Breaking the dimensionality curse in multi-server queues

Ph/Ph/c and and Ph/Ph/c/N queues can be viewed as a common model of multi-server facilities. We propose a simple approximate solution for the equilibrium probabilities in such queues based on a reduced state description in order to circumvent the well-known and dreaded combinatorial growth of the number of states inherent in the classical state description. The number of equations to solve in our approach increases linearly with the number of servers and phases in the service time distribution. A simple fixed-point iteration is used to solve these equations. Our approach applies both to open models with unrestricted buffer size and to queues with finite-size buffers.The results of a large number of empirical studies indicate that the overall accuracy of the proposed approximation appears very good. For instance, the median relative error for the mean number in the queue over thousands of examples is below 0.1% and the relative error exceeds 5% in less than 1.5% of cases explored. The accuracy of the proposed approximation becomes particularly good for systems with more than 8 servers, and tends to become excellent as the number of servers increases.