The stability of open queueing networks

The stability of open Jackson networks is established where service times are i.i.d. general distribution, exogeneous interarrival times are i.i.d. general distribution, and the routing is Markovian. The service time distributions are only required to have finite first moment. The system is modeled (at arrival epochs) as a general state space Markov chain. Explicit regeneration points are found (even in the case when the system never empties) and the chain is shown to be Harris ergodic if standard rate conditions are enforced, that is, if at each node, the long run average amount of work per unit time that arrives exogeneously destined for that node is strictly less than one. In addition, the author proves that if the system is modeled in continuous time when convergence to a steady-state occurs in total variation if the interarrival time distribution is spread-out. Extensions of the results to multi-server nodes, non-Markovian routing and Markov modulated arrivals are given.