Abstract: Inspired by some existing research and based on the background of the order production intelligent manufacturing system, this paper develops a N-policy M/G/1 multiple vacation queueing model with Bernoulli interruption vacation and random start-time. In this model, the server takes a random length of vacation once there is no customer in the system. If customers arrive during this vacation, then the server interrupts vacation immediately with probability p(0≤p≤1) and activates the system. Otherwise, he/she will not interrupt the vacation with probability 1-p until the end of this vacation before returning to the system and starting it. If no customers arrive during this vacation, then the server immediately begins another vacation and repeats this in this way. Meanwhile, it takes a random length of time to start the system. When the system startup is completed, if the number of customers waiting in the system is greater than or equal to a given integer threshold value N(N≥1), then the server begins service immediately until the system becomes empty again. Otherwise, he/she keeps idle but on duty until the number of customers waiting in the system reaches N and immediately begins serving the waiting customers. The queueing model studied in this paper is more in line with the actual situation and this Bernoulli interruption vacation is more flexible, which can be used for modeling and analysis of intelligent manufacturing systems that produce orders.
Firstly, we apply the total probability decomposition technique to obtain the steady probability distribution of queue-length at the beginning of the server’s busy period. Secondly, we obtain the probability generating function of the steady queue-length by applying the stochastic decomposition theorem of the steady queue-length. Meanwhile, some algebraic calculations are used to derive some other queueing performance indicators, such as the average queue-length and the probability distribution of the additional queue-length. Finally, the expression of the long-run expected cost per unit time of the system is obtained by establishing a cost structure model and employing the renewal reward theorem that depends on the update process. Furthermore, the system cost optimization problems with(without) the expected waiting time constraints are respectively discussed.
As is known to all, although setting the threshold N can reduce the cost of the system due to frequent startup, it also increases the customer’s waiting time. As a result, it increases the cost of customer stay, reduces customer satisfaction, and even leads to customer loss. In this issue, the long-term benefits of the system are affected. Therefore, it is of great theoretical importance and application value to consider the optimal control policy and cost optimization problem of the system under the waiting time constraints. Inspired by the above, we characterize the factory’s order processing as the queueing model studied in this paper. Numerical examples are provided to investigate the one-dimensional optimal control policy N^* for economizing the system cost as well as the two-dimensional optimal control policy (N^*, T^*) when the vacation time is a fixed time length T, which provides ideas and theoretical support for the decision-makers to achieve the maximization of the economic benefits. The results show that the optimal control policy N^* of starting the service without the waiting time constraints is larger than that of starting the service under the waiting time constraints, and the smaller the constraint threshold of waiting time is, the smaller the optimal control policy N^* of starting the service is and the larger the corresponding minimum expected cost is. Thus, if the system manager expects to reduce customer’s waiting time and increase customer satisfaction, it needs to start the system earlier and pay more costs. Such a consideration is helpful for balancing the interests of the manager and customer and makes the innovation of this paper clear and the theoretical analysis results have more practical application value.
Therefore, the main innovations of this article are as follows: (1)Based on the background of the order production manufacturing system, this paper develops a new queueing model - a N-policy M/G/1 multiple vacation queueing model with Bernoulli interruption vacation and random start-time,which is more flexible and also is more in line with the actual situation. (2)We use the stochastic decomposition theorem of the steady queue-length to obtain the probability generating function of the steady queue-length, and derive some queueing performance measures such as the average queue-length and so on. (3)In depth, we investigate the cost optimization and the optimal control policy of the system, which will make the theoretical analysis results of this paper have better practical applications.

Key words: M/G/1 queue, Bernoulli interruption vacation, N-policy, stochastic decomposition, optimal control policy

摘要： 基于订单生产智能制造系统的背景,本文考虑一个具有Bernoulli中断休假和带随机启动时间的N-策略M/G/1多重休假排队系统。应用全概率分解技术获得服务员忙期开始时队长的稳态概率分布,然后利用稳态队长的随机分解定理推导出稳态队长的概率母函数,同时使用一些代数运算得到了一些其它的排队性能指标,如平均队长、附加队长的概率分布等。最后我们建立费用结构模型,应用依赖更新过程的更新报酬定理得到系统在长期单位时间内运行的成本费用目标函数,以工厂的订单加工为例,分别在没有平均等待时间限制和有平均等待时间限制下研究了系统的费用优化问题,并通过数值计算例子研究了使系统费用最少的一维最优控制策略N^* 和当休假时间为T时系统的二维最优控制策略(N^*,T^*)。

关键词: M/G/1排队, Bernoulli中断休假, N-策略, 随机分解, 最优控制策略