An index to measure and monitor a system-of-systems' performance risk

Defense AR Journal, Dec, 2005 by Paul R. Garvey, Chien-Ching Cho

COMPUTING TRI FOR SYSTEM OF SYSTEMS

Computing TRI--The TRI of the SoS is computed as a logical combination of the TRIs across the leaf nodes of the tree. Specifically, a [TRI.sub.ti, All] is computed for each leaf node x, in the same way presented in equation 11. Denote the value as [TRI.sub.ti, x], where the subscript x is to represent the set of all TPMs that are applicable to the leaf node x. Next, the [TRI.sub.ti, x] at all leaf nodes are combined to derive the [TRI.sub.ti, SoS] at the SoS level of the tree. To describe this process below, we further generalize the notation [TRI.sub.ti, x] to denote the TRI value for any node x, leaf or parent, in the SoS tree hierarchy and the subscript x now represents all the TPMs that are applicable to the node x, directly (as for a leaf node) or indirectly (as for a parent node).

Combining TRI for a parent node from its children (leaf or lower-level parent nodes) should be done according to the following rule. The overall TRI for a parent node k with M children (nodes k1, ..., kM) at time ti can be written as:

[TRI.sub.ti, k] = ([w.sub.k0][TRI.sub.ti, k0] [w.sub.k1][TRI.sub.ti, k1] ... [w.sub.kM][TRI.sub.ti, kM]) / ([w.sub.k0] [w.sub.k1] ... [w.sub.kM]) Eqt 12

where node k0 is an added child to the parent node k to represent the set of TPMs that are applicable across multiple or all original children of parent node k. Starting at the lowest level of an SoS tree hierarchy, Equation 12 can be used to compute the TRI for all parent nodes, as appropriate to the structure of a given SoS decomposition. Thus, the overall TRI for an SoS tree hierarchy composed of N systems (i.e., with nodes 1, ..., N as children to the top-most node of the tree) is:

[TRI.sub.ti, SoS] = ([w.sub.0][TRI.sub.ti, 0] [w.sub.1][TRI.sub.ti, 1] ... [w.sub.N][TRI.sub.ti, N]) / ([w.sub.0] [w.sub.1] ... [w.sub.N]) Eqt 13

where system 0 is an added child to the top SoS node to represent the set of TPMs that are applicable across multiple or all systems listed as children under the top node.

Suppose the system 1 parent node (k = 1) has just M = 3 subsystems (subsystems 11, 12, and 13) as its children. Besides the TPMs that are to be measured at each of the subsystems, we assume there is also a set of TPMs that are applicable across multiple or all subsystems (e.g., subsystem-to-subsystem integration or system level integration). For notational convenience, we use subsystem 10 to denote the collection of such TPMs and use [TRI.sub.ti, 10] to denote the TRI value computed on those TPMs. Then, the overall TRI of system 1 at time ti is as follows:

[TRI.sub.ti, 1] = ([w.sub.10][TRI.sub.ti, 10] [w.sub.11][TRI.sub.ti, 11] [w.sub.12][TRI.sub.ti, 12]) [w.sub.13][TRI.sub.ti, 13]) / ([w.sub.10] [w.sub.11] [w.sub.12] [w.sub.13]) Eqt 14

Clearly, if the system 1 parent node's TRI is defined solely by its children's TRI values then Equation 14 can be simplified with [w.sub.10] set equal to 0.

Other Rollup Rules--Equations 12, 13, and 14 apply a weighted average rollup rule for determining the TRI values in the SoS tree hierarchy. The rule is appropriate for a parent node when its children's performance levels are considered additive in measuring the parent node's performance level. This implies, with their assigned weights, all children's risk levels directly add to the parent node's risk level. This is probably the most common rule to use in the rollup of TRI values. Other rules may also be defined and applied accordingly. For example, referring to Figure 2 with M = 3, Equation 12 could be rewritten according to the relationship that it is considered to have among the children of the parent node system 1, as follows: