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

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

COMPUTATION EXAMPLE & TIME HISTORY GRAPH

Suppose Table 1 represents a set of Category A and Category B TPMs, along with their hypothetical threshold and raw values for six measurement dates. From these data, what is the overall technical performance risk index? How is it changing over time?

From the data in Table 1 and Equations 9, 10, and 11, we can derive, for each measurement date, the TPM risk indices for the Category A and Category B TPMs, as well as for the overall TPM Risk Index. The results from these derivations are summarized in Table 2.

Note that TRI is a cardinal measure. This means its value is a measure of the "strength" or "distance" that the contributing TPMs are from their individual threshold performance values. A TRI equal to 0.5 is truly twice as "bad" as one equal to 0.25.

Figure 1 presents a time history trend of the TPM risk indices for the data in Tables 1 and 2. Here, the trend is good. All three TRIs are heading toward 0. This means all TPMs defined for the system are converging toward their individual threshold performance values. In practice, management should regularly produce a graphic summary such as this to monitor the extent that each risk index changes over time.

[FIGURE 1 OMITTED]

GENERAL EQUATION SUMMARY

This paper provides an approach and formalism for developing an overall set of quantitative indices that measure a performance risk, as a function of a system's (or system-of-systems') TPMs. Below are the general equations of the three principal risk indices.

Category A: [TRI*.sub.ti, A] = 1 - [([w.sub.A1] [u.sub.ti, A1] [w.sub.ti, A2] ... [w.sub.Am][u.sub.ti, Am]) / [W.sub.A]]

where [W.sub.A] = [w.sub.A1] [w.sub.A2] ... [w.sub.Am]

Category B: [TRI.sub.ti, B] = 1 - [([w.sub.B1]][v.sub.ti, B1] [w.sub.B2]][v.sub.ti, B2] ... [w.sub.Bn]][v.sub.ti, Bn]) / [W.sub.B]

where [W.sub.B] = [w.sub.B1] [w.sub.B2] ... [w.sub.Bn]

Overall Risk Index:

[TRI.sub.ti, All] = [[W.sub.A][TRI*.sub.ti, A] [W.sub.B][TRI.sub.ti, B] / W

where W = [W.sub.A] [W.sub.B]

EXTENSIONS TO SYSTEM Off SYSTEMS

This section extends the general formulation of TRI to a system that is composed of many individual systems that, when connected, provide an overall SoS capability. In this article, we use the following definition of an SoS.

DEFINITION

A system of systems is a set or arrangement of interdependent systems that are related or connected to provide a given capability, as illustrated by Figure 2. The loss of any part of the system will degrade the performance or capabilities of the whole. An example of an SoS could be interdependent information systems. While individual systems within the SoS may be developed to satisfy the peculiar needs of a given user group (like a specific Service or Agency), the information they share is so important that the loss of a single system may deprive other systems of the data needed to achieve even minimal capabilities (Chairman of the Joint Chiefs, 2003).

[FIGURE 2 OMITTED]

SYSTEM-OF-SYSTEMS TREE HIERARCHY

In Figure 2, the SoS is decomposed into its individual systems. Next, these individual systems can be further decomposed into their individual subsystems. Each element in the tree is referred to as a "node." A parent node is a node that has lower level nodes below it as its children. The top-most node represents the SoS level. The bottom leaf nodes are defined as nodes that have no children below them. For instance, in Figure 2, system 2 is a leaf node. System 1 is a non-leaf node. System 1 is a "parent node" composed of M-leaf nodes as its children. They are subsystem 11 through subsystem 1M. A parent node can also have lower-level parent nodes as its children, such as the top-most node in Figure 2. Generally, an SoS tree hierarchy should be decomposed to the level at which the contributions of individual TPMs can be directly evaluated and a TRI for that leaf node, at that level of the tree, can be computed.