Chapter 11: Selected Quantitative Relationships (pt. 2) ISE 443 / ETM 543 Fall 2013

One critical quantitative relationship in SE is error analysis 2 443/543 – 11 A formal error analysis assures that the system meets all requirements.  when this is accomplished prior to the actual building of the system, backtracking and reengineering are avoided, along with the penalties in cost and schedule that are usually involved. 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.

Typically, a 2-sigma error requirement is applied to the error distribution For example, suppose an online transaction processor is required to respond to a request for service with a mean time of 4 seconds.  The error requirement states that “99% of the time, the system must respond to a request for service in less than or equal to seven seconds.”  Using a normal table (or norm.s.inv in Excel), we can determine σ by... Z = ____________________ σ = ____________________  On the other hand, if we use the 2-sigma error requirement and we want 7 seconds to be the upper bound, then the requirement is... σ = ____________________ and P(x < 7) = __________________ 3 443/543 – 11

Note that the error “budget” is generally defined for systems of components, each of which has it’s own error mean and sigma See the example on page 353  The system error, Z, is composed of 2 independent error variables, X and Y  f(Z) = 2X + 3Y  μ(X) = 6, σ(X) = 4  μ(Y) = 5, σ(Y) = 7  So, we can calculate the mean and standard deviation of the system error as: μ(Z) = _____________________ σ 2 (Z) = ____________________ σ(Z) = _____________________ 4 443/543 – 11

There are 2 larger examples in the book that illustrate the use of quantitative analysis in trade-offs An example involving radar detection given specified probabilities of detection and false alarms starts on page 353. An example using thresholds instead of assumed or known probabilities starts on page /543 – 11

Predictions of system reliability and availability are based on the exponential distribution Let’s look at the example on page 358 (assuming serial subsystems). We are interested in the probability that the system will survive without failure for 500 hours is :  R(t) = exp(−λt), where λ = 1/MTBF  R s = R(A)R(B) = exp(−λ a t)exp(−λ b t) = exp[−(λ a + λ b )t] 6 443/543 – 11

If the subsystems are in parallel, on the other hand … Assuming that 2 of the systems from the previous example are placed in a parallel configuration  R s = 1 − [1 − R(A)][1 − R(B)] = 1 − [1 − exp( −λ a t)][1 − exp( −λ b t)] Note that subsystems in parallel are redundant and therefore the reliability of the system is increased 7 443/543 – 11

System availability is calculated based on reliability and the mean down time See the example on page 361 … If the failure rate for a system is 0.01 failure per hour and the mean-time-to-repair distribution is uniform in the range 2 to 8 hours, what is the system availability? MTBF = _________________________ MDT = ________________________________ 8 443/543 – 11