## 3 Model Formulation  
### Two Hospitals with Unbalanced Congestion  
First, we describe the mismatch between supply and demand of two hospitals based on a rational queueing framework. We consider two independent hospitals; each hospital has a M/M/1 queueing system with processing rate \(\mu_{i},i=\{1,2\}\). Their potential patient arrival rate is \(\Lambda_{i}\). Patients must wait when the doctor is busy. Patient's delay cost is proportional to the system delay. The cost of stay per unit time is denoted by \(c_{i}\). Each arriving patient must decide whether to request for the medical service (joining) or not (balking). Patients have a reserved value for each service, which is denoted by \(V_{i}\). The utility function of patients is formulated as \(U_{i}=V_{i}-c_{i}w_{i}-p_{i}\), where \(w_{i}\) is the average sojourn time and \(p_{i}\) is the price. We assume that customers can make decision based on the long-term average sojourn time and system parameters such as stay cost per unit time. In several practical situations, the average stay time is available public information. For instance, in Canada [31] or Hong Kong [32], the average stay time for some non-emergent medical service in public hospitals is posted on the website. A patient purchases the service if the patient's net surplus is positive, that is, \(U_{i}\geq 0\). The effective arrival rate, which represents the patient arrivals with purchase, is denoted by \(\lambda_{i}\). The revenue function of the two independent hospitals can be written as \(\pi_{i}^{0}=p_{i}\lambda_{i}\). We assume that the service capacity is fixed and its investment is sunk cost. Previous literature [27] provides the optimal pricing strategy of a monopoly firm via queueing modeling as follows,  
1. if \(\Lambda_{i}\geq\mu_{i}-\sqrt{c_{i}\mu_{i}/V_{i}}\), then the optimal price is \(p_{i}^{*}=V_{i}-\sqrt{c_{i}V_{i}/\mu_{i}}\), the optimal effective arrival rate is \(\lambda_{i}^{*}=\mu_{i}-\sqrt{c_{i}\mu_{i}/V_{i}}\), and the corresponding optimal revenue is \(\pi_{i}^{0}=(\sqrt{V_{1}\mu_{1}}-\sqrt{c_{1}})^{2}\). 2. if \(\Lambda_{i}<\mu_{i}-\sqrt{c_{i}\mu_{i}/V_{i}}\), then the optimal price is \(p_{i}^{*}=V_{i}-c_{i}/(\mu_{i}-\Lambda_{i})\), the optimal effective arrival rate is \(\lambda_{i}^{*}=\Lambda_{i}\), and the corresponding optimal revenue is \(\pi_{i}^{0}=\Lambda_{2}[V_{2}-c_{2}/(\mu_{2}-\Lambda_{2})]\). This optimal price need to balance the trade off between the revenue from a patient and the externality caused by this patient. The hospital HD, indexed by \(i=1\), which is a large general hospital, attracts a demand which is too large to be met in China because of the medical habits of Chinese patients, the distrust of patients in the treatment capacity of small hospitals, and poor doctors in small hospitals, etc. We define this large demand scenario as the case where the potential patient arrival rate is larger than the effective arrival rate under the optimal pricing strategy, i.e., \(\mu_{1}-\sqrt{c_{1}\mu_{1}/V_{1}}<\Lambda_{1}\). Accordingly, we make the following assumption for the HD. HD is characterized by a limited service rate that satisfies \(\mu_{1}^{0}<\mu_{1}<\mu_{1}^{1}\), where \(\mu_{1}^{0}\) and \(\mu_{1}^{1}\) are defined as \(\mu_{1}^{0}=\frac{c_{1}}{V_{1}}\); and \(\mu_{1}^{1}=\Lambda_{1}+\frac{c_{1}}{2V_{1}}+\sqrt{\frac{c_{1}^{2}}{4V_{1}^{2 }}+\frac{c_{1}\Lambda_{1}}{V_{1}}}\). In Assumption 1, the condition \(\mu_{1}<\mu_{1}^{1}\) is equivalent to \(\mu_{1}-\sqrt{c_{1}\mu_{1}/V_{1}}<\Lambda_{1}\). The other condition \(\mu_{1}^{0}<\mu_{1}\) is equivalent to \(V_{1}-c_{1}/\mu_{1}>0\), which ensures that the HD can attract at least one patient. Accordingly, we can get the optimal revenue of HD when it operates independently, that is \(\pi_{1}^{0}=(\sqrt{V_{1}\mu_{1}}-\sqrt{c_{1}})^{2}\). The hospital HS, indexed by \(i=2\), which is a community hospital, cannot attract patients due to its lack of medicine and poor doctors. However, it can provide alternative services for common diseases as that of the HD.