IE306Lec4 - IE306 SYSTEMS SIMULATION Spring 2008 Ali Rıza...

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Unformatted text preview: IE306 SYSTEMS SIMULATION Spring 2008 Ali Rıza Kaylan kaylan@boun.edu.tr 1 LECTURE 4 OUTLINE Stochastic Components of Simulation Models Bernoulli Process Discrete Random Variables Continuous Random Variables Statistical Analysis Poisson Process 2 RANDOM VARIABLES Discrete : Bernoulli Process Related Distributions Binomial, Geometric, Multinomial, Pascal Poisson Process Continuous : Uniform Exponential Normal 3 BERNOULLI TRIALS Independent trials with two possible outcomes p p( x ) = 1 - p 0 F ( x ) = 1- p 1 if x = 1 if x = 0 if x < 0 if 0 ≤ x p 1 if 1 ≤ x E( x ) = p Var ( x ) = p (1 − p ) 4 BERNOULLI TRIALS Binomial Bernoulli X = {1 0 Success Failure n n trials Y= Σ Xi i=1 First Success Negative Binomial Geometric T k kth success S= Σ Ti i=1 5 BINOMIAL DISTRIBUTION n x p (1 - p) n-x p( x ) = x 0 0 x n i F ( x ) = ∑ p (1 - p) n-i i=0 i 1 if x = 0,1,2,..., n otherwise if x < 0 if 0 ≤ x ≤ n if n ≤ x E ( x ) = np Var ( x ) = np(1 − p ) 6 GEOMETRIC DISTRIBUTION p(1- p) x-1 p( x ) = 0 0 F( x ) = 1- (1 - p) x 1 E( x ) = p 1 Var ( x ) = 2 p if x = 1, 2,... otherwise if x < 1 if 1 ≤ x 7 NEGATIVE BINOMIAL DISTRIBUTION s - 1 k s-k p (1 - p) p( s ) = k - 1 0 0 F (s ) = s i - 1 k p (1- p) s-k ∑ i =k k - 1 if s = k , k + 1,... otherwise if s < k if k ≤ s k E( x ) = p k Var ( x ) = 2 p 8 POISSON DISTRIBUTION e -λ λ x p( x ) = x ! 0 0 F ( x ) = -λ x λ i e ∑ i =0 i ! if x = 0,1,... otherwise if x < 0 if 0 ≤ x E( x ) = λ Var ( x ) = λ 9 UNIFORM DISTRIBUTION f(x) 1 b-a x a b 10 UNIFORM DISTRIBUTION 1 f (x) = b − a 0 0 x−a F( x ) = b − a 1 if a ≤ x ≤ b otherwise if x < a if a ≤ x ≤ b if b < x a+b E( x ) = 2 ( b − a )2 Var ( x ) = 12 11 EXPONENTIAL DISTRIBUTION 1 -x/ β e f (x) = β 0 1- e -x /β F( x ) = 0 if x ≥ 0 otherwise if x ≥ 0 otherwise E( x ) = β Var ( x ) = β 2 12 EXPONENTIAL DISTRIBUTION Memorlyless Property Relationship to other distributions Poisson Weibull Erlang 13 NORMAL DISTRIBUTION f(t) t Density function is the familiar “bell-shaped” curve” Completely described by mean µ and standard deviation σ: N( µ, σ2). f(t) is symmetric around the mean: f(µ + t) = f (µ - t) f(t) is highest at its mean (when t = µ) 14 NORMAL DISTRIBUTION 68.3% 95.4% µ−3σ µ−2σ µ−σ .0228 .1587 µ µ+σ µ+2σ µ+3σ .5 .8413 .9772 15 NORMAL DISTRIBUTION 0.80 X 0.60 Y 1 2 Z 0.40 0.20 0.00 -1 0 3 4 3 4 X 0.80 Y 0.60 0.40 Z 0.20 0.00 -1 0 1 2 16 STATISTICAL ANALYSIS 1. Collect a random sample from the process of interest X={X1,X2,...,Xn} 2. Preliminary statistical analysis Graphical Analysis (Histogram, Box Plot) Calculate Descriptive Statistics (Mean, Median, Mode, Variance, Range, Coef. of Variation) 3. Identify the candidate model 4. Carry out Statistical Analysis (Parameter Estimation, Goodness of Fit Tests) 5. Draw conclusions about the underlying model. If necessary, repeat previous steps. 17 POISSON PROCESS Let {N(t),t>=0} be a counting process where N(t) designates the number of occurrences in the time interval (0,t]. The counting process is said to be Poisson with mean rate λ if i) Arrivals occur one at a time, ii) Process has stationary increments, iii) Process has independent increments, iv) N(0)=0. e − λt (λ t ) n P{N (t ) = n} = n! n = 0,1,2, ,... t ≥ 0 Examples: Arrival of calls to a call center, Arrival of jobs to a job shop, Arrival of customers to a bank. 18 POISSON PROCESS Random Splitting: Consider a Poisson Process {N(t),t>=0} with rate λ. Suppose that each time an event occurs, it is classified as type 1 with probability p or type 2 with probability q=1-p. Let N1(t) and N2(t) be the random variables that denote the type 1 and type 2 events respectively. N1(t) and N2(t) are both Poisson processes with rates λp and λ(1-p) respectively. λp λ Poisson Bernoulli Poisson λ(1-p) Poisson 19 POISSON PROCESS Pooled Process: Let N1(t) and N2(t) be two independent Poisson processes with rates λ1 and λ2 respectively. Then N(t)= N1(t) + N2(t) is a Poisson process with rate λ1 + λ2. Poisson Poisson λ1 λ= λ1+λ2 Poisson λ2 20 ...
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