Chapter 14. Queuing Models

Presentation on theme: "Chapter 14. Queuing Models"— Presentation transcript:

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？
排队模型存在的问题 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

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 排队系统的描述 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 运筹学 排队模型

8 排队系统的描述 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 运筹学 排队模型

9 Table of Contents (主要内容)
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)

10 Table of Contents (主要内容)
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)

11 Table of Contents (主要内容)
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)

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 Properties of the Exponential Distribution
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)

20 指数分布 到达间隔时间服从指数分布则到达过程服从泊松分布

21 泊松分布

22 Properties of the Exponential Distribution
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)

23 Properties of the Exponential Distribution
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)

24 Properties of the Exponential Distribution
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)

25 The Queue (队列) 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)

26 The Queue (队列) 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)

27 The Queue (队列) 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)

28 The Queue (队列) 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)

29 Service (服务) 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)

30 Service (服务) 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)

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).

51 Defining the Measures of Performance
Lq= Expected number of customers in the queue, which excludes customers being served.

52 Defining the Measures of Performance
W = Expected waiting time in the system (including service time) for an individual customer (the symbol W comes from Waiting time).

53 Defining the Measures of Performance
Wq = Expected waiting time in the queue (excludes service time) for an individual customer.

54 Defining the Measures of Performance
These definitions assume that the queuing system is in a steady-state condition. (这些定义假设排队系统处于平稳条件中) 运筹学 排队模型

55 Relationship between L, W, Lq and Wq
并且 知其一即知其四

56 Using Probabilities as Measures of Performance
In addition to knowing what happens on the average, we may also be interested in worst-case scenarios. (除了知道系统性能指标的各种平均值之外，我们还可能对最坏情况感兴趣) What will be the maximum number of customers in the system? (Exceeded no more than, say, 5% of the time.)(在95%的时间里，系统的最大顾客数不会超过某个值?)

57 Using Probabilities as Measures of Performance
What will be the maximum waiting time of customers in the system? (Exceeded no more than, say, 5% of the time.)(在95%的时间里，系统的顾客最大等待时间不会超过某个值?) Statistics that are helpful to answer these types of questions are available for some queuing systems (统计手段在回答这类问题时非常有用，因此它也可用于一些排队系统中):

58 Using Probabilities as Measures of Performance
Pn = Steady-state probability of having exactly n customers in the system.(在系统中有n个顾客的平稳概率) P(W ≤ t) = Probability the time spent in the system will be no more than t.(在系统中的时间消耗不超过t的概率) P(Wq≤ t) = Probability the wait time will be no more than t.(等待时间不超过t的概率)

59 Using Probabilities as Measures of Performance
Examples of common goals(常用目标的一些例子): No more than three customers 95% of the time: P0 + P1 + P2 + P3 ≥ 0.95 (在95%的时间里系统中的顾客数不超过3个) No more than 5% of customers wait more than 2 hours: P(W ≤ 2 hours) ≥ 0.95 (等待时间超过2小时的顾客数不超过总顾客数的5%)

60 The Dupit Corp. Problem The Dupit Corporation is a longtime leader in the office photocopier marketplace. Dupit’s service division is responsible for providing support to the customers by promptly repairing the machines when needed. This is done by the company’s service technical representatives, or tech reps.

61 杜皮特公司在办公复印机市场上长期处于领导地位
The Dupit Corp. Problem 杜皮特公司在办公复印机市场上长期处于领导地位 杜皮特公司的服务部门负责为公司的顾客提供高质量的服务支持，在需要的时候维修公司的设备。这项工作由公司的技术服务代表完成

62 The Dupit Corp. Problem Current policy: Each tech rep’s territory is assigned enough machines so that the tech rep will be active repairing machines (or traveling to the site) 75% of the time. A repair call averages 2 hours, so this corresponds to 3 repair calls per day. Machines average 50 workdays between repairs, so assign 150 machines per rep.

