HMMY ?e???????a ????s

1
HMMY ?e???????a ????sµ??????d?s?????sta?
???t??????????ap?. ?a????t??, ?.?.?
2
S?????

• ????p?st?a ????sµ????
• ?? e??a? ? a???p?st?a s?st?µ?t?? ????sµ?????
• ?as???? a????
• ???t??a

3
????p?st?a ????sµ???? ???sµ?? - ???s?

• ? a???p?st?a e??? s?st?µat?? ????sµ???? ????eta?
  sa? ? p??a??t?ta P(t) ?t? t? s?st?µa de? ?a
  ast???se? st? d??st?µa ??p???? s???e???µ????
  ?????? ?e?t?????a? t?? t, ?a? ded?µ???? ?t? t?
  s?st?µa ?e?t????e? se ??p??? pe??ß????? e
• ? a???p?st?a e??? s?st?µat?? ????sµ????
  ep??e??eta? ?at? t? p?e?st?? ap? t? d?ad??as?a
  p??d?a??af?? / s?ed?as?? / ???p???s?? / e??????
  ?a? ??? ap? t? ????? ???s?? t?? (se a?t??es? µe
  ???a f?s??? s?st?µata p.?. µ?a µ??a??)
• ? ?????a t?? a???p?st?a? e??? s?st?µat??
  ????sµ???? ???s?µ?p??e?ta? ??
• ??a?e???s? ????? e?µaste ?t??µ?? ??a t? ??a
  ??d?s? t?? s?st?µat??? ??e?a??µaste pe??ss?te??
  ??e??? (testing)?
• ?p?????sµ? e????? a??a??? ?a? ep??tas?? ??
  a??a??? st?? ??d??a µp??e? ?a µe??s??? t??
  a???p?st?a t?? s?st?µat??
• ???t???? ??a t?? a???????s? ??p??a? µe??d?????a?
  ? te???????a? ?atas?e??? ????sµ????

http//www.ece.cmu.edu/koopman/des_s99/sw_reliabi
lity http//www.stsc.hill.af.mil/crosstalk/1996/06
/reliabil.asp /
4
?as???? ?????

• ?st???a S?st?µat??, Sf??µa, ???tt?µa
• ???tt?µa (fault) ??p??? e??tt?µa st?? ??d??a p??
  µp??e? ?a ?d???se? t? s?st?µa se ast???a. ??
  e??tt?µa µp??e? ?a e??a? ep??e?t??µ??? se ??a
  s?µe?? ? ?a e??a? d??spa?t? (a????ep?d?as? ap?
  d?af??et????? pa?????te?). ?? e??tt?µa µp??e? ?a
  ?d???se? se sf??µa ? ???
• Sf??µa (error) ? d?af???p???s? t?? es?te?????
  ?at?stas?? (state) t?? s?st?µat?? ?a??? a?t?
  ?e?t????e? ap? t?? p??d?a?e??aµµ??? es?te????
  ?at?stas? p?? ?a ?p?epe ?a ß??s?eta? t? s?st?µa
  ?at? t? ?e?t?????a t??. ?? sf??µa µp??e? ?a
  ?d???se? se ast???a ? ???
• ?st???a S?st?µat?? (failure) ? d?af???p???s? t??
  pa?at????µe??? s?µpe??f???? t?? s?st?µat?? ap?
  a?t? p?? ??e? p??d?a??afe?
• ??????
• ?????? e?t??es?? (se ????? µ??a??? CPU time) (t)
• ?µe??????a??? ?????? e?t??es?? - Calendar time
  (t)
• ? ?????a t?? ast???a? t?? s?st?µat?? se s??s? µe
  t?? ?????a t?? ??????
• ? ast???a e??a? st??µ?a?a
• ???s?µ? µet???? e??a? ? ?????? a??µesa se d??
  ast???e? - Time between failures)
• ???? ???s?µ? µet???? e??a? ? s???????? a???µ??
  ast????? µ???? ??p??? s???e???µ??? ?????
  ?e?t?????a? (p.?. ?p? t? ????? a?????? ??d?s? t??
  s?st?µat??) - Cumulative failures)
• ????? µ?a ???? µet???? e??a? ? a???µ?? ast?????
  se ??p??? s???e???µ??? ??????? ???? (p.?. ?eta??
  500 ?a? 700 CPU hours) - Failures in a time
  interval

5
????p?st?a ????sµ????

