光パケット交換機のトラヒック理論(その1)-指数関数分布パケット長に対する近似解法-
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(2) . శࡄࠤ࠶࠻឵ᯏߩ࠻ࡅ࠶ࠢℂ⺰㧔ߘߩ㧝㧕 ̆ ᜰᢙ㑐ᢙಽᏓࡄࠤ࠶࠻㐳ߦኻߔࠆㄭૃ⸃ᴺ ̆ A Traffic Theory for Optical Packet Switchers (Part 1) – Approximate Solutions for Exponential Distribution Packet Length – ᵏม* Yasuji Murakami Abstract Optical packet switchers are expected for future IP network core nodes, to overcome the throughput bottlenecks and the huge power consumptions of electronics nodes. Various optical packet switch architectures have been proposed, in which fiber delay lines (FDLs) are used for optical buffers.. In this paper, a traffic theory for optical. packet switchers is presented, and especially, approximate solutions for exponential distribution packet length are driven.. The solutions are useful to calculate packet. loss probabilities and mean packet delay times.. 㧝㧚ߪߓߦ ࠗࡦ࠲ࡀ࠶࠻࠻ࡅ࠶ࠢ߇Ფᐕ೨ᐕᲧ 1.5 ߩિ߮₸ߢჇᄢߒߡߊ⁁ᴫߦ߅ߡ㧘 㔚ሶಣℂߦࠃࠆ IP ࡞࠲ߢߪォㅍ⢻ജߦᔅߕ㒢⇇߇ࠇࠆ㧚߹ߚ㧘IP ࡞࠲ߩ㔚ജᶖ⾌ ㊂߽ߎߩ߹߹Ⴧᄢࠍ⛯ߌࠇ߫㧘㔚ജࠦࠬ࠻߇ㆇ↪ࠦࠬ࠻ߩᄢ߈ߥᲧ㊀ࠍභࠆࠃ߁ߦߥࠆ ߣߣ߽ߦ㧘᷷ᥦൻࠍഥ㐳ߔࠆ⚿ᨐߣߥࠆ㧚ᄢ߈ߊߪߎߩ 2 ߟߩ㗴ࠍ⸃ߔࠆᣇᴺߣ ߒߡ㧘శାภߩ߹߹ࡄࠤ࠶࠻ࠍಣℂߔࠆశࡄࠤ࠶࠻឵ᯏߩ⊓႐߇ᦼᓙߐࠇߡࠆ㧚ߔߥ ࠊߜ㧘శࡈࠔࠗࡃㅢାࠪࠬ࠹ࡓߦࠃࠅ 1 ࿁✢ߩવㅍㅦᐲ߇ 100Gbps ߦ㆐ߔࠆ⁁ᴫߦߥࠆ ߣ㧘વㅍ⢻ജߩ㜞శᛛⴚࠍ↪ߚశ឵ᯏ߇㧘ᰴઍߩ IP ࡞࠲ࠍᜂ߁ߎߣߦߥࠆߪ ߕߢࠆߣ߁ᦼᓙߢࠆ㧚 శߪ㧘߽ߣ߽ߣࡆ࠶࠻ᖱႎࠍዊߐࡄࡢߢォㅍߔࠆߎߣߪᓧᗧߢࠆ߇㧘శ⥄りࠍ ᓮߔࠆߎߣߪᓧᗧߢߪߥ㧚శࡄࠤ࠶࠻ࠍశߩ߹߹ォㅍಣℂߔࠆࡄࠤ࠶࠻឵ᯏߪ㧘ᄙߊ ߩ⎇ⓥ⠪ߩᦼᓙࠍ⢛⽶ߘߩታߦะߌߡ♖ജ⊛ߥ⎇ⓥ߇⛯ߌࠄࠇߡࠆ[1-2]߇㧘߹ߛ߹ ߛߪ㆙ߩ߇⁁ߢࠆ㧚ࡔࡕ߿⺰ℂṶ▚ߥߤ㧘ࠄ߆ߦశᛛⴚߢߪਇᓧᗧߥಽ㊁߇ ࠆ߆ࠄߢࠆ㧚ዋߥߊߣ߽㧘⺰ℂṶ▚ߪ㔚᳇⊛ߦⴕ߁ߎߣߢ⸃ࠍ࿑ࠆߩߢࠈ߁㧚 ࡔࡕߦߪ㧘ㆃశ㧔slow lights㧕ߥߤ♖ജ⊛ߦ⎇ⓥߐࠇߡࠆ߽ߩ߇ࠆ߇㧘ታᕈ ߆ࠄߺࠆߣశ శࡈࠔࠗࡃㆃᑧ✢㧔optical fiber delay lines㧦FDL㧕ࠍ↪ߔࠆߎߣ߇ㄭߢ ࠆ㧚FDL ߪ㧘㔚᳇ RAM ߣ⇣ߥࠆᰴߩࠃ߁ߥ․ᓽࠍᜬߟ㧚 (1) RAM ߢߪછᗧߩ⫾Ⓧᤨ㑆ߣછᗧᤨೞߢߩ⺒ߺߒ߇ታⴕߢ߈ࠆ߇㧘FDL ߪߘߩ㐳ߐ ߦᲧߒߚ৻ቯ㊂ߩ⫾Ⓧᤨ㑆ߒ߆ᓧࠄࠇߥ㧚FDL ߢߩ⫾Ⓧᤨ㑆ߪ㧘శࡈࠔࠗࡃ㐳ߦ Ყߔࠆߩߢ㧘FDL 㐳ߩනࠍㆃᑧᤨ㑆ߢ⠨߃ࠆ㧚ߎࠇࠍᤨ ᤨ㑆☸ᐲ㧔time granularity㧕 ߣ߱㧚 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ *ᄢ㒋㔚᳇ㅢାᄢቇ ᖱႎㅢାᎿቇㇱ ㅢାᎿቇ⑼ !. - 1 .
