基于位相型过程的复杂随机系统研究
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摘要
受到现实世界中某些实际问题的启发,学位论文详细研究了几个复杂的随机运筹学模型,其研究内容涉及到随机运筹学中多个不同的领域,包括可靠性模型,排队理论和库存理论.在论文撰写过程中我们尝试着将上述三个领域的某些研究内容加以结合,并在它们的相互融合与借鉴中挖掘出一些具有较强实际应用背景的新型随机模型.具体来讲,论文围绕如下四个模型的研究而展开：
(1)具有位相型部件采购时间的PH退化可修系统.在这一章中,我们将PH分布与几何过程相结合,通过建立高维马氏过程的最小生成元矩阵获得了系统在稳态情形下的状态概率分布向量及其数值解.与此同时,根据上述研究结果我们也得到了系统的几个重要可靠性指标.进一步,我们还详细讨论了基于部件故障次数的订购策略与更换策略.利用更新报酬过程中的相关结论推导出了系统单位时间平均运行成本的解析表达式并给出了一个确定最优(N-1,N)策略的数值算例.
(2)具有新服务设备位相型采购时间的M/PH(M/PH)/1/K可修排队系统.将基于服务设备失效次数的(N-1,N)维护策略引进移植到随机服务系统中.假设服务设备失效后不能修复如新,用于更换老化服务设备的新服务设备须通过订单方式提前采购后才能得到,并且整个采购过程的持续时间服从—PH分布.在上述假设条件下,我们利用矩阵分析技术给出了系统的稳态队长分布,并获得了一些重要的系统性能绩效指标.最后,借助一个关键引理,给出了长时间稳态条件下服务设备单位时间平均运行成本的解析表达式,并通过直接搜索方法确定了使得平均费用率最小化的N的最优取值.
(3)具有非持久重试需求的休假随机库存系统.在连续盘点(s,S)库存控制策略下考虑一个具有服务员多重休假和非持久性重试需求的随机库存系统.假设来自系统外部的初始需求为一马氏到达过程(MAP),库存物品的补货时间服从PH分布.在库存零水平期间,为节约系统运行成本,可安排服务员离开系统进行持续时间长度服从PH分布的休假.在每一次休假结束后,如果库存水平仍然保持空竭状态,则服务员立即开始他的另一次休假历程.在库存空竭或服务员休假期间到达的初始顾客可以选择放弃其需求而离开系统,也可以选择进入一条容量为无限的重试轨道在一段随机长度的时间后对其需求发起重试.类似的,当重试轨道中的一个顾客需求发起重试时,如果服务员仍然处于休假状态或库存仍未能得到有效补充,则该顾客将以概率1-q选择放弃需求并永久的离开系统,或是以概率q选择再次进入轨道拟进行下一次的需求重试.在上述假设条件下,我们分析了该模型中蕴含的一个潜在的水平相依的拟生灭过程(LDQBD),并利用一种简洁明了的方法确定了该LDQBD过程的截断水平.进一步,基于矩阵连分式技术,我们给出了一个计算重试轨道中顾客需求数量与库存水平平稳联合概率分布的有效算法.利用该平稳联合概率分布并辅以数值实验,我们讨论了系统的各种性能指标.最后,使用直接搜索方法在给定的费用结构下通过数值手段找到了库存控制策略s与S的最优取值.
(4)具有插队行为的M/M/c排队系统等待时间分析及其基于IPH分布的均值简单逼近方法.考虑一个具有顾客插队行为的M/M/c排队系统.将到达顾客划分为普通常规顾客和插队顾客.普通常规顾客在队尾排队等候,而插队顾客试图尽量占据队列中靠近队首的位置以减少自身的等待时间.插队行为通过到达顾客的插队概率和队列中等待顾客对插队行为的容忍概率来描述.利用条件Laplace变换论文给出了三种队列等待时间的均值计算公式.进一步,为提高计算效率,我们在无限位相型(IPH)分布的基础上给出了一种简单的等待时间均值逼近计算方法.数值实验结果表明这种逼近方法是精确且行之有效的.
