Abstract: When faced with the randomness of demand and the limitations of their own service capacity, queuing service systems often take the form of cooperation to reduce operating costs and improve service efficiency. For example, in the medical service system, China is actively promoting cooperation between comprehensive hospitals and community health service centers to form a medical consortium. Patients in the medical consortium can be referred to each other among the cooperating hospitals, thereby reducing congestion in both comprehensive hospitals and hospitals with lower service efficiency. This service cooperation is a cooperation of service providers through the redistribution of customers. Meanwhile, the existence of private hospitals and third-party healthcare providers amounts to an outsourcing market. Motivated by the above healthcare system service cooperation, this paper investigates the problem of cooperation in queuing systems based on customer allocation strategies in the existence of an outsourcing market. In addition, successful cooperation requires a well-developed compensation and incentive mechanism that fairly distributes the total cost of cooperation to the participants, so as to maintain the stability of cooperation and achieve the cooperation goals.
   In this paper, the research objectives are twofold. One is to construct the service collaboration problem in healthcare system as an M/G/1 queuing system cooperation problem based on customer allocation, provide a general modelling framework for cooperative queuing systems, and obtain an optimal customer allocation strategy that effectively facilitates the sharing of service capabilities among service providers. The other is to propose a cost allocation scheme that can maintain the stability of cooperation by using cooperative game theory. The cost allocation scheme needs to be not only intuitive and easy to calculate but also easy to be understood and evaluated by the stakeholders.
   First, considering the delay cost and outsourcing cost of the system, an optimization model for cooperative decision making of M/G/1 queuing system based on customer allocation when an outsourcing market exists is developed with the objective of minimizing the total cost, and the properties of the optimal customer allocation solution are obtained by the KKT condition. Second, with the cooperative game theory, the cost allocation problem of service cooperation in queuing system is constructed as a corresponding cooperative game model. It is proved that the game satisfies subadditivity and a core cost allocation scheme based on the market competitive equilibrium price is proposed. Under this cost allocation scheme, each subsystem will not cost more if it participates in the cooperation than if it is operated independently, and the cost borne by the members of the sub-coalition under the grand alliance is not greater than the total cost of the sub-coalition, so that no member has an incentive to leave the grand coalition and act alone or form a minor coalition, so the grand coalition is stable. This scheme is easy to calculate and has good applicability and operability. Finally, the numerical results with numerical simulation show that when an outsourcing market exists, cooperation based on customer allocation strategy can effectively reduce the total system cost, regardless of whether customers are outsourced or not. With respect to service intensity, collaboration becomes more favorable as the number of participants increases. Coalition decision makers should actively seek cooperation with the outsourcing market when faced with high service intensity and high variability in service times, which can result in more cost savings.
   On the one hand, this paper extends the queuing cooperation model and enriches the theoretical study of the queuing cooperation game, and on the other hand, it provides scientific theoretical guidance for real-life decision makers in queuing service system cooperation. In this paper, we assume the outsourcing market as an additional queuing system with no capacity constraints, and do not consider the service capacity constraints and delay cost of the outsourcing market, so our next research can take them into account. In addition, there is competition between the outsourcing market and the queuing service system, and how the competition affects the cooperation mechanism and the outsourcing pricing strategy is an interesting research direction for the future.

Key words: M/G/1 queuing system, customer allocation, outsourcing market, cooperative game, cost allocation

摘要： 服务商为降低排队系统的服务成本、提高服务效率,通过将顾客在各排队系统间重新分配来进行合作。本文针对由多个M/G/1队列组成的排队服务系统,考虑外包市场,以最小化总成本为目标建立了基于顾客分配的排队合作决策优化模型,并利用KKT条件讨论了最优分配解的性质。结合合作博弈理论,构建了相应的排队合作博弈模型,提出了一种基于市场竞争均衡价格的核心成本分配方案。最后通过数值仿真检验了服务系统的合作效率,数值结果表明:当存在外包市场时,无论顾客是否外包,基于顾客分配策略的合作都可以有效地降低系统总成本;在服务强度方面,随着参与者数量的增加,合作变得更为有利;联盟决策者在面临高服务强度和服务时间差异性较大时,应积极寻求外包市场的合作,从而带来更多的成本节约。

关键词: M/G/1排队系统, 顾客分配, 外包市场, 合作博弈, 成本分配