(3) (2) ⴣ⓭࿁ㆱߦ㑆ߦวࠊߥߌࠇ߫㧘ࡄࠤ࠶࠻ߪᑄ᫈ߐࠇࠆ㧚 శࡄࠤ࠶࠻឵ᯏߦ߅ߡࡄࠤ࠶࠻ᑄ᫈₸߿ᐔဋㆃᑧᤨ㑆ߥߤߩ⸳⸘୯ࠍ᳞ࠆ࠻ࡅ ࠶ࠢℂ⺰ߩᄙߊߪ㧘FDL ࠍ↪ߒߚశࡃ࠶ࡈࠔ᭴ᚑࠍᛒߞߡࠆ㧚శࡃ࠶ࡈࠔ᭴ᚑߦߪᄙ ߊߩឭ᩺߇ࠆ߇㧘ㅢᏱߪᤨ㑆☸ᐲߩᢛᢙߩ FDL ࠍᢙᄙߊ↪ᗧߒߡ㧘ᤨ㑆☸ᐲߩᦨㆡ ൻࠍ࿑ࠆ߽ߩߣߥߞߡࠆ[3-6]㧚ߔߥࠊߜ㧘FDL ߩ㐳ߐಽᏓࠍߤߩࠃ߁ߦߔࠇ߫㧘ࡄࠤ࠶ ࠻ᑄ᫈₸ࠍᛥ߃ࠆߎߣ߇ߢ߈ࠆ߆߁ߣޔ㗴ߢࠆ㧚ߎࠇߦኻߒߡ㧘ㄭૃᢙ୯⸃[3-4]߿ ᢙ୯ࠪࡒࡘ࡚ࠪࡦ[5-6]ߥߤߐ߹ߑ߹ߥขࠅ⚵ߺ߇ߐࠇߡࠆ߇㧘ߕࠇ߽➅ࠅߒ⸘ ▚ࠍᔅⷐߣߒ㧘ㅢߒߩࠃℂ⺰ᑼࠍᓧߡࠆࠊߌߢߪߥ㧚 ᧄ⺰ᢥߪ㧘శࡄࠤ࠶࠻឵ᯏߩ࠻ࡅ࠶ࠢℂ⺰ߦ㑐ߔࠆ߹߹ߢߩᚑᨐࠍ〯߹߃㧘ᢥ₂ [5]ߩ⸥ㅀߦᴪౝኈࠍℂ⸃ߒ߿ߔᒻߦᢛℂߔࠆߣߣ߽ߦ㧘ᜰᢙ㑐ᢙಽᏓࡄࠤ࠶࠻㐳ߦኻ ߔࠆᓙߜᤨ㑆ಽᏓߩㄭૃ⸃ࠍਈ߃ߡࠆ㧚ዉߒߚㄭૃ⸃ߩ♖ᐲࠍࠪࡒࡘ࡚ࠪࡦ⚿ᨐ ߣᲧセߒ㧘ߐࠄߦૐ⽶⩄⁁ᘒߦ߅ߌࠆశࡃ࠶ࡈࠔ᭴ᚑߩᬌ⸛ࠍⴕߞߡࠆ㧚. 㧞㧚శࡃ࠶ࡈࠔࡕ࠺࡞ శ឵ᯏߩ᭴ᚑߣߒߡ㧘࿑㧝. శࡃ࠶ࡈࠔ. ߦ␜ߔࠃ߁ߦ㧘ࠬ࡞ࡊ࠶࠻. 1. 1. 2. 2. ⠨߃ࠆ㧚శࠬࠗ࠶࠴ߩታᣇᴺ ߦߪ㧘ⓨ㑆ࠬࠗ࠶࠴[1]ߣᵄ㐳ࠬ. శࠬࠗ࠶࠴ . . 3. 㒢ߩߥജᓙߜⴕᒻ[7]ࠍ. 3. ࠗ࠶࠴[2]߇ឭ᩺ߐࠇߡ߅ࠅ㧘೨ ⠪ߪⓨ㑆ᄙ㊀ᣇᑼࠍᓟ⠪ߪᵄ㐳 ᄙ㊀ᣇᑼࠍ↪ߒߡࠆ㧚ߕ. N. N. ࠇ߽㧘ജᓙߜⴕᒻశ឵ᯏ ࠍታߢ߈ࠆ㧚. ࿑㧝㧚ജᓙߜⴕᒻశࠬࠗ࠶࠴ߩၮᧄ᭴ᚑ. ᓙߜⴕߣߥࠆജࡃ࠶ࡈࠔ ߦ FDL ࠍ↪ߔࠆ᭴ᚑࠍ㧘࿑ 㧞ߦ␜ߔ㧚㧝ߟߩജߦኻߒߡ శࡄࠤ࠶࠻ߩⴣ⓭ࠍ࿁ㆱߔࠆߚ. శࡈࠔࠗࡃㆃᑧ✢. 㧘 B ᧄߩ FDL ࠍㆬᛯߢ߈ࠆ. 0. ᭴ ᚑ ߢ 㧘 i ⇟ ⋡ ߩ FDL ߪ. 1D 2D. ߓࠆ㧚ߎߎߢ ޔD ߪᤨ㑆☸ᐲߢ ࠅ㧘శࡈࠔࠗࡃߩ㐳ߐනࠍ. 3D . శࠬࠗ䏓࠴. i 1
