CS 8CS 85544Computer SimulationComputer Simulation

Lecture Notes 01

Introduction to SimulationIntroduction to Simulation

Simulation refers to a broad collection of methods and applications to mimic the behavior of real systems usually on a computer with appropriate software.

Simulation is the process of Formulating a model of the real system, and Conducting experiments with the model in order to

understand (predict) the systems behavior, and evaluate various alternatives (design or

operational).

Introduction to SimulationIntroduction to Simulation

System: Collection of elements working together toward accomplishment of a specific objective. The facility or process of interest is usually called a

system. Ways to study a system include experimenting with

the actual system, and a model of the system.

Model: Simplified representation or abstraction of a system.

Types of SimulationTypes of Simulation

Iconic (physical) models Precise replica of an object (maybe in different

material or at smaller scale). Used primarily for training and research purposes. Examples:

Flight simulators (how to fly and handle emergency situations)Wind tunnelsDesktop manufacturing labs

Symbolic models The real system is represented by mathematical equations

and logical relations. Basically computer simulation

Types of Symbolic Simulation Types of Symbolic Simulation ModelsModels

Deterministic Stochastic No random variables (no

uncertainty)

Input: random

variable(s) Given input → unique output

(might take a lot of computer time to evaluate)

Output is also a

random variable

Types of Symbolic Simulation Types of Symbolic Simulation ModelsModels

Static Dynamic

No time dimension System behavior

changes over time

Example: Monte Carlo

Simulation

Example: A conveyor

system

Types of Symbolic Simulation Types of Symbolic Simulation ModelsModels

Continuous Discrete

System state changes

continuously

System state

changes at discrete points in time

Example: Altitude of an

airplane (involves differential equations)

Example:

Customers visiting a supermarket

For Dynamic simulationFor Dynamic simulation

Continuous SimulationContinuous Simulation

Discrete SimulationDiscrete Simulation

Monte Carlo (MC) SimulationMonte Carlo (MC) Simulation

A random number (RN) is a random variable (RV) ~

Uniform (0,1)

In Monte Carlo simulation, an iid sample of RNs is

obtained to solve a problem

Static, Stochastic MC Simulation Static, Stochastic MC Simulation Example: Estimation of Example: Estimation of ππ

Area of the circle is π r2 = π since r = 1Area of shaded quarter-circle = π/4

How can we use MC simulation in estimating the value of π ?

Area of small square (in the main quadrant) = 1 (Area of quarter-circle) / (Area of small square) = π/4

Algorithm: S1. Generate two RNs ~ uniform (0,1). S2. Using these as (x, y) coordinates of a point, find

the Euclidean distance, d, from the origin. S3. If d < 1 then the point is inside the quarter-circle,

otherwise it is outside the quarter-circle (but inside the small square).

Static, Stochastic MC Simulation Static, Stochastic MC Simulation Example: Estimation of Example: Estimation of ππ

Static, Stochastic MC Simulation Static, Stochastic MC Simulation Example: Estimation of Example: Estimation of ππ

Repeat (replicate) S1-S3 n times where n is a very large number. (The points will be distributed uniformly inside the small square.) Count the number of times the generated point falls inside the circle, nq. Then, is an estimate for π/4.

95% confidence interval for π is :

nnp q /ˆ

n

ppp

)ˆ1(ˆ)96.1(4ˆ4

Static, Stochastic MC Simulation Static, Stochastic MC Simulation Example: Estimation of Example: Estimation of ππ

Empirical results (Seila et al., 2003, Applied Simulation Modeling)

Static, Stochastic MC Simulation Static, Stochastic MC Simulation Example: Estimation of Example: Estimation of ππ

Plot showing convergence of estimates to π

1. Analytical Models: Represent the system by a set of equations and logical relations. Some can be solved analytically using algebra, calculus, probability theory, e.g. LP models, queuing models. Many real world problems are too complex to be solved analytically.

2. Simulation: Often the only symbolic model representing a complex system sufficiently. Yields a different set of outputs (estimates) for each set of inputs. No optimization in the strict sense.

Means of Analyzing a SystemMeans of Analyzing a System

Cone of AbstractionCone of Abstraction

Advantages and Advantages and Disadvantages of SimulationDisadvantages of Simulation

Advantages: Increased realism: Simulation is closer to reality

than analytical models. Fewer simplifying assumptions are made. Hence, complex systems can be modeled.

Existing or non-existing systems can be studied. New strategies for an existing system or a totally new system can be studied without actually investing in the physical system (great savings).

Hazardous systems can be studied without risks.