Motivated by some practical problems arising from the real world situa-tion, some complex stochastic models in operations research are studied in this thesis. The research contents of our thesis relate to several different fields of the stochastic operations research, including reliability models, queueing theory and inventory theory. During the current research process, we try to combine some contents of the above three areas, and develop some new stochastic models with strong application background in their blend. Specifically, the following four new models are considered in our thesis:
(1) A PH deteriorating repairable system with phase type compo-nent procurement lead-time. In this Chapter, combining PH distribution with the geometric process, and through constructing the infinitesimal generator matrix of a high dimensional Markov process, the stationary probability distri-bution vectors of the system state and their numerical solutions are obtained. Meanwhile, according to the above research results, several important reliabil-ity indices of the system are also reported. Furthermore, an ordering policy and a replacement policy based on the number of failures of the component are considered in detail. Finally, employing the standard results in renewal reward process, the explicit expression for the long-run average cost rate of the system is derived, and a numerical example for determining the optimal (N—1,N) policy is also given.
(2) An M/PH(M/PH)/1/K repairable queue with new service ma-chine procurement lead time. The maintenance policy (N—1, N) based on the number of failures of the service machine is introduced into the stochastic service system. Assuming that a failed service machine after repair will not be "as good as new", and the new spare service machine for replacement is only available by an order. More specifically, we suppose that the procurement lead time for delivering the new spare service machine follows a PH distribution. Under such assumptions, we apply the matrix-analytic method to develop the queue-length distribution of the system, and then we obtain some important performance measures of this queueing system. Finally, using a key lemma, the explicit expression of the long-run average cost rate for the service machine is derived. In addition, the direct search method is also implemented to determine the optimal value of N for minimizing the average cost rate.
(3)Vacation inventory system with non-persistent retrial demand. We consider a continuous review (s, S) inventory system with multiple server vacations and non-persistent retrial demands. We assume that the primary de-mands from outside the system constitute a Markovian arrival process (MAP), and the lead time for inventory replenishment follows a PH distribution. To save on operating costs, when the inventory level reaches zero, the server leaves for a vacation of a phase type distributed duration. At the end of a vacation, if the inventory is still empty, the server immediately takes another vacation. The primary demands that occur during the stock out period or during the server vacation period may leave the system forever or enter a retrial orbit with infinite capacity and repeat their demands after some random time. Similarly, if the server is on vacation or the inventory replenishment had not taken place at the time of arrival of a retrial demand, with probability1—q, it leaves the system by giving up item, and with probability q, it goes back to the orbit again. Under these assumptions the underlying level-dependent quasi birth and death (LDQBD) process is analyzed. Furthermore, we give a simple method to determine the truncation level of the LDQBD process, and based on a matrix continued fraction approach, we also develop an efficient algorithm to com-pute the joint stationary distribution of the number of demands in the orbit and the inventory level. Employing the joint stationary distribution, various performance measures along with some interesting numerical experiments have been discussed. Finally, through direct search method, we numerically find the optimal values of.s and S under a given cost structure.
(4)Waiting time analysis for an M/M/c queue with cutting in line behavior and a simple approximation method for the mean value based on the IPH distribution. An M/M/c queueing system with cutting in line behavior is considered. The arriving customers are dispersed into common customers and queue jumpers. A common customer joins the queue at the end, and for reducing the waiting time in the queue, a queue jumper tries to cut in the queue and occupy a position as close to the head of the queue as possible. The cutting in line behavior can be described in two ways, namely, the percentage of customers interjecting and the tolerance probability of interjection by individual customers who are already waiting in the queue. The formulae for calculation three kinds of mean waiting time are obtained. Furthermore, in order to improve the computational efficiency, based on the infinite phase type distribution, a simple approximation method for the mean value of the waiting time is given. Finally, numerical results show that the approximation method considered here is quite accurate and very effective.
