Ss : ' ' ' '

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?? ?/?/1 S?st?µa ?????
? (? d?ad??as?a af??e?? e??a? Poisson) / ? (??
?????? e??p???t?s?? a????????? e??et??? ?ata??µ?)
/ 1 (e??p??et?t??)
???et???? µe pa??µet?? µ

• S?µß???sµ?? . / . / . / .

??ad??as?a af??e?? ? Poisson D
?tete?µ???st??? G Ge????
?????? ???p???t?s?? ? ???et???? D
?tete?µ???st???? G Ge?????
????µ?? () e??p??et?t??
????st?? a???µ?? pe?at?? st? s?st?µa

• ?p???t??µe ?t? ?? ?????? e??p???t?s?? e??a?
  a?e???t?t?? µeta?? t???, ?a? a?e???t?t?? t??
  interarrival times
• ? e??p??et?t?? p?t? de? µ??e? a?e?e???? ?ta?
  ?p???e? pe??t?? st? s?st?µa (work conserving).
  G?a ?a e?µaste s???e???µ????, ?p???t??µe ?t? ?
  e??p???t?s? e??a? First Come First Serve

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??ad??as?a Poisson µe ???µ? ?

• H d?ad??as?a Poisson ?(t) e??a? µ?a counting
  process. G?a ???e t, t? A(t) e??a? µ?a t??a?a
  µetaß??t?, ? ?p??a ?p?d????e? t?? a???µ? t??
  af??e?? st? ??????? d??st?µa ap? 0 ??? t

?? a???µ?? t?? af??e?? se µ? ep??a??pt?µe?a
??????? d?ast?µata e??a? a?e???t?t?? µeta?? t???
??e???t?t??
??a?t?µ????
? a???µ?? t?? af??e?? se ???e d??st?µa µ????? t
a??????e? t?? ?ata??µ? Poisson µe pa??µet?? ?t
, ?p?? ? ? ???µ?? af??e??
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?d??t?te? µ?a? d?ad??as?a? Poisson
?st? tn ? ?????? t?? n-st?? ?f???? ?a? tntn1
tn ? ?????? µeta?? d?? d?ad?????? af??e??
(interarrival time)
?? ?????? µeta?? d?ad?????? af??e?? e??a?
e??et??? ?ata?eµ?µ???? µe pa??µet?? ? (µ?s? t?µ?
1/? ?a? d?asp??? 1/?2)
(Memoryless property - ????? µ??µ?)

• G?a ???e t ?a? ???e (a??et? µ????) d

PA(td) ?(t) 0 1 ?d ?(d) PA(td)
?(t) 1 ?d ?(d) PA(td) ?(t) 2 ?(d)
?p??
??t? e?????ta? ap? t?? ???sµ?
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• ?a? A1(t), A2(t), .....,??(t) e??a? a?e???t?te?
  d?ad??as?e? Poisson µe ???µ??? ?1,?2,....??
  a?t?st???a, t?te ? ?1(t)A2(t)A?(t) e??a?
  d?ad??as?a Poisson µe ???µ? ?1?2...??

• ?a? ???e ?f??? µ?a? d?ad??as?a Poisson
  ap?st???eta? a?e???t?ta st? s?st?µa 1 µe
  p??a??t?ta p ?a? st? s?st?µa 2 µe p??a??t?ta
  1-p, t?te ?? af??e?? st? ???e s?st?µa e??a?
  Poisson ?a? a?e???t?te? (random splitting).

