Recently, due to the growing demand in public healthcare system subject to congestions, the service speed as well as the service quality have arose an extensive attention of social manager. It is well believed that longer service times generally result in longer waits for patients, which increases the congestion of overall system. For instance, it is reported that the comprehensive hospitals of China were providing services to almost 50% of the total number of patients, even though they only account for 7.7% of all hospitals in China. A typical interaction between service speed and service quality can be found in the healthcare system with comprehensive hospital and primary hospital, where the former provides better service quality (with lower service speed) and the latter provides higher efficiency (with inferior service quality), respectively. The two medical channels mentioned above play different roles in the healthcare system towards different types of patients. The primary hospital is dedicated to provide the preliminary diagnosis and treatment. If the health problem of patients is found to be severe, they will be suggested to transfer to the comprehensive hospital. Therefore, considering the limited budget provided by government, it is imperative to combine comprehensive and primary hospitals to balance the quality and efficiency in healthcare system, i.e., the efficiency-quality trade-off, so as to improve the social welfare.