4通信网理论-排队论基础1.pdf

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๙ྐຩ৘ં
č೘Ďஆؒંა๙ྐຩြༀٳ༅
ஆؒંࠎԤ
ࡀဝ
КႯ໭ཌྍ࠯ඌ࿹࣮൩
Tel:+86-10-62261492
Fax:+86-10-62269455
E-mail:******@
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ஆؒંۀඍ
•ஆؒ༢๤
–ေ౰ڛༀ֥ܤग़
–ิ܂ڛༀ֥ڛༀჴ
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ஆؒંۀඍ
•ஆؒ༢๤֥ܼٗྟ
–ܤग़+ڛༀ¹ஆؒ༢๤
–઱؛๤၂đܼၬ߄ğ
•๙ྐᇏğ
–ޫࢡ——ཌਫ਼
–ྐ༏Ї——ٳቆࢌߐࠏ
–၍׮ุ——ڛༀ౵
•࠹ෘࠏğ
–ሹཌᆷ਷——CPUԩ৘
–ඔऌੀ——թԥఖ
•ః෱ğ
–ֱࠏ——ٝॢഡീ
–ग़ࠏ——஝֡
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ஆؒંۀඍ
•ஆؒགྷའթᄝ֥ࠎԤ
–ሧჷ֥Ⴕཋྟ
–ླ౰֥ෛࠏྟ
•ஆؒ༢๤֥گᄖྟᄝႿෛࠏྟ
–֞ղა৖ಀčڛༀੱĎन҂ಒקđ۽ቔႿෛ
ࠏሑ෿
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ஆؒંۀඍ
•ஆؒ༢๤ᇏ֥ሧჷԊ๬
–ሧჷഒ——ܤग़ஆؒӉ——ڛༀᇉਈ༯ࢆ
–ሧჷ؟——ڛༀ༽ᇂ——ሧჷশٮ
•ஆؒ༢๤ഡ࠹ଢѓ
–ູܤग़ิ܂ડၩڛༀ๝ൈิۚሧჷ০Ⴈੱb
čა๤࠹ҕඔބ۽ቔٚൔႵܱĎ
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ೂޅ૭ඍ၂۱ஆؒ༢๤č1Ď
•ஆؒ༢๤೘ေ෍:mđȯđȰ
–m-Գ१ඔđіൕሧჷ֥ਈ
•ॖ๝ൈཟܤิ܂ڛༀ֥ഡീඔb
•čֆԳ१mĠ؟Գ१mĎ
–ȯܤग़֞ղੱč௜नĎ
–Ȱğ༢๤ڛༀੱč௜नĎ
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ೂޅ૭ඍ၂۱ஆؒ༢๤č1Ď
ȯ¼——ڵހ౞
•Վ೘ਈॖၘᆩࠇॖҩԛđ֌૭ඍஆؒ༢๤đՎ೘ေ෍҂ԉٳ
•ᇶေ౼थႿtބȷ֥๤࠹หྟčٳ҃Ďބஆؒܿᄵ
ii
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ೂޅ૭ඍ၂۱ஆؒ༢๤č1Ď
•Վ೘ਈॖၘᆩࠇॖҩԛđ֌૭ඍஆؒ༢๤đ
Վ೘ေ෍҂ԉٳ
•ஆؒ༢๤ྟିᇶေ౼थႿ
–tބȷ֥๤࠹หྟčٳ҃Ď
ii
–ڛༀܿᄵބஆؒܿᄵ
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֞ղაڛༀ֥ٳ҃ো྘
•M——ᆷඔٳ҃
–௜໗ྟ
–༎ൿྟ
–׿৫ྟ
•ѻ඾ٳ҃
•Er——rࢨιغধٳ҃
•HR——Rࢨᆷඔٳ҃
•Dٳ҃——ಒקྟٳ҃
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༢๤۽ቔٚൔ
•ڛༀܿᄵ
–༵༵֞ڛༀFirstIn,FirstServe
–ު༵֞ڛༀLastIn,FirstServe
–Ⴊ༵ᇅڛༀPriority
•ஆؒܿᄵ
–֩ր྘
•҂ऋध༢๤
–ࢩᇀ྘
•n=m,ࠧൈऋधđ׈߅ຩ
•m<n,࿼ൈऋधđߏԊ౵ඔऌ๙ྐ
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ஆؒݖӱ֥ൌ২
