Q1. The lifetime (measured in years) of a processor is exponentially distributed, with a mean lifetime of 2 years. You are told that a processor failed sometime in the interval [4, 8] years. Given this information, what is the conditional probability that it failed before it was 5 years old?

Q2. Write the expression for the reliability R_system(t) of the series/parallel system shown in Figure 2.2, assuming that each of the five modules has a reliability of R(t).

Q3. Consider a triplex that produces a 1-bit output. Failures that cause the output of a processor to be permanently stuck at 0 or stuck at 1 occur at constant rates λ₀ and λ₁, respectively. The voter never fails. At time t, you carry out a calculation the correct output of which should be 0. What is the probability that the triplex will produce an incorrect result? (Assume that stuck-at faults are the only ones that a processor can suffer from, and that these are permanent faults; once a processor has its output stuck at some logic value, it remains stuck at that value forever).

Hint: Find the probability that a node is stuck by time t at either 0 or 1, and find Pr(stuck at 1 at time t | stuck by time t). Then find Pr(stuck at 1 at time t). Finally, compute the probability of two or more processor stuck at 1 at time t.

Q4. List the conditions under which the processor/memory TMR configuration shown in the figure below (Figure 2.9 in the chapter) will fail, and compare them to a straightforward TMR configuration with three units, in which each unit consists of a processor and a memory. Denote by Rp, Rm and Rv the reliability of a processor, a memory and a voter, respectively, and write expressions for the reliability of the two TMR configurations.

Q5. The system shown in Figure 2.24 consists of a TMR core with a single spare a that can serve as a spare only for module 1. Assume that modules I and a are active. When either of the two modules 1 or a fails, the failure is detected by the perfect comparator C, and the single operational module is used to provide an input to the voter.

(a) Assuming that the voter is perfect as well, which one of the following expressions for the system reliability is correct (where each module has a reliability R and the modules are independent).

(1) R_system = R⁴ + 4R³(1 - R) + 3R²(1 - R)²

(2) R_system = R⁴ + 4R³(1 - R)+ 4R²(1 - R)²

(3) R_system = R⁴ + 4R³(1 - R) + 5R²(1 - R)²

(4) R_system = R⁴ + 4R³(1 - R)+ 6R²(1 - R)²

(b) Write an expression for the reliability of the system if instead of a perfect comparator for modules 1 and a, there is a coverage factor c (c is the probability that a failure in one module is detected, the faulty module is correctly identified, and the operational module is successfully connected to the voter that is still perfect).

Q6. Your manager in the Reliability and Quality Department asked you to verify her calculation of the reliability of a certain system. The equation that she derived is

R_system = R_C [1 - (1 - R_A)(1 - R_B)] [1 - (1 - R_D)(1 - R_E)] + (1 - R_C) (1 - (1 - R_AR_D)(1 - R_BR_E)]

However, she lost the system diagram. Can you draw the diagram based on the expression above?