We propose a simple approximation to assess the steady-stateprobabilities of the number of customers in Ph/Ph/1 andPh/Ph/1/N queues, as well as probabilities found on arrival,including the probability of buffer overflow for the Ph/Ph/1/Nqueue. The phase-type distributions considered are assumed to beacyclic. Our method involves iteration between solutions of anM/Ph/1 queue with state-dependent arrival rate and a Ph/M/1queue with state-dependent service rate. We solve these queuesusing simple and efficient recurrences. By iterating between thesetwo simpler models our approximation divides the state space, andis thus able to easily handle phase-type distributions with largenumbers of stages (which might cause problems for classicalnumerical solutions). The proposed method converges typicallywithin a few tens of iterations, and is asymptotically exact forqueues with unrestricted queueing room. Its overall accuracy isgood: generally within a few percent of the exact values, exceptwhen both the inter-arrival and the service time distributionsexhibit low variability. In the latter case, especially undermoderate loads, the use of our method is not recommended.