Asymptotic methods for analysis of closed queueing networks in heavy traffic

The paper is concerned with closed Markov queueing networks with an infinite server station when the total number of customers increases without a limit. Equations are provided for normalized asymptotical values of mean queue lengths in cases of high loads. For networks with stationary distribution of a multiplicative form the elements of the covariance matrix in the approximating Gaussian distribution and its determinant are explicitly expressed as functions of normalized mean values. Examples are given where the first and, consequently, second moments of queue lengths are explicitly expressed as functions of network parameters. For a network with many types of customers and blocking the proposed approach is shown to be applicable to finding the most probable state of blocking and the mean time of achieving it for rare blocking is provided.