Stochastic Simulation of

Transcription

1 Stochastic Simulation of Communication Networks -WS 2014/2015 Part 2 Random Number Generation Prof. Dr. C. Görg VSIM 2-1

2 Table of Contents 1 General Introduction 2 Random Number Generation 3 Statistical i Evaluation 4 ComNets Class Library (CNCL) 5 OPNET 6 network simulator (ns) 7 SDL 8 Simulation Speed-Up Methods Prof. Dr. C. Görg VSIM 2-2

3 Random Number Generators Stochastic simulation experiments require random number sequences: x 1, x 2, x 3,... of a random variable X Requirements: Distribution Correlation Length of sequence -... Ideas? Which random number generators do you know? How can random numbers be generated? Prof. Dr. C. Görg VSIM 2-3

4 Random Variables: Discrete: Dscrete Uniform uniform(n,k)... Continuous: Examples Uniform: uniform(a,b) Negative Exponential: negexp(λ) Erlang-k: erlang(k,λ) Normal: normal(μ,б) Pareto: pareto(...)... Types of generators? Prof. Dr. C. Görg VSIM 2-4

5 Techniques & Issues Pseudo random numbers Based on an independent d continuous uniform distribution uniform(0,1) Special techniques Physical random numbers Table generators Correlation: Independent random numbers (Defined) Correlated random numbers Prof. Dr. C. Görg VSIM 2-5

6 Base Generator - uniform(0,1) Requirements: Continuous Independent Uniformly distributed Techniques: Discrete random numbers: discrete uniform (1,N) continuous uniform(0,1)? Independent bits: discrete uniform (0,1) continuous uniform(0,1)? Prof. Dr. C. Görg VSIM 2-6

7 Pseudo Random Number Generators u:=uniform(0,1); Pseudo? deterministic calculation following a defined mathematical formula Next value of sequence d n+1 := f(d n ) depends only on parameter d n Or more generally d n+1 := f(d n-k,...,d n ) with k=1..n Sequence depends on the so-called seed or random seed (d 0 ) Prof. Dr. C. Görg VSIM 2-7

8 Generating Random Numbers Many algorithms for random number generators (RNGs) exist: Simple example: Linear Congruential Generator: d n+1 := (a d n + b ) mod m, e.g., d n+1 := (5 d n + 1 ) mod 16 where d n+1 is the new integer random number Example parameters: Initial value or seed: d 0 = 5 Resulting sequence of random numbers: 10, 3, 0, 1, 6, 15, 12, 13, 2, 11, 8, 9, 14, 7, 4, 5 (and here we are back again!, cycle length: 16), 10, 3,... Division by 15 gives numbers between 0 and 1, to be interpreted as random numbers in [0,1], e.g.: uniform(0,1) Prof. Dr. C. Görg VSIM 2-8

9 Multiplicative Linear Congruential Generator MLCG:d n+1 := (a d n ) mod m LEHMER 1949: RANDU a := = m := 2 31 = d 0 := 929 This generator has been used for a long time. It has been shown to have some bad characteristics, see [Knuth81]!!! (CNCL a:= 16807, m:=2 31-1) Prof. Dr. C. Görg VSIM 2-9

10 Proof of RNG Characteristics Cycle Length of MLCGs d n+1 := (a d n ) mod m with m prime has full cycle length m-1 generating every integer in [1,m-1], 1] if a is a primitive element modulo m; that t is if a i - 1 is a multiple ltil of m for i=m-1 but for no smaller i. Why not cycle length m? Prof. Dr. C. Görg VSIM 2-10

11 Proof of RNG Characteristics (cont.) Cycle Length of LCGs d n+1 := (a d n + c) mod m with c > 0 has full cycle length m generating every integer in [0,m-1] 1], if c and m are relatively prime a-1 is a multiple of every prime p which divides m a-1 is a multiple of 4 if 4 divides m e.g. d n+1 := (a d n + c) mod 2 n is full cycle, cl if c is odd and a has the form 4k + 1 Prof. Dr. C. Görg VSIM 2-11

12 Random Generator Tests Tests only show that a generator has passed a test. A random number generator is good until some test proves the opposite! Knuth: innocent, until proven guilty Tests: correlation tests: e.g. one step serial correlation, runs up and down, multidimensional uniformity,... [Knuth81] Prof. Dr. C. Görg VSIM 2-12

13 Mixed Lagged Fibonacci Generator Shown to have good qualities (until now!!) Intermediate values v i and c i are generated to calculate d i v i := ( v i-97 v i-33 ) mod 2 32 c i := ( c i ) mod 2 32 d i := ( v i c i ) mod 2 32 N = ( ) 2 96 N : cycle length (Periodenlänge) Seeds? Prof. Dr. C. Görg VSIM 2-13

14 RNGs based on Physical Phenomena Measurement of physical phenomena E.g., Radioactive i decay (Poisson) E.g., Electronic Semiconductor Noise Use the given distribution as a basis Sequence of 0,1 bits CD with bits [Richter1992] Prof. Dr. C. Görg VSIM 2-14

