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数学代写|运筹学代写Operations Research代考|IMSE560 Basic Queuing Models

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数学代写|运筹学代写Operations Research代考|Basic Queuing Models

In order to avoid discussing only special cases, we will formalize queuing systems as follows. All entities that are in need of service of some kind will be referred to as customers, while the service is performed at service stations. The process can then be thought of as follows. Customers are in a calling population. Once they are in need of service (here, for simplicity, we will consider only a single type of service customers are interested in), they will approach the service system, where they will line up. When it is the customer’s turn, he will be served, after which the customer will leave the service system and rejoin the calling population. The structure of this process can be visualized in Fig. 15.1.

The service system will require some further specifications. First of all, each service system has only a single waiting line. A service system typically consists of a number of $c$ parallel service stations, each assumed to perform the same type of service. Parallel service stations are usually referred to as channels. In some instances, one channel consists of a series of service stations: imagine entering a building, where a potential customer first has to be cleared by security, then being directed to a general secretary, from where the service continues to the department director’s secretary, and finally on to the department director. At each station, the customer may be asked to leave the system, e.g., for not clearing security, the unavailability of a service station, and for other reasons. Multi-phase systems can be very complex and will not be discussed here.

数学代写|运筹学代写Operations Research代考|Optimization in Queuing

While queuing models are primarily designed to compute performance measures, they can also be applied in the context of optimization. As an example, consider a retail establishment. The owner of the store has to decide how many clerks to employ for the cash registers at the checkout counter. Clearly, increasing the number of clerks will increase the costs. However, at the same time, more clerks will result in less waiting time for customers, which, in turn, results in less ill will, lost sales, and other customer behavior detrimental to sales. One of the main problems applying these models is the quantification of the loss due to customer ill will.

The example of a tool crib is much easier to justify. A tool crib is a place in which expensive tools are kept that are not in constant use by the workers. Due to cost considerations, it would not be feasible to provide each worker with one tool, so that a service desk is established, where workers can sign out the tool whenever it is needed. The costs of the system include the costs of the tool crib clerks as well as the costs for the lost time of the workers. If $c$ is the number of clerks and $\$_c$and$\$_w$ denote the hourly wage of a clerk and a worker, respectively, and $L_{\mathrm{s}}^{\prime}$ denoting the number of workers in each of the $c$ service stations, then the costs can then be written as
$$C=(\text { cost of clerks })+(\text { cost of worker s lost time })=c\left(\_c+\_w L_s^{\prime}\right)=$$
$$=c\left(\_c+\_w \frac{\lambda}{c \mu-\lambda}\right) \text {. }$$
The idea is now to determine the optimal number of clerks so as to minimize the overall costs. As an illustration, consider the following

Example: The demand for a specialized tool occurs randomly at a rate of about 100 times per hour. Whenever the need arises, workers walk over to the tool crib, sign out the tool, use it, and then return it to the tool crib. All clerks are equally efficient with a service time of $3 \mathrm{~min}$. For simplicity, we assume that the organization of the signing out follows $c \times M / M / 1$ systems. Assume that the hourly wage of a clerk is $\$_c=10$, while a worker’s lost hour costs$\$_w=25$.

数学代写|运筹学代写OPERATIONS RESEARCH代 考|OPTIMIZATION IN QUEUING

$$C=(\text { cost of clerks })+(\text { cost of worker s lost time })=c\left(\ c+\_w L_s^{\prime}\right)=$$
$$=c\left(\_c+\_w \frac{\lambda}{c \mu-\lambda}\right) .$$

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。