Reliability: Series versus Parallel

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Reliability Series versus Parallel

• Persons 1 3 have R 1.0, 2 4 have R 0.5
• What is the reliability below?

R 1.0 R0.5 R 1.0 R0.5

• R 1.0 b) R 0.5
• c) R 0.25 d) R 0.0

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Reliability Series versus Parallel

• Persons 1 3 have R 1.0, 2 4 have R 0.5

R (1.0) (0.5) (1.0) (0.5) 0.25 or 25
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Reliability Series versus Parallel

• Persons 1 3 have R 1.0, 2 4 have R 0.5
• What is the reliability below?

12 34
R 0.5 R 0.5
R 1.0 R 1.0

• R 1.0 b) R 0.5
• c) R 0.25 d) R 0.0

4
Reliability Series versus Parallel

• Persons 1 3 have R 1.0, 2 4 have R 0.5
• What is the reliability below?

12 34
R 0.5 R 0.5
R 1.0 R 1.0
R12or34 1 (1-1)(1-0.25) 1
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Reliability Situation Analysis

• Your favorite uncle has offered you two free
  tickets to the Hoobastank concert Saturday night
  in Salt Lake City.
• You have three choices for travelfly, drive, or
  train.
• Which one will get you there most reliably???

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Reliability getting to the Hoobastank concert

• Fly two serial flights, each with a reliability
  of 90 of getting you to each destination
• Drive Your car is 80 reliable, so your friend
  offers to drive side-by-side with her car (to get
  the free ticket). Her car has a reliability of
  80.
• Train Reliability of arrival on time of 80.

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Reliability getting to the Hoobastank concert

• Fly R (0.9)(0.9) 0.81 ? 81
• Drive R 1 (1-0.8)(1-0.8) 0.96 ? 96
• Train R 0.80 ? 80
• Drive, baby, drive!

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Situation Analysis Tires

• My F250 has four independent and identical
  tires. If any tire fails, the vehicle cannot be
  driven. Assume the reliability of an individual
  tire is 0.9.
• What is the reliability of the F250 with respect
  to tires?

Most excellent F250 I might add
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Situation Analysis Tires

• Framing the question The truck works if
• tire1 works AND/OR
• tire2 works AND/OR
• tire3 works AND/OR
• tire4 works
• Is it AND or is it OR?

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Situation Analysis Tires

• The truck works if
• tire1 works AND
• tire2 works AND
• tire3 works AND
• tire4 works
• AND ? series equation
• Rs R1R2Rn (0.9)(0.9)(0.9)(0.9) 0.65

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Situation Analysis More Tires

• What is the reliability of an eighteen wheeler
  with tires of reliability equal to 0.9?

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Situation Analysis More Tires

• What is the reliability of an eighteen wheeler
  with tires of reliability equal to 0.9?
• Rs R1R2Rn (0.9)18 0.15

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Increasing Reliability

• A spare tire in a car increases reliability by
  offering a way to fix any one tire. It
  essentially provides a possible parallel
  element to a flat tire.

car
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Situation Analysis Flying

• You are flying on a Boeing 747 with four engines.
  The reliability of each engine is 0.9. The
  plane can fly with as few as one engine working.
• What is the probability of a failed flight?

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Situation Analysis Flying

• Framing the question The plane works if
• engine1 works AND/OR
• engine2 works AND/OR
• engine3 works AND/OR
• engine4 works
• Is it AND or is it OR?

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Situation Analysis Flying

• The plane works if
• engine1 works OR
• engine2 works OR
• engine3 works OR
• engine4 works
• OR ? parallel
• Probability of failure (0.1)4 .0001
• Reliability 1 (0.1)4 0.9999

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Situation Analysis Rock climbing

• Youve climbed up the backside of a cliff, 400
  feet off the valley floor. You and your partner
  each have one rope, each 300 feet long, and each
  with a reliability of not-breaking of 0.95.
  Theres a ledge halfway down, at 250 feet. You
  can get extra help at the top, but not at the
  ledge.
• What is the most reliable way to descend by rope,
  and what is the probability of a fall due to rope
  failure?

