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# 数学代写| 随机过程代考|Queues

A population starts with one individual at time $n=0: Z_{0}=1$.

After one unit of time (at time $n=1$ ) the sole individual produces $Z_{1}$ identical clones of itself and dies. $Z_{1}$ is an $\mathbb{N}_{0}$-valued random variable.

(a) If $Z_{1}$ happens to be equal to 0 the population is dead and nothing happens at any future time $n \geq 2$.

(b) If $Z_{1}>0$, a unit of time later, each of $Z_{1}$ individuals gives birth to a random number of children and dies. The first one has $Z_{1,1}$ children, the second one $Z_{1,2}$ children, etc. The last, $Z_{1}^{\text {th }}$ one, gives birth to $Z_{1, Z_{1}}$ children. We assume that the distribution of the number of children is the same for each individual in every generation and independent of either the number of individuals in the generation and of the number of children the others have. This distribution, shared by all $Z_{n, i}$ and $Z_{1}$, is called the offspring distribution. The total number of individuals in the second generation is now
$$Z_{2}=\sum_{k=1}^{Z_{1}} Z_{1, k}$$
(c) The third, fourth, etc. generations are produced in the same way. If it ever happens that $Z_{n}=0$, for some $n$, then $Z_{m}=0$ for all $m \geq n$ – the population is extinct. Otherwise,
$$Z_{n+1}=\sum_{k=1}^{Z_{n}} Z_{n, k}$$

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## 数学代写| 随机过程代考|Service times

For the moment we shall suppose that the answers to these questions are as follows.
(a) The number $N(t)$ of customers who have entered by time $t$ is a renewal process. That is, if $T_{n}$ is the time of arrival of the $n$th customer (with the convention that $T_{0}=0$ ) then the interarrival times $X_{n}=T_{n}-T_{n-1}$ are independent and identically distributed.
(b) Arriving customers join the end of a single line of people who receive attention on a ‘first come, first served’ basis. There are a certain number of servers. When a server becomes free, she turns her attention to the customer at the head of the waiting line. We shall usually suppose that the queue has a single server only.
(c) Service times are independent identically distributed random variables. That is, if $S_{n}$ is the service time of the $n$th customer to arrive, then $\left{S_{n}\right}$ is a sequence of independent identically distributed non-negative random variables which do not depend on the arriving stream $N$ of customers.
It requires only a little imagination to think of various other systems. Here are some examples.

## 数学代写| 随机过程代考|single-server

(1) Queues with baulking. If the line of waiting customers is long then an arriving customer may, with a certain probability, decide not to join it.
(2) Continental queueing. In the absence of queue discipline, unoccupied servers pick a customer at random from the waiting mêlée.
(3) Airline check-in. The waiting customers divide into several lines, one for each server. The servers themselves enter and leave the system at random, causing the attendant customers to change lines as necessary.
(4) Last come, first served. Arriving documents are placed on the top of an in-tray. An available server takes the next document from the top of the pile.
(5) Group service. Waiting customers are served in batches. This is appropriate for lift queues and bus queues.
(6) Student discipline. Arriving customers jump the queue, joining it near a friend
We shall consider mostly the single-server queues described by (a), (b), and (c) above. Such queues are specified by the distribution of a typical interarrival time and the distribution of a typical service time; the method of analysis depends partly upon how much information we have about these quantities.

## 数学代写| 随机过程代考|SERVICE TIMES

b到达的客户加入了以“先到先得”的方式受到关注的单一人群的末端。有一定数量的服务器。当服务员空闲时，她会将注意力转向排在最前面的顾客。我们通常假设队列只有一个服务器。
C服务时间是独立同分布的随机变量。也就是说，如果小号n是服务时间n客户到达，然后\left{S_{n}\right}\left{S_{n}\right}是一系列独立同分布的非负随机变量，不依赖于到达的流ñ的客户。

## 数学代写| 随机过程代考|SINGLE-SERVER

1排队等候。如果等待顾客的队伍很长，那么到达的顾客可能会以一定的概率决定不加入。
2大陆排队。在没有排队规则的情况下，空置的服务员会从等待的混战中随机挑选一位顾客。
3航空公司办理登机手续。等待的顾客分成几行，每个服务员一个。服务器本身随机进出系统，导致服务员客户根据需要更改线路。
4后到先得。到达的文件被放置在收件盘的顶部。一个可用的服务器从堆的顶部获取下一个文档。
5团体服务。等候顾客分批送达。这适用于电梯队列和公共汽车队列。
6学生纪律。到达的客户插队，在朋友附近加入

## 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。