Estimating a collision probability in Software

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Estimating a collision probability
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For a le as used in Example 3-10 (Chap 3-5-3) n = 1000, m = 1500, we nd with Eq 6-18 P 1u = 049, and expect o = 265 with a of 258 This value can be compared with observed values from operational measurements
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When the buckets contain more than one record (Bfr > 1), the above computation can be carried out using the cumulative Poisson probability for the records 1 to Bfr being in the buckets and the records from Bfr + 1 to n over owing Estimates of the mean total accesses 1 + LcI are used in Knuth73S (page 535) to compute the length for an unsuccessful fetch leading to an insertion to complement Exanple 6-5 The value of 6-21 P 1u = n/m LcI so that the case of Example 6-7 corresponds to LcI = 1000/1500 049 = 018 6-1-6 Other Distributions When a distribution does not seem uniform, normal, exponential, or Poisson-like to an acceptable extent, the behavior may follow one of many other functions which are available to describe distributions An important reciprocal distribution, similar in shape to a negative exponential, is presented by Zipf s law (Eq 13-6) This distribution is used to describe activity ratios of records and type/token ratios of words in text
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Sec 6-1
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Statistical Methods
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Zipf s law can be viewed as a generalization of Heising s rule:
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80 percent of accesses to a le address 20 percent of its records
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An example of its use is given in Chap 14-3-2 When requests arrive from users at terminals, queues can form due to con icting requests of independent users The requests from each single user are mutually dependent, so that the total set of requests is more complex than a Poisson distribution This behavior is modeled well by Erlang distributions; they are presented in Sec 6-4-1 Some cases where the distributions are not simple can be solved by algebraic transformations of the observations until some t to a known distribution is obtained Reciprocal, logarithmic, and hyperbolic functions are used frequently Observations may also be broken down by di erencing or decomposed into sums or products of functions Di erentiation or taking successive di erences helps to remove the e ect of aggregation if the observations are some form of cdf A visual display of the transformed data is helpful to rapidly establish the bene t of a transformation Any transformations are best based on an understanding of the underlying phenomena, so that we have some model of what is happening In the examples shown earlier we tried to nd a causal mechanism; for instance, utility customers pay in regular cycles and the hospital patients in Fig 6-3 pay according to individual pressures If such an empirical model ts the observations well, it can be used with some con dence for extrapolations and the design of databases that work in general and over a long term The algebraic transformations of the observations based on the simple patient-payment model were adequate to exploit the distribution for e ective direct access If transformations make observed data tractable, but the underlying model which generates the observations is still not understood, then much care has to be taken in applying the results The use of many variables or of complex transformations to describe observations is apt to generate very misleading results when the values obtained after the t for these variables are used to extrapolate the behavior of new systems Empirical functions based on a few reasonable variables can be used for interpolation, since this procedure is less risky Since there is a degree of arbitrariness in assigning a transformation function, subsequent testing for validity should be done with fewer degrees of freedom df
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If a distribution appears lopsided or skewed, it is wise to analyze the relationships being considered for nonlinearities It may be that the parameter being measured is some function which distorts the behavior of parameters of the model at a more basic level The skewed distribution in Fig 6-1 occurred because the blocking factor, Bfr, was described for a xed number of variable-length records per block The underlying measure, the distribution of record lengths or the inverse, 1/Bfr, had a normal distribution
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