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Rock climbing Solution one

• Tie ropes together, drop them down, climb down
  all the way.
• Descent works if rope1 ok AND rope2 ok.series
  equation
• Reliability (0.95)2 0.90
• Probability of fall 1 0.9 0.1 ? 10

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Rock climbing Solution two

• Drop two ropes down, climb half-way down drop
  ropes again, climb rest of way.
• Segment one Descent works if rope1 ok OR rope2
  ok . parallel equation
• RS1 1 (1-0.95)(1-0.95) 0.9975
• RS2 1 (1-0.95)(1-0.95) 0.9975
• Overall descent ok if Segment1 AND Segment2 ok
• R (0.9975)(0.9975) 0.995
• Probability of a fall 1 0.995 0.005 ? 0.5

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Situation Analysis Snowballs

• My two sons have a snowball hit-the-target
• reliability of 75, my two daughters 50.
• What is the reliability of one of them hitting me
  with a snowball, given that all four throw?

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Situation Analysis Snowballs

• I get hit if Sue OR Greg OR Kim OR Sue hit me.
• Reliability 1 (1-0.75)(1-0.75)(1-0.5)(1-0.5)
• 1 - (0.25)(0.25)(0.5)(0.5)
• 0.984375 98

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Situation Analysis Subway train

• Tim and Sue must each be on time to meet me and
  catch the subway train. Tims watch reliability
  is 0.90, Sues watch reliability is 0.80, my cell
  phones reliability is 0.99.
• What is the reliability that we will catch
  the train?

23
Situation Analysis Subway train

• Tim and Sue must each be on time to meet me and
  catch the subway train. Tims watch reliability
  is 0.90, Sues watch reliability is 0.80, my cell
  phones reliability is 0.99.
• We are successful ifJim AND SueAND Tim are on
  time.

R R1R2R3 (0.99)(0.9)(0.8) 0.7128
72
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Transformer Failures

• SDGE, an electric utility company serving San
  Diego County, is considering replacing some of
  its equipment at its numerous substations. They
  are trying to decide whether to replace an
  existing PCB transformer (PCB is a toxic chemical
  that meets all current regulations but, if an
  incident occurred, PCB contamination could cause
  harm and SDGE would be liable for damages).
• Decision to be made Replace or Don't Replace
  Transformer.

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Transformer Failures

• If the utility replaces the transformer, the
  total cost would be 10,000 and they would avoid
  any problems. Alternatively, if they choose to
  continue to use the existing transformer for
  another year, the OM Manager estimates that there
  is a 5 chance of a serious incident and a 10
  chance of a minor incident. A serious incident
  has a 10 chance of a fire that could cost the
  company 1 million in damages and a 90 chance of
  damage requiring 5000 clean-up costs. A minor
  incident has a 10 chance of a cleanup cost of
  10,000 and a 90 chance of clean-up costs of
  1,000.
• (a) Perform a decision tree analysis of this
  problem and indicate the recommended choice or
  solution based on cost.

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Transformer Failures

• (b) Is this the decision you believe the company
  should make? Explain your reasons.
• (c) You are promoted to Chief Financial Officer
  (CFO) for SDGE. You now learn that there are at
  least 10,000 such transformers in San Diego
  County alone. Calculate the cost of your
  decision if applied to all 10,000 transformers.
  Does it change your answer to part (b)?

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SDGE Replace or Don't Replace Transformer

• Methodology
• Draw the Decision Tree and Outcomes
• Chose a decision criteria What is the basis upon
  which you will make the decision?
• Calculate Expected Cost of keeping
• Calculate Expected Cost of replacing

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SDGE
10,000
EC (.05)(.1)(1M) 5,000
1M
P.10
replace
fire damage
P.85 No Incident
which path?
EC (.05)(.90)(5000) 225
P.90
5000
P.05
keep
Serious incident
EC (.1)(.1)(10K) 100
10K
P.10
P.10
Minor incident
fire damage
P0.90
EC (.1)(.9)(1K) 90
1000
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SDGE
10,000
EC (.05)(.1)(1M) 5,000
Expected Replace Cost 10,000
1M
P.10
fire damage
which path?
EC (.05)(.90)(5000) 225
P.90
5000
Serious incident
P.05
Expected Keep Cost 5,415
EC (.1)(.1)(10K) 100
10K
P.10
P.10
Minor incident
P0.90
EC (.1)(.9)(1K) 90
1000
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SDGE Expected Costs