(1)具有位相型部件采购时间的PH退化可修系统.在这一章中,我们将PH分布与几何过程相结合,通过建立高维马氏过程的最小生成元矩阵获得了系统在稳态情形下的状态概率分布向量及其数值解.与此同时,根据上述研究结果我们也得到了系统的几个重要可靠性指标.进一步,我们还详细讨论了基于部件故障次数的订购策略与更换策略.利用更新报酬过程中的相关结论推导出了系统单位时间平均运行成本的解析表达式并给出了一个确定最优(N-1,N)策略的数值算例.
(2)具有新服务设备位相型采购时间的M/PH(M/PH)/1/K可修排队系统.将基于服务设备失效次数的(N-1,N)维护策略引进移植到随机服务系统中.假设服务设备失效后不能修复如新,用于更换老化服务设备的新服务设备须通过订单方式提前采购后才能得到,并且整个采购过程的持续时间服从—PH分布.在上述假设条件下,我们利用矩阵分析技术给出了系统的稳态队长分布,并获得了一些重要的系统性能绩效指标.最后,借助一个关键引理,给出了长时间稳态条件下服务设备单位时间平均运行成本的解析表达式,并通过直接搜索方法确定了使得平均费用率最小化的N的最优取值.
(3)具有非持久重试需求的休假随机库存系统.在连续盘点(s,S)库存控制策略下考虑一个具有服务员多重休假和非持久性重试需求的随机库存系统.假设来自系统外部的初始需求为一马氏到达过程(MAP),库存物品的补货时间服从PH分布.在库存零水平期间,为节约系统运行成本,可安排服务员离开系统进行持续时间长度服从PH分布的休假.在每一次休假结束后,如果库存水平仍然保持空竭状态,则服务员立即开始他的另一次休假历程.在库存空竭或服务员休假期间到达的初始顾客可以选择放弃其需求而离开系统,也可以选择进入一条容量为无限的重试轨道在一段随机长度的时间后对其需求发起重试.类似的,当重试轨道中的一个顾客需求发起重试时,如果服务员仍然处于休假状态或库存仍未能得到有效补充,则该顾客将以概率1-q选择放弃需求并永久的离开系统,或是以概率q选择再次进入轨道拟进行下一次的需求重试.在上述假设条件下,我们分析了该模型中蕴含的一个潜在的水平相依的拟生灭过程(LDQBD),并利用一种简洁明了的方法确定了该LDQBD过程的截断水平.进一步,基于矩阵连分式技术,我们给出了一个计算重试轨道中顾客需求数量与库存水平平稳联合概率分布的有效算法.利用该平稳联合概率分布并辅以数值实验,我们讨论了系统的各种性能指标.最后,使用直接搜索方法在给定的费用结构下通过数值手段找到了库存控制策略s与S的最优取值.
(4)具有插队行为的M/M/c排队系统等待时间分析及其基于IPH分布的均值简单逼近方法.考虑一个具有顾客插队行为的M/M/c排队系统.将到达顾客划分为普通常规顾客和插队顾客.普通常规顾客在队尾排队等候,而插队顾客试图尽量占据队列中靠近队首的位置以减少自身的等待时间.插队行为通过到达顾客的插队概率和队列中等待顾客对插队行为的容忍概率来描述.利用条件Laplace变换论文给出了三种队列等待时间的均值计算公式.进一步,为提高计算效率,我们在无限位相型(IPH)分布的基础上给出了一种简单的等待时间均值逼近计算方法.数值实验结果表明这种逼近方法是精确且行之有效的.
