Invariance of Stationary Distributions of Exponential Networks with Prohibitions and Determination of Maximum Prohibitions

The paper considers queuing networks with prohibitions on transitions between network nodes that determine the protocol of their operation. In the graph of transient network intensities, a set of base vertices is allocated (proportional to the number of edges), and we raise the question of whether some subset of it can be deleted such that the stationary distribution of the Markov process describing the functioning of the network is preserved. In order for this condition to be fulfilled, it is sufficient that the set of vertices of the graph of transient intensities, after the removal of a subset of the base vertices, coincide with the set of states of the Markov process and that this graph be connected. It is proved that the ratio of the number of remaining base vertices to their total number n converges to one-half for n→∞. In this paper, we are looking for graphs of transient intensities with a minimum (in some sense) set of edges for open and closed service networks.