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Unformatted text preview: Session 16 Operations Management 1 Operations Management Session 16: Simulation Session 16 Operations Management 2 Class Objectives Generate random numbers. Develop confidence intervals. Simulate an M/M/1 queue. Simulate a call center. The idea is to first set up a simulation for a model with which we are familiar (the M/M/1 queue), and then set up a simulation for a model whose behavior cannot be predicted analytically. This is TN17, pages 693716 in your text. Session 16 Operations Management 3 Generating exponential random numbers Assume U has a uniform distribution, so P(U u) = u. (There are many algorithms, and much previous work on how to generate “truly” uniform random numbers.) Suppose we would like to generate an observation of the random variable X, which has an exponential distribution. Generate samples of the random variable U, and apply the formula ln(1U)/( ). ) exp( 1 ) ( ) ( x x F x X P λ = = ≤ Session 16 Operations Management 4 Why did that work? ( 29 ) ( ) ( ) exp( 1 )) exp( 1 ( )) exp( 1 ( ) 1 ln( ) 1 ln( u X P u F u u U P u U P u U P u U P ≤ = = = ≤ = ≥ = ≥ = ≤ λ λ λ λ λ So, the random variable ln(1U)/(F) has an exponential distribution. Session 16 Operations Management 5 Example 1 0.787458 2 0.432425 3 0.380017 4 0.420585 5 0.315984 6 0.145883 7 0.818392 8 0.860034 9 0.332883 10 0.824931 Uniform random numbers generated in Excel using the command: =RAND(). Note: the average of the ten numbers is 0.532. As more uniform random numbers are generated, that will approach 0.5. Session 16 Operations Management 6 Example Cont. Apply the formula ln(1u)/(1) to generate samples from an exponential distribution with mean 1 (h=1). Note: the average of the 10 numbers in the 3 rd column is 0.950. As more random numbers are generated, this will approach 1. =ln(10.787458)/(1) 1 0.787458 1.548617 2 0.432425 0.566382 3 0.380017 0.478063 4 0.420585 0.545737 5 0.315984 0.379775 6 0.145883 0.157687 7 0.818392 1.705904 8 0.860034 1.966357 9 0.332883 0.404789 10 0.824931 1.742576 Session 16 Operations Management 7 Example Cont. Apply the formula ln(1u)/(2) to generate samples from an exponential distribution with mean ½ (h=2). Note: the average of the 10 numbers in the 3 rd column is 0.475. As more random numbers are generated, this will approach 0.5. 1 0.787458 0.774308 2 0.432425 0.283191 3 0.380017 0.239032 4 0.420585 0.272868 5 0.315984 0.189887 6 0.145883 0.078843 7 0.818392 0.852952 8 0.860034 0.983179 9 0.332883 0.202395 10 0.824931 0.871288 =ln(10.787458)/(2) Session 16 Operations Management 8 What is the general formula?...
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This note was uploaded on 05/08/2008 for the course BUAD 311 taught by Professor Vaitsos during the Spring '07 term at USC.
 Spring '07
 Vaitsos
 Management

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