• ?as??? ???t?s? ??? µp????µe ?a ?p?????s??µe t??
  a???s? t?? a???p?st?a? e??? s?st?µat?? ????sµ????
  ?s? afa????µe e?att?µata ap? t?? ??d??a?
• S?µe?a ???ß??µat?sµ??

• ??e???? - (??s? p??pe? ?a e??????µe? ??te
  µp????µe ?a staµat?s??µe t? ??e????)
• ?at? t? ?e?t?????a ( ??s?? ??e?????ta? ??a t??
  ?p?st????? t?? s?st?µat???)

• ????t??? µ??t??a a???p?st?a? ????sµ???? (S/W
  reliability growth models) ?a?at????µe t?
  s?µpe??f??? ast???a? t?? s?st?µat?? ap? ?st?????
  ded?µ??a ?a? ?p????????µe t? µe????t??? t??
  s?µpe??f??? (se s??s? µe p??ta µe t??
  a?aµe??µe?e? ast???e?). St? ß?ß?????af?a ?????
  p??ta?e? pe??p?? 40 d?af??et??? µ??t??a.

6
???????a
???a? st?? ??d??a... (bug)
???tt?µa (Fault)
?p??e? ?a p???a??se?
?s?te???? ?at?stas? t?? s?st?µat??
Sf??µa (Error)
?p??e? ?a p???a??se?
?a?at????µe??
?st???a (Failure)
7
?as???? ????? S??µat???
S?st?µa ????sµ????
???a?a ep????? e?t??es?? ?? s?st?µa e?te?e?ta?
K f???? µe d?af??et??? ded?µ??a e?s?d?? ???e
f??? se ??a ? pe??ss?te???? (?d????) ?p?????st??
?a?at????µe?e? ?st???e?

• ?? µet??µe
• ??s? s??????? a???µ? ast????? (Average Total
  Number of Failures) µ(t),
• ??tas? ast????? (Failure Intensity) ????µ??
  ast????? st? µ???da t?? ?????? ?(t)
• S???????? µ?s?? ?????? a??µesa se d?ad??????
  ast???e? t?? s?st?µat??
• (Mean Time to Failure) 1/?(t)

8
????p?st?a ?a? ??a?es?µ?t?ta

• ??a ap?? µet???? t?? ?????a? t?? a???p?st?a?
  (reliability) e??a? ?

MTBF MTTF MTTR , ?p??
MTBF S???????? µ?s?? ?????? a??µesa se d?ad??????
ast???e? t?? s?st?µat?? MTTF µ?s?? ??????
a??µesa se d?ad?????? ast???e? t?? s?st?µat??
MTTR µ?s?? ?????? ap??at?stas?? e?att?µat?? ?
MTBF e??a? ?a??te??? e?t?µ?t?? t?? a???p?st?a?
ap? ?t? ? µet???? e?att?µata/KLOC
9
????p?st?a ?a? ??a?es?µ?t?ta

• ??a?es?µ?t?ta (Availability) ????eta? sa? ?
  p??a??t?ta t? s?st?µa ?a s?µpe??fe??e? s?µf??a µe
  t?? p??d?a??af?? t?? ?ta? ??t??e? ? ?e?t?????a
  t?? se ??p??a s???e???µ??? ??????? st??µ?.

• Ge????, ? µet???? t?? ?????a? t?? d?a?es?µ?t?ta?
  e??a? ?a? ?a??te??? e?t?µ?t?? t??
  s??t???s?µ?t?ta? e??? s?st?µat?? ap? t? µet????
  MTTR.

10
??tas? ?st????? se f?s??? S?st?µata ?a?
?atas?e???
http//www.ece.cmu.edu/koopman/des_s99/sw_reliabi
lity/
11
??tas? ?st????? se S?st?µata ????sµ????
http//www.ece.cmu.edu/koopman/des_s99/sw_reliabi
lity/
12
?e??d?????e? ?p?????sµ?? ????p?st?a? ????sµ????

