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Estimation and Output Analysis

# Estimation and Output Analysis - ESTIMATION AND OUTPUT...

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ESTIMATION AND OUTPUT ANALYSIS (L&K Chapters 9, 10) Set up standard example and notation. Review performance measures (means, probabilities and quan- tiles). A framework for conducting simulation experiments and report- ing results. Review point estimators. Measuring and controlling error for absolute and relative perfor- mance. Measuring and controlling error in steady-state simulation. 14

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STANDARD EXAMPLE Public Sub TTF(Lambda As Double, Mu As Double, Sum As Double) ’sub to generate one replication of the ttf for the ’airline reservation system ’variables State = number of operational computers Lambda = computer failure rate Mu = computer repair rate Sum = generate ttf value that is returned from the Call Dim State As Integer Dim Fail As Double Dim Repair As Double State = 2 Sum = 0 While State > 0 If State = 2 Then Fail = Exponential(1 / Lambda) Sum = Sum + Fail State = 1 Else Fail = Exponential(1 / Lambda) Repair = Exponential(1 / Mu) If Repair < Fail Then Sum = Sum + Repair State = 2 Else Sum = Sum + Fail State = 0 End If End If Wend End Sub 15
NOTATION AND PERFORMANCE MEASURES Let Y 1 , Y 2 , . . . , Y n denote outputs across n replications. Thus, Y 1 , Y 2 , . . . , Y n are independent and indentically distributed (i.i.d.) with common, unknown cumulative distribution function (cdf) F . We want to estimate mean μ = E[ Y ] = -∞ y dF ( y ) probability p = Pr { a < Y b } = b a dF ( y ) quantile θ , where for given 0 < γ < 1 Pr { Y θ } = θ -∞ dF ( y ) = γ 16

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Example: In the TTF example, μ could represent the expected time to failure; p the probability failure occurs in less than 700 hours; and θ the number of hours so that failure occurs on or before then 80% of the time. Comments: Means and probabilities are the same problem, since a probability is the mean of an indicator random variable Z = I ( a < Y b ). E[ Z ] = 0 · Pr { Y a or Y > b } + 1 · Pr { a < Y b } Higher moments (variances, skewness, kurtosis, etc.) can also be viewed as means. Quantiles cannot be (conveniently) expressed as means. 17
A FRAMEWORK FOR CONDUCTING EXPERIMENTS AND REPORTING RESULTS We should distinguish between absolute and relative performance measures. The expected TTF of a system is an absolute performance mea- sure. The difference in expected TTF between two proposed systems is a relative performance measure. When we report simulation-based performance measures, the mini- mum we report is either a point estimate (our best guess) and a measure of error, or a point estimate with only significant digits. Without a measure of error, a point estimate may be mean- ingless. 18

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REPORTING POINT ESTIMATES Suppose we estimate the expected TTF to be 997.958. What should we report? point estimate and a measure of error 997 . 958 ± 60 . 96 or [937 . 098 , 1058 . 818] Report 998 ± 61 or [937 , 1059] a point estimate with only significant digits 997 . 958 with standard error 31 . 014.
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