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Unformatted text preview: Lecture 5 Mariana OlveraCravioto Columbia University molvera@ieor.columbia.edu February 1st, 2012 IEOR 4404, Simulation Lecture 5 1/15 The danger of replacing a probability by its mean I Example: Consider a manufacturing system consisting of a single machine. Raw parts arrive to the machine with exponential interarrival times having mean 1 minute. Processing times at the machine are exponentially distributed with mean 0.99 min. I Note: This is an M/M/1 queue with = 0 . 99 . I It can be shown that the average waiting time in queue in the long run is 98.01 minutes. I If we replace each distribution by its corresponding mean, then no part is ever delayed in the queue! I Conclusion: Replacing any random quantity in a simulation by its mean (a deterministic quantity) may result in extremely inaccurate estimates. IEOR 4404, Simulation Lecture 5 2/15 Random number generators I To run a simulation we might need to have observations from a random phenomenon. I If we know the distribution of these observations we might be able to generate them artificially. I We can generate random observations from many distributions by using Uniform[0,1] random numbers. I Historically, random numbers have been generated physically, e.g. by throwing dice, dealing out cards, or drawing numbered balls from an urn. Electronic random number generators came to use in the mid 50s. I There are two kinds of random number generators: physical and numerical (or arithmetic ). IEOR 4404, Simulation Lecture 5 3/15 Arithmetic random number generators I The first arithmetic generator was proposed by von Neuman and Metropolis in the 1940s. I Arithmetic generators follow a sequential rule to compute the numbers, and are therefore NOT random at all. I We also call these generators pseudorandom number generators. I A good pseudorandom number generator is such that it passes several statistical tests designed to determine if the numbers produced are random. I Tests a good random number generator should pass: I Numbers should appear to be distributed uniformly on [0,1] and should not exhibit correlation with each other....
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 Spring '10
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