Todays class

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Todays class

• Review Case Problem 2.1 - Mountain States
  Electric Transformer
• More on Reliability
• Complete chapter 3
• Start on chapter 4

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Case Problem 2.1

• Issue Possibility of faulty transformer, keep or
  replace?
• Methodology
• Draw the 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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Case Problem 2.1 Decision Tree
85,000
BAD FIRE 90M
P.2
replace
bad minor
P.004
which path?
P.8
fire no fire
MINOR FIRE 8M
high likelihood of an incident
P.996
P.5
NO FIRE
keep
BAD FIRE 90M
P.2
P.5
bad minor
P.001
low likelihood of an incident
fire no fire
P.8
MINOR FIRE 8M
P0.999
NO FIRE
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Case Problem 2.1 Decision Tree
85,000
BAD FIRE 90M
P.2
replace
EVbad fire (0.5)(0.004)(0.2)(90M)
(0.0004)(90M) 36,000
bad minor
P.004
which path?
P.8
fire no fire
MINOR FIRE 8M
high likelihood of an incident
P.996
P.5
NO FIRE
keep
BAD FIRE 90M
P.2
P.5
bad minor
P.001
low likelihood of an incident
fire no fire
P.8
MINOR FIRE 8M
P0.999
NO FIRE
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Case Problem 2.1 Expected Costs
EVreplace 85K
EVkeep 36K 12.8K 9K 3.2K 61K
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Case Problem 2.1 Expected Costs
As the CFO of the electric company, I insist that
we base our decision on financial measures.
According to my calculations, we save 24,000 by
not replacing the transformer.
EVreplace 85K
EVkeep 61K
For those of you who chose to replace the
transformer, can you provide me with a
convincing financial argument?
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Case Problem 2.1

• Possible criteria
• Lowest expected cost ? keep transformer
• Minimax regret ? replace transformermaximum
  possible regret (replace) 85,000maximum
  possible regret (keep) 90,000,000
• Decide that loss of goodwill is not included in
  the cost of a failed transformer and include it
  and recalculate
• Decide that social cost is not included in the
  cost of a failed transformer and include it and
  recalculate

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Reliability
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Functional Design(How the Product Performs)

• Reliability
• Probability product performs intended function
  for specified length of time
• Maintainability
• Ease and/or cost or maintaining/repairing product

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Reliability and Maintainability

• Reliability measures enable us to find the
  weakest links in a product or complex system and
  improve it at lowest possible cost.
• Maintainability measures enable us to identify
  the highest maintenance cost items and reduce
  them.

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Computing Reliability
R reliability, expressed as a fraction (0 lt R lt
1)
Probability of failure 1 - R
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Computing Reliability
Components in series
R1
R2
Series the system works only if all components
work
For example, the system works if a works AND b
works AND c works AND
AND ? series
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Computing Reliability
Components in series
R R1R2
R1
R2
Rs R1R2Rn
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Computing Reliability
Components in parallel
R2
R1
Parallel the system works if any components work
For example, the systems works if a works OR b
works OR c works OR
OR ? parallel
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Computing Reliability
Rn
n components in parallel
R2
R1
Parallel the system fails works if any
components work
n
Rp 1 - ? (1-Ri)
I 1
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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?

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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 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 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?

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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.
• 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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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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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.

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Other 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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Other 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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MTBF Examples

• Item MTBF (estimates)
• pencil 30 minutes
• makeup, cologne 8 hours
• ballpoint pen 40 hours
• printer/copier 40 hours
• automobile 500 hours
• commercial jet gt100,000s of hours
• deep space probe gt10s of years

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MTBF Other Examples

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

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MTTR Examples

• Item MTTR
• Broken pencil 30 seconds (sharpen)
• Windows crash several minutes (reboot)
• Flat tire 30 minutes
• Automobile 8 hours (est.)
• House fire damage many months
• Deep space probe many years

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System Availability
tells us what percentage of the time A system is
up or available for our use
Not equal to, but a practical counterpart to
System Reliability
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Wood Pencil Availability
If we assume pencil MTBF 30 minutes MTTR 30
seconds then
30 30 0.5
Pencil Availability
98.3
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Auto Availability
If we assume automobile MTBF 500 hours MTTR 8
hours then
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F250 Availability
When new-to-me, -- MTBF was 1 start (24
hours) -- MTTR was 1 hour (jump start)
Added 2nd battery and solar charger on roof --
MTBF extended to 6 months -- MTTR was 1 hour
(jump start)
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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
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Megans Car Availability

• Megan has 4 keys to her car
• 2 under fenders
• one in her purse
• one on her keychain.
• The average time of her misplacing any given key
  is about one hour once its in her hand.
• Failure is the inability to get into her car,
  having exhausted all the key backups.
• What is her car availability MTBF?

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Megans Car Availability

• Megan has 4 keys to her car
• 2 under fenders
• one in her purse
• one on her keychain.
• The average time of her misplacing any given key
  is about one hour once its in her hand.
• MTBF 4 hours

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Megans Car Availability

• Megan has 2 additional keys to her car one at
  her home, one at our home. When she has lost all
  4 of her keys, she methodically calls her mom
  after she loses that key, she calls us.
• It typically takes an hour to get a key to her
  car, meaning that the MTTR is one hour.
• What is Megans cars Availability?

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Megans Car Availability
MTBF 4 hours MTTR 1 hours 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.
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Informal feedback

• Write a 2 minute journal to be handed in
  immediately
• The journal should briefly summarize
• Major points learned
• Areas not understood or requiring clarification