This study analyzes dynamic pricing in a non-preemptive M/M/1 service system in which customers exhibit patience in regard to waiting time. We use a stylized, single-server queue to incorporate customer patience into the queuing problem, with the assumption that customers will not enter the queue when the waiting time exceeds their inherent patience time, reflecting the decision whether to wait. We focus on queue design, structure, and revenue and consider two homogeneous and heterogeneous customer scenarios. Our findings show that, for homogeneous customers, price-raising and price-discounting strategies can mitigate revenue loss. We examined which policy is better for service providers in different scenarios. For heterogeneous customers, we draw two conclusions. First, a revenue-maximizing service provider should increase her priority fee if customers have patience in regard to waiting time, regardless of whether they are regular customers, priority customers, or a mix of both. Second, contrary to conventional wisdom, we show that the service provider should abandon a portion of his or her regular customers to attract more customers who are willing to pay a higher priority fee for better service when the value of regular customers is small. Under this situation, customers' patience times can bring more revenue to the service provider. If the value of regular customers is high, however, then an increased priority fee cannot compensate for the revenue loss from abandoning these customers and losing their revenue. Overall, our findings confirm that customers' patience times result in a high net monetary cost for the time-sensitive customer.