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?f??e??
??a????se??
? ?at?stas? t? ??????? st??µ? t
Te????µe µ?a s???e???µ??? ??????? st??µ? t ??
µe????t???? af??e?? de? e?a?t??ta? ap? ?t?d?p?te
??e? p??????e? ?a? ?? µe????t???? a?a????se??
e?a?t??ta? ap? t? pa?e???? µ??? µ?s? t?? a???µ??
N(t) t?? pe?at?? st? s?st?µa ?at? t? ???????
st??µ? t. S???e???µ??a, af?? ?? ??????
e??p???t?s?? e??a? e??et????, de? ??e? ?aµ?a
s?µas?a t? ??????? d??st?µa e??p???t?s?? p?? ??e?
?d? ??ße? ? t????? pe??t??. ? ?p??e?p?µe???
?????? µ???? t?? a?a????s? pa?aµ??e? e??et???? µe
t?? ?d?e? pa?aµ?t????. ? µe????t???? a???µ?? t??
pe?at?? st? s?st?µa e?a?t?ta? ap? t???
p??????µe???? a???µ??? µ??? µ?s? t?? t?????t??
a???µ?? N(t) .
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?p??e?t?????µe st?? ???????? st??µ?? 0, d, 2d,
3d, ..., kd, .... (d a?????t?? µ????)
?k ?(kd) ? a???µ?? t?? pe?at?? st? s?st?µa
t? st??µ? kd Pij P Nk1 j ?k i
(p??a??t?te? µet?ßas??) P00 1 ?d ?(d)
(?aµ?a ?f???) Pii 1 ?d µd ?(d) , i 1
(?aµ?a ?f??? ? a?a????s?) Pi,i1 ?d ?(d), i
0 (µ?a ?f???) Pi,i-1 µd ?(d), i 0 (µ?a
a?a????s?) S?µe??s? G?a ???e ?at?stas? n 1, ?
e??p??et?t?? e??a? apas????µ???? ?a? ? p??a??t?ta
a?a????s?? e??a? Pr(X d) 1-e-µd µd
?(d) Pi,j o(d), (?
p??a??t?ta p???ap??? af??e?? ? a?a????se?? e??a?
aµe??t?a)
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?st? pn ? p??a??t?ta t? s?st?µa ?a ß??s?eta? st??
?at?stas? n ?ta? t? s?st?µa e??a? st?? sta?e??
?at?stas? (steady state probabilities) S?µe??s?
G?a ??a a??a??eta µe???? ??????? d??st?µa, ?
a???µ?? t?? µetaß?se?? ap? t?? n st?? n1 e??a?
?d??? µe t?? a???µ? t?? µetaß?se?? ap? t?? n1
st?? n (/- 1). ?p?µ????, ??a ???e n
?s??e? pn-1?d pnµd pn(?/µ) pn-1(?/µ)2
pn-2(?/µ)n p0
??????ta? ? ?/µ (utilization factor) ????µe
pn?np0 , n1,2,3,. G?a ?a ß???µe t? p0
? ? p??a??t?ta t? s?st?µa ?a ??e? t??????st??
??a? pe??t? (1-p0) ? p??a??t?ta ?
e??p??et?t?? ?a e??a? apas????µ???? ?
a?aµe??µe??? a???µ?? pe?at?? (?) st? s?st?µa
e??a?
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? a???µ?? t?? pe?at?? st? s?st?µa a????eta?
d?aµat??? ?ta? ? 1 ?p?te ?
(d??. ?ta? ? t?µ? t?? ? p??s???e? t?? t?µ? t?? µ)
?p? t? ?e???µa t?? Little, ????µe ?t? ? µ?s?
?a??st???s? ? e??? pe??t? e??a?
? µ?s?? ??????, W, st?? ???? e??a?
?a? ? µ?s?? a???µ?? pe?at?? st?? ????, ?Q, e??a?

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?a??de??µa 1? ?p?t????s? e??? ?/?/1 s?st?µat??
for
?p?? ??/µ
?st? ?t? ? ???µ?? af??e?? ? ?a? ? ???µ??
e??p???t?s?? µ p???ap?as?????ta? µe µ?a sta?e??
k. ??te ta ?, ?Q ?a? pn pa?aµ????? aµet?ß??ta.
?1/(µ-?) ?a?
W?/(µ-?) ? ?a??st???s? st? s?st?µa ?, ?a? ?
?a??st???s? st?? ???? W, µetaß?????ta? ?? p???
1/k
????µ?? a?a????se?? ß(t)
????µ?? af??e?? a(t)
?p?ta?????? s?st?µa
????µ?? a?a????se?? ß(t)
????µ?? af??e?? a(t)
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?a??de??µa 2? Stat?st??? p???p?e??a ??a?t? FDM
Te???ste 100 s???d??? (sessions), µe af??e??
Poisson s???????? ???µ?? ? ?a? e??et???
?ata?eµ?µ??a µ??? pa??t??, ?? ?p??e? µ???????ta?
??a? s??desµ? µe ???µ? e??p???t?s?? µ
pa??ta/sec Stat?st??? ????p?e??a
???µ?? e??p???t?s?? µ
????p?e??a µe d?a??es? s????t?ta? (FDM) ??
???s?µ?p???s??µe FDM, t?te ? ???e s???d?? ??e?
???µ? ?/100 ?a? ß??pe? ???µ? e??p???t?s?? µ/100
? ????t???t?ta d?a??e?ta? se 100 ?sa µ???
? 100/(µ-?)
???µ?? e??p???t?s?? µ/100 a?? s???d?
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?/?/m S?st?µa ?????
?? af??e?? e??a? Poisson µe ???µ? ? ?a? ?p??????
m t? p????? e??p??et?t?? (servers). ?? ??????
e??p???t?s?? e??a? e??et??? ?ata?eµ?µ???? µe
pa??µet?? µ.
m e??p??et?t??

