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ENGINEERINGRELIABILITY

INTRODUCTION

PREVENTIVEMAINTE-NANCE

CORRECTIVEMAINTE-NANCE

AVAILABILITY

SUMMARY

ENGINEERING RELIABILITYMAINTAINED SYSTEMS

Harry G. Kwatny

Department of Mechanical Engineering & MechanicsDrexel University


INTRODUCTION



AVAILABILITY

SUMMARY

OUTLINE

INTRODUCTION

PREVENTIVE MAINTENANCE

CORRECTIVE MAINTENANCE

AVAILABILITY

SUMMARY


INTRODUCTION



AVAILABILITY

SUMMARY

MAINTAINED UNITS

Maintenance can be employed in two different manners:

I Preventive maintenanceI employed before failure occursI goal is to improve reliabilityI What is the effect of preventive maintenance on the reliability

function?

I Corrective maintenanceI employed after failure occursI goal is to return item to service as quickly as possibleI How frequently do we anticipate repairs and how much inventory of

replacement parts is required?

Several concepts clarify maintenance issues. Notably:

I AvailabilityI A system is not always available to perform its function – it has

downtime.I Uptime/downtime captures the essence of reliability/unreliability.I Can downtime be minimized? Can downtime be avoided when the

system is most needed?


INTRODUCTION



AVAILABILITY

SUMMARY

IDEALIZED MAINTENANCE

I Consider a system whose reliability withoutmaintenance is R(t).

I Suppose maintenance is performed at regular timeintervals TM. Denote the reliability of the maintainedsystem by RM(t).

I Since no maintenance is performed for 0 ≤ t < TM,

RM (t) = R (t) , 0 ≤ t < TM

I Assume ideal maintenance so that the system isrestored to as-good-as-new condition. Thus,

RM (t) = R (TM) R (t − TM) , TM < t < 2TM...

RM (t) = R (TM)n R (t − nTM) , nTM < t < (n + 1) TM


INTRODUCTION



AVAILABILITY

SUMMARY

MTTF - MAINTAINED UNITS

For a system with ideal preventive maintenance:

MTTF =

∫ ∞0

RM (t) dt =∑∞

n=0

∫ (n+1)TM

nTM

RM (t) dt

MTTF =∑∞

n=0

∫ (n+1)TM

nTM

R (TM)n R (t − TM) dt =∑∞

n=0R (TM)n

∫ TM

0R (τ) dτ

The infinite series evaluates to∑∞

n=0R (TM)n =

11− R (TM)

Consequently,

MTTF =

∫ TM0 R (t) dt

1− R (TM)


INTRODUCTION



AVAILABILITY

SUMMARY

SYSTEMS WITH CONSTANT FAILURE RATE

I Ideal preventive maintenance has no affect on thereliability of systems with constant failure rate!

I RecallR (t) = e−λt

I Thus,

RM (t) =(

e−λTM)n

e−λ(t−nTM) = e−λt = R (t)


INTRODUCTION



AVAILABILITY

SUMMARY

SYSTEMS WITH VARIABLE FAILURE RATE

I Consider a system with a Weibull distribution

R (t) = e−(βt)α

I With ideal preventive maintenance

RM (t) == e−n(βTM)α

e−(β(t−nTM))α

, nTM < t < (n + 1) TM

I ComputeRM (nTm)

R (nTm)= e−n(βTM)α+(nβTM)α

I Preventive maintenance improves reliability if nα−1 − 1 > 0 anddegrades reliability if nα−1 − 1 < 0

I Reliability improves if α > 1 (aging) and degrades if α < 1 (burn-in).


INTRODUCTION



AVAILABILITY

SUMMARY

CORRECTIVE MAINTENANCE – PROBLEM SETUP

I Consider a situation in which a failed unit can berepaired or replaced, so that operation can proceedindefinitely.

I We would like to anticipate the number of repairs thatwill be required in a given time period.

I Assume a constant failure rate λ0.

p (n|t) , probability of exactly n repairs at time t

@t = 0 : p (0|0) = 1, p (n|0) = 0, n = 1, 2, . . .

@t = T > 0 :∑∞

n=0p (n|T) = 1


INTRODUCTION



AVAILABILITY

SUMMARY

MAINTAINED UNITS – PROBABILITY OF ‘0’FAILURES

I The system operates for time period t + dtI At most one failure can occur in the infinitesimal period

dtI 0 failures can occur at t + dt requires 0 failures at time t

and 0 failures during dt

p (0|t + dt) = p (0|t) (1− λ0dt)p (0|t + dt)− p (0|t) = λ0dt

dp (0|t)dt

= −λ0p (0|t)

p (0|t) = e−λ0t


INTRODUCTION



AVAILABILITY

SUMMARY

MAINTAINED UNITS – PROBABILITY OF nFAILURES

I The system operates for time period t + dtI At most one failure can occur in the infinitesimal period

dtI n failures can occur at t + dt in two ways

I n failures at time t and 0 failures during dtI n− 1 failures at time t and 1 failures during dt

p (n|t + dt) = p (n|t) (1− λ0dt) + p (n− 1|t)λ0dt

dp (n|t)dt

= −λ0p (n|t) + λ0p (n− 1|t)

p (n|t) = λ0e−λ0t∫ t

0p (n− 1|τ)e−λ0τdτ


INTRODUCTION



AVAILABILITY

SUMMARY

MAINTAINED UNITS – RECURSIVE SOLUTION

We sequentially solve these equations for n = 0, 1, 2, . . .

p (0|t) = e−λ0t

p (1, t) = (λ0t) e−λ0t

p (2|t) =[(λ0t)2 /2

]e−λ0t

...p (n|t) = [(λ0t)n /n!] e−λ0t

This is the Poisson distribution

p (n;λ) =λn

n!e−λ, n = 0, 1, 2, . . .

with λ = λ0t.


