A Simpler and More Direct Derivation of System Reliability Using Markov Chain Usage Models

Reasoning with Diagrams: Observation, Inference and Overspecificity (Plenary)

“…Assuming no prior information a i,j = b i,j = 1. By (1) and 2each arc reliability has a mean of 2 3 and a variance of 1 18 . One can easily verify that if we plug in r = 2 3 in the formulae we derived above for Examples 1 -3, we get the same single use reliability means as shown in Table 1 (see the two rows for SUR mean).…”

“…By (1) and 2each arc reliability has a mean of 2 3 and a variance of 1 18 . One can easily verify that if we plug in r = 2 3 in the formulae we derived above for Examples 1 -3, we get the same single use reliability means as shown in Table 1 (see the two rows for SUR mean). One can also verify for Example 1 that the single use reliability variance degenerates to the arc reliability variance (as there is only one arc in the path), i.e., 1 18 = 55.5555556E − 3.…”

“…In this section we illustrate a new derivation of single use reliability mean and variance that is simpler, faster, and more intuitive than the old derivation. The new derivation of the mean was published in [2], however, back then it was unclear if there existed an alternative and new derivation of the variance that is similarly simple and intuitive. This is the major contribution of this paper.…”

“…With the same Markov chain we are able to compute E(r 2 ) similarly. Now the (i, j)-th arc is associated with a new random variable (i.e., E[r 2 i,j ]) instead of E[r i,j ]. We can substitute R 1 for R 2 and compute E(r 2 ) similarly as follows, with the reasonable assumption that all r 2 i,j s are independent random variables.…”

“…In this paper we focus on the derivation of a system endto-end reliability estimate, called single use reliability (both mean and variance), driven solely by the test data without any mathematical growth assumptions, given the Markov chain usage model, and improve on an earlier analytical solution described in [8]. Our new derivation of the mean was inspired earlier and published in [2], however, it was not until recently that we figured out a new derivation of the variance that is similarly simpler, faster, more direct, and more intuitive, which arrived through a rather convoluted path. The derivation in [8] follows from the definitions of mean and variance and only first principles, which could be a little counter-intuitive to understand.…”