• ?????t?? ?t? ?p?????? n pe??te? µ?sa st?
  s?st?µa, µe n m, µ?a ??a a?a????s? ?a s?µße?
  µ?sa st? ep?µe?? ??????? d??st?µa d µe p??a??t?ta
  nµd.
• G?a ngtm, µ?a a?a????s? ?a s?µße? µe p??a??t?ta
  mµd.

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??
??
?st?
? p??a??t?ta ??a? pe??t??, ?ta? ft??e? st?
s?st?µa, ?a ß?e? ????? t??? e??p??et?t??
apas????µ????? e??a?
Erlang C formula (???s?µ?p??e?ta? st??
t??ef???a)
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? a?aµe??µe??? a???µ?? pe?at?? st?? ???? e??a?
? ?????? a?aµ???? W e??a?
?a? ? s???????? ?????? st? s?st?µa, e??a?
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?/?/m/m S?st?µa ?????
?? pe??te? p?? ft????? st? s?st?µa ?ta? ???? ??
e??p??et?t?? e??a? apas????µ????, ap????pt??ta?
?a? ?????ta? ???st??? ap? a?t?.
? p??a??t?ta ??a? pe??t?? ?a ß?e? ????? t???
e??p??et?t?? apas????µ????? e??a?
Erlang B Formula ?a ?,? e??a? µ????te?a ap?
?,t? st? ?/?/m s?st?µa, a??? de? ??e? t?s? ???µa
?a µ???µe ??a µ?s? ?a??st???s? se ??a s?st?µa p??
???e? pe??te?. Se ??a t??ef????? d??t??, ?
s?µpe??f??? t?? pe?at?? e??a? ??t? t? e?d??µes?
µeta?? t?? s?µpe??f???? p?? ep?de??????? ??
pe??te? se ??a ?/?/m ?a? se ??a M/M/m/m s?st?µa.
??p???? fe????? ?ta? de µp????? ?a e??p??et?????
?a? ??p???? s??e?????? ?a p??spa????.
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?a??de??µa
?? ?p???s??µe ?t? ?p?????? m100 s???d?? p??
µ???????ta? ??a s??desµ?. ?p?s?? ?p???t??µe ?t?
?p?????? 100 ?a????a s????t?t?? µe ta pa??ta ?a
d??µ???????ta? p??? ???e d?a??s?µ? ?a????. ??t?
e??a? ??a ?/?/100 s?st?µa.
Stat?st??? p???p?e??a µe ???s? m ?a?a????
100 ?p??a????a µe ???µ? e??p???t?s?? µ/100 t?
?a???a
?ta? t? f??t?? e??a? e?af?? t? s?st?µa
s?µpe??f??eta? sa? ??a FDM s?st?µa.
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G?a µe???? f??t?, t? s?st?µa s?µpe??f??eta? sa?
??a s?st?µa p?? ???s?µ?p??e? stat?st???
p???p?e??a
Stat?st??? p???p?e??a ??a ???????? t? ?a????
??a ?a???? µe ???µ? e??p???t?s?? µ
G?a µ???? ?
G?a µe???? ? ( )
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S?st?µa ?????
???a? pa??µ??? µe ??a ?/?/m s?st?µa, µ??? p??
t??a
?? µa??µat???? e?f??se?? µp????? ?a ß?e????, a?
p????µe t? ???? ap? t??
a?t?st???e? e?f??se?? t?? ?/?/m s?st?µat??.

? µ?s?? a???µ?? pe?at?? st? s?st?µa e??a?
?a?