INTRODUCTION



AVAILABILITY

SUMMARY

MEAN TIME BETWEEN FAILURES

From the Poisson distribution p (n;λ), we can compute theexpected number of failures in the time period (0, t]

µN = E {N (t)} =∑∞

n=0np (n;λ0t) = λ0t

The variance is

σ2N = E

{(N − µN)2

}= λ0t

The mean time between failures is

MTBF =tµN

=1λ0

Finally, we may wish to compute the probability that thenumber of failures in time t is greater than the number n

P (N > n) =∑∞

k=n+1

(λ0t)k

k!e−λ0t = 1−

∑n

k=0

(λ0t)k

k!e−λ0t


INTRODUCTION



AVAILABILITY

SUMMARY

DOWNTIME

I The downtime due to a failure is the sum of severalelements: diagnosis time, part(s) access time, repairtime, etc. Thus, we consider downtime, TD to be arandom variable.

I The mean down time is

MDT =∫ ∞

0τ f TD

(τ) dτ

Mean downtime is also called mean time to repair,MTTR.

I Three distributions are commonly used for downtime:I ExponentialI NormalI Lognormal


INTRODUCTION



AVAILABILITY

SUMMARY

EXAMPLE: DOWNTIME

I The downtime associated with a particular failure isassumed to have an exponential distribution.

I The MDT is known to be 5 hours. Hence the repair rateis

µ =1

MDT= 0.2 hr−1

I The probability that the repair will take longer than 5hours is

P (TD > 5) = e−0.2×5 = 0.368


INTRODUCTION



AVAILABILITY

SUMMARY

AVAILABILITY

I Availability, A(t), is the probability that the system isperforming properly at time t. This is sometimes called‘point’ availability.

I The (average) interval availability over the time interval(t1, t2), Aav(t1, t2), is

Aav (t1, t2) =1

t2 − t1

∫ t2

t1A (t)dt

This is sometimes referred to as ‘mission’ availability.I The long run average availability is

Alr = limT→∞

1T

∫ T

0A (t)dt


INTRODUCTION



AVAILABILITY

SUMMARY

NON-REPAIRABLE SYSTEMS

If a system is not repairable, thenI availability is simply reliability,

A(t) = R(t)

I interval availability is

Aav (t1, t2) =1

t2 − t1

∫ t2

t1R (t)dt

I long run availability is

Alr = 0


INTRODUCTION



AVAILABILITY

SUMMARY

CONSTANT REPAIR RATE

Consider an item that has a constant failure rate λ andconstant repair rate µ.

I λ∆t is the conditional probability of failure during ∆t,given availability at t.

I µ∆t is the conditional probability of repair during ∆t,given unavailability at t.

I Consequently,

A (t + ∆t) = A (t)− λ∆t A (t) + µ∆t (1− A (t))A(t+∆t)−A(t)

∆t = − (λ+ µ) A (t) + µ

ddt

A (t) = − (λ+ µ) A (t) + µ

I assuming A(0) = 1,

A (t) =µ

λ+ µ+

λ

λ+ µe−(λ+µ)t


INTRODUCTION



AVAILABILITY

SUMMARY

CONSTANT REPAIR RATE, CONT’D

I Integrate the last expression to obtain

Aav (t1, t2) =µ

λ+ µ+

λ

(λ+ µ)2 (t2 − t1)

(e−(λ+µ)t1 − e−(λ+µ)t2

)I The long run availability is obtained with t1 → 0, t2 →∞:

Alr =µ

λ+ µ

I ordinarily µ >> λ, so that

Alr ≈ 1− λ

µ=

MTTFMTTF + MTTR


INTRODUCTION



AVAILABILITY

SUMMARY

SYSTEM AVAILABILITY

I Consider a system composed of n components.I Suppose the components are independent with

availabilities Ai(t).I The system availability, A(t), can be computed from

RBD’s using the same formulas as for computingsystem reliability.

I For a serial structure, all components must be availablefor the system to be available, so the system availabilityis

A (t) =∏n

i=1Ai (t)

I For a parallel structure all components must beunavailable for the system to be unavailable, so thesystem availability is

A (t) = 1−∏n

i=1(1− Ai (t))


INTRODUCTION



AVAILABILITY

SUMMARY

SYSTEM AVAILABILITY – CONSTANT RATE

COMPONENTS

I The component long run availabilities are

Alr,i =µi

λi + µi≈ 1− µi

λifor µi >> λi

I For a serial configuration

Alr =∏n

i=1

(1− µi

λi

)≈ 1−

∑n

i=1

µi

λi

I For a parallel configuration

Alr = 1−∏n

i=1

(1− µi

λi + µi

)≈ 1−

∏n

i=1

(µi

λi

)


INTRODUCTION



AVAILABILITY

SUMMARY

SUMMARY

I Preventive maintenanceI periodic, ideal maintenanceI improves reliability only when failure rate is increasing

(aging period).I Corrective maintenance

I maintenance is performed upon failureI For constant failure rate λ0, MTBF = 1/λ0I The probability of more than n failures in a a given time

period T is

1−∑n

k=0

(λ0T)k

k!e−λ0T

I AvailabilityI downtime and uptimeI point and interval availabilityI computing MDT (MTTR) and system availability

harry g. kwatny s - information technologyhgk22/courses/mem361/mem361-lecture5.pdf · harry g....

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