Nonergodic Jackson networks with infinite supply – local stabilization and local equilibrium analysis. (English) Zbl 1356.60159

Sommer, Jennifer; Daduna, Hans; Heidergott, Bernd

J. Appl. Probab. 53, No. 4, 1125-1142 (2016).

Summary: Classical Jackson networks are a well-established tool for the analysis of complex systems. In this paper we analyze Jackson networks with the additional features that (i) nodes may have an infinite supply of low priority work and (ii) nodes may be unstable in the sense that the queue length at these nodes grows beyond any bound. We provide the limiting distribution of the queue length distribution at stable nodes, which turns out to be of product form. A key step in establishing this result is the development of a new algorithm based on adjusted traffic equations for detecting unstable nodes. Our results complement the results known in the literature for the subcases of Jackson networks with either infinite supply nodes or unstable nodes by providing an analysis of the significantly more challenging case of networks with both types of nonstandard node present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures.

MSC:

60K25	Queueing theory (aspects of probability theory)
90B15	Stochastic network models in operations research
90B22	Queues and service in operations research

Keywords:

Jackson network; stability; instability; product-form solution; bottleneck analysis; shortest path