle time, given the state of system at the beginning of the cycle. We also find delay time and cycle time distributions for systems both in and out of equilibrium.