Stability of Jackson type network output

We consider an open Jackson type queueing network N with input epochs sequence I = {T-n((0)), n greater than or equal to 0}, T-0((0)) = 0, assume another input (I) over tilde = {(T) over tilde ((0))(n)} and denote delta(k) = |(T) over tilde ((0))(k) -\(T) over tilde ((0))(k)|, Delta(0) = 0, Delta(n) = max(1less than or equal tokless than or equal ton) delta(k), n greater than or equal to 1. Let {T-n} and {(T) over tilde (n)} be the output points in network N and in modified network, (N) over tilde with input (I) over tilde, accordingly. We study the long-run stability of the network output, establishing two-sided bounds for output perturbation via input perturbation. In particular, we obtain conditions that imply max(k)less than or equal ton\T-k - (T) over tilde (k)\ = o(n(1/r)) with probability 1 as n --> infinity for some r > 0. This result is also extended to continuous time. We consider successively separate station (service node), tandem and feedforward networks. Then we extend stability analysis to general (feedback) networks and show that in our setting these networks can be reduced to feedforward ones. Similar stability results are also obtained in terms of the number of departures. Application to a tandem network with the overloaded stations is considered.