Abstract
We continue the study of zero-automatic queues first introduced in Dao-Thi and Mairesse (Adv Appl Probab 39(2):429–461, 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 stable 0-automatic queues have an explicit “multiplicative” stationary distribution and a Poisson departure process (Dao-Thi and Mairesse, Adv Appl Probab 39(2):429–461, 2007). 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 contents has a “product-form” and can be explicitly determined. Furthermore, the departure process out of the network is Poisson.
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Dao-Thi, TH., Mairesse, J. Zero-Automatic Networks. Discrete Event Dyn Syst 18, 499–536 (2008). https://doi.org/10.1007/s10626-008-0048-1
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DOI: https://doi.org/10.1007/s10626-008-0048-1