The reliability of two-terminal parallel-series networks subject to two kinds of failure

Two-terminal parallel-series networks formed from m identical and independent components may be subject to two kinds of failure. Any component can either operate or idle and so can fail to operate or ‘fail to idle’. For any value of q, the proportion of failures which are ‘failures to idle’, the optimum two-terminal parallel-series network formed from m components is taken to be that network which maximises the reliability. Considering all values of q, for 0 < q < 1, the group of networks, which are optimum for some value of q, is found for values of m up to 5. Other criteria used for finding optimum networks are discussed, such as maximising the expected time to failure (E.T.T.F.). Also the special limiting cases as c, the reliability of a component, tends to zero and unity are examined. Using the result that if a network has a probability of failure which is a convex combination of the probabilities of failure of other networks then that network is not an optimum network, the optimum groups of networks were found in the cases when m = 6, 7 and 8 using linear programming and graph plotting techniques, for the limiting case as c tends to zero.