Many real-world systems (e.g., electric power transmission, transportation, telecommunications, and manufacturing systems) can be regarded as capacitated-flow networks, whose arcs have independent but random capacities. A team of Chinese university professors '

investigated the reliability of several flow networks in the journal Networks (May 1995). One network examined in the article, and illustrated below, is a bridge network with arcs a,, a,, a3, a,, a,, and a,. The probability distribution of the capacity x for each of the six arcs is provided in the table on p. 180.

a. Verify that the properties of discrete probability distributions are satisfied for each arc capacity distribution.

b. Find the probability that the capacity for arc

a, will exceed 1.

c. Repcat part b for each of the remaining five arcs.

d. One path from the source node to the sink node is through arcs a,and a,. Find the probability that the system maintains a capacity of more than 1 through the a,-a, path. (Recall that the arc capacities are independent.)