Performance distribution of a fault-tolerant system in the presence of failure correlation

IIE Transactions, June, 2006 by Gregory Levitin, Min Xie

It follows from Proposition 2 that Equation (20) is algebraically equivalent to Equation (5). Thus, the operator that is Equation (20) can be realized numerically by handling triplets ([a.sub.f],[g.sub.f], k) for each term of the function [H.sub.[phi]](z). For each combination of two terms ([a.sub.f],[g.sub.f], k) and ([a.sub.e],[g.sub.e], h) from [H.sub.[phi]](z) and [H.sub.[pi]](z) respectively the corresponding term of [H.sub.[phi][union][pi]](z) takes the form ([a.sub.f][a.sub.e],[g.sub.f][union][g.sub.e], k h). Different terms of [H.sub.[phi][union][pi]](z) that have the same g and k can be collected by summing their coefficients a.

Note that any triplet ([a.sub.f],[g.sub.f], k) in [H.sub.[phi]](z) corresponds to the term [a.sub.f] [[product].sub.s=1.sup.Y] [x.sub.s.sup.[g.sub.f](s)] in the multi-linear form [L.sub.[phi],k](X). If the function [L.sub.{1,..., j},M](X) is represented by the triplets ([a.sub.f.sub.1], [g.sub.f.sub.1], M),..., ([a.sub.f.sub.n], [g.sub.f.sub.n], M), the probability [p.sub.j] can be obtained as:

[p.sub.j] = [n.summation over (k=1)][a.sub.[f.sub.k]][Y.[product].[s=1]][q.sub.s.sup.[g.sub.f.sub.k](s)]. (21)

The operator of Equation (20) over the UGF representing functions [H.sub.[phi]](z) and [H.sub.{i}](z) takes the following form:

[H.sub.[phi][union]{i}](z) = [H.sub.[phi]](z) [cross product] [H.sub.{i}](z) = [K.summation over (k=0)][[F.sub.k].summation over (f=1)][a.sub.f][z.sup.[g.sub.f],k] [cross product] ([z.sup.[g.sub.i],1] [z.sup.0,0] - [z.sup.[g.sub.i],0]) = [K.summation over (k=0)][[F.sub.k].summation over (f=1)][a.sub.f][z.sup.[g.sub.f][union][g.sub.i],k 1] [K.summation over (k=0)][[F.sub.k].summation over (f=1)][a.sub.f][z.sup.[g.sub.f],k] - [K.summation over (k=0)][[F.sub.k].summation over (f=1)][a.sub.f][z.sup.[g.sub.f][union][g.sub.i],k]. (22)

The UGF technique presented in this section allows us to obtain the component performance distribution using the numerical realization of the algorithm, presented in Section 4.2. The numerical procedure is as follows:

Step 1. Determine the function [H.sub.{i}](z) for each version of the component according to Equation (18).

Step 2. Define [H.sub.[null]](z) = [z.sup.0,0].

Step 3. For j = 1, 2,..., N:

3.1. Assign [p.sub.j] = 0.

3.2. Obtain [H.sub.{1,..., j}](z) = [H.sub.{1,..., j-1}](z) [cross product] [H.sub.{j}](z) using Equation (22).

3.3. For any term [a.sub.f][z.sup.[g.sub.f],M] add [a.sub.f] [[product].sub.s=1.sup.Y] [q.sub.s.sup.[g.sub.f](s)] to [p.sub.j] and remove the term from [H.sub.{1,..., j}](z)

Step 4. Determine the system performance distribution as ([p.sub.j], t(j)) for M [less than or equal to] j [less than or equal to] N.

An example of the numerical procedure is presented in Section 7.2.

6. Performance distribution of the entire system

Having the distributions of random execution times [[PSI].sub.c] for each component ([p.sub.ci],[t.sub.c](i)) for 1 [less than or equal to] c [less than or equal to] C, 1 [less than or equal to] i [less than or equal to] [N.sub.c] one can represent these distributions in the UGF form as follows: