Add to ORKGIn this paper, by the singular-perturbation technique, we investigate the heavy-traffic behavior of a priority polling system consisting of three M/M/1 queues with threshold policy. It turns out that the scaled queue-length of the critically loaded queue is exponentially distributed, independent of that of the stable queues. In addition, the queue lengths of stable queues possess the same distributions as a priority polling system with N-policy vacation. Based on this fact, we provide the exact tail asymptotics of the vacation polling system to approximate the tail distribution of the queue lengths of the stable queues, which shows that it has the same prefactors and decay rates as the classical M/M/1 preemptive priority queues. Finally, a ...