Session16

# Session16 - Operations Management Session 16: Simulation...

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Session 16 Operations Management 1 Operations Management Session 16: Simulation

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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 693-716 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(1-U)/(- z ). ) exp( 1 ) ( ) ( x x F x X P λ = =

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Session 16 Operations Management 4 Why did that work? () ) ( ) ( ) 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 So, the random variable ln(1-U)/(- z ) 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.

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Session 16 Operations Management 6 Example Cont. Apply the formula ln(1-u)/(-1) to generate samples from an exponential distribution with mean 1 ( z =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 =ln(1-0.787458)/(-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.
Session 16 Operations Management 7 Example Cont. Apply the formula ln(1-u)/(-2) to generate samples from an exponential distribution with mean ½ ( z =2). =ln(1-0.787458)/(-2) 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 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.

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Session 16 Operations Management 8 What is the general formula? ± Step 1: solve for the inverse cdf F -1 , F(F -1 (x))=x. ² Note that for an exponential random variable, F(x) = 1-exp(- z x) and F -1 (x)=ln(1-x)/(- z ) so that F(F -1 (x))=x. ± Step 2: for each sample u i of a uniform random variable, apply the formula F -1 (u i ).
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## Session16 - Operations Management Session 16: Simulation...

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