EQUIVALENCE, REVERSIBILITY, SYMMETRY AND CONCAVITY PROPERTIES IN FORK-JOIN QUEUING-NETWORKS WITH BLOCKING

In this paper, we study quantitative as well as qualitative properties of Fork-Join Queuing Networks with Blocking (FJQN/Bs). Specifically, we prove results regarding the equivalence of the behavior of a FJQN/B and that of its duals and a strongly connected marked graph. In addit ion, we obtain general conditions that must be satisfied by the servic e times to guarantee the existence of a long-term throughput and its i ndependence on the initial configuration. We also establish conditions under which the reverse of a FJQN/B has the same throughput as the or iginal network. By combining the equivalence result for duals and the reversibility result, we establish a symmetry property for the through put of a FJQN/B. Last, we establish that the throughput is a concave f unction of the buffer sizes and the initial marking, provided that the service times are mutually independent random variables belonging to the class of PERT distributions that includes the Erlang distributions . This last result coupled with the symmetry property can be used to i dentify the initial configuration that maximizes the long-term through put in closed series-parallel networks.