###### Abstract  
The medical alliance has developed rapidly in recent years. This kind of alliance established by multiple hospitals can alleviate the imbalance of medical resources. We investigate the benefit of demand sharing between a hospital with large demand (HD ) and another hospital with large supply (HS). Two hospitals are modeled as queueing systems with finite service rates. Both hospitals set prices to maximize the revenues by serving their time-sensitive patients. We adopt a cooperative game theoretic framework to determine when demand sharing is beneficial. We also propose an optimal allocation of this benefit through a commission fee, which makes the alliance stable. We find that demand sharing may not be beneficial even if HS has a low capacity utilization. Demand sharing  
Forming an alliance is a reasonable strategy for hospitals to solve the imbalance between supply and demand. Such a strategy would be feasible if the HD shares a part of its demand with the HS who pays an appropriate commission fee. It is called as demand sharing strategy, and both HD and HS can benefit from this strategy. Evidently, the HD can generate extra revenue by collecting the commission fee, and HS can utilize its idle capacity to serve more patients. It seems that the demand sharing strategy is a win-win thing for the two hospitals. However, as shown in this study, this intuitive win-win situation can only occur under some conditions. These conditions are very important for practitioners considering this demand sharing strategy by forming an alliance of hospitals facing demand and supply mismatch. In the cross-regional specialized medical alliance, since each member of the alliance is decentralized managed, a reasonable method of profit distribution is one of the key factors to achieve cooperation. Otherwise, the HD may prefer to expand its facilities and serve more patients, the HS may tends to reduce its investment, and the patients are more likely to go to the HD because they do not believe in the diagnosis and treatment ability of HS. The mismatch problem becomes more serious, and enter a virtuous circle. Therefore, without a reasonable method of profit distribution, it is difficult to achieve cooperation. This problem also shows in many other service markets due to asymmetric information or other reasons. Owing to some barriers to market entry (such as exclusive licenses, anti-competitive subsidization, and tariff protection), a service provider (monopoly) with limited service capacity in the market can be faced with a potential demand that is larger than what it can handle. In this paper, we try to study the alliance, where some participants are often in short supply, while others are facing an oversupply situation. For example, the demand of hospitals in remote areas exceeds supply, or the demand of core hospitals of the medical alliance exceeds supply. HD will transfer part of their demand to HS, which can improve the efficiency of HS. There are several main research questions to answer; they are as follows: (1) When does demand sharing benefit both hospitals? (2) With demand sharing, what are the optimal pricing decisions for both hospitals? (3) How to divide the revenue gains from the demand sharing alliance between two hospitals? Two hospitals serve their own stream of delay-sensitive patients in two different regions. Each service provider is modeled as an M/M/1 queueing system with different potential Poisson arrival rate and exponential service rate. If the HS participates in this alliance, then it should serve two classes of patients from two different regions. These two classes differ in their arrival rates and delay sensitivities. Based on the price and delay information, each arriving patient in HD is faced with three options--joining the line of HD, switching to the line of HS, and balking. Patients in Region-2 (or HS region) decide whether to join the line of HS or balk. The term 'balk' means that the patient will not join the system and leave without service.