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There are numerous mathematical models for these types of distributions. Four of these models cover the overwhelming majority of user delay scenarios: Linear or uniform distribution Normal distribution Negative exponential distribution Double hump normal distribution
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Linear or Uniform Distribution
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A uniform distribution between a minimum and a maximum value is the easiest to model. This distribution model simply selects random numbers that are evenly distributed between the upper and lower bounds of the range. This means that it is no more likely that the number generated will be closer to the middle or either end of the range. The figure below shows a uniform distribution of 1000 values generated between 0 and 25. Use a uniform distribution in situations where there is a reasonably clear minimum and maximum value, but either have or expect to have a distinguishable pattern between those end points.
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Figure 13.3 Uniform Distribution
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Normal Distribution
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A normal distribution, also known as a bell curve, is more difficult to model but is more accurate in almost all cases. This distribution model selects numbers randomly in such a way that the
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frequency of selection is weighted toward the center, or average value. The figure below shows a normal distribution of 1000 values generated between 0 and 25 (that is, a mean of 12.5 and a standard deviation of 3.2). Normal distribution is generally considered to be the most accurate mathematical model of quantifiable measures of large cross-sections of people when actual data is unavailable. Use a normal distribution in any situation where you expect the pattern to be shifted toward the center of the end points. The valid range of values for the standard deviation is from 0 (equivalent to a static delay of the midpoint between the maximum and minimum values) and the maximum value minus the minimum value (equivalent to a uniform distribution). If you have no way to determine the actual standard deviation, a reasonable approximation is 25 percent of (or .25 times the range) of the delay.
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Figure 13.4 Normal Distribution
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Negative Exponential Distribution
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Negative exponential distribution creates a distribution similar to that shown in the graph below. This model skews the frequency of delay times strongly toward one end of the range. This model is most useful for situations such as users clicking a play again link that only activates after multimedia content has completed playing. The following figure shows a negative exponential distribution of 1000 values generated between 0 and 25.
Figure 13.5 Negative Exponential Distribution
Double Hump Normal Distribution
The double hump normal distribution creates a distribution similar to that shown in the graph below. To understand when this distribution would be used, consider the first time you visit a Web page that has a large amount of text. On that first visit, you will probably want to read the text, but the next time you might simply click through that page on the way to a page located deeper in the site. This is precisely the type of user behavior this distribution represents. The figure below shows that 60 percent of the users who view this page spend about 8 seconds on the page scanning for the next link to click, and the other 40 percent of the users actually read the entire page, which takes about 45 seconds. You can see that both humps are normal distributions with different minimum, maximum, and standard deviation values.
Figure 13.6 Double Hump Normal Distribution To implement this pattern, simply write a snippet of code to generate a number between 1 and 100 to represent a percentage of users. If that number is below a certain threshold (in the graph above, below 61), call the normal distribution function with the parameters to generate delays with the first distribution pattern. If that number is at or above that threshold, call the normal distribution function with the correct parameters to generate the second distribution pattern.