## 数学代写|排队论代写Queueing Theory代考|IE522

statistics-lab™ 为您的留学生涯保驾护航 在代写排队论Queueing Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写排队论Queueing Theory方面经验极为丰富，各种代写排队论Queueing Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|排队论代写Queueing Theory代考|Performance measures

We are usually only interested in queue systems that reach a steady-state, meaning that eventually the probability of any particular system size no longer changes over time. For example, suppose $\lambda=2$ and $\mu=1$ with just one server. On the average, two new customers will arrive for every customer who completes service, so the system size will just keep growing. In contrast, if $\rho=0.9$, the service capacity is larger than the expected rate of customer arrivals. This means that there will be times when the system is empty because random chance has provided a period between arrivals long enough for the servers to empty the system. Given enough time, there will be some probability $P_0$ that the system is empty at any particular time. Similarly, there will be probabilities $P_1, P_2$, and so on, that indicate the probabilities of the system having a total of 1,2 , and so on customers. It is possible that a queue system can achieve steady state even when $\rho>1$; this requires some additional mechanism, such as a limited system size or a limited calling population, so that the arrival rate falls to 0 when the system reaches its capacity.

Given a system that reaches steady-state for whatever reason, the set of values $P_n$ is an emergent property; that is, it is a property that comes about in the running of the system and requires analysis to determine. Queueing theory is largely about how to determine these steady-state probabilities and some important performance measures. Two of these involve the numbers of customers.

1. The mean number of customers in the system over time, including those who are in the queue as well as those being served. This quantity is usually designated as $L$.
2. The mean number of customers in the queue, usually denoted $L_q$. This quantity is seldom of special interest in modeling, but it is mathematically important because it is usually the easiest of the four performance measures to determine.

The other two performance measures involve the average amount of time spent by customers. There are two common choices for terminology and notation, so one must be careful to identify which system is being used, both when reading what others have written and when writing for the benefit of others.

1. The mean amount of time that a customer spends in the system. Some authors call this the “sojourn” time and denote it with the symbol $S$. Others call it the “waiting” time and use the symbol $W$.
2. The mean amount of time that a customer spends in the queue. Authors who use “sojourn” time for mean time in the system usually call this the “waiting” time and denote it as $W$, while authors who use “waiting” time for mean time in the system usually call this the “waiting time in the queue” and denote it as $W_q$.

## 数学代写|排队论代写Queueing Theory代考|M/G/1/∞/∞ Results

Usually, the performance measures of a system can only be determined after the steady-state probabilities are known, and this in turn can be done only for an $\mathrm{M} / \mathrm{M}$ system. However, there is a simple formula for the queue length $L_q$ that works for any $\mathrm{M} / \mathrm{G} / 1 / \infty / \infty$ system; ${ }^{-5}$ that is, systems for which

1. There are no limits to the number of potential customers or the number of customers who can be in the system at any one time;
2. Arrival times are exponentially distributed with mean rate $\lambda$;
3. Service times have a mean of $\mu_T=1 / \mu$ and a standard deviation $\sigma$, but no specific service distribution. $^6$
4. There is one server.
We present this formula here without derivation, as the derivation is beyond the scope of this presentation:
$$L_q=\frac{\rho^2\left(1+\mu^2 \sigma^2\right)}{2(1-\rho)}, \quad \rho=\frac{\lambda}{\mu},$$
6. $$7. L=\frac{2 \rho-\rho^2\left(1-\mu^2 \sigma^2\right)}{2(1-\rho)} . 8.$$
9. Formula (2.7) is called the Pollaczek-Khintchine formula.
10. For the specific case where the service times are exponentially distributed $(\mathrm{M} / \mathrm{M} / 1)$, the standard deviation is $\sigma=1 / \mu$ and the result reduces to
11. $$12. L=\frac{\rho}{1-\rho}, 13.$$
14. while the assumption of uniform service time $1 / \mu(\mathrm{M} / \mathrm{D} / 1)$ yields
15. $$16. L=\frac{\rho(2-\rho)}{2(1-\rho)} . 17.$$
18. Note that formula (2.10) has an extra factor $(1-\rho / 2)$ compared to formula (2.7). Given that this factor is less than one, we see that greater uniformity in service times improves system performance by as much as $50 \% .^7$ See Figure 2.1. This is a general characteristic of queue systems (although the maximum amount of improvement might be different from 50\%). Less variability with the same mean service rate is always better. There are no obvious design implications of this result, since we don’t get to choose the characteristics of service jobs. It may influence the decision to add a server, as additional servers have more benefit when service times have a higher variability.