• Possible criteria
• Lowest expected cost ? keep transformer
• Minimax regret ? replace transformermaximum
  possible regret (replace) 10,000maximum
  possible regret (keep) 1M
• Decide that loss of goodwill, social cost,
  cost of possible lawsuits, etc. are not included
  in the cost of a failed transformer and include
  it and recalculate

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SDGE Expected Costs
As CFO, you learn that there are at least 10,000
such transformers in San Diego County alone. 
Calculate the cost of your decision if applied to
all 10,000 transformers.   Does it change your
answer to part (b)?
EVreplace 10,000 x 10K 100 million
EVkeep 10,000 x 5K 54 million
The system cost of replacing each transformer is
now 54 million. Does this change your mind?
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If we add other possible paths between home
and school, we can increase the reliability.
Internet access Cal State from home occurs by
traversing 5 linked computers Each link has a
reliability of 0.99
RHgtS (0.99)5 0.95
RHgtS 1 (1-0.95)10
RHgtS 0.9999999999999
Home
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The Internet
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• The Internet
• Each color represents a region
• North America
• Europe/Central Asia/Africa
• Latin America
• Asia Pacific
• Unknown

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Internet Reliability
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Increasing Reliability

• Many of todays systems that we take for granted
• Internet
• Telephones
• Computers
• Automobiles
• Copiers
• are the direct result of investments in
  understanding
• and improving reliability. Lets see one example

37
Increasing reliability transistors to ICs

• Individual transistors(shown at right) have a
  typical reliability of 0.999999999

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Increasing reliability transistors to ICs

• Sophisticated electronics require interconnects
  of thousands if not millions of such transistors,
  which result in unreliable systems. This led to
  the development of integrated circuits.
• A typical integrated circuit (shown at right) has
  50,000 transistors, and is 100-500 times more
  reliable than 50,000 transistors would be alone.
• A modern day integrated circuit (Pentium IV) has
  tens of millions of transistors, a direct result
  of reliability analysis

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number of transistors
Speed in MegaHertz
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A complex real-world example
Reliability equation has greater than 15,000
terms
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Consumer-Oriented Reliability Measures

• Mean Time Between Failure
• Mean Time To Repair
• System Availability

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Consumer-Oriented Reliability Measures

• MTBF
• Mean Time Between Failure
• MTTR
• Mean Time to Repair
• Both are statistical estimates gathered
• from large numbers of measurements

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Onsumr-Oriented Reliability Measures

• Mean Time Between Failure gives a concrete
  measure of reliabilitytells us the expected
  failure time.
• Mean Time to Repair gives a concrete measure of
  the indicated repair interval.

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What is the MTBF (Mean Time Between Failure) of...

• Your car
• Your cell phone
• Your water heater
• Your UPS delivery truck
• The Jupiter probe

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What is the MTTR (Mean Time To Repair) of...

• Your car
• Your cell phone
• Your water heater
• Your UPS delivery truck
• The Jupiter probe

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Reliability Examples

• Item MTBF
• PC (hardware) 10,000 hours 1 year
• Disk drive 150,000 hours 15 years
• Windows 98 8 hours
• Windows XP gt 40 hours (est.)
• Linux gt 1000 hours (est.)
• Automobile 500 hours (est.)

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Reliability Examples

• Item MTTR
• Broken pencil 30 seconds (sharpen)
• Windows crash several minutes (reboot)
• Flat tire 30 minutes
• Automobile 8 hours (est.)
• Jupiter probe many years

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System Availability

• Wed like a measure that tells us what fraction
  of the time when we go to use a system that it is
  actually available

0 lt SA lt 1 fraction of time a system is
available for use
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Wood Pencil Availability
If we assume pencil MTBF 30 minutes MTTR 30
seconds then
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Auto Availability
If we assume automobile MTBF 500 hours MTTR 8
hours then
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Laptop Availability
If we assume laptop MTBF 2 years MTTR 2
weeks then
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Windows Availability
Windows ME MTBF 2 hours MTTR 10
minutes then Windows XP MTBF 40 hours MTTR
10 minutes then
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System Availability
System availabilities greater than 98-99 are
generally acceptable to a consumer.
Systems that support business processes (e.g.
manufacturing, etc.) generally require much
higher availabilities.
Systems that support extreme environments (e.g.
military vehicles, space vehicles, etc.) require
extremely high availabilities.