# Questions: A Production Line Has Three Machines

*Register to read the introduction…*A production line has three machines A, B, and C, with reliabilities of .99, .96, and .93, respectively. The machines are arranged so that if one breaks down, the others must shut down. Engineers are weighing two alternative designs for increasing the line’s reliability. Plan 1 involves adding an identical backup line, and plan 2 involves providing a backup for each machine. In either case, three machines (A, B, and C) would be used with reliabilities equal to the original three. j. Which plan will provide the higher reliability? k. Explain why the two reliabilities are not the same. l. What other factors might enter into the decision of which plan to adopt?

*…show more content…*

In #1 the system will fail if any one original and any one backup fails. In #2 the system will fail only if a component and its backup fails. c. Space for a line versus space for individual backups, ease of shifting to backups when needed, possible cost differences. 8. Refer to the previous problem. m. Assume that the single switch used in plan 1 is 98 percent reliable, while reliabilities of the machines remain the same. Recalculate the reliability of plan 1. Compare the reliability of this plan with the reliability of plan 1 calculated in solving the original problem. How much did the reliability of plan 1 decrease as a result of a 98 percent reliable switch? n. Assume that the three switches used in plan 2 are all 98 percent reliable, while reliabilities of the machines remain the same. Recalculate the reliability of plan 2. Compare the reliability of this plan with the reliability of plan 2 calculated in solving the original problem. How much did the reliability of plan 2 decrease? 8. a. RL1 = .8839 RL2 = .8839 FL1 = .1161 FL2 = .1161 If the switch is 100% reliable: Rsystem = 1 – (.1162)2 = .98652 If the switch is 98% reliable and the switch fails: Pfailure = (FL1)(Fswitch)(RL2) Pfailure = (.1161)(.02)(.8839) = .0021 Therefore,

*…show more content…*

A Web server has five major components that must all function in order for it to operate as intended. Assuming that each component of the system has the same reliability, what is the minimum reliability each one must have in order for the overall system to have a reliability of .98? 10. Repeat Problem 9 under the condition that one of the components will have a backup with a reliability equal to that of any one of the other components. 11. Hoping to increase the chances of reaching a performance goal, the director of a research project has assigned three separate research teams the same task. The director estimates that the team probabilities are .9, .8, and .7 for successfully completing the task in the allotted time. Assuming that the teams work independently, what is the probability that the task will not be completed in time? 12. An electronic chess game has a useful life that is exponential with a mean of 30 months. Determine each of the following: o. The probability that any given unit will operate for at least (1) 39 months, (2) 48 months, (3) 60 months. p. The probability that any given unit will fail sooner than (1) 33 months, (2) 15 months, (3) 6 months. q. The length of service time after which the percentage of failed units will approximately equal (1) 50 percent, (2) 85 percent, (3) 95 percent, (4) 99 percent. 13. A manufacturer of programmable calculators is attempting to determine a reasonable free-service period for a model it