ܤ
ग़
ඔ
tttU
12n
֞ղ
τττ
12n
৖ಀ
ફൈ5༽ൈ*
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ೂޅٳ༅၂۱ஆؒ༢๤
•ஆؒؒӉk֥ۀੱٳ҃aनᆴaٚҵ
•֩րൈࡗw֥नᆴaٚҵ
•ڛༀൈࡗτ
•༢๤ൈࡗS
•༢๤ིੱη
–Գ१ᅝႨੱđηĢSN
•໗קྟ
–ρȯµ
•JG
ρN

҂ऋध༢๤đOPU
TUBCMF
•JG
ρN

ࢩᇀ྘༢๤đTUBCMF

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ؒӉL
ஆؒӉ؇—Uටࡗ༢๤ଽ֥ܤग़ඔčݣᄝԳ१֥Ď
L—৖೛ෛࠏэਈ
೘ᇕܴҳğ
d—ܤग़֞ղൈܴҳؒӉູk֥ۀੱ
k
r—ܤग़৖ಀൈܴҳؒӉູk֥ۀੱ
k
čၛഈູႵ่ࡱԎဢĎ
Q—čڛༀჴĎෛࠏܴҳؒӉູL֥ۀੱ
L
ᄝቋࡥ༢๤ᇏđp=r=d
kkk
ᇿၩğః෰֥༢๤ᇏᆃࠫᇕ
—௜नؒӉđႻӫ༢๤ඔܴҳ֥ࢲݔ൞҂๝֥b
k
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֩րൈࡗXބڛༀൈࡗτ
֩րൈࡗX
Ֆ֞ղ→ष൓ڛༀ֥ൈࡗđ൞৵࿃ෛࠏэਈ
w—௜न֩րൈࡗ
֩րൈ࿼൞ྐ༏ᄝຩ઎ᇏ௜नൈ࿼֥ᇶေ҆ٳ
ః෰ൈ࿼đೂԮൻൈ࿼aԩ৘ൈ࿼၂Ϯ൞ࢠཬ֥
ڛༀൈࡗτğ
ष൓ࢤ൳ڛༀ→ڛༀи৖ಀ
τ—௜नڛༀൈࡗ
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༢๤ൈࡗT
•ࠇӫ༢๤๔਽ൈࡗ֞ղ→৖ಀ
ཁಖႵsw=+τ,
Ⴟ൞Ⴕsw=+τ
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༢๤ིੱη
•༢๤ིੱηğ௜नԳ१ᅝႨੱ
–܋N۱Գ१đଖൈख़ႵೂS۱ФᅝႨđᄵ
ֆԳ१,ࠧફ֥ۀੱ
r
η=r
m؟Գ१,ླ౰
m
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໗קྟᆷѓ
•໗קྟᆷѓğρȯµ
҂ऋध༢๤ğρ>m҂໗ק.
(ֆԳ१)ρ>1w↑ࠇw→∞
ࢩᆸ྘༢๤ğρ>mđಯಖ໗ק
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Little܄ൔ
ᄝ၂۱௜न֞ղੱູλ֥ஆؒ༢๤đ
ᄝ௜नၩၬഈđႵλ⋅=Tk
Littleק৘ൡႨႿ಩ޅஆؒ༢๤
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ೂޅ૭ඍ၂۱ஆؒ༢๤č2Ď
•Kendall࠺ݼ——A/B/m(N,n)
–A——֞ղܿੰđٳ҃a(t)
–B——ڛༀܿੰđٳ҃C U


–N—Գ१ඔ
–/—ܤग़ჷđమᄝܤग़ඔđസ੻ູ∞
–O—ࢩᆸؒӉđസ੻ູ∞đ҂ऋध
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ೂޅ૭ඍ၂۱ஆؒ༢๤č2Ď
•֞ղაڛༀ֥ٳ҃ো྘
–M——ᆷඔٳ҃
–ѻ඾ٳ҃
–Er——rࢨιغধٳ҃
–HR——Rࢨᆷඔٳ҃
–Dٳ҃——ಒקྟٳ҃
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ଢѓ໙ี
•ࠎЧ໙ี
–M/M/1
–M/M/mčnĎ
•ᇏࠩ໙ี
–M/G/1
–G/M/1
•ۚࠩ໙ี
–G/G/1