63 目前的政策：每一位技术服务代表的地域应当有足够多的机器设备，使得技术服务代表在75%的时间里都处于维修工作状态(或在前往维修地点的路上)
The Dupit Corp. Problem 目前的政策：每一位技术服务代表的地域应当有足够多的机器设备，使得技术服务代表在75%的时间里都处于维修工作状态(或在前往维修地点的路上) 每台设备的平均维修时间是2小时，一天平均要接到3个维修电话 机器平均50个工作日需要维修一次，因此为每位技术服务代表分配150台机器

64 目前存在的问题 刚买了一款新式彩色 打印-复印机，现在出 了问题，等了这么久 还没有人来维修，服 务水平太差了，我要 投诉！！

65 The Dupit Corp. Problem Proposed New Service Standard: The average waiting time before a tech rep begins the trip to the customer site should not exceed two hours. (建议的新服务标准：在技术服务代表开始前往顾客所在地修理设备之前，顾客的平均等待时间不应超过两个小时)

66 Alternative Approaches to the Problem
Approach Suggested by John Phixitt: Modify the current policy by decreasing the percentage of time that tech reps are expected to be repairing machines. (约翰建议的做法：修改目前的政策，降低期望技术服务代表进行维修工作的时间百分比。减少分配数量，增加服务代表)

67 Alternative Approaches to the Problem
Approach Suggested by the Vice President for Engineering: Provide new equipment to tech reps that would reduce the time required for repairs. (工程副总裁建议的做法：为技术服务代表提供新的装备，以减少设备修理时间)

68 Alternative Approaches to the Problem
Approach Suggested by the Chief Financial Officer: Replace the current one-person tech rep territories by larger territories served by multiple tech reps. (首席财务总监建议的做法：将现在的单个技术服务代表负责区域转变为较大的区域，由多个技术服务代表提供服务)

69 Alternative Approaches to the Problem
Approach Suggested by the Vice President for Marketing: Give owners of the new printer-copier priority for receiving repairs over the company’s other customers. (营销副总裁建议的做法：授予这种新的打印-复印机的所有者比公司的其他顾客优先接受维修服务的权利)

70 The Queuing System for Each Tech Rep
The customers: The machines needing repair.(顾客：需要修理的设备) Customer arrivals: The calls to the tech rep requesting repairs.(顾客到达：打给每一个技术服务代表要求修理的电话) The queue: The machines waiting for repair to begin at their sites.(队列：顾客所在地等待修理的所有设备) The server: The tech rep.(服务者：技术服务代表)

71 The Queuing System for Each Tech Rep
Service time: The total time the tech rep is tied up with a machine, either traveling to the machine site or repairing the machine. (Thus, a machine is viewed as leaving the queue and entering service when the tech rep begins the trip to the machine site.)

72 The Queuing System for Each Tech Rep
服务时间：技术服务代表花在一台设备上的总时间，包括到待修设备所在地的行进时间和维修时间(因此当技术服务代表开始前往设备所在地时，这台设备就可以视作已经离开队列进入服务系统了)

73 Notation for Single-Server Queuing Models

74 Notation for Single-Server Queuing Models
运筹学 排队模型

75 Assumptions(假设) The M/M/1 Model
Interarrival times have an exponential distribution with a mean of (到达间隔时间服从均值为 的指数分布) Service times have an exponential distribution with a mean of 1/m. (服务时间服从均值为1/m 的指数分布) The queuing system has one server. (排队系统有一个服务台)

76 The M/M/1 Model 运筹学 排队模型

77 The M/M/1 Model 运筹学 排队模型

78 The M/M/1 Model The probability of having exactly n customers in the system is (系统中刚好有n个顾客的概率为) The probability that the waiting time in the system exceeds t is (在系统中的等待时间超过t的概率是)

79 The M/M/1 Model The probability that the waiting time in the queue exceeds t is (在队列中的等待时间超过t的概率是) 运筹学 排队模型

80 M/M/1 Queuing Model for the Dupit’s Current Policy

81 John Phixitt’s Approach (Reduce Machines/Rep)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day). (建议的新服务标准是将每一个服务开始前的平均等待时间降为2小时)

82 John Phixitt’s Approach (Reduce Machines/Rep)
John Phixitt’s suggested approach is to lower the tech rep’s utilization factor sufficiently to meet the new service requirement. (约翰 建议的方法是降低技术服务 代表的有效因子以适应新的 服务要求)

83 John Phixitt’s Approach (Reduce Machines/Rep)
运筹学 排队模型

84 M/M/1 Model for John Phixitt’s Suggested Approach (Reduce Machines/Rep)

85 指定给每一位技术服务代表的设备数从150下降到100 现有的技术服务代表人数10000人，工资约6亿美元/年
方案1的结果 指定给每一位技术服务代表的设备数从150下降到100 现有的技术服务代表人数10000人，工资约6亿美元/年 公司大约要增加5000位技术服务代表，增加的费用大约为3亿美元/年

