1-Handout 1_2010 - 1 IE 372 Simulation Handout 1...

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IE 372 Simulation - Handout 1 INTRODUCTION TO SIMULATION 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 mode l 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). 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. The state of a system is the collection of variables necessary to describe a system at a particular time. Types of Simulation Models 1. 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 tunnels Desktop manufacturing labs 2. Symbolic models 1
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The real system is represented by mathematical equations and logical relations. Basically computer simulation. Types of Symbolic Simulation Models 1. Deterministic vs Stochastic - No random variables (no uncertainty) - Given input unique output (might take a lot of computer time to evaluate) - Input: random variable(s) - Output is also a random variable 2. Static vs Dynamic - No time dimension - Example: Monte Carlo simulation - System behavior changes over time - Example: A conveyor system For Dynamic simulation: 3. Continuous vs Discrete - System state changes continuously - Example: Altitude of an airplane (involves differential equations) altitude of airplane time - System state changes at discrete points in time - Example: Customers visiting a supermarket Nunmer of customers in system time 3 1 2 In IE 372, we will cover Stochastic , Dynamic , Discrete simulation. 2
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Monte 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 Example: Estimation of π Area of the circle is π r 2 = π since r =1 Area 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).
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