On the arrivai theorem for product form queueing networks with blocking

The arrival theorem is known to be valid for product form queueing networks that do not have capacity constraints at the queues. This paper investigates the validity of the arrival theorem for open and closed queueing networks with finite queues under product form conditions, and derives a necessary and sufficient condition for the validity of the arrival theorem under product form conditions. It will be shown that a product form alone is not a guarantee for the validity of the arrival theorem. Further conditions are required to retain its validity. The paper provides an extensive set of examples and counterexamples, and can therefore also be considered as a survey to provide insight for the validity of the arrival theorem.