The extended song rule as validators of analytical results — A note correcting “system reliability results” in a review of the literature

An important performance measure for stochastic networks is so-called “system reliability θ”, defined as the probability that the output meets demand for rework-networks with n workstations, which have positive defective rates and random capacities X _i following a discrete distribution. Many analytical results in computing θ of stochastic networks widely exist in more than 18 related publications, but their papers' calculated values are incorrect. The most recent work - the Song rule - proposed in Song [19] indicated that “in general, θ can not be expressed as “Π ⁿ _i=1 P(X _i ≥ x _i )” for some real values x _i .” Motivated by the problem that the implementation of the Song rule used to solve for θ is time-consuming for networks with rework when n is large, we propose an extended Song rule to improve the computational efficiency. The computational efficiency for extended Song rule is increased by more than 90 percent of that for the Song rule in terms of CPU time for the two examples considered in this paper. The extended Song rule is a more analytical computational-efficient validator than the song rule of computing system reliability θ.