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Bimodal distributions occur when the events being measured are due to two separate, underlying phenomena An example is the response time for queries, where some can be answered by using an index and others require an exhaustive reading of the le or a subset of the le Often a frequency distribution is used to obtain an estimate of the fraction of cases exceeding a certain limit For each distribution shown a cumulative distribution function (cdf) can be obtained by summing or integrating the frequency distribution from left to right over time The height of the curve is directly proportional to the number of events occurring with values less than the corresponding value A few cdfs are shown with their source functions in Fig 6-4 Table 6-7 and Fig 6-15 use cdfs to predict the fraction of desirable or undesirable events These cumulative distributions may also appear in the system measurements, since often only the aggregate e ect is observed Di erencing or di erentiation of these functions can be used to re-create frequency distributions Skew, multiple modes, etc, are more di cult to recognize when they occur in cdfs instead of in frequency distributions
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6-1-2 Describing Distributions When the result of a measurement generates a distribution, a similarity to one of the distribution types shown will provide some clues about the process causing this distribution As a next step, some quantitative measures can be obtained Two basic parameters useful with nearly any observed distribution are: the mean xbar and the standard deviation sigma of the observations, given f samples x(1),x(2),,x(f) In order to have an unbiased estimate, these sample observations should be a random sample of the events For the standard deviation we will use the symbol in equations, but sd in program examples For any collection of observed events we can compute the mean and parameters: Mean and Standard Deviation of an array of values /* 6-1 */
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Table 6-1
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/* Mean (xbar) and Standard Deviation (sd) of array x */ xsum,xsqsum = 0; DO i = 1 TO f; xsum = xsum+x(i); xsqsum = xsqsum+x(i)**2; END; xbar = xsum/f; sd = SQRT((xsqsum-xsum**2/f)/(f-1));
The number of samples, f, should be large enough so that we can have con dence that the sample set used is representative of the events occurring within the system Any of the statistics texts referenced can be consulted for tests which provide measures of con dence A chi-square test will be used in an example in Sec 6-1-5 6-1-3 Uniform Distribution If a distribution looks at or uniform over a range of values, many estimation tasks are simpli ed A uniform distribution of events has often been assumed in examples in the earlier chapters, since the uniform distribution often gives a relatively poor, and therefore conservative result A uniform distribution of addresses of seek requests to a disk will cause a higher average seek time than all but a bimodal distribution The probability that a next record is not available in the current block is greater for a uniform distribution of requests than for any other The average cost, however, of following over ow chains is lower when update insertions were distributed uniformly and greater for nonuniform update distributions Parameters of the uniform distribution /* 6-2 */
Table 6-2
/* Parameters of the uniform distribution */ height = f/number of categories; range = MAX(x) - MIN(x);
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A uniform distribution is described by its height, that is, the frequency of occurrence of categorized events, and by its range, which has to be nite for a nite number of events as shown in the program segment in Table 6-2 To visualize the uniformity of the distribution the size and value for each category icn = 1, , number of categories is computed in Table 6-3 CEIL and FLOOR functions are used to improve the presentation of the information in a histogram
Table 6-3
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