Sojourn time distributions in non-product-form queueing networks

From a customer’s point of view, the most important performance measure of a queueing network is his sojourn time. This paper unifies and extends the results presently found in the literature, which provide explicitly the joint distribution of a customer’s successive sojourn times over a prescribed path. The central feature for obtaining these results is the quasi overtake-free property of the path traversed by the customer. The paper concentrates on closed networks and sketch the cases of open and mixed networks. The present setting generalizes both the product-form networks of Kelly with general servers and the networks with overall-congestion dependent transition rates of Serfozo. The paper obtains equilibrium probabilities, prove an ‘arrival theorem’, and compute joint sojourn time distributions for quasi overtake-free paths. The main conclusion is that general distributional results are rare.