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We propose a simple approximation to assess the steady-state probabilities of the number of customers in Ph/Ph/1 and Ph/Ph/1/N queues, as well as probabilities found on arrival, including the probability of buffer overflow for the Ph/Ph/1/N queue. The phase-type distributions considered are assumed to be acyclic. Our method involves iteration between solutions of an M/Ph/1 queue with state-dependent arrival rate and a Ph/M/1 queue with state-dependent service rate. We solve these queues using simple and efficient recurrences. By iterating between these two simpler models our approximation divides the state space, and is thus able to easily handle phase-type distributions with large numbers of stages (which might cause problems for classical numerical solutions). The proposed method converges typically within a few tens of iterations, and is asymptotically exact for queues with unrestricted queueing room. Its overall accuracy is good: generally within a few percent of the exact values, except when both the inter-arrival and the service time distributions exhibit low variability. In the latter case, especially under moderate loads, the use of our method is not recommended.

An approximate solution for Ph/Ph/1 and Ph/Ph/1/N queues