Exponential queuing network with dependent servicing, negative customers, and modification of the customer type

The open exponential queuing network with negative customers (G-network) was considered.For each arriving customer, given was a set of its random parameters such as the route defining the sequence of network nodes passed by the customer, route length, size, and servicing duration at each stage of the route. It was assumed that the negative customer arriving to the $s$th node with the probabilities $\omega_s$ and $\omega^+_s$ either kills the positive customer in a randomly chosen server or does not affect it at all and with the probability $\omega^-_s=1-\omega_s-\omega^+_s$ transforms it into a negative customer which after an exponentially distributed time arrives to the $s'$th node with the given probability. The multidimensional stationary probability distribution of the network states was proved to be representable in the multiplicative form.