1. Calculating End-to-End Series and Parallel Solution-System Availability

2. page 2 End-to-End Solution Availability End-to-End Solution Availability is calculated by modeling the solution as an interconnection of components in SERIES and in PARALLEL. Use these rules to decide if components should be modeled in SERIES or in PARALLEL: – If the un-availability of one component leads to the combination of components becoming un-available, the components are operating in SERIES. – If the un-availability of one component leads to the other components taking over the operations of the un-available component, the components are operating in PARALLEL.

3. page 3 Series Availability

4. page 4 Calculating Series Availability Components X, Y and Z are considered to be operating in SERIES if the un-availability of any one results in the un- availability of the combination. The combined system is available only if components X, Y, and Z are all available. From this it follows that the combined availability is a product of the availability of the components: – (combined availability) = (X availability) x (Y availability) x (Z availability) The implication of the above equation is that the combined availability of the components in series is always lower than the availability of every one of its individual components.

5. page 5 Series Availability Example Example – (X availability) = 75% – (Y availability) = 80% – (Z availability) = 85% – (combined availability) = (X availability) x (Y availability) x (Z availability) – (combined availability) = 0.75 x 0.80 x 0.85 – (combined availability) = 0.51 – (combined availability) = 51%

6. page 6 Series Un-Availability Example Example (2 components) – (X availability) = a (X un-availability) = 1-a – (Y availability) = b (Y un-availability) = 1-b – (combined availability) = ab (combined un-availability) = 1-ab – (combined un-availability) = (1-a)+(1-b) – ((1-a)(1-b)) Proof –1-ab = (1-a)+(1-b) – ((1-a)(1-b)) – = 2-a-b – (1-a-b+ab) – = 2-a-b – 1+a+b-ab – = 1-ab

7. page 7 Series Un-Availability Example Example (3 components) – (X availability) = a (X un-availability) = 1-a – (Y availability) = b (Y un-availability) = 1-b – (Z availability) = c (Z un-availability) = 1-c – (combined availability) = abc (combined un-availability) = 1-abc – (combined un-availability) = (1-a)+(1-b)+(1-c) – ((1-a)(1-b)+(1-a)(1-c)+(1-b)(1-c)+((1-a)(1-b)(1-c))) Proof –1-abc = (1-a)+(1-b)+(1-c) – ((1-a)(1-b)+(1-a)(1-c)+(1-b)(1-c)+((1-a)(1-b)(1-c)))

8. page 8 Series Availability Calculator Try it ! – Double-click on the table and change the X, Y, or Z availability and see the new combined availability. Note: Entering an availability value of 100% is equivalent to saying that that component does not exist. So if there are only 2 components, enter 100% for one of the components to simulate it not being there.

9. page 9 Parallel Availability

10. page 10 Calculating Parallel Availability Components X, Y, and Z are considered to be operating in PARALLEL if the combination is considered unavailable only when all the components are unavailable. The combined system is available if any component is available. From this it follows that the combined availability is 1 - (the probability that all components are unavailable at the same time) : – (combined availability) = 1- ( (1 - X availability) x (1 - Y availability) x (1 - Z availability) ) The implication of the above equation is that the combined availability of components in parallel is always higher than the availability of every one of its individual components. Thus, parallelism is a powerful mechanism for making a highly reliable system from low reliability components.

11. page 11 Parallel Availability Example Example – (X availability) = 75% – (Y availability) = 80% – (Z availability) = 85% – (combined availability) = 1- ((1 - X availability) x (1 - Y availability) x (1 - Z availability)) – (combined availability) = 1- ((1 – 0.75) x (1 – 0.80) x (1 – 0.85)) – (combined availability) = 1- (0.25 x 0.20 x 0.15) – (combined availability) = 1- (0.0075) – (combined availability) = 0.9925 – (combined availability) = 99.25%

12. page 12 Parallel Availability Calculator Try it ! – Double-click on the table and change the X, Y, or Z availability and see the new combined availability. Note: Entering an availability value of 0% is equivalent to saying that that component does not exist. So if there are only 2 components, enter 0% for one of the components to simulate it not being there.

13. page 13 Parallel Availability Calculator II If all the parallel components are identical (have the same availability value) use this calculator. Try it ! – Double-click on the table and change the components’ availability and the number of identical components to see the new combined availability.