"Simulation Input and OutputAnalysis 1Why Input and Output Data Analysis? 2 Input Data Output Data Simulation Model Random Random ? Analysis of input data ? Necessary for building a valid model ? Three aspects ? Identification of (time) distributions Integrated into Extend ? Random number generation ? Generation of random variates • Analysis of output data – Necessary for drawing correct conclusions – The reported performance measures are typically randomvariables!Capturing Randomness in Input Data 3 1. Collect raw field data and use as input for the simulation + No question about relevance ? Expensive/impossible to retrieve a large enough data set ? Not available for new processes ? Not available for multiple scenarios ? No sensitivity analysis + Very valuable for model validation 2. Generate artificial data to use as input data ? Must capture the characteristics of the real data1. Collect a sufficient sample of field data 2. Characterize the data statistically – Distribution type and parameters 3. Generate random artificial data mimicking the real data? High flexibility – easy to handle new scenarios ? Cheap ? Requires proper statistical analysis to ensure model validityProcedure for Modeling Input Data 4 ? Plot histograms of the data 1. Gather data from the real? Compare the histogram graphicallysystem (“eye-balling”) with shapes of wellknown distribution functions ? How about the tails of the2. Identify an appropriatedistribution, limited or unlimited? distribution family? How to handle negative outcomes? • Informal test – “eye-balling” 3. Estimate distributionparameters and pick an • Formal tests, for example 2 “exact” distribution – ? - test – Kolmogorov-Smirnov test 4. Perform Goodness–of–fittest If a known distribution is rejected ? Use an empirical distribution (Reject the hypothesis that thepicked distribution is correct?) Distribution hypothesis rejectedDistribution Choice in Absence of Sample Data 5 ? Common situation especially when designing new processes ? Try to draw on expert knowledge from people involved in similar tasks ? When estimates of interval lengths are available ? Ex. The service time ranges between 5 and 20 minutes ? Plausible to use a Uniform distribution with min=5 and max=20 ? When estimates of the interval and most likely value exist ? Ex. min=5, max=20, most likely=12 ? Plausible to use a Triangular distribution with those parameter values ? When estimates of min=a, most likely=c, max=b and theaverage value=x-bar are available ? Use a ?-distribution with parameters ? and ? ? ?b ? x ? (x ? a)(2c ? a ? b) ? ? ? ? (x ? a) (c ? x)(b ? a)Random Number Generators 6 ? Needed to create artificial input data to the simulationmodel ? Generating truly random numbers is difficult ? Computers use pseudo-random number generators based onmathematical algorithms – not truly random but good enough ? A popular algorithm is the “linear congruential method” 1. Define a random seed x from which the sequence is started 0 2. The next “random” number in the sequence is obtained from theprevious through the relation x ? (a ? x ? c) mod m n ?1 n where a, c, and m are integers > 0Example – The Linear Congruential Method 7 ? Assume that m=8, a=5, c=7 and x =4 0 ? x ? (5 ? x ? 7) mod 8 n ?1 n n x 5x +7 (5x +7)/8 x n n n n+1 0 4 27 3 +3 /8 3 1 3 22 2 +6 /8 6 2 6 37 4 +5 /8 5 3 5 32 4 +0 /8 0 4 0 7 0 +7 /8 7 5 7 42 5 +2 /8 2 6 2 17 2 +1 /8 1 7 1 12 1 +4 /8 4 Larger m? longer sequence before it starts repeating itselfGenerating Random Variates 8 ? Assume random numbers, r, from a Uniform (0, 1)distribution are available? Random numbers from any distribution can be obtained byapplying the “inverse transformation technique” The Inverse Transformation Technique 1. Generate a U[0, 1] distributed random number r 2. T is a random variable with a CDF F (t) from which we would likeT to obtain a sequence of random numbers? Note: 0 ? F (t) ?1 for all values of t T ?1 Let F (t) ? r and solve for t ? t ? F (r) T T ? t is a random number from the distribution of T, i.e., a realizationof T "