[Information] Consider a process of sampling the part 3 pieces at the same time

Question

[Information]
Consider a process of sampling the part 3 pieces at the same time for every hour of production. The quality engineer wants to determine if the quality of the sampled product pass or fail. The specification is the "good product" has deviation not greater than 4 micron in magnitude.
The distribution of deviation is as follows
X deviation P(x)*100 percentages
-2.5 0.88
-1.0      1.77
-0.5       2.65
0      11.50
0.5       9.73
1.0       13.27
1.5       16.81
2.0       18.58
2.5       13.27
3.0       6.19
3.5       3.54
4.0       0.88
4.5       0.88
The sample inspection results will be labeled as (1,0,0) or (1,1,1), etc. depending on the test results ("1" means good product)

[Question]

Consider the pervious problem. Suppose we have four production lines. Each line an inspector will take sample and judge if the production line is in-control. In the past, the production line once found out-of-control, will be stopped and repair. The company has only 2 repair man. Only one repair man is needed to fix the problem line. The time to failure of each production line is exponential distributed with the rate of 2 times a week. This can be considered the arrival rate of the production line to repair shop. The repairing time is also exponentially distributed with the rate of 8 times a week. Let X(t) denote the number of production line in good status.

a) Draw diagram of birth and death process for X(t) with complete identification of the rates in each state

b) Compare the long run probability that these will be at least 1 production line waiting for repair

c) Compute the average production line that is in good status

d) Compute the average time in the system (W)

e) Compute the average waiting time in the system (Wq)

Explanation / Answer

3x+y=21

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