ARENA INPUT ANALYZER

7.4 ARENA INPUT ANALYZER

The Arena Input Analyzer functionality includes fitting a distribution to sample data. The user can specify a particular class of distributions and request the Input Analyzer to recommend associated parameters that provide the best fit. Alternatively, the user can request the Input Analyzer to recommend both the class of distributions as well as associated parameters that provide the best fit. Table 7.2 displays the distributions supported by Arena and their associated parameters.

Table 7.2 Arena-supported distributions and their parameters

Arena Distribution

parameters Exponential

Arena name

Mean Normal

EXPO

Mean, StdDev Triangular

NORM

Min, Mode, Max Uniform

TRIA

Min, Max Erlang

UNIF

ExpoMean, k Beta

ERLA

Beta, Alpha Gamma

BETA

Beta, Alpha Johnson

GAMM

G, D, L, X Log-normal

JOHN

LogMean, LogStdDev Poisson

LOGN

Mean Weibull

POIS

Beta, Alpha Continuous

WEIB

P1, V1, . . . a Discrete

CONT

P1, V1, . . . a The parameters P1, P2, . . . are cumulative probabilities.

DISC

Input Analysis 131 Note, however, that the parameters in Table 7.2 may differ from those in the corres-

ponding distributions of Sections 3.7 and 3.8. We next illustrate distribution fitting via the Input Analyzer. Suppose the analyst would like to fit a uniform distribution to the data of Table 7.1. To this end, the Repair_Times.dft file (see Section 7.2) is opened from the File menu of the Input Analyzer. The Fit menu displays all Arena-supported distributions (see Table 7.2), and upon selection of a particular distribution, the Input Analyzer computes the associated best-fit parameters and displays the results. In our case, Figure 7.2 depicts the output for the best-fit uniform distribution, including the following items:

Graph of the best-fit distribution (pdf ) superimposed on a histogram of the empirical data Parameters of the best-fit distribution Statistical outcomes of the goodness-of-fit tests employed (see Section 7.5) Summary information on the empirical data and the histogram constructed from it.

Figure 7.2 Best-fit uniform distribution for the repair time data of Table 7.1.

132 Input Analysis In our case (best-fit uniform distribution), the associated parameters are just the

minimum and maximum of the sample data. The Curve Fit Summary option of the Window menu provides a detailed summary of any selected fit.

Next, suppose that the analyst changes her mind and would like instead to find the best-fit beta distribution for the same data. Figure 7.3 depicts the resulting Input Analyzer output.

Note that the state space of the corresponding random variable, generated via the expression 10 þ 20 (Beta(0.988, 1.02)) is S ¼ [10, 30] rather than the standard

Figure 7.3 Best-fit beta distribution for the repair time data of Table 7.1.

Input Analysis 133 state space S ¼ [0, 1]. Thus, the beta distribution was transformed from the interval

[0, 1] to the interval [10, 30] by a linear transformation in the expression above. In a similar vein, Figure 7.4 depicts the results of fitting a gamma distribution to

a sample of manufacturing lead-time data (not shown). Note the Square Error field appearing in each summary of the distribution fit. It provides an important measure, e 2 , of the goodness-of-fit of a distribution to an empirical data set, defined by

j ¼1

where J is the number of cells in the empirical histogram, ^ p j is the relative frequency of the j-th cell in the empirical histogram, and p j is the fitted distribution's (theoretical)

Figure 7.4 Best-fit gamma distribution for a sample of lead time data.

134 Input Analysis

Figure 7.5 Fit All Summary report for a sample of lead-time data.

probability of the corresponding interval. Obviously, the smaller the value of e 2 is, the better the fit. Finally, suppose the analyst would like the Input Analyzer to determine the best-fit distribution over all distribution classes supported by Arena as well as the associated parameters. To this end, the Fit All option is selected from the Fit menu. In this case, the Fit All Summary option may be selected from the Window menu to view a list of all

classes of fitted distributions and associated square errors, e 2 , ordered from best to worst (see Figure 7.5).