Queues served by a rotating ring

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Abstract
A ring of N cells rotates in discrete steps past N queues, moving customers from their queues of arrival to randomly chosen destinations. The model has applications in communication systems, processor interconnection networks, and flexible manufacturing. The arrivals to the queues are independent and stochastically identical. The total numbers of arrivals to the system during successive steps are independent, identically distributed random variables with mean λ and finite second and third moments. A greedy policy governs the insertion of customers on the ring: A customer waiting at the head of a queue enters the next unoccupied cell to appear at that queue. The customer then remains on the ring a random travel time d, leaves, and frees its cell for another customer A necessary condition for the system to be stable is . If no customer travels further than once around the ring is sufficient for stability. Other results assume d to be stochastically bounded by an exponentially distributed travel time with mean . Then is sufficient for stability. In the limit of large N, stable systems with fixed λ and μ have expected numbers of waiting customers per queue; then a customer's wait in a queue is usually negligible compared with his travel time. Simulations suggest that the mean number waiting in a queue may even be .

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