Abstract: This paper develops an M/G/1 repairable queue with single vacation and variable failure rates under N-policy control, in which the server takes an uninterrupted vacation once the system becomes empty. When the server returns from vacation and finds that at least N customers are in the system, he/she immediately begins serving the waiting customers until the system becomes empty again. Otherwise, the server keeps idle but on duty until the number of customers waiting in the system reaches N and immediately begins serving the waiting customers. In addition, the service station has variable failure rates during its busy and idle periods. Such queueing model considers not only the random failures of the service station (service facility) that occur during its working periods but also the random failures of the service station that can also happen in its non-working periods due to environmental changes. Further, the random failures of the service station that occur during its non-working periods can be found only when the service station is activated. Hence, the idle failures of the service station can occur at most once in a busy cycle. The queueing model studied in this paper is more in line with the actual situation.
Firstly, we apply the stochastic decomposition property of the steady-state queue size to derive its probability generating function of the system, and obtain some performance measures by some algebraic operations, such as the average queue size, the average length of the busy cycle and the average waiting time of an arbitrary customer. Secondly, we use the renewal process theory, the total probability decomposition technique and Laplace transform to discuss some critical reliability measures of the system, including the unavailability and failure frequency.
Although setting the threshold N can reduce the cost of the system due to frequent startup, it also increases the customer's waiting time. Therefore, it is of great theoretical importance and application value to consider the cost optimization problem of the system under the expected waiting time constraints. Inspired by the above, we establish a cost model and a cost objective function to separately discuss the cost optimization problems with (without) the expected waiting time constraints under the widely-applied PH distribution. Several numerical examples are presented to determine the one-dimensional optimal threshold N^* that minimizes the long-run expected cost of the system as well as the two-dimensional optimal threshold (N^*,T^*) when the vacation time is fixed as T, which provides ideas and theoretical support for the decision-makers to achieve the maximization of the economic benefits. Moreover, we compare the results of the unconstrained and constrained scenarios. The results show that the N^* of starting the service without the expected waiting time constraints is larger than that of starting the service under the expected waiting time constraints, and the smaller the constraint threshold of waiting time is, the smaller the N^*of starting the service is. 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 from the manager's and customer's point of view to determine the optimal threshold, which is helpful for balancing the interests of the manager and customer, so as make the innovation of this paper clear. The theoretical analysis results have more practical application value.
For future research, the continuous-time queueing system studied in this paper can be further extended to the corresponding discrete-time queueing system, and the non-Markovian arrival processes of customers and uninterrupted multiple vacations can also be considered.

Key words: N-policy, single vacation, variable failure rates, performance measures, optimal control policy

摘要： 本文研究在N-策略控制下有单重休假和可变失效率的M/G/1可修排队系统,其中每当系统清空顾客,服务员就立刻开启一次不中断休假。当服务员休假回来若发现系统中至少有N个顾客时便立即开始服务,否则便待在系统中直到有N个顾客再提供服务,而且服务台在其忙期和闲期中的失效率不同。首先应用稳态队长的随机分解性质推导出了稳态队长的概率母函数,然后通过一些代数运算得到了系统一些排队性能指标,如平均队长、忙循环的平均长度和顾客的平均等待时间等。其次,我们还得到了服务台的可靠性指标,如不可用度和故障频度。最后建立费用目标函数,用数值实例讨论了有(没有)等待时间约束下的费用优化问题,并得到了使得费用最小的一维最优控制策略N^*和当休假时间长度固定为T时的二维最优控制策略(N^*,T^*)。

关键词: N-策略, 单重休假, 可变失效率, 性能指标, 最优控制策略