• ?a p?? ???st? µ??t??a / µe??d?????e? ?p?????sµ??
  a???p?st?a? s?st?µ?t?? ????sµ???? eµp?pt??? se
  d?? ßas???? ?µ?de?
• ???t??a sp???? e?att?µ?t?? (error seeding)
• ????t??? µ??t??a a???p?st?a? (reliability growth
  models)

• error seeding ???a? µ???d?? ?p?????sµ?? t??
  a???µ?? t?? e?att?µ?t?? p?? ß??s???ta? st??
  ??d??a. Se a?t? t? µe??d?????a ta e?att?µata
  eµp?pt??? se d?? ?at?????e? a?t? p?? ?p??????
  st?? ??d??a ?a? a?t? p?? eµe?? ??e??µ??a
  t?p??et??µe (?a? f?s??? ????????µe p?sa e??a?). ?
  a???µ?? t?? e?att?µ?t?? p?? ?p?????? st?? ??d??a
  ?p??????eta? ap? t?? (?e????µe?? sta?e??) ????
  t?? t?? a???µ?? t?? d?? t?p?? e?att?µ?t??, ?p??
  a?t? a?a?a??pt??ta? ?at? t?? ??e??? (testing) t??
  s?st?µat??.

13
?e??d?????e? ?p?????sµ?? ????p?st?a? ????sµ????

• ????t??? ???t??a
• ??t? ta µ??t??a ?p????????? t? ße?t??s? t??
  a???p?st?a? e??? s?st?µat?? ?a??? e?att?µata
  a?a?a??pt??ta? ?a? d????????ta? sa? ap?t??esµa
  t?? d?ad??as?a? e?????? (testing) ???s?µ?p????ta?
  µ?a s????t?s? ße?t??s?? (growth function).
• ?? e?e??e?e? µetaß??t?? (µetaß??t?? e?s?d??)
  e??a? ? ?????? (time) ?a? ?? desµe?µ??e?
  µetaß??t?? (ap?t??esµa) e??a? ? a???p?st?a
  (reliability), ? ???µ?? ast???a? t?? s?st?µat??
  (system failure rate),? ? a????st???? a???µ??
  e?att?µ?t?? p?? ????? e?t?p?ste? µ???? ??p??a
  ??????? st??µ?

14
?et??se?? ????p?st?a? sta ????t??? ???t??a

• ??a t?p??? µ?t??s? ??a t?? ?p?????sµ? t??
  a???p?st?a? e??? s?st?µat?? (a???µ?? ast????? st?
  µ???da t?? ??????) e??a? ? ??tas? ast?????
  (failure intensity) p?? ????eta? ap? t?? ???µ?

• ?p?? ? ?????? CPU (se ??a multitasking
  µ?????µa) ? ? p?a?µat???? ?????? se ??p???
  embedded s?st?µa).

15
????t??? ???t??a ????p?st?a?

• ????? µ??t??a ????? p??ta?e? st? ß?ß?????af?a
  a??? ta p???? ???st? e??a? ta
•          G?aµµ??? (ßas???) ???t??? Musa (Musa
  Basic model)
• ???a???µ??? ???t??? Musa/Okomoto (Musa/Okomoto
  Logarithmic model)

16
?as???? ?p???se?? t?? ???t???? ????p?st?a?
????sµ???? Musa

1. ?a e?att?µata st?? ??d??a e??a? a?e???t?ta t? ??a
   ap? t? ???? ?a? ?ata??µ??ta? µe ??a sta?e?? ???µ?
   eµf???s??.
2. ? ?????? e?t??es?? t?? s?st?µat?? e??a?
   µe?a??te??? ap? t? ????? e?t??es?? µ?a? e?t????
   t?? p?????µµat??.
3. ? ??e???? t?? ?e?t?????a? t?? s?st?µat?? ?a??pte?
   ??e? t?? ?e?t?????e? p?? t? s?st?µa ?a p??sf??e?
   ?at? t?? ???s? t?? (test space covers use
   space).
4. ?a ded?µ??a e?s?d?? ?a? ?? pe??pt?se?? e??????
   (test cases) ep??????ta? t??a?a.
5. ??e? ?? ast???e? pa?at?????ta?.
6. ?a e?att?µata p?? d?µ???????? t?? ast???a
   ep?d????????ta? aµ?s??. ? eµf???s? t?? ?d???
   e?att?µat?? de? p??sµet?e?ta? d?? f????.

