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lecture9p - 1.225J (ESD 205) Transportation Flow Systems...

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Lecture 9 Lecture 9 Simulation Models Simulation Models Prof. Prof. Ismail Chabini and Prof. and Prof. Amedeo Amedeo R. R. Odoni 1.225 1.225 J (ESD 205) Transportation Flow Systems J (ESD 205) Transportation Flow Systems 1.225, 11/28/02 Lecture 9, Page 2 Lecture 9 Outline Lecture 9 Outline ± About this lecture: • It is based on R16. Only material covered in class must be read • Download a spreadsheet model from the 1.225F01 website ± Introduction to simulation models and a simple example ± Random numbers ± Simulation models for M/M/1 queueing systems ± Random observations ± An event-driven simulation model for an M/M/1 queueing system ± A spreadsheet implementation of an event-driven simulation model for M/M/1 ± A discrete-time simulation model for an M/M/1 queueing system ± Lecture summary 1
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1.225, 11/28/02 Lecture 9, Page 3 Introduction to Simulation Models Introduction to Simulation Models ± Types of models covered in this subject: • Analytical deterministic models (Lectures 1-7) • Analytical probabilistic models (Lecture 8) • Simulation models for stochastic systems (This lecture) ± Stochastic systems: • Systems that evolve probabilistically over time • Example: a queueing system ± We are interested in determining estimates of quantities such as true mean and variance for a given random variable of the physical stochastic system ± A computer simulation model is a computer representation that mimics the behavior of the physical system ± Simulation models can be used to obtain virtual statistical samples to estimate the performance of stochastic systems 1.225, 11/28/02 Lecture 9, Page 4 Examples Studied in this Lecture Examples Studied in this Lecture ± Coin-Flipping Game: • Repeatedly flip a coin until the difference between the number of heads tossed and the number of tails tossed is ± 3 • You pay $1 per flip of the coin • You receive $8 at the end of each play of the game • You cannot quit when you start a play How would you decide whether you play the game or not ? ± Queueing Models: How to determine the steady-state variables of a queueing system with general interarrival and service time distributions ? ± Simulation can answer questions such as the above two questions 2
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1.225, 11/28/02 Lecture 9, Page 5 Coin Coin -Flipping Game: A Simulated Play Flipping Game: A Simulated Play 1 2 3 4 5 6 7 8 9 10 11 Number of heads & tails Number of coin-flips Cumulative number of heads Cumulative number of tails 8, 1, 3, 7, 2, 7, 1, 6, 5, 5, 7 Random number: T , H , H , T , H , T , H , T , T , T , T Head or Tail: Number of tails = number of heads + 3 Stop simulated play Outcome: $8 - $11 = $ -3 Sample average of number of coin-flips per play: (11 + 5 + L + 7) / 14 = 7 (see R16) ; True mean is 9 1.225, 11/28/02 Lecture 9, Page 6 Simulation Model for Coin Simulation Model for Coin
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This note was uploaded on 12/06/2011 for the course ESD 1.225j taught by Professor Ismailchabini during the Fall '02 term at MIT.

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lecture9p - 1.225J (ESD 205) Transportation Flow Systems...

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