# 排队论代写

## 数学代写|排队论代写Queueing Theory代考|Performance measures

1. 一段时间内系统中的平均客户数，包括排队中的客户和正在接受服务的客户。这个数量通常指定为大号.
2. 队列中顾客的平均数，通常表示为大号q. 这个量在建模中很少引起特别的兴趣，但它在数学上很重要，因为它通常是四种性能度量中最容易确定的。

1. 客户在系统中花费的平均时间。一些作者称此为“逗留”时间并用符号表示小号. 其他人称之为“等待”时间并使用符号在.
2. 客户在队列中花费的平均时间。在系统中使用“逗留”时间作为平均时间的作者通常将其称为“等待”时间并将其表示为在，而在系统中使用“等待”时间作为平均时间的作者通常将其称为“队列中的等待时间”并将其表示为在q.

## 数学代写|排队论代写Queueing Theory代考|M/G/1/∞/∞ Results

1. 潜在客户的数量或任何时候可以进入系统的客户数量没有限制；
2. 到达时间以平均速率呈指数分布升;
3. 服务时间的平均值为米吨=1/米和一个标准差p，但没有具体的服务分布。6
4. 有一台服务器。
我们在这里不推导这个公式，因为推导超出了本演示文稿的范围：
大号q=r2(1+米2p2)2(1−r),r=升米,
5. 导致
6. $$7. L=\frac{2 \rho-\rho^2\left(1-\mu^2 \sigma^2\right)}{2(1-\rho)} 。 8.$$
9. 公式(2.7)称为Pollaczek-Khintchine公式。
10. 对于服务时间呈指数分布的特定情况(米/米/1), 标准差是p=1/米结果减少到
11. $$12. L=\frac{\rho}{1-\rho}, 13.$$
14. 而假设统一服务时间1/米(米/丁/1)产量
15. $$16. L=\frac{\rho(2-\rho)}{2(1-\rho)} 。 17.$$
18. 注意公式（2.10）有一个额外的因素(1−r/2)与公式（2.7）相比。鉴于这个因素小于 1，我们看到服务时间的一致性提高了系统性能50%.7见图 2.1。这是队列系统的一般特征（尽管最大改进量可能不同于 50%）。具有相同平均服务率的较小可变性总是更好。这个结果没有明显的设计含义，因为我们无法选择服务工作的特征。它可能会影响添加服务器的决定，因为当服务时间具有更高的可变性时，添加服务器会带来更多好处。

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|排队论代写Queueing Theory代考|COE755

statistics-lab™ 为您的留学生涯保驾护航 在代写排队论Queueing Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写排队论Queueing Theory方面经验极为丰富，各种代写排队论Queueing Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|排队论代写Queueing Theory代考|Defining a Queue System

A number of elements must be prescribed in order to define a queue system, including

1. The type of probability distribution used for the arrival process, with one or more parameters.
2. The type of probability distribution used for the service process, with one or more parameters.
3. The number of servers in the service station (for example, the number of check-out stations in service at a grocery store).
4. The maximum number of customers that can be in the system, if limited.
5. The size of the population of potential customers, if limited.
Some of this information is presented in compact form as an identifier of the form A/S $/ s / K / N$ , where
6. A designates the type of distribution used for arrival times,
7. S designates the type of distribution used for service times,
8. $s$ designates the number of servers,
9. $K$ is the maximum number of customers that can be in the system at any one time, and
10. $N$ is the size of the population of potential customers.
Typical designators for the distributions are
11. M for the exponential distribution (Markovian),
12. $\mathrm{E}_k$ for an Erlang distribution (which we will use later), and
13. D for the deterministic distribution (constant times),
14. G for a general distribution (unspecified except for mean and standard deviation).
The size of the calling population is important because customers that enter a queue system should be removed from the list of potential customers. This decreases the mean arrival rate, but usually the decrease is too small to worry about. Unless the calling population is small enough for the decrease to matter, it is best to consider it to be infinite. In this case, the fifth designator $N$ is often omitted. Similarly, the number of customers that can be in the system at any one time is usually limited, but most of the time the capacity is large enough that it is never actually reached. In this case, it is best to consider the maximum system size to be infinite. As with infinite calling population size, it is common to omit the fourth designator $K$ when the queue size is unlimited.
15. The most commonly used systems are of the form $\mathrm{M} / \mathrm{M} / \mathrm{s}$, meaning that the arrival and service processes are exponentially distributed and the system size and calling population are unlimited. We will study these systems in Section 4.