17
?as??? ???t??? Musa

• ? ??tas? ?st???a? (Failure Intensity) - (FI)
  ????eta? ?p?? e?paµe sa? ? a???µ?? ast????? st?
  µ???da t?? ??????.
• ?? ?as??? ???t??? Musa ?e??e? ?t? ? e??tt?s? t??
  ??tas?? ?st???a? (pa??????? se s??s? µe t?
  s??????? a???µ? ast????? µ) e??a? sta?e??.
• ??t? s??ep??eta? ?t? ? ??tas? ?st???a? e??a? µ?a
  s????t?s? t?? µ?s?? s???????? a???µ?? ast?????
  p?? ????µe pa?at???se? µ???? ??p??a ???????
  st??µ? (CPU time ? calendar time).

Reference Musa, Iannino, Okumoto, Software
Reliability Measurement, Prediction,
Application, McGraw-Hill, 1987.
18
?as??? ???t??? Musa

• ? ??tas? ?st???a? ????eta? ??

• ?p?? ?0 e??a? ? a????? ??tas? ?st???a? st?? a???
  e?t??es?? t?? s?st?µat?? (p??? ???e? ?p??ad?p?te
  d?????s? e?att?µ?t??).
• ? e??a? ? (µ?s??) s???????? a???µ?? ast?????
  µ???? ??p??a s???e???µ??? st??µ?
• ?0 e??a? ? s???????? a???µ?? ast????? p??
  pe??µ????µe ?a ????µe se ape?????st? ?????
  e?t??es?? t?? p?????µµat??

• ??t?st???a, ? µ?s?? s???????? a???µ?? ast?????
  sa? s????t?s? t?? ?????? e?t??es?? e??a?

19

• ?a??de??µa
• ?? ?e???s??µe ?t? ??a p????aµµa ?a ??e? 100
  ast???e? se ??? t? ?e?t??????? t?? ???, ?a? ?t?
  µ???? st??µ?? ????µe pa?at???se? 50 ast???e?, ?a?
  ?t? ? a????? ??tas? ast???a? e??a? 10 ast???e?
  a?? CPU hour.
• ??te ? ?p???????µe?? pa???sa ??tas? ast???a?
  e??a?

• ? (µ?s??) s???????? a???µ?? ast????? µet? ap? 10
  cpu ??e? ?e?t?????a? ?a e??a?

• ?a? ??a 100 CPU ??e? ?e?t?????a?

20
???a???µ??? ???t???

• ? e??tt?s? t?? ??tas?? ?st???a? ???eta?
  µ????te?? (e??et??? µe??s?) ?s? e?att?µata st??
  ??d??a a?a?a??pt??ta? ?a? d????????ta?.

• ?(?) ?0 exp( -?? )
• ?p?? ? e??a? µ?a pa??µet??? e?as????s?? t??
  ??tas?? ?st???a?
• ??t?st???a ? (µ?s??) s???????? a???µ?? ast?????
  se ??p??a ??????? st??µ? t e??a?

21

• ?a??de??µa a? ?e???s??µe ?t? ?0 10
  failures/cpu hour, ? 0.02/failure, ?a? ?t?
  µ???? t??a ????µe pa?at???se? 50 ast???e?.
• ??te

