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Chapter 14. Queuing Models

Chapter 14. Queuing Models. 第十四章. 排队模型. 运筹学 排队模型. 现实生活中的排队模型. Bank Airport Hospital Road Manufacturing Hotel Restaurant WC ……. 运筹学 排队模型. 如何以最经济的方式控制排队系统,使其达到特定的要求? 提供过多的服务能力来控制排队系统将会造成过量的成本 提供的服务能力不足将会导致过多的等待,降低顾客满意度并造成顾客流失,减少收益. 排队模型存在的问题. 什么是排队论. 什么是 排队论 ?. 什么是排队论.

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Chapter 14. Queuing Models

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  1. Chapter 14. Queuing Models 第十四章. 排队模型 运筹学 排队模型

  2. 现实生活中的排队模型 • Bank • Airport • Hospital • Road • Manufacturing • Hotel • Restaurant • WC • …… 运筹学 排队模型

  3. 如何以最经济的方式控制排队系统,使其达到特定的要求?如何以最经济的方式控制排队系统,使其达到特定的要求? 提供过多的服务能力来控制排队系统将会造成过量的成本 提供的服务能力不足将会导致过多的等待,降低顾客满意度并造成顾客流失,减少收益 排队模型存在的问题

  4. 什么是排队论 什么是 排队论?

  5. 什么是排队论 排队论是研究排队系 统的理论,又称随机服 务系统理论,它提供了 很多不同的排队模型, 通过这些排队模型能够 找到服务成本和服务水 平之间较好的平衡。

  6. A Basic Queuing System

  7. 涉及的要素 顾客 队列 服务台 到达间隔时间 服务时间 排队规则 排队系统的描述 运筹学 排队模型

  8. 绩效测度 等待顾客数 顾客等待时间 服务台忙期 服务台闲期 服务台利用率 排队系统的描述 运筹学 排队模型

  9. Elements of a Queuing Model (Section 14.1)(排队模型的要素) Some Examples of Queuing Systems (Section 14.2)(排队系统的一些实例) Measures of Performance for Queuing Systems (Section 14.3)(排队系统的绩效测度) Table of Contents (主要内容)

  10. A Case Study: The Dupit Corp. Problem (Section 14.4)(案例研究:杜皮特公司问题) Some Single-Server Queuing Models (Section 14.5)(一些单服务台排队模型) Some Multiple-Server Queuing Models (Section 14.6)(一些多服务排队模型) Table of Contents (主要内容)

  11. Priority Queuing Models (Section 14.7)(有优先权的排队模型) Some Insights about Designing Queuing Systems (Section 14.8)(关于设计排队系统的一些启示) Economic Analysis of the Number of Servers to Provide (Section 14.9)(服务台数量的经济分析) Table of Contents (主要内容)

  12. Herr Cutter’s Barber Shop • Herr Cutter is a German barber who runs a one-man barber shop. (赫尔卡特先生是一个德国理发师,开了一家单人理发店) • Herr Cutter opens his shop at 8:00 A.M. (赫尔卡特每天早上8点开始工作) • The table shows his queuing system in action over a typical morning. (下表显示了一个典型的上午排队系统情况)

  13. Herr Cutter’s Barber Shop

  14. Arrivals(到达) • The time between consecutive arrivals to a queuing system are called the interarrival times. (连续两个顾客到达排队系统的时间间隔称为到达间隔时间) • The expected number of arrivals per unit time is referred to as the mean arrival rate. (单位时间到达的期望数量称为平均到达率)

  15. Arrivals(到达) • The symbol used for the mean arrival rate is (平均到达率的符号如下) • The mean of the probability distribution of interarrival times is (间隔时间的概率分布均值是)

  16. Arrivals(到达) • Most queuing models assume that the form of the probability distribution of interarrival times is an exponential distribution. (大多数排队模型假设到达间隔时间的概率分布形式是指数分布)

  17. Evolution of the Number of Customers

  18. The Exponential Distribution for Interarrival Times

  19. There is a high likelihood of small interarrival times, but a small chance of a very large interarrival time. This is characteristic of interarrival times in practice. (小间隔时间出现的可能性很大,而大间隔时间出现的机会则很小,这个间隔时间的特征能在实践中观察到) Properties of the Exponential Distribution

  20. 到达间隔时间服从指数分布则到达过程服从泊松分布到达间隔时间服从指数分布则到达过程服从泊松分布 指数分布

  21. 泊松分布

  22. For most queuing systems, the servers have no control over when customers will arrive. Customers generally arrive randomly. (对于大多数排队系统,服务台无法控制顾客何时到达,顾客的到达一般是随机的) Properties of the Exponential Distribution

  23. Having random arrivals means that interarrival times are completely unpredictable, in the sense that the chance of an arrival in the next minute is always just the same. (随机到达意味着到达时间完全是不可预测的,下一分钟有顾客到达的概率和其他分钟相同) Properties of the Exponential Distribution

  24. The only probability distribution with this property of random arrivals is the exponential distribution. (唯一符合随机到达的到达间隔时间分布是指数分布) The fact that the probability of an arrival in the next minute is completely uninfluenced by when the last arrival occurred is called the lack-of-memory property. (下一分钟到达的概率完全不受上一次到达的影响的事实称为无记忆性) Properties of the Exponential Distribution

  25. The number of customers in the queue (or queue size) is the number of customers waiting for service to begin. (队列中的顾客数也称队列大小或队列长度,是等候服务的顾客数量) The number of customers in the system is the number in the queue plus the number currently being served. (系统中的顾客数是队列中的顾客数加上正在接受服务的顾客数量) The Queue (队列)

