Simulation

Simulation involves using a computer to imitate (simulate) the operation of an entire process or system.

For example, simulation is frequently used to perform risk analysis on financial processes.

Simulation also is widely used to analyze stochastic systems that will continue operating indefinitely.

Simulation model

To prepare for simulating a complex system, a detailed simulation model needs to be formulated to describe the operation of the system and how it is to be simulated.

A simulation model has several basic building blocks:

1. A definition of the state of the system (e.g., the number of customers in a queueing system).

2. Identify the possible states of the system that can occur.

3. Identify the possible events (e.g., arrivals and service completions in a queueing system) that would change the state of the system.

4. A provision for a simulation clock, located at some address in the simulation program, that will record the passage of (simulated) time.

5. A method for randomly generating the events of the various kinds.

6. A formula for identifying state transitions that are generated by the various kinds of events.

Discrete – Event versus Continuous Simulation

A discrete-event simulation is one where changes

in the state of the system occur instantaneously at

random points in time as a result of the occurrence

of discrete events.

A continuous simulation is one where changes in

the state of the system occur continuously over

time.

Generation of Random Numbers

Implementing simulation sometimes requires random numbers to obtain random observations from probability distributions.

The procedure used by a computer to obtain random numbers is called a random number generator.

A random number generator is an algorithm that produces of numbers that follow a specified probability distribution and possess the appearance of randomness.

Random numbers can be divided into two main categories.

(1) random integer number:

A random observation from a discretized uniform distribution over some range

The probabilities for this distribution are

.,,1, nnn

.1

1)()1()(

nnnPnPnP

Usually, .1or 0n

(2) uniform random number:

A random observation from a (continuous) uniform distribution over some interval [a, b].

The probability density function of this uniform distribution is

otherwise. 0

ba if 1

)(x

abxf

When a and b are not specified, they are assumed to be a = 0 and b = 1.

For a given random integer number in the

range 0 to , dividing this number by

yields (approximately) a uniform random

number.

(If is small, this approximation should be

improved by adding to the random integer

number and then dividing by instead.)

n

n

1n2

1

Strictly speaking, the numbers generated by the computer should not be called random numbers because they are predictable and reproducible, given the random number generator being used.

Therefore, they are sometimes given the name pseudo - random numbers.

However, the important point is that they satisfactory play the role of random numbers in the simulation if the method used to generate them is valid.

Congruential Method for Random Number Generation

Mixed congruential method:

The mixed congruential method generates a sequence of random integer numbers over the range from 0 to m-1.

The method always calculates the next random number from the last one obtained, given an initial random number .0x

), modulo)((1 mcaxx nn

where a, c, and m are positive integers ).,( mcma

N nx0

1

2

3

4

5

6

7

4

3

6

5

0

7

2

1

27

22

37

32

7

42

17

12

75 nx 8/)75( nx 1nx3

6

5

0

7

2

1

4

3

2

4

4

0

5

2

1 84

81

82

87

80

85

86

83

Illustration of the mixed congruential method.4,7,5,8 where 0 xcam

The cycle length is 8.

Outline of a Major Simulation Study

Step 1: Formulate the Problem and Plan the Study

Step 2: Collect the Data and

Formulate the Simulation Model

Step 3: Check the Accuracy of the Simulation Model

Step 4: Select the Software and

Construct a Computer Program

Step 5: Test the Validity of the Simulation Model

Step 6: Plan the Simulations to Be Performed

Step 7: Conduct the Simulation Runs and Analyze the Results

Step 8: Present Recommendations to Management

Step 1: Formulate the Problem & Plan the Study

1. What is the problem that management wants studied?

2. What are the overall objectives for the study?

3. What specific issues should be addressed?

4. What kinds of alternative system configurations should be considered?

5. What measures of performance of the system are of interest to management?

6. What are the time constraints for performing the study?

Step 2: Collect the Data and Formulate the Simulation Model

The types of data needed depend on the nature of the system to be simulated.

In each cases, the probability distributions of the relevant quantities are needed.

In order to generate representative scenarios of how a system would perform, it is essential that a simulation generate random observations from these distributions rather than simply using averages.

Step 3: Check the Accuracy of the Simulation Model

Before constructing a computer program, the OR team should engage the people most intimately familiar with how the system will operate in checking the accuracy of the simulation model.

Then several erroneous model assumptions will be discovered and corrected, a few new assumptions will be added.

Step 4: Select the Software and Construct a Computer Program

There are four major classes of software used for computer simulations.

(1) spreadsheet software, e.g., Excel

for more extensive applications

(2) general-purpose programming languages,

e.g. C, FORTRAN, PASCAL, BASIC

(3) general-purpose simulation languages

(4) applications-oriented simulators

Another key development in recent years

the development of animation capabilities for displaying computer simulations in action, including three-dimensional capabilities.

Key elements of a system are represented in a computer display by icons that change shape, color, or position when there is a change in the state of the simulation capabilities.

Step 5: Test the Validity of the Simulation Model

The next step is to test whether the simulation model incorporated into the program is providing valid results for the system it is representing.

Typically, the purpose of simulation is to investigate and compare various proposed system configurations to help choose the best one.

Step 6: Plan the Simulations to Be Performed

At this point, you need to begin making

decisions on which system configurations to

simulate.

This often is an evolutionary process, where the

initial results for a range of configurations help

you to hone in on which specific configurations

warrant detailed investigation.

Step 7: Conduct the Simulation Runs and Analyze the Results

The output from the simulation runs now provides

statistical estimates of the desired measures of

performance for each system configuration of

interest.

In addition to a point estimate of each measure, a

confidence interval normally should be obtained

to indicate the range of likely values of the

measure.

Step 8: Present Recommendations to Management

After completing its analysis, the OR team needs to present its recommendations to management.

This usually would be done through both a written report and a formal oral presentation to the managers responsible for making the decisions regarding the system under study.