Reliability Engineering

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Presentation transcript:

Reliability Engineering Redundant Structure and Standby Units

Cold Standby, Perfect Switching, With Repairs

Possible States of a 2-Unit System with Cold Standby and Perfect Switching Unit A Unit B 4 O S 3 F 2 1

State Space Diagram 4 3 2 1

State Equations

Eliminating the Failed State

Laplace Transform Substitute s=0 Note that

Solution

Mean Time to Failure

Mean Time to Failure Take Laplace transform of R(t) Substitute s=0

Mean Time to Failure

Cold Standby, Perfect Switching, With Repairs, A Main Operating Unit

Possible States System Unit A (Main Unit) Unit B 4 O S 3 F 2 1

State Space Diagram 4 3

State Equations Where

Steady State Probabilities

Availability and Unavailability

Eliminate Failed State from State Equations Where

Treating State 0 as An Absorbing State Take Laplace transform and let s=0 Solution

Mean Times to Failure and to Repair Mean time to failure Mean time to repair

Cold Standby, Imperfect Switching, With Repairs, A Main Operating Unit

State Space Diagram 4 3

Steady State Probabilities

Availability and Unavailability

Mean Time to Failure

Partly-Loaded Standby, Perfect Switching, With Repairs, A Main Operating Unit

Possible States of a 2-Unit System with Partly-Loaded Standby and Perfect Switching Unit A Unit B 4 O S 3 F 2 1

State Space Diagram 4 3 1

Steady State Probabilities

L Spares, With Replacements and Repairs

State Space Diagram 2 2j 2L 1

Notation State 2j (j = 0, 1, …,L): A total of j spare units are in a repair queue, and (L-j) spares are normal. A failed unit in the system is being replaced by a normal spared unit, the system is working. State 2j+1 (j = 0, 1, …, L-1): A total of j spare units are in a repair queue, and (L-j) spares are normal. A failed unit in the system is being replaced by a normal spared unit, the system does not work. State 2L+1: All spares are in a repair queue. A failed unit in the system is under priority repair. This is a type of quasi-replacement.

Notation λ: Constant failure rate μ: Constant repair rate ε: Constant replacement rate

Steady-State State Equations

Steady-State Availability