Reliability of a flow network. The journal Networks periodically publishes studies on the reliability of flow networks. For example, Networks (September 2007) provided applications in mobile ad hoc and sensor networks. Consider a similar network with four activities, called arcs. The probability distribution of the capacity x for each of the four arcs is provided in the following table. Arc 1 x 0 1 2 3 p1x2 .05 .10 .25 .60 Arc 2 x 0 1 2 3 p1x2 .10 .30 .60 0 Arc 3 x 0 1 2 3 p1x2 .05 .25 .70 0 Arc 4 x 0 1 2 3 p1x2 .90 .10 0 0 Source: Adapted from J. Lin, “On Reliability Evaluation of CapacitatedFlow Network in Terms of Minimal Pathsets.” Networks, Vol. 25, No. 3, May 1995 (Table 1).

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 1 will exceed 1.

c. Repeat part b for each of the remaining three arcs.

d. Compute m for each arc and interpret the results.

e. Compute s for each arc and interpret the results.