Question

4. A system consists of three independent components connected in perallol (i.e., the system is operational if at least one component is operational). The status of each component is checked at the beginning of every day. An opcrational component will break down at the end of a day with probability 02.Its repir takes the entire next day after which the component becomes operational again. The daily cost of lost production due to non-working components is k, where k is the number of non-working components on that day. The system starts with all three components operational Pind the long-run expected average cost of lost production per day. 25 points) COS

Answer 1

Let X be the number of non-working components on a particular day

P(X = 0) = (1 - 0.2)³ = 0.512

P(X = 1) = 3C1 * 0.2 * 0.8² = 0.384

P(X = 2) = 3C2 * 0.2² * 0.8 = 0.096

P(X = 3) = 0.2³ = 0.008

Cost of lost production when X is 0 = 0

Cost of lost production when X is 1 = 1 * e^2*1 = 7.389

Cost of lost production when X is 2 = 2 * e^2*2 = 109.196

Cost of lost production when X is 3 = 3 * e^2*3 = 1210.286

Expected average cost of lost production per day = 0 * 0.512 + 7.389 * 0.384 + 109.196 * 0.096 + 1210.286 * 0.008 = 23 (ans)