In this paper, we study the stability of light traffic achieved by a scheduling algorithm which is suitable for heterogeneous traffic networks. Since analyzing a scheduling algorithm is intractable using the conventional mathematical tool, our goal is to minimize the largest queue-overflow probability achieved by the algorithm. In the large deviation setting, this problem is equivalent to maximizing the asymptotic decay rate of the largest queue-overflow probability. We first derive an upper bound on the decay rate of the queue overflow probability as the queue overflow threshold approaches infinity. Then, we study several structural properties of the minimum-cost-path to overflow of the queue with the largest length, which is basically equivalent to the decay rate of the largest queue-overflow probability. Given these properties, we prove that the queue with the largest length follows a sample path with linear increment. For certain parameter value, the scheduling algorithm is asymptotically optimal in reducing the largest queue length. Through numerical results, we have shown the large deviation properties of the queue length typically used in practice while varying one parameter of the algorithm.
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