Combinatorial model is a class of dependability models. Combinatorial models assume that the failures of individual components are mutually independent, and it includes reliability block diagrams, fault trees, and reliability graphs, etc. Please give an example about combinatorial dependability model and analyze its reliability.

In contrast with state-space models, combinatorial models do not enumerate all possible system states to obtain a solution. Instead, simpler approaches are used to compute system dependability measures. We present a brief overview of one of the combinatorial models (Nicol, Sanders, & Trivedi, 2004).

Reliability Block Diagrams (RBD)

Reliability block diagrams are diagrammatic methods of showing how component reliabilities relate to system reliability through the existence of ‘success paths.’ An RBD is a graphical structure with two types of nodes: blocks representing system components and dummy nodes for connections between the components. Edges and dummy nodes model the operational dependency of a system on its components. At any instant of time, if there exists a path in the system from the start dummy node to the end dummy node, then the system is considered operational; otherwise, the system is considered failed. A failed component blocks all the paths on which it appears. RBDs thus map the operational dependency of a system on its components and not the actual physical structure of the system. The best way of communicating how an RBD works is through applying it to an example system (Jackson, 2018). 

Consider the secondary cooling loop of a nuclear power plant illustrated in the figure below. The primary cooling system transfers heat from the radioactive core to a heat exchanger. The heat exchanger then transfers heat to the secondary cooling loop within the reactor’s containment vessel. The secondary cooling loop transports steam to a turbine (to generate electricity), a condenser, and then through a parallel arrangement of pumps and valves back into the containment vessel to again receive heat from the primary cooling loop.

Figure 1: A simplified schematic of a secondary cooling loop for a nuclear power plant

The RBD of the system illustrated in figure 1 above can be illustrated as in the figure below. We can immediately identify two sub-systems: one in series and one in parallel.

Figure 2: A reliability block diagram of the secondary cooling loop above

Each ‘block’ of the RBD represents a single component, a sub-system, software or human element. Each block is analyzed with reliability life models, which allows reliability engineers to use the RBD to produce a system reliability model. A RBD can indicate when a system will be functional through the success path. A success path is a line that traces the reliability block diagram from one side to the other through functional components only. If any success path exists, the system is functional. If no success path exists, there is at least one failed component that prevents at least one success path from being drawn and the system has failed (Jackson, 2018). 

The parallel subsystem in figure 2 above is so called because there are two visibly parallel success paths. This means that one of the components on each of the parallel paths needs to fail for the system to fail. 

RBDs are used to model the system logically- but not necessarily in the physical way its components are connected. Notwithstanding, they can (in many instances) be constructed to visually approximate the physical layout of the system.  For this reason, they are a popular visualization methodology. 

Consider the pumping system in the figure below. The system is required to pump water at a rate of 50 gallons per minute (GPM). The system comprises of two pumps in parallel- each of which can pump 50 GPM.

Figure 3: Pumping system

The way that failure is defined in this system means that only one pump is needed to function for the system to be functional. This means that one of the pumps is redundant, and the system is a parallel system with a corresponding RBD illustrated below:

Figure 4: 50gallon per min pumping system ‘parallel’ RBD

If we consider the same system (structurally) that is now required to pump 100GPM. This means that both pumps need to be working for the system to be functional. This also means that none of the pumps are redundant, and the system becomes a series system with a corresponding RBD shown below:

Figure 5: 100 gallon per minute pumping system ‘parallel’ RBD

When the system is required to pump 50 GPM, the RBD looks like the parallel physical layout of the pumps. When the system is required to pump 100GM, the RBD then becomes a series RBD even though the physical layout remains unchanged. 

A series system is a system comprising of a number of elements (components or sub-systems) that requires each element to be functional for the system to be functional. It is also referred to as the weakest link arrangement as the system will fail when the first component fails.

A parallel system is a system comprising of a number of elements that requires at least one element to be functional for the system to be functional. Parallel systems are the most common form of redundant systems. This is because the system will fail when all the components fail.  

In conclusion, dependability analysis involves a set of methodologies dealing with the reliability aspects of large, safety critical systems. Combinatorial methods requires a description of the system to be analyzed in terms of components and their interactions (Portinale & Bobbio, 1998). In particular, components are modeled as binary events corresponding to component up and component down respectively

References

Jackson, C. (2018). Reliability engineering and management. 

Nicol, D. M., Sanders, W. H., & Trivedi, K. S. (2004). Model-Based Evaluation: From Dependability to Security. IEEE TRANSACTIONS ON DEPENDABLE AND SECURE COMPUTING, 48-65.

Portinale, L., & Bobbio, A. (1998). Bayesian Networks for Dependability Analysis: an Application to Digital Control Reliability.