关键词: (p,N)-策略, 不中断多重休假, 随机分解, 更新报酬定理, 最优控制策略

Abstract: In order to effectively control the queue size and reduce the cost caused by the frequent switching, the investigation concerning queueing models with control policy and server vacation is worth doing. The earliest works on the control policy of queueing systems were the N-policy, T-policy and D-policy. As to the study of vacation models, most works concentrated on studying models with multiple vacations, single vacation and multiple adaptive vacations. Based on the actual background of the production plant, this paper characterizes the acceptance of orders and their production process by a queueing service process, and studies the performance indicators of the system based on an M/G/1 queueing model with (p,N)-policy and uninterrupted multiple vacations. Here, the (p,N)-policy and uninterrupted multiple vacations mean that once the system becomes empty, the server immediately turns the system off and takes an uninterrupted vacation. When the server returns from vacation and finds the number of customers waiting in the system is greater than or equal to the threshold N, he/she immediately begins serving the waiting customers until the system becomes empty again. If there are arrivals in the vacation but the number of customers arriving in the system is less than N, the server is activated for work with probability p(0p1) or the server is idle but on duty with probability (1-p) until the number of arriving customers in the system reaches N and immediately begins serving the waiting customers. If there is no arrival in the vacation, the server begins another vacation at once.
Firstly, applying the well-known stochastic decomposition theorem of the steady-state queue size of the queueing system, we obtain the steady-state stochastic decomposition result of the system's order number and present the expression of the expected order number through direct algebraic operations. Meanwhile, we use the Little's formula to obtain the expression of the expected waiting time for any order in the system. Secondly, under a given cost structure model, we derive the explicit expression of the long-run expected cost per unit time of the system by using the renewal reward theorem. Without the constraint of average waiting time, we discuss the cost optimization problem of the system. Several numerical examples are presented to investigate the one-dimensional optimal threshold N^* for starting production to minimize the long-run expected cost as well as the two-dimensional optimal threshold (N^*,T^*) when the vacation time is fixed as T. Obviously, if the waiting time is too long, it will reduce customer satisfaction and may even lead to system congestion and customer loss. Therefore, it is of great theoretical importance and application value to consider the cost optimization problem of the system under the expected waiting time constraint. Inspired by the fact above, we further discuss the cost optimization problem of the system under the constraint of average waiting time, and also numerically determine the one-dimensional optimal control policy N^* and the two-dimensional optimal control policy (N^*,T^*). At last, the effects of the probability p on the optimal threshold and the minimum cost are also discussed.
From our study, it is suggested that in some systems where customers are not sensitive to waiting time, system managers can disregard the waiting time constraint and directly choose the threshold value that minimizes the cost of starting the service. In systems such as hospitals and transportation, where customers are more sensitive to waiting time, managers adopt a control policy that minimizes the cost of starting the service within the acceptable waiting time of customers. For future research, the continuous time queueing model studied in this paper can be extended to discrete-time queueing model.

Key words: (p,N)-policy, uninterrupted multiple vacations, stochastic decomposition, renewal reward theorem, optimal control policy