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RELIABILITY AND DIAGNOSTIC

Faculty of Electronic Engineering and

Technologies

Anna Andonova

E-mail: [email protected]


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Definition for repairable systems that

are required to operate continuously i.e. round the clock

are at any random point in time either operating or down because of failure

are being worked upon so as to restore their operation in minimum time

Possible states of the system:

operating

in repair

Definition - probability that a system is operating satisfactorily at any random

point in time t, when subject to a sequence of up and down cycles which

constitute an alternating renewal process

Combination of

reliability parameters

maintainability parameters

AVAILABILITY THEORY


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AVAILABILITY THEORY

a single equipment which is to be operated continuously

a record is kept on when the equipment is operating or down overa period of time

describe its availability as a random variable defined by

a distribution function H(A)

expected value availability is simply the average value

of the function over all possible values of the variable

a systems steady state availability ensemble of equipments

at any particular time the number of equipments that are in state 0

(available) to be NP0 NP0/N = P0N = total number of equipments and P0 = total equipment (N) in state 0


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System availability can be defined in the following ways:

Instantaneous Availability: A(t)

Mission Availability: A m(t2 - t1) average availability, AAV

Steady State of Availability: AS

Basic Concepts


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System availability

Basic Concepts

Fig. 6. The relationship between

instantaneous, mission, and

steady state availabilities as

a function of operating time


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System availability

Achieved Availability: AA

includes all

repair time (corrective and preventive maintenance time)

administrative time

logistic time

Intrinsic Availability: Ai function of the basic equipment/system design

does not include

administrative time

logistic time

Basic Concepts


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System availability

Achieved Availability: AA

includes all

repair time (corrective and preventive maintenance time)

administrative time

logistic time

Intrinsic Availability: Ai function of the basic equipment/system design

does not include

administrative time

logistic time


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The basic concepts of the Markov process

state of the system (e.g., operating, nonoperating)

state transition (from operating to nonoperating due to failure, or from

nonoperating to operating due to repair)

Markov process is defined by a set of probabilities pij which define the probability of transition from any

state i to any state j

The most important features of any Markov model

transition probability pij depends only on states i and j and is completely

independent of all past states except the last one, state I

pij does not change with time

Availability Modeling (Markov Process Approach)


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Fig. 6. The relationship betweeninstantaneous, mission, and

steady state availabilities as

a function of operating time


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Assumptions in system availability modeling utilizing the Markov process

The conditional probability of a failure occurring in time (t, t + dt) is l dt

The conditional probability of a repair occurring in time (t, t + dt) is m dt

The probability of two or more failures or repairs occurring simultaneously

is zeroThe most important features of any Markov model

Each failure or repair occurrence is independent of all other occurrences

(failure rate) and (repair rate) are constant (e.g., exponentiallydistributed)



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Single Unit Availability Analysis (Markov Process Approach)

Markov graph for single unit

S0 = State 0 = the unit is operating and available for use

S1 = State 1 = the unit has failed and is being repaired

= failure rate

= repair rate

transition matrix

the conditional probability of failure in (t, t + dt) is dt

the conditional probability of completing a repair in (t, t + dt) is dt


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For example the probability that the unit was in state 0 (operating) at time t and remained

in state 0 at time t + dt is the probability that it did not fail in time dt, or (1 - ) dt

the probability that the unit transitioned from state 0 (operating) to state 1

(failed) in time t + dt is the probability of systems failure in time dt, or dt

the probability that it was in state 1 (failed) at time t and transitioned to state 0

(operating) in time dt is the probability that it was repaired in dt, or dt

the probability that it was in state 1 (failed) at time t and remained in state 1 at

time t + dt is the probability that it was not repaired in dt, or (1 - ) dt

The single units availability


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The differential equations describing the stochastic behavior of this system canbe formed by

the probability that the system is in state 0 at time t + dt is derived from the

probability that it was in state 0 at time t and did not fail in (t, t + dt)

the probability that it was in state 1 (failed) at time t and transitioned to state 0

(operating) in time dt is the probability that it was repaired in dt, or dt

the probability of being in state 1 at time t + dt is derived from the probability

that the system was in state 0 at time t and failed in (t, t + dt); or it was in

state 1 at time t, and the repair was not completed in (t, t + dt)

the coefficients of these equations represent the columns of the transition matrix


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find the differential equations by defining the limit of the ratio

differential - difference equations

which yields

If the system was initially in operation,the initial conditions are P0(0) = 1, P1(0) = 0,

and the solutions are

(23)

1. Transforming Equation (23) into LaPlace transforms under the initial conditions

2. Simplifying (24)


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3.S

olving these simultaneously for P0(s) yields

4. Therefore

5. Taking the inverse Laplace transform

(25)


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If the system was initially failed,the initial conditions are P0(0) = 0, P1(0) = 1,

and the solutions are

As t becomes very large, Eqs. (25) and (26) become equivalent.This indicates that after the system has been operating for some

time its behavior becomes independent of its starting state.

(26)

The transient term becomes negligible when

For a mission of (t1 - t2 ) duration, the mission availability is


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The steady state availability, As , is

Therefore Eq. (26) becomes

as

The steady state availability becomes

Usually is much larger in value than , and As may be written as

The transient part decays relatively fast and becomesnegligible before

If is substantially greater than , then the transient part

becomes negligible before


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Fig.8. Single unit availability with repair


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The same technique described for a single unit canbe applied to different equipment/system reliability

configurations, e.g., combinations of series and

parallel units.

As the systems become more complex, the

mathematical manipulations can be quite laborious.

The important trick is to set up the Markov graph

and the transition matrix properly; the rest is just

mechanical.For example

For the most general case of n equipments and r repairmen where r = n,

the steady state availability, As , is


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parallel units.









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parallel units.









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Reliability vs. Maintainabilitytradeoff example

Figure 9 represents a system consisting of five major subsystems in aseries arrangement. The MTBF andMTTR of this system are

Fig.9. Block

diagram of a

series system


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Reliability vs. Maintainability

the maintenance time ratio equals

which is the sum of the maintenance ratios of the five serial subsystems

then(33)

highest maintenance time ratios

the first recourse could be the adding of a parallel redundant subsystem

to Subsystem 3

two cases may have to be considered:

1) the case where no repair of a failed redundant unit is possible until both fail

and the system stops operating

2) repair is possible while the system is operating

for a single repair crew


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The value of A3 = 0.946 of the redundant configuration corresponds to a

maintenance time ratio of

The whole system maintenance time ratio now becomes

System availability A as compared with 0.752 without redundancy is

Redundancy will not increase availability and may even reduce it,

even though it increases system reliability

To achieve gains in availability, repair of failed redundant units must be possible

while the system is operating. This is called availability with repair.


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A method of straightforward trade-off between reliability and maintainability

based on a requirement that the inherent availability of the

system must be at least A = 0.99,

the MTBF must not fall below 200 hr,

and the MTTR must not exceed 4 hr

Fig.9. Reliability-maintainability

trade-offs

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