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Catenations of records whose lengths are symmetrically distributed according to any patter will rapidly become normally distributed If the record lengths are already normally distributed, their catenations will be always normally distributed Exponential distributions occur when the probability of values of a series of event types increases or decreases rapidly as the event values increase Increasing exponentials are di cult to manage in models and in reality An example of such disconcerting behavior in a database is the number of interconnection possibilities among its elements, and hence the time required for thorough checking of a complex database Negative exponentials decay initially quite rapidly, but then reach their asymptotic value of zero slowly An example is seen in Fig 6-3 Distributions of a similar shape arise when measurements of terminal requests and of access frequencies to records of les are made These are best described by the Erlang and Zipf distributions noted in Sec 6-1-6
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Figure 6-3 shows the distribution by date of charges for hospital services which have not yet been settled Bills older than 18 months (August a year past) are either turned over to a collection agency or written o A total of 94 048 bills were outstanding The distribution has the form of a negative exponential, as shown by the curve tted to the data using visual comparison on a display screen The long tail of this distribution is handled through the imposition of the arbitrary cuto date
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Sec 6-1
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A negative exponential distribution is typically generated by a succession of mutually dependent events of the same probability In Figure 6-3 it appears that the probability of a customer paying an outstanding bill is about 20% every month Negative exponential distributions are also found when the times between random arrivals of service requests are displayed Exponential and Erlang distributions are important when queues of service requests are analyzed Poisson distributions occur when important contributing events have a small probability p This distribution is the limit of the binomial distribution for the case where the likelihood of a particular event is small The Poisson distribution becomes a valid approximation when the number of events f > 25 while f p = mean < 5 When the mean becomes large (f p > 30), the peak is su ciently away from the left boundary that a normal distribution can be used to approximate the Poisson distribution For a moderate sample size, the Poisson distribution diminishes more rapidly than the exponential to the asymptotic value of zero A Poisson distribution is typically caused by independent events, whereas an exponential distribution occurs more often where successive events are related Poisson distributions which have a mean value of less than one do not show the initial rise shown in Fig 6-1 but decline immediately and hence look similar to exponential distributions
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Example 6-1
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Using a Poisson distribution
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We wish to model over ows due to insertions in an indexed-sequential le organization for a le of 400 blocks We expected to insert 550 records between reorganizations The probability of a block being updated is only 1/400 per insertion, and f p is 550/400 = 138 With this information we know that a Poisson distribution is likely This means the distribution curve will be as shown in Fig 6-1c The intersection at freq 10 gives the relative count f r0 of 0 insertions in a block About the same number will receive two insertions (f r2 ), but most blocks (f r1 = 138f r0 ) will receive one insertion Some blocks will receive up to 6 insertions The speci c details are worked out in Example 6-5
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It can be shown that, for events which have an exponential interarrival-time distribution, the number of expected arrivals in a given time period has a Poisson distribution Skewed distributions occur whenever nonlinear transformations a ect the outcome of events A distribution as shown in Fig 6-1d can represent the number of records, Bfr, in a block Since the operation Bfr = B/R is not linear for R, the normal distribution of record sizes, R, is distorted Skewed distributions are characterized by the fact that their mean occurs at a di erent point than their median; the median is the value with an equal number of observations to either side; in Fig 6-1d the mean is at 063, but the median is 044 Yet another measure of central tendency is the mode, the position of the most frequent value, here at zero For moderately skewed distributions, (mean mode) 3(mean median) 6-1
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This relationship can provide a quick estimate of the median from the mode and the mean, which are easier to compute
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