86 Assumptions(假设)： The M/G/1 Model
Interarrival times have an exponential distribution with a mean of 1/l. (到达间隔时间服从均值为1/l 的指数分布) Service times can have any probability distribution. You only need the mean (1/m) and standard deviation( s). (服务时间可以为任何概率分布，仅需要知道均值和标准差) The queuing system has one server.(排队系统有一个服务台)

87 The probability of zero customers in the system is (系统中没有顾客的概率是)
The M/G/1 Model The probability of zero customers in the system is (系统中没有顾客的概率是) The expected number of customers in the queue is (队列中的顾客期望数是)

88 The expected number of customers in the system is (系统中的顾客期望数是)
The M/G/1 Model The expected number of customers in the system is (系统中的顾客期望数是) The expected waiting time in the queue is (队列中的期望等待时间是)

89 The expected waiting time in the system is (在系统中的期望等待时间是)
The M/G/1 Model The expected waiting time in the system is (在系统中的期望等待时间是) 运筹学 排队模型

90 运筹学 排队模型

91 VP for Engineering Approach (New Equipment)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day).(建议的新服务标准是将每一个服务开始前的平均等待时间降为2小时)

92 VP for Engineering Approach (New Equipment)
The Vice President for Engineering has suggested providing tech reps with new state-of-the-art equipment that would reduce the time required for the longer repairs. (工程副总裁建议为技术服务代表提供新式的设备以大大降低需要较长时间的修理工作)

93 VP for Engineering Approach (New Equipment)
After gathering more information, they estimate the new equipment would have the following effect on the service-time distribution: (收集了更多的信息之后，他们估计新设备对服务时间分布将产生如下影响) Decrease the mean from 1/4 day to 1/5 day. (均值减少) Decrease the standard deviation from 1/4 day to 1/10 day. (标准差减少)

94 M/G/1 Model for the VP of Engineering Approach (New Equipment)
方案2的结果：可以满足要求，增加的费用大约5亿美元(一次性投入)

95 Any number of servers (denoted by s). (任意数量的服务台，用s表示)
The M/M/s Model Assumptions (假设)： Interarrival times have an exponential distribution with a mean of 1/l. (到达间隔时间服从均值为 的指数分布) Service times have an exponential distribution with a mean of 1/m. (服务时间服从均值为 的指数分布) Any number of servers (denoted by s). (任意数量的服务台，用s表示)

96 The M/M/s Model With multiple servers, the formula for the utilization factor becomes (对于多服务台系统，有效性因子的公式变为) but still represents that average fraction of time that individual servers are busy. (仍然代表服务台处于繁忙状态下的时间比例)

97 Values of L for the M/M/s Model for Various Values of s

98 CFO Suggested Approach (Combine Into Teams)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day). (建议的新服务标准是将每一个服务开始前的平均等待时间降为2小时)

99 CFO Suggested Approach (Combine Into Teams)
The Chief Financial Officer has suggested combining the current one-person tech rep territories into larger territories that would be served jointly by multiple tech reps. (首席财务官建议将目前的单人技术服务代表区域合并为一些较大的区域，由多位技术服务代表一起提供服务)

100 CFO Suggested Approach (Combine Into Teams)

101 M/M/s Model for the CFO’s Suggested Approach (Combine Into Teams of Two)

102 CFO Suggested Approach (Teams of Three)

103 M/M/s Model for the CFO’s Suggested Approach (Combine Into Teams of Three)

104 Comparison of Wq with Territories of Different Sizes

105 方案3的结果 如果将三个连续的单人技术服务代表区域合并为一个大区域，由三个相同的技术服务代表提供服务，能够满足要求
没有直接增加额外更多的成本 缺点 增加技术代表前往设备所在地的时间 团队作业的协调困难 个性化服务减弱 个体自豪感下降 团队成员的摩擦问题

106 Values of L for the M/D/s Model for Various Values of s

107 Values of L for the M/Ek/2 Model for Various Values of k

108 Priority Queuing Models
General Assumptions: There are two or more categories of customers. Each category is assigned to a priority class. Customers in priority class 1 are given priority over customers in priority class 2. Priority class 2 has priority over priority class 3, etc.

