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Output Analysis for Simulation. Written by: Marvin K. Nakayama. Presented by: Jennifer Burke MSIM 752. Outline. Performance measures Output of a transient simulation Techniques for steady-state simulations Estimation of multiple performance measures

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## Output Analysis for Simulation

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**Output Analysis for Simulation**Written by: Marvin K. Nakayama Presented by: Jennifer Burke MSIM 752**Outline**• Performance measures • Output of a transient simulation • Techniques for steady-state simulations • Estimation of multiple performance measures • Other methods for analyzing simulation output**Example**• Automatic Teller Machine (ATM)**Performance Measures**• Measure how well the simulation runs • Different types of simulations require different statistical techniques to analyze the results • Terminating (or transient) • Steady-state (or long run)**Terminating Performance Measures**• Terminating simulation • Simulation will finish at a given event • Initial conditions have a large impact • Ex: Queue starts with no customers present**ATM example (Terminating)**• Open 9:00am – 5:00pm • X = # of customers using ATM in a day • E(X) • P(X 500) • C = queue is empty**Output of a Terminating Simulation**• Goal: • calculate E(X) • Approach: • n 2 i.i.d duplications • X1,X2,…,Xn • find the average of those duplications**Output of a Terminating Simulation**• calculate the sample variance of X1,X2,…,Xn and the sample standard deviation**Output of a Terminating Simulation**• Central Limit Theorem • confidence interval for E(X)**Output of a Terminating Simulation**• the confidence interval provides a form of error bound Hn is the half-width of the confidence interval**ATM example (Terminating)**• Expected daily withdraw • within $500 • ε = 500 • S(n) = sample standard deviation**Steady-state Performance Measures**• Steady-state simulation • Simulation that stabilizes over time • Initial condition C • Fi(y|C) • Fi(y|C) → F(y) as i → **ATM example (Steady-state)**• Open 24 hours a day • Yi = number of customers served on the ith day of operation • E(Y) • P(Y 400) • C = queue is empty**Output of a Steady-state Simulation**• Case 1: • discrete-time process Y1,Y2,…,Yn • estimate v, as m → • Case 2: • continuous-valued time index Y(s) • estimate v, as m → **ATM (Continuous)**• Y(s) = number of customers waiting in line at time s • Assume Y(s) has a steady-state • Calculate v**Difficulties of Steady-state analysis**• Discrete-time process • if m is large, then is a good approximation of v • Confidence interval**Simplifications to Steady-state Analysis**• Multiple replications • Initial-data deletion • Single-replicate algorithm**Method of Multiple Replications**• Estimate • r i.i.d replications, length k = m/r • 10 r 30**Method of Multiple Replications**• Average of jth row • Using find the sample mean**Method of Multiple Replications**• Sample variance • Confidence interval**Problem with Multiple Replication Method**• Simple estimation of variance • can be contaminated by initialization bias**Initial-Data Deletion**• Partial solution • Delete first c observations • Replication mean sample mean sample variance confidence interval**Single-Replicate Algorithm**• Single simulation of length m + c • Divide the m observations into n batches • 10 n 30 • Batch mean**Single-Replicate Algorithm**• Sample mean • Sample variance • Confidence interval**Estimating Multiple Performance Measures**• Terminating simulations • Confidence interval for each performance measure • Joint confidence interval**ATM example (Terminating)**• Open 9:00am – 5:00pm • μ1 = expected # of customers served in a day • μ2= probability # served in a day is at least 1000 • μ3 = expected amount of $ withdrawn in a day**Conclusions**• Basics of analyzing simulation output • Application potential is high • Not state of the art • Benefit • Lacked comparison

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