Reliability of a manufacturing network. A team of industrial management university professors investigated the reliability of a manufacturing system that involves multiple production lines (Journal of Systems Sciences & Systems Engineering, Mar. 2013). An example of such a network is a system for producing integrated circuit (IC) cards with two production lines set up in sequence. Items (IC cards)

first pass through Line 1 and then are processed by Line 2.

The probability distribution of the maximum capacity level of each line is shown in the next column. Assume the lines operate independently.

Line Maximum Capacity, x p(x)

1 0 .01 12 .02 24 .02 36 .95 2 0 .002 35 .002 70 .996

a. Verify that the properties of discrete probability distributions are satisfied for each line in the system.

b. Find the probability that the maximum capacity level for Line 1 will exceed 30 items.

c. Repeat part b for Line 2.

d. Now consider the network of two production lines. What is the probability that a maximum capacity level of 30 items is maintained throughout the network?