Ananlysis
help_req
emse_6770_sp_2014_on_campus_final_exam.docDepartment of Engineering Management and Systems Engineering Management
School of Engineering and Applied Science
The George Washington University
Final Examination
EMSE 280
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 rules of academic integrity will be strictly enforced. You are not to discuss the exam with anyone. If you have questions email me and copy the class, I will respond to all so that everyone benefits. I expect an MS Word .doc or .docx file or a pdf. In the file embed pictures from the excel sheets to back up your claims. The exam is due Sunday May 11 at 5:00pm. Enjoy!
1. Consider the following system of components
1
2
3
The pdf of the component distributions are given below.
2
3/2
(ln0.25)
8
1.5
/10
3/2
1/2
10
2
1
Component A ()
2
Component B () (1/10)
1
Component C ()
96
3
Component D ()
2010
1
Component X ()
50
x
x
x
x
x
fxe
x
fxe
fxxe
x
fxe
fxxe
p
-
-
-
-
æö
-
ç÷
èø
-
=
=
=
æöæö
=
ç÷ç÷
èøèø
=
2
/200
a. Find the mission length that corresponds to a probability of mission success (that is the system continues to function) of .9
b. Approximate and plot the failure rate curve for the system and discuss what it says about the system.
2. Consider the following system of components
where the failure data for component 1 and 2 are given as (where “+” indicates a
censored test)
|
Component |
|
|
1 |
2 |
|
14.2 |
15.+ |
|
7.7 |
1.8 |
|
12.0+ |
8.0+ |
|
9.7 |
17.0+ |
|
2.3 |
5.2 |
|
4.6 |
3.5 |
|
3.1 |
18.+ |
|
10.0+ |
8.6 |
|
10.0+ |
3.9 |
|
9.4 |
6.9 |
|
7.3 |
19.0+ |
|
16.0+ |
11.0+ |
|
13.0+ |
16.0+ |
|
6 |
0.8 |
|
1.8 |
4.3 |
Unfortunately there are only two data points, failure at 2.8 and a censored value at 10 for component 3. This is not enough data, so in addition to the data, experts specified that an exponential distribution would be appropriate distribution for component 3 and specified a 3 point prior distribution for as follows
|
|
Probability |
|
0.05 |
0.5 |
|
0.1 |
0.4 |
|
0.2 |
0.1 |
What is the probability of this system surviving a mission of length 10?
3. Consider the following system of components
A
B
C
D
E
where for a fixed mission time we have the following data. For 1000 missions the following failures were observed
|
Failures of A |
11 |
|
Failures of B |
15 |
|
Failures of C |
9 |
|
Failures of D |
11 |
|
Failures of E |
7 |
|
Failures of A and B |
3 |
|
Failures of C and D |
5 |
|
Failures of C and E |
7 |
|
Failures of D and E |
5 |
|
Failures of C, D, and E |
2 |
What is the probability of system failure assuming a multivariate exponential model?
4. A component life time, T, is known to have a failure distribution (pdf) of the form
(
)
q
q
q
t
e
t
f
-
=
1
)
|
(
for t>0Since ( is unknown a prior distribution is specified as
(
)
q
b
a
a
q
a
b
q
-
+
G
=
e
g
1
1
)
(
)
(
for (>0(this is not a distribution you have seen before)
with (=2 and ( = 10
a. What does ( represent, i.e. what is your interpretation of (?
b. Based on the prior, what is a point estimate for (?
c. Derive the predictive distribution for T.
d. If 5 systems are tested until time t=10 and three failures are observed at 5, 7, and 8, what would be your estimate that a new system would survive for a mission of length 10?
NOTE: This is not a spreadsheet problem!
5. (20 points) Consider the NHPP with mean value function
M(t) = (1/()Ln(((t+1).
Given the following interarrival time data
|
5.5 |
|
1.83 |
|
2.75 |
|
17.89 |
|
3.94 |
|
14.98 |
|
3.47 |
|
9.96 |
|
11.39 |
|
19.88 |
|
7.81 |
|
14.6 |
|
11.41 |
|
18.94 |
|
25.3 |
|
75.67 |
|
82.69 |
|
31.61 |
|
129.31 |
|
47.6 |
a. What is the probability that the next arrival time will be more than 10 time units?
b. Predict the number of arrivals in the next interval of size 10 time units.
c. What is the probability of no arrivals in the next 10 time units and then more than 2 arrivals in the next 10 time units after that?
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