• St?? 10 cpu hours

• Smaller than the Basic model

• ?a? st?? 100 cpu hours

(pe??ss?te?e? ast???e? ap? t? ?as??? ???t???)
22
?a ???t??a Musa - S???pt???
?as??? ???t???
???a???µ??? ???t???
m(t) v01 exp(-l0t/v0)
m(t) (1/q).ln(l0qt 1)
l(t) l0/(l0qt 1)
l(t) l0exp(-l0t/v0)
l
m
Log.
Log.
v0
Basic
Basic
t
t
23
?a??de??µa
??. ?? ?e???s??µe ?t? ??a s?st?µa ?a ??e? 100
ast???e? se ??? t? ?e?t??????? ??? ???. ??
?p???s??µe ?t? ? a????? ??tas? ?st???a? e??a? 10
ast???e? / CPU hour ?a? ?t? ? pa???sa ??tas?
?st???a? e??a? 3.68 ast???e? / CPU hour. ???
T????µe ?a pet????µe ??tas? ?st???a? t??
s?st?µat?? ?a e??a? 0.000454 ?st???e? / CPU hour.
??se? ??e? a??µ? ?a p??pe? ?a e??????µe t?
s?st?µa G?a ?a pet????µe t? ep???µ?t? ep?ped?
??tas?? ?st???a??
?p. ??? ?e???s??µe ?t? ????µe t? ?as???
???t???, t?te
l(t) l0exp(-l0t/v0)
St? ??????? st??µ? t1, l(t1) l0exp(-l0t1/v0)
lp
St? ??????? st??µ? t2, l(t2) l0exp(-l0t2/v0)
lf
t2 - t1 (v0/ l0).ln(lp/ lf)
v0 100 faults, l0 10 failures/CPU-hr
lp 3.68 failures/CPU-hr, lf 0.000454
failure/CPU-hr
(t2 - t1 ) 90 CPU-hr
S?µp????µat???? ?????? e??????
24
?µe??????a??? ?????? ??t??es?? S?st?µat??

• ? ?µe??????a??? ?????? (calendar time)
  ?p??????eta? ????????ta? t?? t?µ? t?? ?????

• ?a? t? t?µ? t?? ?????? e?t??es?? t??
  p?????µµat?? se CPU hours.
• ? ?µe??????a??? ?????? e??a? ???s?µ?? ??a t??
  ?e???? p????aµµat?sµ? t?? p????? ??a t? d???es?
  t?? s?st?µat?? st??? ???ste? t?? (p.?.
  ?p????????µe t?? ?µe??µ???a st?? ?p??a ?a ????µe
  pet??e? ??p??a s???e???µ??? t?µ? st? ??tas?
  ?st???a?).

25

• ? ?µe??????a??? ?????? st? p?? s???et? t?? µ??f?
  ßas??eta? ?a? st? d?ad??as?a p?? a??????e?ta?
  ??a t? d?????s? t?? e?att?µ?t?? ?a??? ?a? t??
  pa?a??t? pa?aµ?t????

• ???? d?a??s?µ??? ?p?????st????? p?????
• ?? ßa?µ? µe t?? ?p??? ??a p???? ???s?µ?p??e?ta?

26

• G?a pa??de??µa, st?? a??? t?? d?ad??as?a?
  e?????? pa?at????µe p????? ast???e? se µ????
  ??????? d?ast?µata µeta?? t???
• ??te ? ??e???? staµat? ??a ?a d?????s??µe ta
  e?att?µata st?? ??d??a.
• ?s? ? ??e???? p??????,
• ?a ??????? d?ast?µata a??µesa se ast???e?
  ?????ta? µe?a??te?a,
• St? µeta?? t? p??s?p??? p?? d??????e? ta
  e?att?µata de? d???e?e?
• ? ?µ?da e?????? e??a? t??a ? ?µ?da p??
  d?µ??????e? t? µp?t??????sµa
• ?e???? ?? ?p?????st???? p???? e??a? ? pa?????
  p?? ?a?????e? t?? ap?d?t???t?ta t?? ????
  d?ad??as?a?.

