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Random-Number Generation

Random-Number Generation. Andy Wang CIS 5930-03 Computer Systems Performance Analysis. Generate Random Values. Two steps Random-number generation Get a sequence of random numbers distributed uniformly between 0 and 1 Random- variate generation

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Random-Number Generation

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  1. Random-Number Generation Andy Wang CIS 5930-03 Computer Systems Performance Analysis

  2. Generate Random Values • Two steps • Random-number generation • Get a sequence of random numbers distributed uniformly between 0 and 1 • Random-variate generation • Transform the sequence to produce random values satisfying the desired distribution

  3. Background • The most common method • Use a recursive function xn = f(xn-1, xn-2, …)

  4. Example • xn = (5xn-1 + 1) %16 • Suppose x0 = 5 • The first 32 numbers are between 0 and 15 • Divide xnby 15 to get numbers between 0 and 1

  5. Basic Terms • x0 = seed • Given a function, the entire sequence can be regenerated with x0 • Generated numbers are pseudo random • Deterministic • Can pass statistical tests for randomness • Preferred to fully random numbers so that simulated results can be repeated

  6. Cycle Length • Note that starting with the 17th number, the sequence repeats • Cycle length of 16

  7. More Terms • Some generators do not repeat the initial part (tail) of the sequence • Period of a generator = tail + cycle length tail cycle length period

  8. Question • How to choose seeds and random-number generation functions? • Efficiently computable • Heavily used in simulations • The period should be large • Successive values should be independent and uniformly distributed

  9. Types of Random-Number Generators • Linear-congruential generators • Tausworth generators • Extended Fibonacci generators • Combined generators • Others

  10. Linear-Congruential Generators • In 1951, Lehmer found residues of successive powers of a number have good randomness properties xn = an % m = aan-1 % m = axn-1 % m • Lehmer’s choices of a and m a = 23 (multiplier) m = 108 + 1 (modulus) • Implemented on ENIAC

  11. (Mixed) Linear-Congruential Generators (LCG) • xn = (axn-1 + b) % m • xn is between 0 and m – 1 • a and b are non-negative integers • “Mixed”  using both multiplication by a and addition by b

  12. The Choice of a, b, and m • m should be large • Period is never longer than m • To compute % m efficiently • Make m = 2k • Just truncate the result by k bits

  13. The Choice of a, b, and m • If b > 0, maximum period m is obtained when • m = 2k • a = 4c + 1 • b is odd • c, b, and k are positive integers

  14. Full-Period Generators • Generators with maximum possible periods • Not equally good • Look for low autocorrelations between successive numbers • xn = ((234 + 1)xn-1 + 1) % 235 has an autocorrelation of 0.25 • xn = ((218 + 1)xn-1 + 1) % 235 has an autocorrelation of 2-18

  15. Multiplicative LCG • xn = axn-1 % m, b = 0 • Can compute more efficiently when m = 2k • However, maximum period is only 2k-2 • Problem: Cyclic patterns with lower bits

  16. Multiplicative LCG with m = 2k • When a = 8i ± 3 • E.g., xn = 5xn-1 % 25 • Period is only 8 • Which is ¼ of 25 • When a ≠ 8i ± 3 • E.g., xn = 7xn-1 % 25 • Period is only 4

  17. Multiplicative LCG with m ≠ 2k • To get a longer period, use m = prime number • With proper choice of a, it is possible to get a period of m – 1 • a needs to be a prime root of m • If and only if an % m ≠ 1 for n = 1..m - 2

  18. Multiplicative LCG with m ≠ 2k • xn = 3xn-1 % 31 • x0 = 1 • Period is 30 • 3 is a prime root of 31

  19. Multiplicative LCG with m ≠ 2k • xn = 75xn-1 % (231 – 1) • 75 is a prime root of 231 – 1 • But watch out for computational errors • Multiplication overflow • Need to apply tricks mentioned in p. 442 • Truncation due to the number of digits available

  20. Tausworthe Generations • How to generate large random numbers? • The Tausworthe generator produces a random sequence of binary digits • The generator then divides the sequence into strings of desired lengths • Based on a characteristic polynomial

  21. Tausworthe Example • Suppose we use the following characteristic polynomial x7 + x3 + 1 • The corresponding generation function is • bn+7 bn+3bn = 0 Or • bn = bn-4 bn-7 • Need a 7-bit seed

