A method of approximate analysis of an open exponential queuing network with losses due to finite shared buffers in multi-queue nodes

We consider a model of an open exponential queuing network where each node comprises several multi-class MR/M/1 queues that share a common waiting space (a buffer) of limited capacity. A customer arriving to a node with fully occupied buffer is lost. An assumption is made that each class input traffic to a node, which is a superposition of the class external Poisson flow and the class flows coming from other nodes, is a Poisson process. Under this assumption a method of an approximate analysis is presented. It is based on solving iteratively a system of non-linear equations for the unknown nodal flow rates. It is shown that the gradient iterations solve the multi-class network equations. For the single-class model we use the direct substitution iterations. In the latter case existence and uniqueness of the solution, obtained by the iterative algorithm, is rigorously proven. It is demonstrated for a few network configurations that the network and node performance characteristics received by analytic approach are close to those obtained by simulation method. Our contribution is a performance evaluation methodology that could be usefully employed in queuing network design.