# 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|STATS783

## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|SIMULATION MODELS

By a system we mean a collection of related entities, sometimes called components or elements, forming a complex whole. For instance, a hospital may be considered a system, with doctors, nurses, and patients as elements. The elements possess certain characteristics or attributes that take on logical or numerical values. In our example, an attribute may be the number of beds, the number of X-ray machines, skill level, and so on. Typically, the activities of individual components interact over time. These activities cause changes in the system’s state. For example, the state of a hospital’s waiting room might be described by the number of patients waiting for a doctor. When a patient arrives at the hospital or leaves it, the system jumps to a new state.

We will be solely concerned with discrete-event systems, to wit, those systems in which the state variables change instantaneously through jumps at discrete points in time, as opposed to continuous systems, where the state variables change continuously with respect to time. Examples of discrete and continuous systems are, respectively, a bank serving customers and a car moving on the freeway. In the former case, the number of waiting customers is a piecewise constant state variable that changes only when either a new customer arrives at the bank or a customer finishes transacting his business and departs from the bank; in the latter case, the car’s velocity is a state variable that can change continuously over time.

The first step in studying a system is to build a model from which to obtain predictions concerning the system’s behavior. By a model we mean an abstraction of some real system that can be used to obtain predictions and formulate control strategies. Often such models are mathematical (formulas, relations) or graphical in nature. Thus, the actual physical system is translated – through the model – into a mathematical system. In order to be useful, a model must necessarily incorporate elements of two conflicting characteristics: realism and simplicity. On the one hand, the model should provide a reasonably close approximation to the real system and incorporate most of the important aspects of the real system. On the other hand, the model must not be so complex as to preclude its understanding and manipulation.

There are several ways to assess the validity of a model. Usually, we begin testing a model by reexamining the formulation of the problem and uncovering possible flaws. Another check on the validity of a model is to ascertain that all mathematical expressions are dimensionally consistent. A third useful test consists of varying input parameters and checking that the output from the model behaves in a plausible manner. The fourth test is the so-called retrospective test. It involves using historical data to reconstruct the past and then determining how well the resulting solution would have performed if it had been used. Comparing the effectiveness of this hypothetical performance with what actually happons then indicates how woll the model predicts reality. However, a disadvantage of retrospective testing is that it uses the same data as the model. Unless the past is a representative replica of the future, it is better not to resort to this test at all.

## 数学代写|模拟和蒙特卡洛方法作业代写simulation and monte carlo method代考|Classification of Simulation Models

Computer simulation models can be classified in several ways:

1. Static versus Dynamic Models. Static models are those that do not evolve over time and therefore do not represent the passage of time. In contrast, dynamic models represent systems that evolve over time (for example, traffic light operation).
2. Deterministic versus Stochastic Models. If a simulation model contains only deterministic (i.e., nonrandom) components, it is called deterministic. In a deterministic model, all mathematical and logical relationships between elements (variables) are fixed in advance and not subject to uncertainty. A typical example is a complicated and analytically unsolvable system of standard differential equations describing, say, a chemical reaction. In contrast, a model with at least one random input variable is called a stochastic model. Most queueing and inventory systems are modeled stochastically.
3. Continuous versus Discrete Simulation Models. In discrete simulation models the state variable changes instantaneously at discrete points in time, whereas in continuous simulation models the state changes continuously over time. A mathematical model aiming to calculate a numerical solution for a system of differential equations is an example of continuous simulation, while queueing models are examples of discrete simulation.

This chapter deals with discrete simulation and in particular with discrete-event simulation (DES) models. The associated systems are driven by the occurrence of discrete events, and their state typically changes over time. We shall further distinguish between so-called discrete-event static systems (DESS) and discrete-event dynamic systems (DEDS). The fundamental difference between DESS and DEDS is that the former do not evolve over time, whereas the latter do. A queueing network is a typical example of a DEDS. A DESS usually involves evaluating (estimating) complex multidimensional integrals or sums via Monte Carlo simulation.

# 模拟和蒙特卡洛方法代考

## 数学代写|模拟和蒙特卡洛方法作业代写仿真与蒙特卡罗方法代考|仿真模型分类

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1. 静态与动态模型。静态模型是那些不随时间演进的模型，因此不代表时间的流逝。相反，动态模型表示随着时间发展的系统(例如，红绿灯操作)。确定性与随机模型。如果一个仿真模型只包含确定性(即非随机)组件，则称为确定性的。在确定性模型中，元素(变量)之间的所有数学和逻辑关系都是预先确定的，不受不确定性的影响。一个典型的例子是描述化学反应的复杂的、解析上不可解的标准微分方程组。相比之下，至少有一个随机输入变量的模型称为随机模型。大多数排队和库存系统都是随机建模的。
2. 连续与离散仿真模型。在离散仿真模型中，状态变量在离散时间点上瞬时变化，而在连续仿真模型中，状态随时间连续变化。一个旨在计算微分方程组数值解的数学模型是连续模拟的一个例子，而排队模型是离散模拟的一个例子本章讨论离散模拟，特别是离散事件模拟(DES)模型。相关的系统是由离散事件的发生驱动的，它们的状态通常随时间而变化。我们将进一步区分所谓的离散事件静态系统(DESS)和离散事件动态系统(DEDS)。DESS和DEDS之间的根本区别在于前者不会随着时间的推移而发展，而后者会。排队网络是DEDS的一个典型例子。DESS通常涉及通过蒙特卡罗模拟计算(估计)复杂多维积分或和

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