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The time with two caches is 21.2 milliseconds lower than that on an average processor, and the time with one cache is 20.2 milliseconds lower than that on an average processor. The time without a cache is 41.4 milliseconds higher than the average. This is equivalent to saying that the mean difference between a two-cache processor and a one-cache processor is 1 millisecond. Similarly, the difference between a one-cache processor and a no-cache processor is 41.4 - 20.2, or 21.2, milliseconds.
The workloads also affect the processor time required. An average workload on an average processor takes 72.2 milliseconds. The ASM workload takes 0.5 millisecond less than the average. TECO takes 8.8 milliseconds higher than the average, and so on.

21.3 ESTIMATING EXPERIMENTAL ERRORS

Having computed the model parameters, the estimated response in the (i, j)th experiment is given by

The difference between the estimated response and the measured response yij is attributed to experimental errors. In other words,

The variance of errors can be estimated from the Sum of Squared Errors (SSE):

Example 21.2 For the cache comparison study of Example 21.1, the errors in each of the 15 observations are shown in Table 21.3. To see how the entries are computed, consider the first experiment (with ASM workload on a two-cache processor). The estimated processor time is

TABLE 21.3 Error Computation for the Cache Comparison Study

Workload Two Caches One Cache No Cache

ASM 3.5 3.5 -7.1
TECO 0.2 -0.8 0.6
SIEVE -4.5 -5.5 9.9
DHRYSTONE -0.5 1.5 -1.1
SORT 1.2 1.2 -2.4

The measured processor time is 54 milliseconds. The difference 54 - 50.5 = 3.5 is the error. The sum of squared errors is

SSE = (3.5)2 + (0.2)2 + ... + (-2.4)2 = 2368.00

21.4 ALLOCATION OF VARIATION

As in the case of one-factor designs and in 22r designs, the total variation of y in a two-factor design can be allocated to the two factors and to the experimental errors. To do so, we square both sides of the model equation and add across all observations. The cross-product terms cancel, and we obtain

SSY = SS0 + SSA + SSB + SSE

where various sums of squares have been appropriately placed below their corresponding terms. The total variation (SST), as before, is

SST = SSY - SS0 = SSA + SSB + SSE

Thus, the total variation can be divided into parts explained by factors A and B and an unexplained part due to experimental errors. This equation can also be used to compute SSE, since the other sums can be easily computed. The percentage of variation explained by a factor can be used to measure the importance of the factor.

Example 21.3 For the cache comparison study of Example 21.1, the sums of squares are

The percentage of variation explained by the caches is

The percentage of variation due to workloads is

The unexplained variation is

Looking at these percentages, we conclude that the choice of caches is an important parameter in the processor design.

21.5 ANALYSIS OF VARIANCE

To statistically test the significance of a factor, we divide the sum of squares by their corresponding degrees of freedom to get mean squares. The degrees of freedoms (DF) for various sums are as follows:

SSY = SS0 + SSA + SSB + SSE

ab = 1 + (a - 1) + (b - 1) + (a - 1)(b - 1)

The errors have only (a - 1)(b - 1) degrees of freedom since only (a - 1)(b - 1) of the ab errors can be independently chosen. This is because the errors in each column should add to zero. Similarly, errors in each row should also add to zero. Once the errors in the first a - 1 columns and first b - 1 rows have been chosen, those in the last column and the last row can be computed using the constraints. The degrees of freedom for other sums can be similarly justified.

The mean squares are obtained by dividing the sum of squares by their corresponding degrees of freedom:

The F-ratio to test the significance of the factor A is

This ratio has an F distribution with a - 1 numerator degrees of freedom and (a - 1)(b - 1) denominator degrees of freedom. Thus, the factor A is considered significant at level α if the computed ratio is more than F[1-α;a-1,(a-1)(b-1)] obtained from the table of quantiles of F-variates (Tables A.6 to A.8 in the Appendix).

TABLE 21.4 ANOVA Table for Two Factors without Replications

Component Sum of Squares Percentage of
Variation
Degrees of Freedom Mean Square F-Computed F-Table

y ab
SSO = ab µ2 1
SST = SSY - SS0 100 ab - 1
A a - 1 F[1 - α,a - 1],(a - 1)(b - 1)]
B b - 1 F[1 - α,b - 1,(a - 1)(b - 1)]
e SSE = SST - (SSA + SSB) (a - 1)(b - 1)

The F-ratio for factor B, similarly, is

A convenient tabular arrangement to conduct Analysis Of Variance (ANOVA) for a two factor design without replications is shown in Table 21.4.

Example 21.4 The ANOVA for the cache comparison study of Example 21.1 is shown in Table 21.5. Notice that the F-ratio for caches is more than that obtained from the table. This reconfirms our conclusion that the choice of the cache would make a significant difference in the performance of the processor. The F-ratio for the workloads is less than that obtained from the table. Thus, in this set of experiments, the workloads did not make any significant impact on the performance. Actually, this is a real case study in which the experimenter had taken precautions in choosing workloads that had comparable run times. This helped ensure that the effect of caches, if any, is not overshadowed by that due to differences in the workloads. This is an important point that is commonly missed by inexperienced analysts.


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