109 Priority Queuing Models
一般假设： 有两类或更多类的顾客。每一类都具有一个优先级。优先级1的顾客将比优先级2的顾客优先接受服务。如果优先级多于两个，优先级2的顾客将具有优先于优先级3的顾客的权利

110 Priority Queuing Models
General Assumptions(一般假设): After deferring to higher priority customers, the customers within each priority class are served on a first-come-fist-served basis. (除了在较高优先级的顾客后面接受服务外，同一优先级内的顾客服从先到先服务的规则)

111 Priority Queuing Models
Two types of priorities Nonpreemptive priorities: Once a server has begun serving a customer, the service must be completed (even if a higher priority customer arrives). However, once service is completed, priorities are applied to select the next one to begin service.

112 Priority Queuing Models
两类优先权 无强制的优先规则：当一个服务台开始为一个顾客服务时，服务必须在不被打断的条件下完成，即使在服务过程中有更高优先级的顾客到来。然而，这个服务一完成，如果队列中有顾客，将根据优先权从中选出一个顾客进行服务。被选中的顾客是在具有最高优先级的顾客中等待时间最长的一个

113 Priority Queuing Models
Two types of priorities Preemptive priorities: The lowest priority customer being served is preempted (ejected back into the queue) whenever a higher priority customer enters the queuing system.

114 Priority Queuing Models
两类优先权 强制优先规则：当一个具有更高优先权的顾客进入排队系统时，正在接受服务的最低优先级的顾客被挤出(退回到队列中)。当一个服务台成功完成了一个服务工作后，下一个开始接受服务的顾客将以上面无强制优先规则描述的相同规则选出

115 Preemptive Priorities Queuing Model
Additional Assumptions (附加假设)： Preemptive priorities are used as previously described.(强制优先级如上所述) For priority class i (i = 1, 2, … , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/li. (对于优先级i，这个优先级顾客的到达间隔时间服从均值为 的指数分布)

116 Preemptive Priorities Queuing Model
Additional Assumptions (附加假设)： All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved. (不管优先级是多少，所有服务时间服从均值为 的指数分布) The queuing system has a single server. (排队系统有一个服务台)

117 Preemptive Priorities Queuing Model
The utilization factor for the server is (服务台的有效因子是) 运筹学 排队模型

118 Nonpreemptive Priorities Queuing Model
Additional Assumptions (附加假设)： Nonpreemptive priorities are used as previously described. (无强制优先级如上所述) For priority class i (i = 1, 2, … , n), the interarrival times of the customers in that class have an exponential distribution with a mean of 1/li. (对于优先级i，这个优先级顾客的到达间隔时间服从均值为 的指数分布)

119 Nonpreemptive Priorities Queuing Model
Additional Assumptions (附加假设)： All service times have an exponential distribution with a mean of 1/m, regardless of the priority class involved. (不管优先级是多少，所有服务时间服从均值为 的指数分布) The queuing system can have any number of servers. (排队系统可以有任意数量的服务台)

120 Nonpreemptive Priorities Queuing Model
The utilization factor for the servers is (服务台的有效因子为)： 运筹学 排队模型

121 VP of Marketing Approach (Priority for New Copiers)
The proposed new service standard is that the average waiting time before service begins be two hours (i.e., Wq ≤ 1/4 day). The Vice President of Marketing has proposed giving the printer-copiers priority over other machines for receiving service. The rationale for this proposal is that the printer-copier performs so many vital functions that its owners cannot tolerate being without it as long as other machines.

122 VP of Marketing Approach (Priority for New Copiers)
建议的新服务标准是将每一个服务开始前的平均等待时间降为2小时 营销副总裁建议给打印-复印机高于其他设备的接受服务的优先权。理由是打印-复印机有很多的关键功能，如果这台设备的停机等待维修时间与其他设备一样长，其拥有者是无法忍受的

123 VP of Marketing Approach (Priority for New Copiers)
The mean arrival rates for the two classes of copiers are (这两类复印机的平均到达率分别是)： The proportion of printer-copiers is expected to increase, so in a couple years (据估计在几年里这种打印-复印机的比例会增加，于是)

