Calculating Availability for a Time-Varying Multi-Path Network

1. Calculating Availability for a Time-Varying Multi-Path Network Dr. Richard C. Mayer Boeing Technical Fellow 27 October 2005

2. Outline • What Is Availability? • Availability for Series Architecture • Availability for Parallel (Redundant) Architecture • Availability for Time-varying Architecture • Multi-level Availability Model • Cost of Incremental Availability

3. A B B A System Availability The probability that a system, when used under stated conditions in an ideal support environment (i.e., readily available tools, spares, maintenance, personnel, etc.), will operate as required at any point in time. In other words: “I want to call from A to B now. What are my chances of getting through?”

4. MTBF MTTR MTBF MTTR t = 0 Failure #1 Failure #2 time MTBF = Mean Time Between Failures MTTR = Mean Time to Restore

5. Relationship to Reliability • Availability is not a function of time. • Reliability is a function of time. Definitions: • The probability that a system will accomplish its designated mission in a satisfactory manner. Mission duration gives the time. • The probability that a system will perform in a satisfactory manner for a given period when used under specified operating conditions. • Reliability function (Survival function): R(t) = probability that the system will be successful for at least time t • Exponential model of reliability: where q = MTBF Reliability and availability are related through the MTBF.

6. C1 C2 Ci CM Availability of Components in Series Series availability is the product of the individual availabilities. For small unavailabilities, series unavailability is just the sum of the individual unavailabilities.

7. C1 C2 CN Availability of Components in Parallel -Only One Required for Block to Be Available Redundant unavailability is the product of the individual unavailabilities plus the fraction of time spent in switching from bad components to good (which may be zero in some cases.) E.g., airplane engines, dual data streams

8. Knowing the Status of Redundant Paths Affects Availability • Is the backup component in working order? Switching to a backup component and finding that it has failed impacts availability. • How and how often should backup paths be tested? More frequent sampling reduces the probability of switching to a failed component, but increases complexity and cost. • Timely replacement/repair decreases the probability of not having redundant path when it is needed. These factors can be included in availability models to do cost/benefit analysis of a maintenance/sparing strategy.

9. C1,1 CN,1 C1 Ci CM C1,2 CN,2 C1,K1 CN,KN Availability of Arbitrary Configuration M simple blocks + N cross-strapped blocks all in series kJ = number of units in jth cross-strapped block for small unavailabilities

10. A B B A Time-Varying Configurations: Examples Path availability affected by external conditions ORCLE (DARPA) COTM • RF Downlinks • Lasercom Downlinks • NASA Mars Link • Japanese Space Agency

11. p1 C1 CU S1 CU p2 S2 C2 CU CU p3 C0 C3 CU S3 CU pN CN All units are identical pHO CHO Modeling a Time-Varying Configuration • The configuration has a different redundancy in each possible state, S1 through SN. • Each state occurs with probability pi, which is known or estimated. • The availability, Ai, for each state is worked out per the preceding models. • Add a state for switching or handover. • The system availability is a weighted average of the state availabilities:

12. Availability Allocation Tree Sources of unavailability Possible states Yellow cells give probability of each type of failure in each state, weighted by probability of being in that state. Cyan cells give sum of unavailabilities in each state. Tan cells give sum of unavailabilities by cause. Input allocations to sources (in bold)

13. Availability Allocation Tree (Expanded-1) Almost all loss of availability occurs when there is only one path, despite the fact that this happens only 7.5% of the time Input allocations to sources (in bold) • Node Decision Errors • Knowing status of redundant paths • Assess impact of bad or missing information • Assess a penalty: time to correct

14. Availability Allocation Tree (Expanded-2) • Switching state: • How often? • How long does it take? • Relying on one path during switch • (vs. hot backup at other times) Diminishing returns with greater redundancy

15. C1,1 C1,1 CN,1 C1 Ci CM C1,2 C1,2 CN,2 C1,K1 C1,K1 CN,KN C1 C2 Multi-Level Models • Each component in the previous example can be a configuration of parallel and serial components. • A model can be made to compute the availability of each top-level and lower-level component. • The lower-level availabilities become inputs for the equation for the next level up.

16. Availability Model: Top Level

17. Availability Model: Level Two

18. Cost Performance The Cost of Incremental Availability • More assets => Greater availability performance • More assets => Greater cost • Cost as an Independent Variable (CAIV): typical cost vs. performance curve How much are you willing to pay for incremental improvements?

19. Contact Information • Richard Mayer, Boeing IDS: Saint Louis • Richard.c.mayer@boeing.com • 314-232-1268