Note--Some Results on Sojourn Times in Acyclic Jackson Networks

Lemoine (Lemoine, A. 1977. Networks of queues--a survey of equilibrium analysis. Management Sci. 24 464-481.) studies sojourn times in acyclic, Jackson networks as part of his survey of equilibrium results. In his paper, he argues that a particular customer has a sojourn time at node i that is independent of his remaining sojourn time in the network given that the customer transfer from node i to node j. This assumption is then used to derive a set of equations involving the Laplace-Stieltjes transform of the sojourn times at each node from which several properties of sojourn times are determined. In a private communication, Mitrani (Mitrani, I. 1978. A critical note on a result of Lemoine. Private communication.) argues that the independence assumption is in error and provides a heuristic argument based on a four node, acyclic, Jackson network. From this, Mitrani concludes that the sojourn time in such networks is still an unsolved problem. We make Mitrani's argument rigorous, using some new results on sojourn times in Jackson networks.