  26. The queue capacity is the maximum number of customers that can be held in the queue. (队列容量是队列所能容纳的最大顾客数量) An infinite queue is one in which, for all practical purposes, an unlimited number of customers can be held there. (无限队列是为了应用方便,可以容纳无限顾客数量的队列) The Queue (队列)

  27. When the capacity is small enough that it needs to be taken into account, then the queue is called a finite queue. (当容量很小,需要将其考虑在内时,这个队列就称为有限队列) The Queue (队列)

  28. The queue discipline refers to the order in which members of the queue are selected to begin service. (排队规则指选择队列中的成员接受服务的顺序) The most common is first-come, first-served (FCFS). (最普通的排队规则是先到先服务) Other possibilities include random selection, some priority procedure, or even last-come, first-served. (其他的可能规则包括随机选择、有优先权的服务甚至后到先服务) The Queue (队列)

  29. When a customer enters service, the elapsed time from the beginning to the end of the service is referred to as the service time. (当一个顾客接受服务时,从服务开始到服务结束经过的时间称为服务时间) Basic queuing models assume that the service time has a particular probability distribution. (基本的排队模型假设服务时间是一个特定的概率分布) Service (服务)

  30. The symbol used for the mean of the service time distribution is (用于表示服务时间分布均值的符号是) • The interpretation of itself is the mean service rate. ( 就是平均服务率) • = Expected service completions per unit time for a single busy server (一个连续工作的服务台单位时间完成的服务量的期望值) Service (服务)

  31. Some Service-Time Distributions • Exponential Distribution (指数分布) • The most popular choice. (最常用的分布形式) • Much easier to analyze than any other. (比其它分布更容易分析) • Although it provides a good fit for interarrival times, this is much less true for service times. (尽管指数分布很适合描述到达间隔时间,但是却不太符合服务时间的特点)

  32. Some Service-Time Distributions • Provides a better fit when the service provided is random than if it involves a fixed set of tasks. (服务是随机时,指数分布比较适合;服务包含一系列固定任务时,指数分布不太适合) • Standard deviation: s = Mean (标准差等于均值) • Constant Service Times (固定服务时间) • A better fit for systems that involve a fixed set of tasks. (更适合于包含一系列固定任务的系统) • Standard deviation (标准差): s = 0.

  33. Some Service-Time Distributions • Erlang Distribution (爱尔朗分布) • Fills the middle ground between the exponential distribution and constant. (服务时间的波动量介于指数分布和常量之间) • Has a shape parameter, k that determines the standard deviation. (有一个形状参数k决定其标准差)

  34. 爱尔郎分布

  35. Erlang Distribution (爱尔朗分布)

  36. Standard Deviation and Mean for Distributions

  37. Labels for Queuing Models • To identify which probability distribution is being assumed for service times (and for interarrival times), a queuing model conventionally is labeled as follows: (为了表示服务时间[以及到达间隔时间]服从什么概率分布,基本排队系统的排队模型通常用如下的符号表示)

  38. Labels for Queuing Models • The symbols used for the possible distributions are (用于表示可能的分布的符号是) • M = Exponential distribution (Markovian) 指数分布(马尔科夫) • D = Degenerate distribution (constant times) 退化分布(固定时间) 运筹学 排队模型

  39. Labels for Queuing Models • Ek = Erlang distribution (shape parameter = k) (爱尔朗分布[形态参数=k]) • GI = General independent interarrival-time distribution (any distribution) (独立的到达间隔时间的一般分布[允许任意类型的分布]) • G = General service-time distribution (any arbitrary distribution) (服务时间的一般分布[允许任意类型的分布])

  40. Summary of Usual Model Assumptions Interarrival times are independent and identically distributed according to a specified probability distribution. (到达间隔时间独立同分布) All arriving customers enter the queuing system and remain there until service has been completed. (所有到达的顾客都进入排队系统,并待在那里直到服务结束)

  41. Summary of Usual Model Assumptions The queuing system has a single infinite queue, so that the queue will hold an unlimited number of customers (for all practical purposes). (排队系统有一个无限队列,因此队列将可以容纳无限量的顾客[出于应用的原因]) The queue discipline is first-come, first-served. (排队规则是先到先服务)

  42. Summary of Usual Model Assumptions The queuing system has a specified number of servers, where each server is capable of serving any of the customers. (排队系统拥有特定数量的服务台,每一个服务台能够为任意一位顾客提供服务)

  43. Summary of Usual Model Assumptions Each customer is served individually by any one of the servers. (每一位顾客由一个服务台单独提供服务) Service times are independent and identically distributed according to a specified probability distribution. (服务时间是独立的,服从特定的概率分布)

  44. Examples of Commercial Service Systems that Are Queuing Systems

  45. Examples of Internal Service Systems That Are Queuing Systems

  46. Examples of Transportation Service Systems That Are Queuing Systems

  47. Choosing a Measure of Performance • Managers who oversee queuing systems are mainly concerned with two measures of performance: (检查排队系统的管理人员主要考虑两种绩效测度) • How many customers typically are waiting in the queuing system? (排队系统中有多少顾客在等待) • How long do these customers typically have to wait? (这些顾客要等待多少时间)

  48. Choosing a Measure of Performance • When customers are internal to the organization, the first measure tends to be more important. (当顾客是提供服务的组织[内部服务系统]的内部顾客时,第一个测度比较重要) • Having such customers wait causes lost productivity. (让内部服务系统的顾客等待会导致损失生产力)

  49. Choosing a Measure of Performance • Commercial service systems tend to place greater importance on the second measure. (商业服务系统[外部顾客接受商业组织的服务]会认为第二个测度更加重要) • Outside customers are typically more concerned with how long they have to wait than with how many customers are there. (比起已经有多少顾客在等待,顾客更关心他们会等待多久)

  50. Defining the Measures of Performance L = Expected number of customers in the system, including those being served (the symbol L comes from Line Length).

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