标题: 在N方策和T方策下具故障及启动服务者M/G/1排队之研究
Analysis of an M/G/1 Queue with Sever Breakdowns and Startup Times under the N Policy and the T Policy
作者: 王琮胤
Tsung-Yin Wang
彭文理
W. L. Pearn
工业工程与管理学系
关键字: N方策;T方策;一般修理时间;一般启动时间;M/G/1排队;敏感度分析;最大熵值;N policy;T policy;general repair time;general startup time;M/G/1 queue;sensitivity analysis;maximum entropy
公开日期: 2007
摘要: 可控制排队系统已被广泛的应用于各领域,诸如制造/存货系统、通讯及电脑网路系统等等,配合成本分析可帮助决策者在舜息万变的资讯时代做出最佳决策,追求最大的利润与效益。本论文主要是探讨在N方策和T方策下服务者会故障及需一般启动时间的M/G/1排队系统,所谓N方策是指服务者一停止服务,要再重新开始提供服务,完全取决于等候线的顾客数是否达到N人,当顾客数达到N人时,服务者马上开始提供服务直到系统中的所有顾客服务完成才停止服务;而所谓T方策是指服务者一停止服务,在一定时间T内至少有一位顾客到达,服务者才会开始提供服务直到系统中的所有顾客服务完成才停止服务,若在间T内无顾客到达,则必需等待直到下一个时间T内至少有一位有顾客到达。我们分别针对N方策和T方策,利用M/G/1排队系统的随机分解性质求出系统中的期望顾客数及系统期望的闲置、启动、忙碌、故障期间长度,并推导出系统闲置、启动、忙碌、故障状态的机率,利用上述的期望值及机率,我们建构成本函数,分别求得最佳的N方策和T方策,并针对最佳的方策做敏感度分析,提供决策者在各种不同的参数下选择最佳的方策。
此外,鉴于在N方策和T方策下服务者会故障及需一般启动时间的M/G/1排队系统中顾客数的机率分配无法确切求出,我们利用最大熵值法,在有限有用的资讯下诸如排队系统中的期望顾客数、闲置、启动、忙碌、故障状态的机率,在最少偏误的资讯下,求出估计的系统中顾客数的机率及近似的平均等候时间,并在各种不同分配下,比较最大熵值法解得近似的平均等候时间与真正的平均等候时间两者之间的误差。我们验证得到最大熵值法是一个有用且够精确的方法,可用来解决复杂的排队问题。
The controllable queueing systems have been done by a considerable amount of work in the past and successfully used in various applied problems such as production/inventory systems, communication systems, computer networks and etc. To cooperate with the cost analysis, it can help the decision maker to make the optimal decision to obtain the maximal profit and efficiency for use. In this dissertation, we investigate an M/G/1 queue under the policy and the policy with sever breakdowns and general startup times. The N policy means the server returns to provide service only when the number of customers in the system reaches N (N>=1) until there are no customers present. The T policy means the server is turned on after a fixed length of time T repeatedly until at least one customer is present in the waiting line. Using the stochastic decomposition property of the M/G/1 queue, we derive various system performance measures, such as the expected number of customers, the expected length of the turned-off, startup, busy, and breakdown periods under the N policy and T policy, respectively. Then we deduce the probabilities of turned-off, startup, busy, and breakdown periods. We also construct the total expected cost function per unit time to determine the optimal threshold N and T , respectively, in order to minimize the cost function for the both policies. Sensitivity investigations on the optimal value of N and T for the both policies, respectively, are studied. Some numerical investigations are presented to demonstrate the analytical results obtained, and show how to make the decision based on minimizing the cost function.
In addition, it is extremely difficult, not impossible, to obtain the explicit formulas such as the steady-state probability mass function of the number of customers and the expected waiting time for the N policy and the T policy M/G/1 queues with repair times and startup times are generally distributed. Under the given available information such as the queue length and the probabilities of idle, startup, busy and breakdown period, we use the maximum entropy principle to derive the approximate formulas for the steady-state probability distributions. We perform a comparative analysis between the approximate waiting time with established waiting time for various distributions, such as exponential (M), k-stage Erlang (Ek), and deterministic (D). We demonstrate that the maximum entropy approach is accurate enough for practical purposes and is a useful method for solving complex queueing systems
URI: http://140.113.39.130/cdrfb3/record/nctu/#GT009233817
http://hdl.handle.net/11536/77146
显示于类别:Thesis


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