标题: | 具有第二次可选择服务、服务者故障和启动时间特征之M/G/1排队系统 M/G/1 Queues with Second Optional Service, Server Breakdowns and Startup |
作者: | 杨东育 Yang, Dong-Yuh 彭文理 Pearn, W. L. 工业工程与管理学系 |
关键字: | 比较分析;第一必要服务;一般启动时间;最大熵值原理;<N, p>-方策;联合最佳门槛值;第二次可选择服务;<T, p>-方策;Comparative analysis;first required service;general startup times;maximum entropy principle;<N, p>-policy;optimal threshold values;second optional servic;<T, p>-policy |
公开日期: | 2009 |
摘要: | 本论文主要研究 <N, p>-方策 和 <T, p>-方策 M/G/1 排队系统具有第二次可选择服务、服务者会故障以及启动时间。所有抵达的顾客必须接受第一必要服务,而服务者在提供第一必要服务前,需要一启动时间方可提供服务,当顾客接受完第一必要服务时,部分的顾客会继续选择接受第二种服务。所谓 <N, p>-方策是指,当系统顾客数到达某一数目 N 时,则服务者开始提供服务,直到系统中没有顾客时,服务者拥有 p 的机率停止服务,亦有1- p 的机率服务者无法停止服务,即服务者保持着随时可以提供服务状态。而 <T, p>-方策则是指当系统中没有人时,服务者会进行休假,假期时间长度为T,假期结束后服务者会返回系统察看是否有顾客,当系统至少有一个顾客时,服务者拥有 p 的机率对顾客提供服务,亦有 的机率服务者继续离开系统,并进行另外一个时间长度 T 的休假,直到假期结束后才对顾客提供服务。对于 <N, p>-方策以及 <T, p>-方策的差异性,<N, p>-方策是服务者在服务结束阶段时,对服务者随机控制。而<T, p>-方策则是服务者在假期结束后返回系统,并发现系统内至少有一个顾客,服务者要准备提供服务阶段时,对服务者随机控制。 对于这两个排队系统,我们利用凸组合性质 (convex combination property) 以及 更新报酬定理 (renewal reward theorem) 来推论不同的系统绩效,并建构成本函数来找寻联合最佳门槛值 (N, p) 以及 (T, p),经由推论后我们可以得到明确的封闭解 (closed-form) 以使得成本函数最小。由于排队系统进行敏感度研究,可以提供系统分析者了解输入参数对系统影响,因此,我们也将对最佳的门槛值进行敏感度分析,藉此分析来了解系统参数 (或是成本元素) 的变动后,对于最佳门槛值之影响,最后,我们有提供数值结果并讨论之。 除此之外,要推论出这两个排队系统确切的稳态机率分配是相当困难的,然而在排队理论中,最大熵值原理 (Maximum Entropy Principle) 已经被广泛地利用来分析不同种类的排队模型。在本篇论文里,我们将利用最大熵值原理来近似这两个排队系统队伍长度的稳态机率分配,然后利用推导出来的结果求取近似的队伍中平均等候时间。给定特定的系统参数下,在不同的服务时间、修理时间以及启动时间分配下,我们将对近似以及精确的队伍中平均等候时间进行比较分析,藉由数值结果可以得知相对误差是非常小的,因此,我们可以进一歩说明最大熵值法是相当准确地,而且处理较复杂的排队模型时,确实是一个有用的方法。 In this dissertation, we analyze the <N, p>-policy and the <T, p>-policy M/G/1 queues with second optional service, server breakdowns and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service which follows a general distribution. By <N, p>-policy we mean that the server is turned on when N customers have accumulated, but the server is turned off with probability p as the system becomes empty. The so-called <T, p>-policy queue is characterized by the fact that if at least one customer is present in the queue after T time units have elapsed since the end of the busy period, the server can be switched on with probability p or leaves for a vacation of the same length T with probability (1-p). The main difference between the two randomized policies is that for an <N, p>-policy a decision maker selects actions randomly at completion epoch when there are no customers in the system, whereas for a <T, p>-policy a decision maker selects actions randomly at the beginning epoch of the service when at least one customer appears. For those two queueing systems, we develop various system performances by the convex combination property and the renewal reward theorem. The expected cost functions are established to determine the joint optimal threshold values of (N, p) and (T, p), respectively. Then we obtain the explicit closed-form of the joint optimal solutions for those two policies. Because of sensitivity investigation on the queueing system with critical input parameters may provide some answers for the system analyst. A sensitivity analysis is provided to discuss how the system performances can be affected by the input parameters (or cost parameters) in the investigated queueing service model. For illustration purpose, numerical results are also presented. Furthermore, it is rather difficult to derive the steady-sate probability explicitly for those two queueing systems. The maximum entropy approach has been widely applied in queueing theory to analyze different queueing models. Here, we use this approach to approximate the steady-state probability distributions of the queue length for those two queues. Then the approximate expected waiting time in the queue can be obtained by the maximum entropy solutions. Subsequently, we perform comparative analysis between the approximate results with established exact results for various service time, repair time and startup time distributions with specific parameter values. Our numerical investigations showed that the relative error percentages of the approximate method are quite small. Based on the numerical results, one can demonstrate that the maximum entropy approach is accurate enough and provides a helpful method for analyzing more complicated queueing models. |
URI: | http://140.113.39.130/cdrfb3/record/nctu/#GT009433812 http://hdl.handle.net/11536/81673 |
显示于类别: | Thesis |