Tail Asymptotics of Two Parallel Queues with Transfers of Customers

This paper studies a system consisting of two parallel queues with transfers of customers. In the system, one queue is called main queue and the other one is called auxiliary queue. The main queue is monitored at exponential time instances. At a monitoring instant, if the number of customers in main queue reaches L \((>K)\), a batch of \(L-K\) customers is transferred from the main queue to the auxiliary queue, and if the number of customers in main queue is less than or equal to K, the transfers will not happen. For this system, by using a Foster-Lyapunov type condition, we establish a sufficient stability condition. Then, we provide a sufficient condition under which, for any fixed number of customers in the auxiliary queue, the stationary probability of the number of customers in the main queue has an exact geometric tail asymptotic as the number of customers in main queue increases to infinity. Finally, we give some numerical results to illustrate the impact of some critical model parameters on the decay rate.