Abstract: Queueing phenomena is very common in our daily life. For various reasons, the customers waiting in the queue may have different kinds of behaviors. We mainly focus on three different customer behaviors here. When an arriving customer finds the queue in the system is too long, this customer may decide not to join the queue anymore, which is called “balking”. For example, during rush hours, the arriving customer who finds a long queue at the bus stop may choose to take the next bus. When an arriving customer finds that the queue is too long, this customer may consider cutting in the queue to reduce the waiting time, which is called “interjection”. For instance, during peak time, the customer in the canteen may try to find a friend in the front of the queue to cut in the queue to obtain the food sooner. For the customers already waiting in the queue, after waiting for some time, some customers may lose patience and leave the queue before receiving any service, which is called “reneging”. For instance, in the inventory system where perishable goods are stored, while waiting for sale in the inventory system, if the goods decay then they have to exit the queue for the original sales plan. The initial investigation of the customer behaviors in the queue considered the impact of only single customer behavior mentioned above on the performance of the queueing systems. Recently, the influence of the combination of two customer behaviors mentioned beforehand on the performance of the queueing systems has been investigated. However, the theoretical exploration of the comprehensive influence of customer balking, interjections and reneging on the performance of the queueing system, possibly with priority customers, is still absent. Meanwhile, the combination of these three behaviors may occur in a real priority queueing system. For example, in the hospital, compared with ordinary patients, the emergency patients have priority, when an ordinary patient finds the hospital is too crowded, this patient may choose to visit another hospital, which is “balking”. Moreover, the ordinary patient may try to cut in the queue to receive the treatment in a shorter time, which is “interjection”. Meanwhile, after waiting too long in a waiting room in the hospital, an ordinary patient may decide to leave the hospital, which is “reneging”. Therefore, from both theoretical and practical aspects, we call for the investigation of multi-server priority queueing systems with customer balking, interjections and reneging.
Firstly, by considering the aforementioned three customer behaviors which are customer balking, interjections and reneging, we develop a multi-server queueing system with three-stage input rates and service rates which depend on the system state. Moreover, we determine the threshold levels of the input and service rates that are based on the revenue & cost structure function. In particular, when there is no priority in the system, i.e., there are only ordinary customers in the system, we have obtained the explicit expressions for the stationary distribution of the system with customer balking, interjections and reneging and the corresponding performance measures. Secondly, we investigate the priority queueing system with customer balking, interjections and reneging. We construct a quasi-birth and death process to model the problem, obtain the stationary distribution of the system via the matrix analytical method, and compute the corresponding performance measures. Finally, we conduct numerical analysis to illustrate that the impact of customer balking, interjections and reneging on the system performance is profound. In the future, we may extend our work to systems with more than two types of customers, where more types of customers have priority over other types. Moreover, apart from preemptive queueing systems, we could also investigate non-reemptive queueing systems in our future work.

Key words: customer behavior, priority, threshold segmentation, quasi birth-and-death process, matrix analytical method

摘要： 在实际排队系统中,顾客可能会出现各种不同的行为,本文主要研究了同时具有顾客止步、插队和中途退出三种行为的优先权排队系统。首先,本文基于顾客的止步、插队和中途退出行为构建了依赖系统状态的三段式输入率和服务率的多服务台排队模型,且采用收益-费用结构函数确定分段阈值。其次,本文研究具有顾客止步、插队和中途退出行为的普通排队系统和强占优先权排队系统。本文利用拟生灭过程对问题建模并使用矩阵分析法对模型进行求解,推导了两个排队系统的稳态概率的表达式并计算了相关的性能指标。通过数值分析,本文说明了顾客的止步、插队和中途退出三种行为对系统性能带来的影响是不容忽视的。

关键词: 顾客行为, 优先权, 阈值分段, 拟生灭过程, 矩阵分析法