  22. Tausworthe Example • The bit stream sequence 1111111000011101111001011001…. • Convert to random numbers between 0 and 1, with 8-bit numbers x0 = 0.111111102 = 0.9921910 x1 = 0.000111012 = 0.1132810 x2 = 0.111001012 = 0.8945310 …

  23. Tausworthe Generator Characteristics • For the L-bit numbers generated +E[xn] = ½ +V[xn] = 1/12 +The serial correlation is zero + Good results over the complete cycle - Poor local behavior within a sequence

  24. Tausworthe Example • If a characteristic polynomial of order q has a period of 2q – 1, it is a primitive polynomial • For x7 + x3 + 1 • q = 7 • Sequence repeats after 127 bits = 27 - 1 • A primitive polynomial

  25. Tausworthe Implementation • Can be easily generated via linear-feedback shift-registers • For x5 + x3 + 1  bn bn-1 bn-2 bn-3 bn-4 bn-5

  26. Extended Fibonacci Generators • xn = (xn-1 + xn-2) % m • Does not have good randomness properties • High serial correlation • An extension • xn = (xn-5 + xn-17) % 2k

  27. Combined Generations • Add random numbers by two or more generators • Can considerably increase the period and randomness xn = 40014xn-1 % 2147483563 yn = 40692yn-1 % 2147483399 wn = (xn - yn) % 2147483562 • This generator has a period of 2.3 x 1018

  28. Combined Generators wn = 157wn-1 % 32363 xn = 146xn-1 % 31727 yn = 142yn-1 % 31657 vn = (wn - xn + yn) % 32362 • This generator has a period of 8.1 x 1012 • Can avoid the multiplication overflow problem

  29. Combined Generators • XOR random numbers by two or more generators

  30. Combined Generators • Shuffle • One sequence as an index • To an array filled with random numbers generated by the second sequence • The chosen number in the second sequence is replaced by a new random number • Problem • Cannot skip to the nth random number

  31. A Survey of Random-number Generators • Some published generator functions xn = 75xn-1 % (231 – 1) • Full period of 231 – 2 • Low-order bits are randomly distributed • Many others (see textbook) • All have problems • General lessons: Use established ones; Do not invent your own

  32. Seed Selection • If the generator has a full period • Only one random variable is required • Any seed value is good • However, with more than one random variable, the story is different for multistream simulations • E.g., random arrival and service times • Should use two streams of random numbers

  33. Seed Selection Guidelines • Do not use zero • Not good for multiplicative LCGs and Tausworthe generators • Avoid even values • Not good if a generator does not have a full period • Do not use one stream for all variables • May yield strong correlations among variables

  34. Seed Selection Guidelines • Use nonoverlapping streams • Each stream requires a separate seed • Otherwise… • A long interarrival time may correlate with a long service time • Suppose we need 10,000 random numbers for interarrival times; 10,000 for service times, use seeds 1 and 10,001 • xn = [anx0 + c(an – 1)/(a – 1)] % m • For multiplicative LCGs, c = 0

  35. Seed Selection Guidelines • Not to reuse seeds in successive simulation runs • No point to run a simulation again with the same seed • Just continue with the last random number as the seed for the successive runs

  36. Seed Selection Guidelines • Do not use randomrandom-number generator seeds • E.g., do not use the time of day, or /dev/random to seed simulations • Simulations should be repeatable • Cannot guarantee that multiple streams will not overlap • Do not use numbers generated by random-number generators as seeds

  37. Myths About Random-number Generation • A complex set of operations leads to random results • Hard to guess does not mean random • Random numbers are not predictable • Given a few successive numbers from an LCG • Can solve a, c, and m • Not suitable for cryptographic applications

  38. Myths about Random- number Generation • Some seeds are better than others • True • Avoid generators whose period and randomness depend on the seed • Accurate implementation is not important • Watch out for overflows and truncations

  39. Myths about Random- number Generation • Bits of successive words generated by a random-number generator are equally randomly distributed • Nope

  40. Myths about Random- number Generation • xn = (25173xn-1 + 13849) % 216 • x0 = 1 • Least significant bit is always 1 • Bit 2 is always 0 • Bit 3 has a cycle of 2 • Bit 4 has a cycle of 4 • Bit 5 has a cycle of 8

  41. Myths about Random- number Generation • For all multiplicative LCGs • The Lth bit has a period that is at most 2L • For LCGs, with the form xn = axn-1 % 2k • The least significant bit is always 0 or 1 • High-order bits are more random

  42. More on Random Number Generations • Mersenne twister • Period =~ 219937-1 • /dev/random • Extract randomness from physical devices • Truly random

  43. White Slide

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