Zero-automatic networks
Résumé
We continue the study of zero-automatic queues first introduced in "Zero-automatic queues and product form", T.-H. Dao-Thi and J. Mairesse, Adv. Appl. Probab., vol. 39, n. 7, 2007. These queues are characterized by a special buffering mechanism evolving like a random walk on some infinite group or monoid. The simple M/M/1 queue and Gelenbe's G-queue with positive and negative customers are the two simplest 0-automatic queues. All 0-automatic queues are quasi-reversible. In this paper, we introduce and study networks of 0-automatic queues. We consider two types of networks, with either a Jackson-like or a Kelly-like routing mechanism. In both cases, and under the stability condition, we prove that the stationary distribution of the buffer content has a "product-form" and can be explicitly determined.