## 数学代写|排队论代写Queueing Theory代考|Properties of Queue Systems

The number of customers in a queue system changes over time. When the system first begins to operate, there are generally no customers. Those customers who arrive before the servers are all occupied get to begin service immediately, while customers who arrive later only get to begin service when they get to the front of the queue. Thus the probability that there are 4 customers in the system is initially 0 , but it rises as the system remains open.

Suppose a queue system has the property that the mean arrival rate does not change as the system size changes. This requires that both the maximum system size $K$ and the calling population $N$ are infinite and that there are no unusual features that can lower the arrival rate as the system size increases. Given a mean rate of service completions of $\mu$ for each of $s$ servers, the total service capacity is a mean rate of $s \mu$. The ratio of mean arrival rate to mean total service completion rate, given by
$$\rho=\frac{\lambda}{s \mu}$$ then represents the fraction of the service capacity that is used. For this reason, it is called the utilization factor for the queue system. The quantity is often used even in cases where $\lambda$ is not fixed, but the interpretation as utilization factor no longer holds.
For both modeling and computation, it is also helpful to define the arrival-service ratio
$$\gamma=\frac{\lambda}{\mu}$$
This parameter represents the expected number of arrivals during the average amount of time for a service completion, which we might call the “load” of the system. In modeling, the most frequent scenario is one in which the rates $\lambda$ and $\mu$ are fixed and the problem is to choose the optimal number of servers. The parameter $\gamma$ is much more useful than $\rho$ in this context because it is strictly a property of the scenario while $\rho$ combines elements of the scenario data $(\lambda$ and $\mu$ ) with the independent variable of the optimization problem $(S)$. Computationally, we’ll find $\gamma$ more useful that $\rho$ in cases where the arrival rate depends on the system state.

# 排队论代写

## 数学代写|排队论代写Queueing Theory代考|Defining a Queue System

1. 用于到达过程的概率分布类型，具有一个或多个参数。
2. 用于服务过程的概率分布类型，带有一个或多个参数。
3. 服务站中服务器的数量（例如，杂货店服务中的结账站数量）。
4. 系统中可以容纳的最大客户数（如果有限制）。
5. 潜在客户的人口规模（如果有限）。
其中一些信息以紧凑形式作为 A/S 形式的标识符呈现/秒/钾/否， 在哪里
6. A 指定用于到达时间的分布类型，
7. S 指定用于服务时间的分布类型，
8. 秒指定服务器的数量，
9. 钾是任一时刻系统中可以存在的最大客户数，并且
10. 否是潜在客户的人口规模。
分布的典型指示符是
11. M 为指数分布（马尔可夫），
12. 和k对于 Erlang 发行版（我们稍后会用到），以及
13. D 为确定性分布（恒定时间），
14. G 表示一般分布（除均值和标准差外未指定）。
呼叫人口的规模很重要，因为进入队列系统的客户应该从潜在客户列表中删除。这会降低平均到达率，但通常降幅很小，无需担心。除非呼叫人口足够小以至于减少很重要，否则最好将其视为无限大。在这种情况下，第五个指示符否经常被省略。同样，在任何时候系统中可以容纳的客户数量通常是有限的，但大多数时候容量足够大，实际上永远不会达到。在这种情况下，最好将最大系统规模视为无限大。与无限呼叫人口规模一样，通常会省略第四个指示符钾当队列大小不受限制时。
15. 最常用的系统具有以下形式米/米/秒，这意味着到达和服务过程呈指数分布，系统规模和呼叫人口是无限的。我们将在第 4 节中研究这些系统。

## 数学代写|排队论代写Queueing Theory代考|Properties of Queue Systems

$$\rho=\frac{\lambda}{s \mu}$$

$$\gamma=\frac{\lambda}{\mu}$$

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 数学代写|排队论代写Queueing Theory代考|MATH895

statistics-lab™ 为您的留学生涯保驾护航 在代写排队论Queueing Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写排队论Queueing Theory方面经验极为丰富，各种代写排队论Queueing Theory相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|排队论代写Queueing Theory代考|A Note on Notation