Advantages and Advantages and Disadvantages of SimulationDisadvantages of Simulation

Advantages: Time compression or expansion is possible Bottleneck analysis can be performed “What if” questions can be answered Results are reproducible. No measurement errors

except round off. Same random variates → same results.

Random variates: Realizations of an input random variable generated during simulation.

Advantages and Advantages and Disadvantages of SimulationDisadvantages of Simulation

Advantages:

Good control on experimental conditions to eliminate undesired sources of variability, and to try alternatives under identical conditions

Ease of communication with the management, especially with the help of animation.

Advantages and Advantages and Disadvantages of SimulationDisadvantages of Simulation

Disadvantages:

Model building requires special training and data collection

Statistical analysis: Stochasticity requires more and longer runs and special knowledge for analysis

Expensive and time consuming to develop, code and analyze

No exact solutions are obtained. Results are only estimates

Advantages and Advantages and Disadvantages of SimulationDisadvantages of Simulation

Disadvantages:

Lack of generality: Results are valid only for a particular configuration of a particular system (given results for 3 servers, we cannot extrapolate for n servers)

Lack of optimization

Don’t Simulate When…Don’t Simulate When…

(Banks and Gibson, 1997, IIE Solutions)

The problem can be solved using “common sense analysis” (simple calculations), e.g. if customers arrive at an average rate of 100/hr and they are served at a mean rate of 12/hr, then we need 100/12 = 8.33 ~ 9 servers

The problem can be solved analytically, e.g. average waiting time of customers in system can be found analytically for M/M/s queueing systems

Direct experimentation in real system is easier, e.g. direct observation in a drive-in fast food restaurant

Don’t Simulate When…Don’t Simulate When…

(Banks and Gibson, 1997, IIE Solutions)

Cost of simulation exceeds possible savings Resources are not available The model cannot be verified or validated Project expectations cannot be met (do not

overpromise) The system behavior is too complex or cannot be

defined (ill-structured problem)

Don’t Simulate When…Don’t Simulate When…

Simulation projects cost money, but can produce a significant return on investment when conducted properly. If the project is successful then the money is well spent, and this will promote the technology. If the project is not successful, it hurts the reputation of simulation and, by association, each of us.

Steps in a Simulation StudySteps in a Simulation Study

Collect data and build a conceptual model

(flowchart? Pseudo code?)

Collect data and build a conceptual model

(flowchart? Pseudo code?)

Formulate the problemand plan the study

Formulate the problemand plan the study

Yes

NoValid?Valid?

Construct computer program and verify

Construct computer program and verify

• Define the problem (system and environment) nad objectives• Plan the study (team, cost, time)

• Probability distribution of input random variables• Start with simple, build up as necessary

• Does the conceptual model represent reality? (Talk to system people)

• Choice of language• Test if the program does what you intend it to do

Steps in a Simulation StudySteps in a Simulation Study• Preliminary runs for validation purposes

• Which input conditions to consider?• How to conduct runs based on the output analysis technique

• Statistical outputs

• Compare output to real system, if any

Document, present and implement

Document, present and implement

Make pilot runsMake pilot runs

Yes

NoValid?Valid?

Design experiment(s)Design experiment(s)

Make production runsMake production runs

Analyze ouputAnalyze ouput

Elements in a Simulation Elements in a Simulation StudyStudy

Entities: Objects of interest (customers, jobs, etc.) that flow through the system.

An entity is characterized by its attributes and usually interacts with resources.

Events: An occurrence (arrival, service completion, etc.) that changes the system state.

Input parameters (constants or random variables): Uncontrollable, e.g. time between customer arrivals Controllable, e.g. number of parallel servers

Elements in a Simulation Elements in a Simulation StudyStudy

State variables: Variables that define system state at any given time, e.g. number of customers in system, number of busy servers.Performance measures: Output statistics used to evaluate objectives, e.g. average waiting time of customers in system, percentage utilization of servers.Alternative strategies to be evaluated.

In developing our conceptual model, we must identify/determine the above.

Examples of System and Examples of System and ComponentsComponents

SystemSystem EntitiesEntities AttributesAttributes ActivitiesActivities EventsEvents State VariablesState Variables

Banking CustomersChecking account balance

Making deposits

Arrival, departure

Number of busy teller, number of waiting customer

Rapid rail RidersOrigination, destination

TravelingArrival at station

Number of riders waiting at each station, number of riders in transit

Production JobsArrival time, process time(s)

Welding, stamping

Completion of welding, breakdown

Status of machines (busy, idle or down)

Communications MessagesMessage length, destination

TransmittingArrival at destination

Number of messages waiting to be transmitted

Inventory OrdersOrder quantity, due date

Withdrawing renewing

DemandLevels of inventory, backlogged demand