emdad and lol stochastic simulation

7/26/2019 Emdad and Lol Stochastic Simulation

1/3

Stochastic Simulation ~ 1 of 3

Stochastic Simulation

Simulation:

A simulation is the imitation of the operation of a real-world process or system over time. Whether done by hand or

on a computer, simulation involves the generation of an artificial history of a system and the observation of that

artificial history to draw inferences concerning the operating characteristics of the real system. From the simulation

data are collected as if a real system were being observed. This simulation generated data is used to estimate the

measure of performance of the system.

Uses of simulation:

Simulation can be used for the following purpose:

Simulation enables the study of and experimental with the internal interactions of a complex system or of a

subsystem within a complex system

Informational, organized, and environmental changes can be simulated and the effect alterations of the

models behavior can be observed. Simulation can be used to experiment with new designs or policies before implementation, so as to prepare

for what might happen.

Simulation can be used to verify analytic solutions.

Simulating different capabilities for a machine can help determine the requirements on it.

Simulation models designed for training make learning possible without the cost disruption of on-the-job

instruction

Situation when simulation is not appropriate:

First, simulation should not used when the problem can be solved by common sense.

Second, the simulation should not be used if the problem can be solved analytically.

Third, the simulation should not be used if it is easier to perform direct experiments.

Fourth, simulation should not be used if the cost exceeds the savings.

Fifth, simulation should not be performed if the resources or time are not available.

Advances of simulation:

Some advances of simulation which are listed by Pegden, Shanon, and, Sdadowski[1995] are:

New policies, operating procedures, design rules, information flows, organizational procedures, and so on

can be explored without disrupting ongoing operations of the real system.

New hardware designs, physical layouts, transportation systems, and so on can be tested without

committing resources for their acquisition.

Hypothesis about how or why certain phenomena occur can be tested for feasibility.

Insight can be obtained about the interaction of variables.

Insight can be obtained about the importance of variables to the performance of the system.

A simulation study can help in understanding how the system operates rather than how individuals think the

system operates.

Some disadvantages of simulation:

Model building requires special training. It is an art that is learned over time and through experience.

Furthermore, if two models are constructed by different component individuals, they might have similarities,

but it is highly unlikely that they will be the same.


2/3


Simulation results can be difficult to interpret. Most simulation outlets are essentially random variables (they

are usually based on random inputs), so it can be hard to distinguish whether an observation is a result of

system interrelationships or of randomness.

Simulation modeling and analysis can be time consuming and expensive.

Areas of application:

The applications of simulation are these:

Manufacturing applications.

Semiconductor manufacturing.

Construction Engineering and project management.

Military applications.

Logistics, supply chain and distribution applications.

Transportation models and traffic.

Business process simulation.

Health care.

Explain the meaning of stochastic simulation and Monte-Carlo simulation.

When we use the word simulation, we refer to any analytical method meant to imitate a real-life system, especially

when other analyses are too mathematically or too difficult to reproduce.

Thus by the stochastic simulation we meant that , when the physical problems is not always possible to specify the

transition probability matrix but we can analysis this using simulation is known as stochastic simulation.

On the other hand, Monte-Carlo simulation is one type of spreadsheet simulation, which randomly generates values

of uncertain variables and over to simulate a model.

What are the roles of limiting or initial probabilities in stochastic simulation?

In stochastic process, limiting probabilities or initial probabilities plays an important role. By using the given initial

probability we can simulate the given transition probability matrix and we can test whether the transition probability

matrix comes from a Marcov chain, or stationary of the transition probability matrix and order of a Marcov Chain.

Metropolis-Hastings Marcov chain Monte-Carlo:

Metropolis-Hastings Marcov Chain Monte-Carlo is a certain type of algorithm which generates a MC with

equilibrium distribution .

The Algorithm is as follows:

Let 1, nn XiX is determined in the following way:

1. Generation Step:

Generate a candidate state j from i with some distribution |g j i . Now |g j i is a fixed distribution that we are

free to choose, so long that it satisfies the conditions

(a) | 0 | 0g j i g i j [ can go forward implies can go back]

(b) |g j i is the transition matrix of an irreducible Marcov chain on .


3/3


2. Acceptance step:

With probability

( | )( | ) min 1, .... ( )

( | )

j

i

g i jj i i

g j i

Set 1 ( . ., )nX j i e accept j

Otherwise set ( . ; )nX i i e reject j

Since the general and acceptance steps are independent, the transition probability to go from i to j is

1Pr , | |n nX j X i g j i j i provided that i j

This says that the probability to land in state j from the current state i is equal to the probability to generate j

from the current state i times the probability that the new state j is accepted. The probability iiP can be from the

requirement that 1ijj

P .

Theorem:

Let be a given probability distribution. The Marcov Chain simulated by the Metropolis-hasting algorithm is

reversible with respect to . If it is also irreducible and aperiodic, then it defines all ergodic Marcov chain.

Proof:

We have to show that the transition matrix P determined by the Metropolis-Hasting algorithm satisfies-

... ( )i ij j jiP P i

For all ji . If this is the case, then the chain is reversible and the rest of the assertion follow ergodicity.

Assume without loss of generality that

( | ) ( | ) ... ( )j ig i j g j i ii

Since,

( | )( | ) ( | ) ( | ) min 1, ( | ) ... ( )

( | )

j

ij

i

g i jP g j i j i g j i g j i iii

g j i

By the assumption,

( | ) ( | )

( | ) ( | )( | ) min 1, ( | ) ( | ) ... ( )

( | ) ( | )

ji

i i i

j j j

P g j i i j

g j i g j ig j i g i j g j i iv

g i j g i j

again by assumption. From these expressions, it is clear that i ij j jiP P as required.

emdad and lol stochastic simulation

Documents