We present a method for deriving the normalizing constant for the state space probabilities in a closed Markovian queueing network with constant service rates. Our method produces a much simpler expression than any obtained previously. This expression takes the form of a sum of only M terms (if the visitation rate to service rate ratios are distinct), and is easily generalized to the degenerate case. The expression also yields a closed form solution for time delay distributions in cyclic networks of any length and with any number of customers.