关键词: 多服务员排队, 同步多重休假, 队长不可视, 止步行为, 定价策略

Abstract: Due to the randomness of both the arrival intervals of customers and the service times of servers, queueing phenomena is inevitable in our daily life. When a customer arrives at a certain queueing system, he/she always estimates his/her personal expected residual utility according to the system information he/she gets first, and then decides whether to join the queue or not. Regardless of the visible queueing phenomena like going to the hospital for treatment, purchasing tickets at the train station and so on, or the invisible queueing phenomena like shopping online, consulting business by telephone and so on, there are always several servers in most queueing scenarios. Thus, the research on the balking behavior of customers in the multi-server queueing systems has important practical significance in improving service efficiency, reducing service cost and improving resource utilization and many other aspects.
Considering two information levels: Nearly unobservable case and entirely unobservable case, we study the customers' balking behavior and the pricing strategy in an M/M/c queueing system with synchronous multiple vacations. Once the system is empty, all servers will enter the vacation state synchronously. After a vacation, if there is still no customer in the system, all servers will continue to the next vacation, otherwise they will start to work meanwhile. The nearly unobservable case means that the servers' state is disclosed to the customer but the queue length is not, and the entirely unobservable case means that customers don't know both the servers' state and the queue length. Based on the two information levels, we deduce the stationary distribution of the queue length and some performance indicators like the mean sojourn time of customers by using the quasi birth and death process theory and the matrix geometric method. Next, we obtain the customers' equilibrium balking strategy and socially optimal balking strategy by formulating the personal expected residual utility and the social welfare per unit time from the view of the individual optimization and the system optimization respectively, on which we set the pricing strategy for social welfare optimization. Finally, we numerically make sensitivity analysis of the equilibrium arrival rate,the optimal arrival rate, the optimal social welfare and the optimal price with respect to c, then observe the trend and their relationship as well as the influence of system information on customer behavior.
As a result, we get the following conclusions. Firstly, regardless of the nearly unobservable case or the entirely unobservable case, the number of servers always has the positive influence on both the equilibrium arrival rate and the socially optimal arrival rate of customers, and the optimal behavior of customers in the entirely unobservable case is always the comprehensive reflection of that in the nearly unobservable case where the servers' state is on vacation and working. Next, the amount of system information held by customers has an impact on both the equilibrium behavior and the socially optimal behavior of them, but the information of the servers' state has more and more limited influence on the customers' behavior as the number of the servers increases. Besides, no matter what information level the system is at, there always exists inconsistence between the equilibrium and the socially optimal balking behaviors of customers, and their selfishness will make the system overcrowded. To regulate the customers' behavior, the pricing strategy can be made for the system, i.e., charging customers for optimizing social welfare. It's found that the service price always increases with the increase of the number of servers, but the increase rate slows down when the number of servers increases to a certain number, and eventually, the service price stabilizes. When the number of servers exceeds a certain threshold, the pricing strategy for the entirely unobservable case is consistent with that for the nearly unobservable case when the servers are working. Last but not least, it's advisable for the system manager to set the number of the servers when the service price basically tends to be steady, which can not only achieve the purpose of optimizing the social welfare, but also save the operating costs as much as possible.
In this paper, we mainly analyze the customers' balking behavior and pricing strategy in an M/M/c queue with synchronous multiple vacations based on the nearly unobservable case and the entirely unobservable case. In the future, we can also consider introducing some threshold strategies like the N-policy into the multi-server queueing systems and analyze the customers' balking behavior and the pricing strategy in view of various information levels.

Key words: multi-server queue, synchronous multiple vacations, unobservable queue length, balking behavior, pricing strategy