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ೂޅٳ༅၂۱ஆؒ༢๤
•၂Ϯটࢃđઔغॖڏ྘ஆؒ༢๤౰ࢳ಩
ၩൈख़֥ሑ෿ۀੱࢠಸၞུĠط٤ઔغ
ॖڏஆؒ༢๤ᄝҐႨళೆઔغॖڏ৽֥
൭مٳ༅ൈ౰ࢳ֞ղൈख़ࠇ๼ಀൈख़ሑ
෿ۀੱбࢠಸၞུđၹՎ೘ᆀᆭࡗೂି
ࡹ৫ఃଖᇕܱ༢ࡼޓႵϺᇹ
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ೂޅٳ༅၂۱ஆؒ༢๤
•௜नؒஆ൞ࡗൈր֩न௜ބ؇Ӊਙؒ༢๤֥ਆ
۱ቋࠎЧ֥ྟିҕඔ
–.,ॉ੮಩ၩ၂۱ֆ၂ڛༀᆀஆؒ༢๤bࡌഡܤग़
֥௜न֞ղࡗۯູ4ٳᇒđ௜नڛༀൈࡗູ3ٳᇒđ
ѩ౏ܴࣜҳᄝ಩ၩൈख़௜न֩րؒਙӉ؇ູ5ದb
൫໙ğč1Ďᆞᄝࢤ൳ڛༀ֥ܤग़ඔ֥௜नĤč2Ď
ஆؒ༢๤ᇏ֥௜नሹದඔčЇওᆞᄝڛༀᇏ֥ܤग़Ď
ູٳ15čഒ؟ູࡗൈր֩न௜֥ग़ܤĎ3čĤഒ؟
ᇒĤ18ٳᇒĤ20ٳᇒĤĎ
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ೂޅٳ༅၂۱ஆؒ༢๤
•ࢳمğ
ಒקሑ෿эਈđೂL
ਙሑ෿ሇ၍๭
ਙሑ෿ሇ၍ٚӱ
౰ࢳଢѓҕਈ
•ࠫ۱܄৘ൔࡌഡ
–ਆ۱ෛࠏэਈ๝ൈؿള֥ۀੱູۚࢨ໭౫ཬđ҂Ⴭ
ၛॉ੮
–၂Ϯ؟ॉ੮ஆؒ༢๤ղ֞໗෿౦ঃ༯֥หྟ
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M/M/1໙ี
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•UູL෿đȐUଽ֞ղದđ໭ದ৖ಀ
ۀੱ
p()tt⋅∆λ(1−µλ∆t)≅p(tt)⋅∆
kk−−11
•UູL෿đȐUଽ৖ದđ໭֞ղ
ۀੱ
p(t)⋅µ⋅∆t(1−λ∆t)≅p(t)⋅µ∆t
k+1k+1
•UູL෿đȐUଽ֞ղದđ৖ಀದ
2
ۀੱ
p(t)⋅λµ∆t
k
•UູL෿đȐUଽ໭֞ղđ໭৖ಀ
ۀੱ
p()tt(1−λ∆−)(1µλ∆t)≅p()t⋅(1−∆t−µ∆t)
kk
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֤:(pt+∆t)=λ∆tp(t)+µλ∆tp(t)+p(t)(1−∆t−µ∆t)
kk−+11kk
pt()+∆t−p(t)
kk
ࠧ=+λµp()tp()t−(λ+µ)p()t
kk−+11k
∆t
∆t→0֤֝ඔ
dp(t)
k
=λp(t)+µp(t)−(λ+µ)p(t)
k−1k+1k
dt
ࠧॏٚڏઌۢ଍غӱ
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dp()t
k
໗෿༯đ=0
dt
pk()ࣉೆ෿=p(৖षk෿)
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——๙ݖଛݦඔ౰֥֝ٚمট࠹ෘ
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2
σk
ρ
2
ೂ๭ρ>
k
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֩րൈࡗw
೏༢๤ၘႵkದ,ট၂ದ֥֩րൈࡗwູkದ֥ڛༀൈࡗࠧ:
k
w=τ+τ+"+τ=τ
12k∑i
i=1
τ֥หᆘݦඔ:(a(t)֥ڱ൦эߐ)
i
∞µ
−µτjzτ
Z(z)=µeedτ=
