Pay‐for‐priority system is believed to be an efficient service mechanism in congested systems since it introduces service differentiation that prioritizes those who are more delay‐sensitive. However, in practice, not all customers are aware of the provision of such auxiliary service (i.e., priority access). Does the lack of awareness or ignorance of priority service make the social welfare or customer surplus worsen off? To answer this question, we establish a queueing‐game‐theoretic model by capturing the strategic interactions between service provider and customers to examine the effect of customer awareness on the priority queues. Our main results are as follows. First, we confirm that increasing the level of customer awareness indeed improves the revenue of service provider, and it triggers a higher priority premium price. Second, perhaps surprisingly, we find that under the profit‐maximizing priority price, the social welfare as well as the customer surplus are both nonmonotone in the level of customer awareness, that is, full or no customer awareness can be suboptimal for the social welfare and customer surplus. Third, despite the common belief that priority is socially efficient in congested systems, we demonstrate that the optimal information levels for social welfare and customer surplus are decreasing in the congestion level, and the full customer awareness is optimal only when the system load is relatively low. Finally, to reach the maximal social welfare or customer surplus, some regulation policies are proposed, whereby the social planner can provide advertising subsidy to (levy tax on) the service provider (advertising agency) when the system load is low (high).