Motivated by some practical problems arising from the real world situa-tion, some complex stochastic models in operations research are studied in this thesis. The research contents of our thesis relate to several different fields of the stochastic operations research, including reliability models, queueing theory and inventory theory. During the current research process, we try to combine some contents of the above three areas, and develop some new stochastic models with strong application background in their blend. Specifically, the following four new models are considered in our thesis:
(1) A PH deteriorating repairable system with phase type compo-nent procurement lead-time. In this Chapter, combining PH distribution with the geometric process, and through constructing the infinitesimal generator matrix of a high dimensional Markov process, the stationary probability distri-bution vectors of the system state and their numerical solutions are obtained. Meanwhile, according to the above research results, several important reliabil-ity indices of the system are also reported. Furthermore, an ordering policy and a replacement policy based on the number of failures of the component are considered in detail. Finally, employing the standard results in renewal reward process, the explicit expression for the long-run average cost rate of the system is derived, and a numerical example for determining the optimal (N—1,N) policy is also given.
(2) An M/PH(M/PH)/1/K repairable queue with new service ma-chine procurement lead time. The maintenance policy (N—1, N) based on the number of failures of the service machine is introduced into the stochastic service system. Assuming that a failed service machine after repair will not be "as good as new", and the new spare service machine for replacement is only available by an order. More specifically, we suppose that the procurement lead time for delivering the new spare service machine follows a PH distribution. Under such assumptions, we apply the matrix-analytic method to develop the queue-length distribution of the system, and then we obtain some important performance measures of this queueing system. Finally, using a key lemma, the explicit expression of the long-run average cost rate for the service machine is derived. In addition, the direct search method is also implemented to determine the optimal value of N for minimizing the average cost rate.
(3)Vacation inventory system with non-persistent retrial demand. We consider a continuous review (s, S) inventory system with multiple server vacations and non-persistent retrial demands. We assume that the primary de-mands from outside the system constitute a Markovian arrival process (MAP), and the lead time for inventory replenishment follows a PH distribution. To save on operating costs, when the inventory level reaches zero, the server leaves for a vacation of a phase type distributed duration. At the end of a vacation, if the inventory is still empty, the server immediately takes another vacation. The primary demands that occur during the stock out period or during the server vacation period may leave the system forever or enter a retrial orbit with infinite capacity and repeat their demands after some random time. Similarly, if the server is on vacation or the inventory replenishment had not taken place at the time of arrival of a retrial demand, with probability1—q, it leaves the system by giving up item, and with probability q, it goes back to the orbit again. Under these assumptions the underlying level-dependent quasi birth and death (LDQBD) process is analyzed. Furthermore, we give a simple method to determine the truncation level of the LDQBD process, and based on a matrix continued fraction approach, we also develop an efficient algorithm to com-pute the joint stationary distribution of the number of demands in the orbit and the inventory level. Employing the joint stationary distribution, various performance measures along with some interesting numerical experiments have been discussed. Finally, through direct search method, we numerically find the optimal values of.s and S under a given cost structure.
(4)Waiting time analysis for an M/M/c queue with cutting in line behavior and a simple approximation method for the mean value based on the IPH distribution. An M/M/c queueing system with cutting in line behavior is considered. The arriving customers are dispersed into common customers and queue jumpers. A common customer joins the queue at the end, and for reducing the waiting time in the queue, a queue jumper tries to cut in the queue and occupy a position as close to the head of the queue as possible. The cutting in line behavior can be described in two ways, namely, the percentage of customers interjecting and the tolerance probability of interjection by individual customers who are already waiting in the queue. The formulae for calculation three kinds of mean waiting time are obtained. Furthermore, in order to improve the computational efficiency, based on the infinite phase type distribution, a simple approximation method for the mean value of the waiting time is given. Finally, numerical results show that the approximation method considered here is quite accurate and very effective.
引文
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[88]P. Zeephongsekul, A. Bedford, Waiting time analysis of the multiple pri-ority dual queue with a preemptive priority service discipline, European Journal of Operational Research,2006,172(3):886-908.
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