GEOMETRY OF LATTICES FOR MULTICLASS MARKOVIAN QUEUEING NETWORKS.

Multiclass Markovian queueing networks are classified according to the geometric structure of the state transition diagram. Two types of networks are distinguished. For networks of the first class the queueing parameters ( lambda , mu ) and the routing matrix tau do not depend on the order of the packets at the queueing nodes. By contrast, the parameters ( lambda , mu ) and tau of the queueing networks of the second class are functions of the exact order of the packets at the queueing nodes. In the former case the state transition diagram can be identified with simplicial complexes and in the latter, a different replication pattern is discerned. For both network types, necessary and sufficient conditions for the existence of a product form solution are given. The significance of our results is manifested through the class of networks with blocking, state dependent routing and different service and queueing disciplines.