Lumpability and nearly-lumpability in hierarchical queueing networks

Hierarchical extended queueing networks (HQNs) have been proposed recently as a model class which allows a very efficient analysis based on an exploitation of the hierarchical model structure in the analysis of the underlying Markov chain (MC). Furthermore the hierarchical structure of the model enables the definition of symmetries due to identical subnets on model level and the automatic generation of a reduced MC resulting from exact or ordinary lumpability. In this paper the notion of symmetries is extended to nearly symmetries at model and submodel level resulting in nearly lumpability of the underlying MC. Bounds on performance quantities for nearly lumpable Markov chains are computed using bounded aggregation in the stationary case