Commonly Used Distributions • Random number generation algorithms for distributions commonly used by computer systems performance analysts. • Organized alphabetically for reference • For each distribution: – Key characteristics – Algorithm for random number generation – Examples of applications c 1994 Raj Jain 29.1
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Commonly Used Distributionsjain/books/ftp/ch5f_slides.pdfCommonly Used Distributions † Random number generation algorithms for distributions commonly used by computer systems performance
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Commonly Used Distributions
• Random number generation algorithms fordistributions commonly used by computersystems performance analysts.
(b) If the sum of geometric RNs so far isless than or equal to n, go back to theprevious step. Otherwise, return thenumber of RNs generated minus one.If ∑m
i=1 Gi(p) > n, return m− 1.
3. Inverse Transformation Method:Compute the CDF F(x) forx = 0, 1, 2, . . ., n and store in an array.For each binomial variate, generate aU(0,1) variate u and search the array tofind x so that F (x) ≤ u < F (x + 1);return x.
• Discrete equivalent of the exponentialdistribution
• Key characteristics:
1. Parameters: p = Probability of success,0 < p < 1.
2. Range: x = 1, 2, . . . ,∞3. pmf: f (x) = (1− p)x−1p
4. CDF: F (x) = 1− (1− p)x
5. Mean: 1/p
6. Variance:1−pp2
• memoryless
• Applications: Number of trials up to andincluding the first success in a sequence ofBernoulli trialsNumber of attempts between successivefailures (or successes)
• Applications: The product of a largenumber of positive random variables tendsto have an approximate lognormaldistributionTo model multiplicative errors that are aproduct of effects of a large number offactors
• Generation: Log of a normal variateGenerate x ∼ N(0, 1) and return eµ+σx.
Particularly appropriate if the arrivals arefrom a large number of independentsources
• Generation:
1. Inverse Transformation Method:Compute the CDF F(x) forx = 0, 1, 2, . . . up to a suitable cutoffand store in an array.For each Poisson random variate,generate a U(0,1) variate u, and searchthe array to find x such thatF (x) ≤ u < F (x + 1), return x.
2. Starting with n = 0, generateun ∼ U(0, 1) and compute the product∏ni=0 ui. As soon as the product
becomes less than e−λ, return n as thePoisson(λ) variate.Note that n is such thatu0u1 · · ·un−1 > e−λ ≥ u0u1 · · ·un
1. Parameters:a = Scale parameter a > 0b = Shape parameter b > 0
2. Range: 0 ≤ x ≤ ∞3. pdf: f (x) = bxb−1
ab e−(x/a)b
4. CDF: F (x) = 1− e−(x/a)b
5. Mean: abΓ(1/b)
6. Variance: a2
b2
[2bΓ(2/b)− {Γ(1/b)}2
]
• If b = 3.602, the Weibull distribution isclose to a normal. For b > 3.602, it has along left tail. For b < 3.602, it has a longright tail.For b ≤ 1, the Weibull pdf is L-shaped,and for b > 1, it is bell-shaped.
For large b, the Weibull pdf has a sharppeak at the mode.
• Applications: To model lifetimes ofcomponents.b < 1 ⇒ failure rate increasing with timeb > 1 ⇒ failure rate decreases with timeb = 1 ⇒ failure rate is constant⇒ life times are exponentially distributed.
• Generation: Inverse transformationGenerate u ∼ U(0, 1) and return
• Books on simulations: Law and Kelton(1982) and Brately, Fox, and Schrage(1986)
• Lavenberg (1983): transient removal,variance estimation, and random-numbergeneration.
• Languages: GPSS in O’Donovan (1980)SIMSCRIPT II in CACI (1983)SIMULA by Birtwistle, Dahl, Myhrhaug,and Nygaard (1973)GASP by Pritsker and Young (1975)
• Sherman and Browne (1973): trace-drivencomputer simulations
• Adam and Dogramaci (1979) includepapers describing the simulation languagesSIMULA, SIMSCRIPT, and GASP bytheir respective language designers.
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