∫0
µ−jz
k۱τބ֥หᆘݦඔµk
iw=()
k
µ−jz
ၹູkູ৖೛эਈ,ܣw֥หᆘݦඔ:
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ၹູkູ৖೛эਈ,ܣw֥หᆘݦඔ:
∞
kk
w(z)=wp
∑
k=0
∞
µkkkµρ
=()p`p=(1−ρ)ρ,਷x=()
∑k
k=0µ−jzµ−jz
∞
k(1−ρ)x
=(1−ρ)x==(1−ρ)(1+)
∑
k=01−x1−x
µρ
=(1−ρ)(1+)
µ(1−ρ)−jz
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w֥ૡ؇:1
−jzw
f(w)=w(z)edz
∫
2π
−µ(1−ρ)w
=(1−ρ)δ(w)+(1−ρ)λe
௜न֩րൈࡗ:−µ(1−ρ)w
w=wf(w)dw=0+(1−ρ)λwedw
∫∫
1ρ1
=(1−ρ)λ=
22
µ(1−ρ)1−ρµ
w֥ٚҵ:
222ρ(2−ρ)
σw=wf(w)dw−w=
∫22
µ(1−ρ)
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༢๤ൈࡗ(௜न๔਽)
1
s=w+τ=
µ(1−ρ)
ّဒLittle܄ൔ
λρ
λs===k
µ(1−ρ)1−ρ
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ફ௹ა༽௹
•༽௹֥ٳ҃൞༢๤ԩႿ໭ܤग़ሑ෿ު֞Ⴕ၂۱ܤग़֞ղ
ᆭࡗ֥ൈࡗđၹՎᄝѻ඾༢๤ᇏđაܤग़֥֞ղܿੰཌྷ๝b
−λI
fI()=λeఃनᆴູI=1/λ
I
ફ௹֥ٳ҃ࢠ଴౰֤bܤग़֞ղൈࡗაڛༀປиൈࡗླ
ေડቀ၂ק่֥ࡱđٚିྙӮ၂۱ܤग़ඔູn֥ફ௹Tb
֌ೂݔᆺܱྏఃनᆴط҂ܱྏఃٳ҃đᄵбࢠಸၞb
I11
༢๤ॢ༽֥ۀੱ
p=−1ρ===
0
IT++T1λT
1+
I
1
֤T=
µ−λ
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ફ௹ᇏ֥௜नದඔ္ࣼ൞ફ่֥ࡱ༯֥௜नؒӉđطફ
֥ۀੱູρ,෮ၛρ
ρnk==
1−ρ
1
n=
1−ρ
M/M/1ᄠ෿ࢳ(੻)
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ᇀՎ,֤M/M/1ࢲંೂ༯
k
kpۀੱ=−(1ρρ)
k
ρ2ρρ(2−)
௜नؒӉk==σ
w2
1(−−ρ1ρ)
11ρ
௜न֩րൈࡗwk=⋅=⋅
1−ρ
µµ
1
༢๤ൈࡗsw=+τ=
µρ(1−)
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1
௜न༽௹I=
λ
1
௜नફ௹T=
µ−λ
ॢ༽ੱp=1−ρ
0
ફۀੱρ
ࠣ۲эਈٳ҃ࢳ༅ࢲݔ
௜नᆴࢲݔन౼थႿρ
⎧ིੱᆷѓ:Գ१ᅝႨੱη=ρ
⎪
ρ֥೘ק໗⎨ၬၩ۱ྟᆷѓ:ρ>1҂໗
⎪
ܤग़ܴׄρ↑→k↑
⎩
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M/M/1༢๤ཬࢲ
•෮ၛ,M/M/1ᇶေ઱؛ࠢᇏᄝρ֥࿊౼
ڛༀᇉਈა༢๤ིੱᆭࡗ֥઱؛
•ࢳथଢѓ
–Ќᆣ໗קᄎྛ่ࡱ༯ิིۚੱ:
•ծീ:
–ᄹࡆԳ१ඔ
–ࢩᆸ྘
–ࢩᆸ؟Գ१
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