Ergodicity of Jackson-type queueing networks
The paper contains stability and convergence theorems for Jackson-type queueing networks under the general assumption that the input process, the service sequences, and the routing mechanism are jointly stationary and ergodic. These results are obtained with the help of the subadditive ergodic theorem and basic theorems on monotone stochastic recursive sequences. All the results are divided into two groups. The first group contains so-called first-order ergodic theorem which is simply the strong law of large numbers. For this purpose, the authors use the graph theoretic considerations which generalize the technique developed by Baccelli et al. for the stability of event graphs. The second group of results is centered around the second order ergodic theorem of Loynes-type which is a sort of convergence to a stationary ergodic regime. A coupling with a stationary ergodic regime and weak convergence to such a regime are considered. As a consequence, new results or new proofs of known results are obtained. In particular, a minimal stationary regime for all models under consideration is constructed. In addition, some restrictions imposed on networks in earlier works are relaxed (for example, the Cramer-type conditions imposed on service times by Borovkov).
stability and convergence theorems for Jackson-type queueing networks
second order ergodic theorem of Loynes-type
scientific article; zbMATH DE number 698850
scientific article