Department of Engineering Management and Systems Engineering Management

School of Engineering and Applied Science

The George Washington University

Examination IV

EMSE 4770

Directions: Answer every question. For full credit, clearly show ALL work and indicate how the solution was determined. The exam should reflect you own work. The exam is due to me by email by 12 noon tomorrow.

1. (20 pts) Consider the following system

where for a fixed mission time, the following states for the 4 components have been assessed as (where 0 indicates that the component functions for the entire mission length and 1 indicates that the component fail before the mission time)

1. What is the probability of system failure during the mission?

1. Which component is the weakest in the system?

1. What is the probability that components 2 and 3 fail during the mission?

1. What is the probability that the system will fail during the mission given that

components 2 and 3 fail during the mission?

1. Given the system fails, what is the probability that components 2 and 3 has failed?

1. Calculate the importance measure I_B for component 3?

2. (20 points) Consider the following system WHICH IS NOT A K-OUT-OF-N system

The system seems to have common cause failures as when 1000 systems were tested, the following failure data was obtained

6. Using an approach similar to what we did with k-out-of-n systems, develop an expression for system failure in terms of Q_i, i=1,….,6.

1. Now calculate the failure probability using the Alpha Factor Model

3. (20 points) Consider the following event tree

where A = a·b + a·c and B = b·c, C = a+b·c. Given the following table for component states a,b,c

1. Determine the split fractions.

1. Determine the probability of each scenario.

1. Determine the probability of system failure.

4. (20 points) Consider the following system of components

Write down the expression for system reliability function using a Marshal-Olkin Model where only individual failure rates and group failure rates for {2,3}, {2,4},  and {2,3,4} are nonzero (that is there is 0 chance of any other group failure).

5. (20 points) Consider a system with cut sets {A,B,C}, (A,C,D} and {B,C,D. Assuming components are independent.

1. Using F_A(t), F_B(t), F_C(t), F_D(t), as the CDF for components A, B, C, and D,  respectively and assuming component independence, determine the system CDF in its most reduced form.

1. What is the probability that at the end of a 10 year mission, component A and B have failed but component C and D are operational.

1. If all components have an exponential time to failure with l_A=.1, l_B=.2,

l_C=.5, l_D=.4, and failure is in terms of years what is the probability of successfully completing a mission of length 10 years?

1. Given that the system has survived for 10 years, what is the probability of failure before year 15?