This document contains an introduction to queueing theory with emphasis on using queueing theory models to make design decisions. It therefore combines probability with optimization. These concepts are contrasted in a statement I once heard in a talk:
Probability is the study of the typical for issues of chance.
Optimization is the study of the exceptional for issues of choice. ${ }^1$
The first five sections of these notes develop the concepts and results of queueing theory. This topic is about issues of chance, such as the amount of time one must wait in line before reaching the checkout counter in a store. Hence, our goal will be to characterize the typical behavior of such systems, and we must keep in mind that the actual behavior in any one instance will not necessarily be close to the typical behavior. Sections 6 and 7 use the results of queueing theory in the context of optimization problems. Our goal in these sections will be to identify the choices that are available in a given setting and determine which choice produces the exceptional result. We must keep in mind that choices involving design of probabilistic systems are less certain than choices involving design of deterministic systems. In a deterministic system, we can say exactly what will be the result of any design choice. In a probabilistic system, the best we can say is what will be the average result of a design choice. The optimal decision may not yield the best result in a particular instance, but it will be the decision that results from rational analysis of the options and therefore the best decision that can be made with the information at hand.

These notes assume familiarity with basic calculus and probability theory. In particular, queueing theory makes extensive use of probability distributions and expected value. These concepts are reviewed here to some extent, but the reader may need to look elsewhere for more information.

The enormous number of quantities in all of mathematics have to be represented with only a handful of symbols-generally the Latin and Greek alphabets with subscripts. Inevitably this means that many symbols get used differently in different contexts. The symbol $\lambda$ is used to denote Lagrange multipliers in optimization, adjoints in control theory, eigenvalues in linear algebra and partial differential equations, and the mean customer arrival rate in queueing theory. Clearly, symbols do not have fixed meanings; rather, they mean what we define them to mean. This has implications for both the reading and writing of mathematics. When reading mathematics, one has to be careful to look for information about symbol meanings and be aware that one author’s $W$ and $W_q$ might be another author’s $S$ and $W ;$; not only are the symbols for a given quantity different, but the symbol ” $W$ ” has different meanings in the different systems. This point is particularly important when reading supplementary material on the internet. It is very likely that the material you are reading has some notational differences from your textbook or lecture notes, and you have to be able to translate from one system to the other. When writing mathematics, it is necessary to be clear about terms and notation to spare your reader any confusion. In general, it is a good idea to define every symbol other than $\pi$ or $e$ when writing about mathematics and modeling. This is the only way to make sure that the reader will understand what you have written.

## 数学代写|排队论代写Queueing Theory代考|Learning Objectives

1. Know the goals of queueing theory.
2. Be able to identify the defining characteristics of a queue system from the standard 5character identifiers.
3. Be able to calculate the arrival-service ratio $\gamma$ and the utilization factor $\rho$ from a given narrative and explain their significance.
4. Know the four principal performance measures of a queue system and be able to calculate them from the steady-state probabilities.
5. Be able to calculate the four performance measures for an $M / G / 1 / \infty / \infty$ system using $\lambda$, $\mu$, and $\sigma$.

A queue system is a system characterized by a bank of parallel service channels with a stream of “customers” who enter at distinct times and receive service, possibly waiting in a queue if all servers are busy. The line of customers in a convenience store is a nice example. Our ultimate goals are

1. To identify the features needed in the specification of a queue system;
2. To develop methods for determining the performance of a system; and
3. To develop protocols that allow system design decisions to be made so as to optimize the overall cost associated with the system.

# 排队论代写

## 数学代写|排队论代写Queueing Theory代考|Learning Objectives

1. 了解排队论的目标。
2. 能够从标准的 5 字符标识符中识别队列系统的定义特征。
3. 能够计算到达服务比C和利用率r从给定的叙述中解释它们的意义。
4. 了解队列系统的四个主要性能指标，并能够根据稳态概率计算它们。
5. 能够计算四个绩效指标米/G/1/∞/∞系统使用升, 米， 和p.

1. 确定队列系统规范中所需的功能；
2. 开发确定系统性能的方法；和
3. 开发允许做出系统设计决策的协议，以优化与系统相关的总体成本。

## 广义线性模型代考

statistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。