Networks with bypasses and negative customers

An open-loop queueing network with two independent inputs, a flow of positi ve customers and a flow of negative customers, is investigated. A negative customer upon arrival annihilates exactly one positive customer (if any) in the queue. Positive customers join the queue at a server with probability dependent on the state of the server or bypass the server with additional p robability and thereafter behave as though they have been served. Service i s exponential and routing of positive customers in the network is a Markov process and a positive customer may become a negative customer if jockeyed to a different server. The stationary distribution of the states of the net work is shown to be multiplicative.