15 Other Generator Principles Transcendental Numbers: e.g. = = Inversive Congruential Generator ICG: x -1 * -1 n :=f(x n-1 ) [Hel95] ( x x ) mod m = 1 Cryptography RSA-Generator [Ric92] (Rivest, Shamir, Adleman) inefficient as online generator used as table generator x d n :=x n-1d mod m with m:= p q (p and q large prime numbers) 1=gcd[d,(p-1)(q-1)] )] last log 2 (m) bit pseudo random long cycle length... Prof. Dr. C. Görg VSIM 2-15

16 Discrete RNGs RV X: parameter p (0 p 1) start; generate u ; if u < (1-p) then x := 0 else x := 1 end; return (x); (Electronic Wheel, Comm. Netw. II) Prof. Dr. C. Görg VSIM 2-16

17 Geometrically Distributed RV Number of trials i until success (success in each trial with probability p) p Probability Mass Function (pmf) P(i)? start ; x := 1; Which random variable do you get for X:=0? generate u ; while u < (1-p) do x := x+1 ; generate u ; end do; return (x); pmf? Prof. Dr. C. Görg VSIM 2-17

18 Negative Binomial Distribution X: geometrically distributed n trials: Y = i=1..n X i result in the (neg.) binomial distribution pmf? How many y( (0,1)-numbers are needed? Prof. Dr. C. Görg VSIM 2-18

19 General Discrete Distribution RV R pmf: P(r) = p r, r = 0, 1,.. k CDF: F R (r) = P{ R r } = i=0..r p i F R (k)=1, F R (-1)=0 start; generate u; r:=0; while u > F R (r+1) then r:=r+1 end; return (r); Prof. Dr. C. Görg VSIM 2-19

20 Random Variates by Inverse Transformation Given RV X with cumulative distribution function F X (x) Claim: u F (x) is uniformly distributed between 0 and 1 X Proof: F U ( u) PU 1 1 u P( F ( X) u) P( X F ( u)) F ( F ( u)) u X Hence, if F -1 X is easy to compute, derive RV X with distribution F X (x) by computing a random number u uniformly distributed between 0 and 1, and apply F -1 X (u) to obtain the random number x Same principle as the General Discrete Distribution X X X Prof. Dr. C. Görg VSIM 2-20

21 Inverse Transformation (cont.) Example: X exponentially distributed F (x) u X 1 ) ( e x F u x X ) ( 1 u F x X ) 1 ln( ) / 1 ( 1 ) ( u x e x F u X (x) F x 1 x u 0 Prof. Dr. C. Görg VSIM x 0

22 Normal Distribution Inverse not available Special generation procedure after Box and Muller [Devroye86] Pair of independent d normally distributed RV X 1 and X 2 with mean μ=00 and variance σ Interpret pair as vector in the complex plane: x 1 + i x 2 = γ [cos (φ) + i sin (φ)] γ follows Rayleigh distribution: CDF 1 - e-x²/ 2v Prof. Dr. C. Görg VSIM 2-22

23 Normal Distribution Inverse available for Rayleigh distribution: γ (u :=[-2 (1-u 1/2 1 ) ν ln 1 )] φ assumes all values uniformly distributed ibut d between 0 and 2 Generate φ := 2 u 2 Generate x 1 (u 1,u 2 ):= γ (u 1 ) cos (2 u 2 ) x 2 (u 1,u 2 ):= γ (u 1 ) sin (2 u 2 ) Need to prove independence and normal distribution Prof. Dr. C. Görg VSIM 2-23

24 G/G/1 Model: Random Number Generators Arrival (λ a ) Server (λ b ) b 1 b FIFO - Queue Prof. Dr. C. Görg VSIM 2-24

25 e.g. G/G/1: Some Pitfalls use one generator for interarrival i time another generator for service time ensure independence: use the same base generator use a new seed known to be far enough away from the original seed use another type of generator Prof. Dr. C. Görg VSIM 2-25

26 How to choose input distributions Traces of a real system can be used directly to drive a simulation e.g.: take a set of recorded interarrival times and service times from a computer network to drive simulation program Traces can be aggregated into an empirical distribution function Or an analytical distribution function is chosen on a hypothetical/principal basis: several rules of thumb exist on how to choose one or several good candidates (use mean, coefficient i of variation, skewness, etc. as a guideline). Generate random numbers from this distribution Important: Perform a goodness-of-fit test to check whether the chosen distributions does indeed match the empirical data with sufficient accuracy, e.g. Kolmogorov-Smirnov test Prof. Dr. C. Görg VSIM 2-26

27 CNCL: Communication Networks Class Library Base Generators: Prof. Dr. C. Görg VSIM 2-27

28 CNCL Examples CNACG: Additive Congruential Generator (GNU-library libg++ ) CNFiboG: Fibonacci Generator CNFileG: File Generator CNLCG: Linear Congruential Generator CNMLCG: Multiple LCG CNTausG: Tausworthe Shift Register Generator Prof. Dr. C. Görg VSIM 2-28

29 CNCL: Communication Networks Class Library Random Number Generators: Prof. Dr. C. Görg VSIM 2-29

30 CNCL Examples e.g. CNHyperExp is a mixed distribution of two negative exponential distributions (H 2 ) CNHyperExp (double p, double m1, double m2, CNRNG *gen) Initializes a CNHyperExp distribution ib ti with a base random number generator gen, the mix probability bilit p and the intensity parameters m1 and m2. Prof. Dr. C. Görg VSIM 2-30