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# Getting Started

## A • STATISTICAL

A • STATISTICAL DISTRIBUTIONS The cumulative distribution function F(x) is piecewise linear with “corners” defined by F(x j ) = c j for j = 1, 2, . . ., n. Thus, for j > 2, the returned value will be in the interval (x j–1 , x j ] with probability c j – c j–1 ; given that it is in this interval, it will be distributed uniformly over it. You must take care to specify c 1 and x 1 to get the effect you want at the left edge of the distribution. The CONTINUOUS function will return (exactly) the value x 1 with probability c 1 . Thus, if you specify c 1 > 0, this actually results in a mixed discretecontinuous distribution returning (exactly) x 1 with probability c 1 , and with probability 1 – c 1 a continuous random variate on (x 1 , x n ] as described above. The graph of F(x) above depicts a situation where c 1 > 0. On the other hand, if you specify c 1 = 0, you will get a (truly) continuous distribution on [x 1 , x n ] as described above, with no “mass” of probability at x 1 ; in this case, the graph of F(x) would be continuous, with no jump at x 1 . As an example use of the CONTINUOUS function, suppose you have collected a set of data x 1 , x 2 , . . ., x n (assumed to be sorted into increasing order) on service times, for example. Rather than using a fitted theoretical distribution from the Input Analyzer, you want to generate service times in the simulation “directly” from the data, consistent with how they’re spread out and bunched up, and between the minimum x 1 and the maximum x n you observed. Assuming that you don’t want a “mass” of probability sitting directly on x 1 , you’d specify c 1 = 0 and then c j = (j – 1)/(n – 1) for j = 2, 3, . . ., n. Range [x 1 , x n ] Applications The continuous empirical distribution is often used to incorporate actual data for continuous random variables directly into the model. This distribution can be used as an alternative to a theoretical distribution that has been fitted to the data, such as in data that have a multimodal profile or where there are significant outliers. 155

GETTING STARTED WITH ARENA Discrete: DISCRETE(CumP 1 , Val 1 , . . ., CumP n , Val n ) (c 1 , x 1 , . . ., c n , x n ) Probability Mass Function p(x j ) = c j – c j-1 where c 0 = 0 Cumulative Distribution Function Parameters The DISCRETE function in Arena returns a sample from a user-defined discrete probability distribution. The distribution is defined by the set of n possible discrete values (denoted by x 1 , x 2 , . . . , x n ) that can be returned by the function and the cumulative probabilities (denoted by c 1 , c 2 , . . . , c n ) associated with these discrete values. The cumulative probability (c j ) for x j is defined as the probability of obtaining a value that is less than or equal to x j . Hence, c j is equal to the sum of p(x k ) for k going from 1 to j. By definition, c n = 1. Range {x 1 , x 2 , . . ., x n } 156

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