A ring of I cells rotates past I queues, carrying customers from their origins to their destinations. The system is modelled as a Markov chain, and the exact ergodicity conditions are given. They are shown to depend on the precise travel lengths distributions, that is, not only on their means. Ergodicity is proven through the stability analysis of the associated fluid limits. The arrivals distributions, which in the ergodicity conditions appear only through their means, are more subtly involved in the fluid limits behaviour, in that they determine the probabilities of random bifurcations that occur infinitely often in a simple system of I=2 queues.
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