Stat 567:Statistical Reliability
Twenty Fourth Class (December 8, 1997)
Availability
Class Objectives:
- Show how Markov chain methods are used in reliability engineering
- Show how more general state space methods can be used in reliability
engineering
- Learn about some stochastic processes useful in reliability analysis
- Know the assumptions underlying reliability system modeling methods
- Expose the students to some mathematical techniques useful in reliability
modeling
Homework Assignments:
- Use Markov chain methods to calculate the availability of a system
with independent, identically distributed (i.i.d.) exponential up times
and i.i.d. exponential down times. Assume the the up and down times are
independent.
- Draw the state diagram for a three component system with one repair
person. Derive the steady state equations. Assume independent exponential
life and repair times.
Class Outline and Main Points:
- Reliability figures of merits for repairable systems
- Availability
- Transient
- Steady state
- Availability "unqualified" refers here to steady state availability
- Alternating "Up" and "Down" process
- Steady state availability = (Expected Up time)/(Expected Cycle time)
- Special case of ergodic theorem
- Reliability Block Diagrams (RBB) method
for calculating availability
- Two subsystems in parallel example
- Key assumption: independent maintenance
- Two subsystems in parallels
- Two repair persons
- Equivalent to assuming independent subsystem maintenance
- Availability can be calculated using RBD methods
- One repair person
- Subsystem maintenance is dependent
- Availability cannot be calculated using RBD methods
- Availability can be calculated using Markov Chain methods
- Assuming exponential distributions for
- Subsystem lifes
- Repair times
- Markov chain method for calculating availability
- State diagrams for two subsystems in parallel
- Two repair persons
- One repair person
- Steady steady-state equations
- Probability flowing in = probability flowing out as each state
- Two repair persons
- One repair person
- Solution of steady-state equations
- Conclusion
- Availability with two repair persons is higher than with one repair
person
- Transient availability
- Can only be done "easily" for simple systems
- Example: one subsystem with one repair person
- Differential equations involving time-dependent state probabilities
- Initial conditions for the differential equations
- Solution: transient state probabilities formulas
- Demonstration that transient state probabilities converge to the steady
state probabilities as time goes to infinity
- Appendix: Solving the differential equations of transient state
probabilities
- Laplace transform method
- Laplace transforms and some its simple properties involving derivatives
- Transformation of differential equations into system of linear equations
involving Laplace transforms
- Solving the linear equations involving Laplace transforms
- Inverting Laplace transforms to get the transient state probabilities
- Approximate solution method
- Steady-state availability of a two subsystem system with one
or two repair persons not assuming exponential repair times - but
still assuming exponential life times
- Unavailability with two repair persons
- Unavailability with one repair person
- Regenerative processes
- Sample path of stochastic process
- Exact solution
- Approximate unavailability with one repair person
- Approximate relationship (involving the coefficient of variation of
the repair times) of the unavailabilites with one and two repair persons
- Conclusions
- The more likely "long" repair times are the more useful it
is to have a second repair person involved
- With constant repair times two repair persons are not more efficient
than one
- With exponential repair times two repair persons are twice as efficient
as one
- With a coefficient of variation of 1.67 fund empirically for a duplex
processor two repair persons are almost four times as efficient as one
Study Questions
- What is the difference between transient availability and steady-state
availability? How are they related?
- What is one important assumption of the reliability block diagram approach
for calculating availabilities?
- What do we assume about the distributions of life and repair times
when using the Markov state diagram approach for calculating availabilities
- Why are "long tails" in the repair distribution harmful?
References
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