具有不完全信息的M/M/1延迟工作休假系统中的社会最优策略

Social Optimal Strategy in M/M/1 Delayed Working Vacations with Incomplete Information

摘要: 本文将延迟假期和工作休假相结合，研究了具有不完全信息精度的M/M/1延迟工作休假系统的社会最优策略和定价策略。本文给予普通休假模型一个延迟时间，即当系统为空时，服务器进入延迟期，如果顾客在这段时间内到达，则服务器立即转入工作期以正常速率为顾客服务。否则，延迟时间过后，服务器会马上进入休假模式，开始进入缓慢工作的状态，如果服务器在工作休假期为一个顾客服务完成后，系统内仍然有剩余没有接受服务的顾客，那么排队系统将会结束慢速工作的休假状态，转而开始正常工作；若服务的顾客是最后一位顾客，且服务完成后再没有顾客进入系统，服务器则继续保持休假模式，直到休假时间结束或又有新顾客进入系统等待服务。本文考虑两类不完全信息精度情形：1) 几乎不可见情形：顾客在到达的瞬间只能观察到服务器的状态，不知道在场的顾客数量；2) 完全不可见情形：顾客既在到达的瞬间既不能观察到服务器的状态，也不知道在场的顾客数量。本文在这两种情形下利用矩阵几何解得到了平稳概率分布和平均队列长度。根据系统信息精度和顾客的预期收益及等待成本的线性收益–成本结构，确定了社会收益函数，得到了社会最优策略。最后，通过具体例子比较说明主要参数对社会最优策略的影响。

Abstract: This paper studies the social optimal strategy and pricing strategy under the background of M/M/1 delayed working vacations with incomplete information accuracy. This paper gives a delay time to the ordinary leave model, that is, when the system is empty, the server will stay in the system for a period of time. If the customer arrives during this period of time, the server enters the delay period. Otherwise, the server will turn on the working leave mode. If there are customers waiting in the system at the moment of service completion in the working leave period, the system will end the working leave and enter the normal working state. Otherwise, the server will continue to leave until a customer appears or the leave ends. This paper considers two kinds of incomplete information accuracy cases: 1) Almost invisible situation: the customer can only observe the state of the server at the moment of arrival and does not know the number of customers present; 2) Completely invisible situation: customers can neither observe the sta-tus of the server nor know the number of customers present at the moment of arrival. In these two cases, we obtain the stationary probability distribution and the mean queue length by using the matrix-geometric solution method. Based on the linear reward-cost structure of system information accuracy, customer expected benefits, and waiting costs, the social welfare function is determined, and the social optimal strategy is obtained. Finally, numerical examples are given to illustrate the impact of main parameters on social optimal strategy.