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lecturenotes06 - MS&E 223 Simulation Peter J. Haas...

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MS&E 223 Lecture Notes #6 Simulation Uniform Variate Generation Peter J. Haas Spring Quarter 2009-10 Uniform Variate Generation Refs : Law Ch. 7; Simulation (Ed. Henderson et al.) Ch . 3.; Knuth, The Art of Computer Programming , Vol. 2, Ch. 3. We now discuss the question of how to generate i.i.d. uniform random variables on a computer. As we have seen, uniform randomness is the basis of all the randomness that one simulates on a computer. 1. Pseudo-Random Numbers Perhaps surprisingly, the basis of most RNG algorithms is a deterministic recurrence relation: the sequence starts with an initial “seed”, and then each “random number” in the sequence is a deterministic function of the previous random number. Although these numbers form a deterministic dependent sequence, if the generation algorithm is well designed the numbers will appear to be random and independently distributed. For this reason such numbers are called pseudo-random numbers. An important property of pseudo-random numbers is that a given sequence can be repeated again at a later time. This repeatability property is useful because it facilitates debugging of simulation programs it permits an efficiency improvement technique called “common random numbers” (to be discussed later) 2. Linear Congruential Generators The most commonly used generators, in practice, are linear congruential generators (LCG’s). A sequence of integers is generated via the recursion 01 x,x, n1 n x (ax c) mod m + =+ where a, c, and m are integers also known, respectively, as the multiplier, increment and modulus. Remark : k mod m is the integer obtained by dividing k by m and computing the remainder (e.g., 14 mod 5 = 4, 26 mod 21 = 5). It follows that the are integers in the set {0, 1, . .., m −1 }. To obtain the pseudo-random uniform sequence , one sets n x' s U,U, nn Ux / m = . Clearly, a LCG produces periodic output, with period m. Full period is desirable because: It increases the potential number of random variates one can reliably use in a simulation. If the generator is not of full period, then there are gaps in the output sequence. The nature of such gaps for an LCG is not well understood, and may create distinctly non-random behavior in the output. Page 1 of 8
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MS&E 223 Lecture Notes #6 Simulation Uniform Variate Generation Peter J. Haas Spring Quarter 2009-10 So, at a minimum, one should choose a, c and m to have full period. (Though a full period by itself does not guarantee that the sequence has nice statistical properties). There are known sufficient conditions for this. LCG’s run faster if the increment c is set equal to 0. Such an LCG is called a multiplicative generator. In practice, most LCG’s are multiplicative generators (also called Lehmer generators). 3. Multiplicative Generators For a multiplicative generator, 0 is a “trap” state, for if n x0 = for some n, then all future values of will be zero. So, for a multiplicative generator, one says that the generator is of full period if for any initial seed k x { } 0 x1 , , m 1 , the sequence visits all m 1 integers in the set {1, . .., m
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lecturenotes06 - MS&E 223 Simulation Peter J. Haas...

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