Dependability

1
Dependability Maintainability Theory and
MethodsPart 2 Repairable systems Availability

• Andrea Bobbio
• Dipartimento di Informatica
• Università del Piemonte Orientale, A. Avogadro
• 15100 Alessandria (Italy)
• bobbio_at_unipmn.it - http//www.mfn.unipmn.it/bob
  bio/IFOA/

IFOA, Reggio Emilia, June 17-18, 2003
2
Repairable systems
X 1
X 2
X 3
UP

DOWN
t
Y 1
Y 2
X 1, X 2 . X n Successive UP times Y1, Y 2
. Y n Successive DOWN times
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Repairable systems

• The usual hypothesis in modeling repairable
  systems is that
• The successive UP times X 1, X 2 . X n are
  i.i.d. random variable i.e. samples from a
  common cdf F (t)
• The successive DOWN times Y1, Y 2 . Y n are
  i.i.d. random variable i.e. samples from a
  common cdf G (t)

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Repairable systems
X 1
X 2
X 3
UP

DOWN
t
Y 1
Y 2

• The dynamic behaviour of a repairable system is
  characterized by
• the r.v. X of the successive up times
• the r.v. Y of the successive down times

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Maintainability

• Let Y be the r.v. of the successive down times
• G(t) Pr Y ? t (maintainability)
• d G(t)
• g (t) (density)
• dt
• g(t)
• h g (t) (repair rate)
• 1 - G(t)
• MTTR ? t g(t) dt (Mean Time To
  Repair)

?
0
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Availability
The measure to characterize a repairable system
is the availability (unavailability)
The availability A(t) of an item at time t is the
probability that the item is correctly working at
time t.
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Availability

• The measure to characterize a repairable system
  is the availability (unavailability)
• A(t) Pr time t, system UP
• U(t) Pr time t, system DOWN
• A(t) U(t) 1

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Definition of Availability

• An important difference between reliability and
  availability is
• reliability refers to failure-free operation
  during an interval (0 t)
• availability refers to failure-free operation at
  a given instant of time t (the time when a
  device or system is accessed to provide a
  required function), independently on the number
  of cycles failure/repair.

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Definition of Availability
I(t)
1
Failed and being restored
Operating and providing a required function
Operating and providing a required function
0
t
1 working 0 failed
I(t) indicator function
System Failure and Restoration Process
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Availability evaluation

• In the special case when times to failure and
  times to restoration are both exponentially
  distributed, the alternating process can be
  viewed as a two-state homogeneous Continuous Time
  Markov Chain

Time-independent failure rate
? Time-independent repair rate ?
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2-State Markov Availability Model

• Transient Availability analysis
• for each state, we apply a flow balance equation
• Rate of buildup rate of flow IN - rate of flow
  OUT

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2-State Markov Availability Model
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2-State Markov Availability Model
1
A(t)
Ass
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2-State Markov Model
1) Pointwise availability A(t)
2) Steady state availability limiting value as

• If there is no restoration (?0) the
  availability
• becomes the reliability A(t) R(t)

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Steady-state Availability

• Steady-state availability
• In many system models, the limit
• exists and is called the steady-state availability

The steady-state availability represents the
probability of finding a system operational after
many fail-and-restore cycles.
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Steady-state Availability
1
0
UP
DOWN
t
Expected UP time EU(t) MUT MTTF
Expected DOWN time ED(t) MDT MTTR
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Availability Example (I)
Let a system have a steady state availability Ass
0.95 This means that, given a mission time T,
it is expected that the system works correctly
for a total time of 0.95T. Or, alternatively,
it is expected that the system is out of service
for a total time Uss T (1- Ass) T
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Availability Example (II)
Let a system have a rated productivity of W
/year. The loss due to system out of service can
be estimated as Uss W (1- Ass) W The
availability (unavailability) is an index to
estimate the real productivity, given the rated
productivity.
Alternatively, if the goal is to have a net
productivity of W /year, the plant must be
designed such that its rated productivity W
should satisfy Uss W W
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Availability
We can show that This result is valid without
making any assumptions on the form of the
distributions of times to failure times to
repair. Also
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Motivation High Availability
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Maintainability

• MDT (Mean Down Time or MTTR - mean time to
  restoration).
• The total down time (Y ) consists of
• Failure detection time
• Alarm notification time
• Dispatch and travel time of the repair person(s)
• Repair or replacement time
• Reboot time

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Maintainability

• The total down time (Y ) consists of
• Logistic (passive) time
• Administrative times
• Dispatch and travel time of the repair
  person(s)
• Waiting time for spares, tools
• Effective restoration (active) time
• Access and diagnosis time
• Repair or replacement time
• Test and reboot time

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Logistics

• Logistic times depend on the organization of the
  assistance service
• Number of crews
• Dislocation of tools and storehouses
• Number of spare parts.

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The number of spares
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Maintenance Costs

• The total cost of a maintenance action consists
  of
• Cost of spares and replaced parts
• Cost of person/hours for repair
• Down-time cost (loss of productivity)

The down-time cost (due to a loss of
productivity) can be the most relevant cost
factor.
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Maintenance Policy

• Is the sequence of actions that minimizes the
  total cost related to a down time
• Reactive maintenance
• maintenance action is triggered by a failure.
• Proactive maintenance
• preventive maintenance policy.

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Life Cycle Cost