124 Nonpreemptive Priorities Model for VP of Marketing’s Approach (Current Arrival Rates)

125 Nonpreemptive Priorities Model for VP of Marketing’s Approach (Future Arrival Rates)

126 Expected Waiting Times with Nonpreemptive Priorities

127 方案4的结果 可以满足目前的要求 将优先权和多服务台单队列相结合可以满足将来的要求 无需增加更多额外费用 缺点
降低没有优先权的顾客的服务水平 和多服务台单队列模型相同的其它缺点

128 The Four Approaches Under Considerations

129 The Four Approaches Under Considerations
Decision: Adopt fourth proposal (except for sparsely populated areas where second proposal should be adopted). (决策：采用第四个方案[对于人口分散的区域则采用第二个方案])

130 Some Insights About Designing Queuing Systems
When designing a single-server queuing system, beware that giving a relatively high utilization factor (workload) to the server provides surprisingly poor performance for the system. (当设计一个单服务台排队系统时，注意相对较高的服务台有效因子[工作强度]将使得系统的绩效测度令人吃惊地低下)

131 Some Insights About Designing Queuing Systems
Decreasing the variability of service times (without any change in the mean) improves the performance of a queuing system substantially. (降低服务时间的波动[不改变均值]可以大大改进单服务台排队系统的绩效[对于多服务台排队系统也是这样，特别是有较高有效因子的系统])

132 Some Insights About Designing Queuing Systems
Multiple-server queuing systems can perform satisfactorily with somewhat higher utilization factors than can single-server queuing systems. For example, pooling servers by combining separate single-server queuing systems into one multiple-server queuing system greatly improves the measures of performance.

133 Some Insights About Designing Queuing Systems
具有较高有效因子的多服务台排队系统能够比单服务台排队系统表现得更令人满意。例如，通过将分立的单服务台排队系统组合为一个多服务台排队系统产生的联合服务台大大改善了系统的绩效

134 Some Insights About Designing Queuing Systems
Applying priorities when selecting customers to begin service can greatly improve the measures of performance for high-priority customers. (应用优先级选择顾客服务可以大大改善对高优先级顾客服务的绩效测度)

135 Effect of High-Utilization Factors (Insight 1)

136

137 Economic Analysis of the Number of Servers to Provide
In many cases, the consequences of making customers wait can be expressed as a waiting cost. (在许多情况下，一个组织让顾客等待的结果可以称为等待成本) The manager is interested in minimizing the total cost. (管理人员对最小总成本很感兴趣)

138 Economic Analysis of the Number of Servers to Provide
TC = Expected total cost per unit time (单位时间期望总成本) SC = Expected service cost per unit time (单位时间期望服务成本) WC = Expected waiting cost per unit time (单位时间期望等待成本) The objective is then to choose the number of servers so as to (目标是选择服务台的数量，使得) Minimize TC = SC + WC

139 Economic Analysis of the Number of Servers to Provide
When each server costs the same (Cs = cost of server per unit time，单位时间的服务成本), (当每一个服务台成本相同时，则服务成本为) SC = Cs s When the waiting cost is proportional to the amount of waiting (Cw = waiting cost per unit time for each customer，每一位顾客单位时间等待成本), (当等待成本与等待量成比例时，等待成本可以表达为) WC = Cw L

140 Two clerks run the tool crib. (有两个管理员在管理这个工具架)
Acme Machine Shop The Acme Machine Shop has a tool crib for storing tool required by shop mechanics. (艾可米机械制造厂有一个存放工厂技术人员所用工具的工具架) Two clerks run the tool crib. (有两个管理员在管理这个工具架)

141 Acme Machine Shop The estimates of the mean arrival rate l and the mean service rate (per server) m are (估计平均到达率 和平均服务率[每个服务台] 分别为) l = 120 customers per hour (120位顾客每小时) m = 80 customers per hour (80位顾客每小时)

142 Choose s so as to Minimize TC = $20s + $48L. (确定服务台数量以最小化TC)
Acme Machine Shop The total cost to the company of each tool crib clerk is $20/hour, so Cs = $20. (每一个工具架管理员给公司带来的总成本大约是每小时20美元) While mechanics are busy, their value to Acme is $48/hour, so Cw = $48. (当技工很忙时，他们为公司带来的产出价值平均为每小时48美元，等待工具的成本) Choose s so as to Minimize TC = $20s + $48L. (确定服务台数量以最小化TC)

143 Excel Template for Choosing the Number of Servers

144 研究排队系统最好的方法 仿真 SIMULATION

145 The end of chapter 14