27

• ???s? ?????
• ? Musa ?de??e ?t? ? ???s? t?? p???? e??a?
  ??aµµ??? a?????? t?? ?????? e?t??es?? t??
  s?st?µat?? ?a? t?? µ?s?? s???????? a???µ??
  ast?????.
• ??? ?e???s??µe ?t? ? µetaß??t? ?r ????e? t?
  ???s? (usage) ??p???? p???? r, t?te

?p?? ?r e??a? ? ?????? ???s? t?? p???? a?? CPU
hour ?a? ?r e??a? ? ?????? ???s?? t?? p???? a??
ast???a
28

• S???pt??? ????µe

????? ???s? a?? CPU hour ???s? a?? ast???a ??a??s?µ?? ????? ????p???s? ?????
???s?p??? ???es?? ??att?µ?t??
???s?p??? ??????s?? ??att?µ?t??
?p?????st???? ?????
29
?a??de??µa

• ??a d?ad??as?a ??e???? t???e? ??a 10 CPU hours
  ?a? a?a??????e? 34 ast???e? ??? ? p??sp??e?a
  e?????? ??a ???e ??a e?t??es?? t?? s?st?µat??
  e??a? 5 person hours, ?a? ???e ast???a apa?te? 2
  hours ??a ?a e??e?e? t? a?t?st???? e??tt?µa,
  t?te.
• ??te ? s???????? ?????? e??es?? t?? e?att?µat??
  e??a?

5(10) 2(34) 118 person hr
30
F?s? ?e?t?????a? t?? S?st?µat??

• ?ta? t? s?st?µa µpa??e? p???? st? ?e?t???????
  t?? f?s? (released, operational phase), ?a? de?
  ????µe ??p??a a??a?? st?? ??d??a, t?te ? ??tas?
  ast????? pa?aµ??e? sta?e??.
• ?a? ta d?? µ??t??a p?? e?daµe ?a a???????s???
  ??a t?? ?p?????sµ? t?? a???p?st?a? t?? s?st?µat??
  µ?a d?ad??as?a Poisson µe pa??µet?? t?? ??tas?
  ????p?st?a? ?.

31
F?s? ?e?t?????a?

• ? ????p?st?a R ?a? ? ??tas? ????p?st?a?
  s?et????ta? ap? t?? e??s?s?
• R(?) exp( ???)
• ?p?? a?aµ????µe ? p??a??t?ta ?a µ??
  pa?at???s??µe ast???a se ??p??a ??????? st??µ?
  (a???p?st?a) e??a? ?aµ???te?? ??a µe?a??te?a
  ??????? d?ast?µata e?t??es?? t?? s?st?µat??.

32
F?s? ?e?t?????a?

• Se p???? p?????ta ????sµ???? ? f?s? ?e?t?????a?
  e??? s?st?µat?? ap?te?e?ta? ap? µ?a se???
  e?d?se?? (releases)

33
????p?st?a S?st?µ?t?? (?a??????a S?st?µata)

• ?? ?e???s??µe ?t? ????µe QP ?p?-s?st?µata/??f?de?
  (components) µe sta?e?? de??t? ??tas?? ?st???a?
  t? ?a???a ?a? ? a???p?st?a t??? ?p??????eta? se
  ????? ?µe??????a?? ?????. ?p???t??µe ep?s?? ?t?
  ??a ?a ?e?t?????se? s?st? t? s?st?µa, ??a ta
  (pa??????a) ?p?-s?st?µata ?a p??pe? ?a
  ?e?t??????? s?st?. ??te ? ????? ??tas? ?st???a?
  e??a?

?p?? ?k e??a? ? ??tas? ?st???a? ???e
?p?-s?st?µat??.
34
????p?st?a S?st?µ?t?? (?a??????a S?st?µata)

• ?p?? ????µe de? R(?) exp( ???).
• ?p?te ??a ????? CPU (????? e?t??es??) ????µe

??? ? µ?s? a???p???s? t?? ?p?????st???? p????
st?? ?/? G?a t? s???e???µ??? s?st?µa e??a? ?c
t?te
Se ?µe??????a?? ?????
35

• ?e???? ? a???p?st?a t?? s?st?µat?? (se s??s? µe
  ?µe??????a?? ?????) e??a?

36

• ???a? ep?s?? p??a?? sta pa?ap??? µ??t??a ?a
  ?e???s??µe ?t? t? s?st?µa p?? ?t?e?e st?? ?/? 1
  p?? ??e? ???µ? e?t??es?? e?t???? r1 µetaf??eta?
  st?? ?/? 2 p?? ??e? ???µ? e?t??es?? e?t???? r2
• ??te ? ??tas? ast???a? st? µ??a?? 2 e??a?