Local Stabilization of Non-Ergodic Jackson Networks with Unreliable Nodes

This thesis contributes to a better understanding of the behavior of queueing networksthat cannot approach a classical equilibrium state.We consider Jackson networks with unreliable nodes which randomly break down andare under repair for a random time. Our networks are described by Markov processesthe states of which incorporate the availability status and the queue-lengths vector.For Jackson networks with unreliable nodes, it is known that, under ergodicity condi-tions, the Markovian availability queue-lengths process has a stationary product-formdistribution. Ergodicity conditions can be expressed as local rate conditions: The totalarrival rate at each node has to be strictly less than its maximal service rate. If forsome nodes the ergodicity condition is violated, the network process is not ergodicand there cannot exist a stationary distribution. Nevertheless, we are able to obtainthe complete asymptotics for non-ergodic Jackson networks with unreliable nodes andshow that the state distribution of the stable subnetworks, i.e., the set of nodes wherethe local rate condition is ful lled, converges to a Jackson-type product-form distribution.The characterization of the asymptotic behavior of non-ergodic Jackson networks withunreliable nodes strongly relies on a detailed investigation of another class of generalizedJackson networks, which is of interest for its own. In these networks some stationsmay have an additional bu er with an in nite supply of lower priority jobs (customers)served at that station whenever the station runs out of standard customers. Networksincorporating such bu ers are called Jackson networks with in nite supply. We analyzethe stationary and limiting behavior of these networks with reliable nodes as well as withunreliable nodes.Our results o er a new way to measure and assess the performance of non-ergodicJackson networks with unreliable nodes where steady-state methods cannot be appliedbecause no steady state exists. Using the obtained limiting distributions we are ableto compute long-time averages of the standard performance and availability metricsexplicitly.The limiting distribution of the stable subnetworks of a non-ergodic network process has aproduct form which resembles the product-form limiting and stationary distribution of anergodic network process. The observed structural similarities of the asymptotic results forergodic network processes and non-ergodic network processes on the stable subnetworkspose the following question: Assume that the non-ergodic network is started with aninitial distribution which has the marginal limiting product-form distribution. Will thisdistribution be preserved over time? It turns out that the answer is in general negative.219