Class aggregation for multi-class queueing networks with FCFS multi-server stations

The analytical solution of a queueing network is an appreciable first option in performance studies of service systems. Under the so-called product form conditions, Mean Value Analysis (MVA) is the standard algorithm still adopted, but the user has to face the exponential computational complexity in the number of customer classes. In the last three decades, some (pseudo) polynomial approximated variants to MVA have been proposed in literature. These approximations are based on the transformation of the recursive MVA equations into a system of nonlinear equations to be solved iteratively. They are consolidated only with reference to (fixed-rate) single-server stations and are used in practice even though theoretical convergence remains an open problem. In this paper we exploit the possibility of aggregating customer classes in order to replace the exact multi-class MVA by new approximated procedures where MVA has to be run under at the most two customer classes. The resulting procedures are developed around a nested fixed-point iteration schema and are especially suitable for solving large size multi-class networks with multi-server stations under a first-come-first-served discipline. Convergence and accuracy of our procedures are numerically assessed through a very large set of experiments against the exact solution by the multi-class MVA.