Simulation and the Song Rule as Spotters and Validators of Analytical Results - A Note Correcting 'System Reliability Results' in a Review of the Literature

摘要

An important performance measure for stochastic networks without or with reworking is so-called "system reliability θ", defined as the probability that the output meets demand for a network with n workstations, which have positive defective rates and random capacities X_i following multinomial cdf F_(X_i)(.), i = 1, 2,..., n. Many analytical results in computing θ of stochastic networks widely exist in related publications. One critical contribution of this paper is the work indicating "common errors" from previous papers. The above-mentioned "common errors" are not subjective due to different misinterpretations; instead, they are serious substantive errors that render previously analyses meaningless. This paper first illustrates how Monte Carlo simulation is used as an initial spotter and validator of analytical results to solve for θ. Correct versions of the analytical results of computing θ for networks without and with reworking are also provided. The proposed analytical approach, named the Song Rule, is a valuable spotter and validator of the correctness of any future approaches to such "system reliability" problems. An important finding, based on the Song rule, is that "in general, θ can not be expressed as Π_(i=1)~n F_(X_i)(.)".