## Error Analyses

The purpose of error analyses is to identify all critical sources of error and be in a position to control the magnitudes of such errors in all cases for which they may significantly detract from system performance. An error analysis example was set forth in Chapter 8 with respect to pointing at a target in a shipboard environment.

The sequence of steps in an error analysis is as follows:

1. Identify all significant error sources.

2. Develop a computational ''model'' that relates the errors to one another.

3. Estimate the magnitudes of the significant errors.

4. Allocate error budgets, where necessary.

5. Continue to estimate, predict, and control errors throughout the project.

The ultimate benefit of a formal error analysis is to assure that the system meets all requirements. The greatest leverage is obtained when this is accomplished prior to the actual building of the system. In that way, backtracking and reengineering are avoided, together with the penalties in cost and schedule that are usually involved.

Errors are often associated with the standard deviation of some error distribution. Unless otherwise specified, the systems engineer must decide how to relate the error requirement (e.g., pointing error) to the error distribution. A ''5% solution'' is often adopted, which means that the error corresponds to the two-sigma value. In more concrete terms, if the pointing error requirement was stated in terms of, for example, 1 degree, then this requirement would be associated with the two-sigma value of the error distribution. The standard deviation of that distribution would then be limited to at most 0.5 degree. If the design were to be even more rigorous, three- and four-sigma values might be used, but these choices impact the design and might be difficult to achieve.

A brief example illustrates some of the issues involved in the preceding. Let us assume that an on-line transaction processor (OLTP) requirement is stated as ''99% of the time, the system must respond to a request for service in less than or equal to seven seconds.'' After some analysis, it is concluded that:

• The error distribution in response time may be well described by the normal distribution.

• Analysis shows that the average response time has been calculated to be four seconds.

From a normal probability table, the value 0.4900 (yielding a 0.99 probability) corresponds to 2.33 sigma from the mean. This constrains the ''distance'' from seven to four seconds to be equal to 2.33 sigma, that is, 2.33 sigma is set equal to 7 - 4 = 3 seconds. From this data, we conclude that the value of sigma should be no greater than 3/2.33 = 1.29 seconds. This establishes an ''error budget'' for the further design of the system.

The same problem can be viewed somewhat differently if we keep the seven-second requirement but the 99% is not part of the stated requirement. In such a situation, we may conclude that we wish the range from the mean of four seconds to the constraint of seven seconds to correspond to the two-sigma value of the distribution. That implies that 2ct = 3 seconds and ct = 1.5 seconds. Two sigma from the mean, from the normal table, yields the value 0.4772. When this is added to the left half of the distribution, we obtain the probability of 0.5 + 0.4772 = 0.9772. Our conclusion is that having a sigma of 1.5 leads to a probability of 0.9772, somewhat smaller than the previous case described earlier. In other words, if the error budget were not exceeded, we would satisfy the required system response time 97.7% of the time.

Example. If Z = 2X + 3Y where X and Y are independent error variables, and m(X = 6, ct(X) = 4, m(Y) = 5, and ct(Y) = 7, find the mean value of the random variable Z and the allowable error variance of Z. The mean value of Zis found simply as (2)(6) + (3)(5) = 27. The allowable variance of Z is equal to (22)(16) + (32)(49) = 505.