Nonergodic Jackson networks with infinite supply–local stabilization and local equilibrium analysis

Abstract

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 instable nodes. Our results complement the results known in the literature for the sub-cases 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 nodes present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures.

Analysis of Jackson networks with infinite supply and unreliable nodes

“…The interplay of nodes with infinite supply and local stability issues has been studied in depth in [18].…”

“…We start the proof with evocation of the subnetwork argument from the proof of Theorem 13 in [18]. It guarantees that the subnetwork W constitutes a Jackson network where the source and sink represent {0} ∪ V .…”

“…Thus, even if the subnetwork V of nodes with infinite supply is not in equilibrium, the equilibrium on the subnetwork W of nodes without infinite supply is preserved, if the initial distribution has the joint marginal (18). This joint queue-length processX is coupled with an availability process Y which only depends on the interaction of the nodes in D ⊆J but not on their load.…”

“…(iii) Under the condition that all nodes j ∈J are in up status, we start the proof with evocation of the M/M/1 argument from the proof of Theorem 13 in [18].…”