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We will evaluate each of these cases using probabilities of the events which lead to their occurrence The following notation is chosen to de ne the more likely condition of each event: P d : The current record is in a primary data block = 1 Pov P b : The successor record is in the same block = 1 1/Bfr P m : There has been no insertion into the block = 1 Bfr Pov P c : The next block is in the same cylinder = 1 1/ P l : The current over ow record is not the last of the chain = 1 1/Lc These values can now be used for the cases shown in Fig 3-12
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Figure 3-12 indicates the conditions based on Fig 3-11 By applying the appropriate combinations of probabilities to their costs, we obtain for the cases considered TN = (P d)(P b)(c)+ (P d)(1 P b)(P m)(P c)(r + btt)+ (P d)(1 P b)(P m)(1 P c)(s + r + btt)+ (P d)(1 P b)(1 P m)(s + r + btt)+ (1 P d)(P l)(r + btt)+ (1 P d)(1 P l)(s + r + btt) / case a / / case b / / case c / / case d / / case e / / case f /
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The seek terms in cases d and f, marked with a , disappear when the over ow areas are kept on the same cylinders as the corresponding data blocks The probabilities P m and P l can be rewritten in terms of n and o using Eqs 3-35 and 3-36
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Equation 3-40 was obtained by a detailed case analysis, but as is typical, the result can be made practical by consideration of the relative importance of the terms If we neglect the memory search time, c, for records within the block, if cylinder seeks can be ignored (ie, is large, so that the value of P c 1), and if the over ow areas are on the same cylinders as the data blocks (s = 0, so that cases b and d, as well as e and f combine), then TN 1 Pov n + o Bfr + Pov (r + btt) = (r + btt) Bfr (n + o )Bfr 3-41
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We note that the chain length Lc does not a ect this approximation for TN , but over ows still increase the estimate Adding a record to the le will always cause an addition to the over ow chain, either because of push-through or because the new record follows serially a record already in the over ow chain Each insertion requires the reading and rewriting of a predecessor data or over ow block, since a pointer will have to be inserted or changed and also a READ and REWRITE of the over ow block for the pushed or inserted record The probability of using the same over ow block for both is small if records are randomly inserted into a large data le We avoid a detailed case analysis now by using the previous result for TF and making some further assumptions The fetch time for the predecessor is equal to TF , the over ow block is on the same cylinder and requires r +btt to be reached, and each REWRITE will take one revolution TRW = 2r (given the conditions of Eq 2-30), so that 3-42 TI = TF + TRW + r + btt + TRW = TF + 5r + btt
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The e ect of the length of the over ow is incorporated into TF Note that the index is not a ected when a record is inserted The cost of insertion is only due to appending the record to the chain and linking it to the predecessor record Alternative issues of chain management, discussed earlier, may be considered An updated record of equal size and identical key can be placed into the place of the previous version of the record, so that the process can be evaluated as a fetch followed by a REWRITE with a cost of TRW : in place 3-43 TU = TF + TRW = TF + 2r