Reliability acceptance testing for new series systems

The mean time between failures (MTBF) of a newly minted series system, all of whose components exhibit wear, will tend to be much larger than the MTBF of the same system after it has become fully aged. When fully aged systems are used for the testing, acceptance tests with a criterion regarding the MTBF of a well-aged system can be based on the assumption that times between system failures are independent, with identical exponential distributions. However, these tests are shown to offer essentially no consumer protection when applied to new systems. Tests are derived which are correct when new systems are under test but the acceptance criterion refers to the MTBF of a well-aged system. The derivation uses an approximate Poisson distribution which is valid if the total number of systems on test is sufficiently large.     

Modified fictitious play

We describe a modification of Brown's fictitious play method for solving matrix (zero-sum two-person) games and apply it to both symmetric and general games. If the original game is not symmetric, the basic idea is to transform the given matrix game into an equivalent symmetric game (a game with a skew-symmetric matrix) and use the solution properties of symmetric games (the game value is zero and both players have the same optimal strategies). The fictitious play method is then applied to the enlarged skew-symmetric matrix with a modification that calls for the periodic restarting of the process. At restart, both players' strategies are made equal based on the following considerations: Select the maximizing or minimizing player's strategy that has a game value closest to zero. We show for both symmetric and general games, and for problems of varying sizes, that the modified fictitious play (MFP) procedure approximates the value of the game and optimal strategies in a greatly reduced number of iterations and in less computational time when compared to Brown's regular fictitious play (RFP) method. For example, for a randomly generated 50% dense skew-symmetric 100 × 100 matrix (symmetric game), with coefficients |a_ij| ≤ 100, it took RFP 2,652,227 iterations to reach a gap of 0.03118 between the lower and upper bounds for the game value in 70.71 s, whereas it took MFP 50,000 iterations to reach a gap of 0.03116 in 1.70 s. Improved results were also obtained for general games in which the MFP solves a much larger equivalent symmetric game. © 1996 John Wiley & Sons, Inc.