(4) D 㧘 1 d i d B ߩㆃᑧࠍ↢. L ߣߔࠆߣ㧘 D. B 1
(5) D. nL c 㧘 n 㧦. శࡈࠔࠗࡃߩታലዮ᛬₸㧘 c 㧦. 㨯㨯㨯. ⌀ⓨਛߩశㅦߢࠆ㧚ߒߚ߇ߞ ߡ㧘ߎߩశࡃ࠶ࡈࠔߢߪ㧘 0 㧘 ࿑㧞㧚శࡈࠔࠗࡃㆃᑧ✢ߦࠃࠆశࡃ࠶ࡈࠔ. 2- - .
(6) t. t. 0. t2 శࡄࠤ࠶࠻ߩㅌ. శࡄࠤ࠶࠻ߩ⌕. 0. D. s1. s1 ⓨᦼ㑆. Ԙ. s2. Ԙ. s2. t2. w2. ԙ. t2 0 w2 d D ߩߣ߈. ԙ. s3. W2. t3. W2. Ԛ. D w2. శࡃ࠶ࡈࠔജ. శࡃ࠶ࡈࠔജ (a). ࡄࠤ࠶࠻Ԙߩ⌕㧔 t. 0㧕. ࡄࠤ࠶࠻ԙߩ⌕㧔 t. (b). t. t2 㧕. t3 శࡄࠤ࠶࠻ߩㅌ. 3D. 2D. D. 0. s1 Ԙ. t3. s2. ⓨᦼ㑆. ԙ. s3. 2 D w3 d 3D ߩߣ߈. w3. W3. Ԛ. 3D w3. W3 (c). ࡄࠤ࠶࠻Ԛߩ⌕㧔 t. ⓨᦼ㑆 Ԛ. ⓨᦼ㑆 ԙ. t3㧕. శࡃ࠶ࡈࠔജ Ԙ. ❥ᔔᦼ㑆 (d) ࡄࠤ࠶࠻Ԙ㨪Ԛߩജ. ࿑ 3 శࡈࠔࠗࡃㆃᑧ✢ࡃ࠶ࡈࠔߦ߅ߌࠆࡄࠤ࠶࠻ߩᵹࠇ. 3- - .
(7) 1D 㧘 2 D 㧘㨯㨯㨯㧘 T. B 1
(8) D ߩ㔌ᢔ⊛ߥㆃᑧᤨ㑆ߣߥࠆ⌕㗅ಣℂ㧔first. service㧦FCFS㧕߇ⴕࠊࠇࠆ㧚 T. come first. B 1
(9) D એߩㆃᑧᤨ㑆߇ᔅⷐߣߥࠆࡄࠤ࠶࠻ߪ⎕᫈. ߐࠇࠆ㧚 ࿑㧞ߩశࡃ࠶ࡈࠔߦ߅ߡ㧘శࡄࠤ࠶࠻߇ᵹࠇࠆ᭽ሶࠍ࿑㧟ߦ␜ߔ㧚㧘࿑(a)ߦ␜ߔࠃ ߁ߦ㧘Ԙ㧘ԙ㧘߅ࠃ߮Ԛߩ 3 ߩశࡄࠤ࠶࠻߇㧘శࡃ࠶ࡈࠔߩജㇱߦߘࠇߙࠇᤨೞ 0 㧘. t 2 㧘߅ࠃ߮ t 3 ߦ⌕ߔࠆߎߣߣߔࠆ㧚߹ߚ㧘శࡄࠤ࠶࠻Ԙߪචಽߦ㐳ߩߢ㧘ᓟ⛯ߔࠆశ ࡄࠤ࠶࠻ԙߣԚߣߪജߢⴣ⓭ߔࠆߣߔࠆ㧚ߘߎߢߪ㧘ⴣ⓭ࠍ࿁ㆱߔࠆߚᰴߩࠃ߁ߥಣ ⟎߇ขࠄࠇࠆ㧚 ࿑(b)ߦ␜ߔࠃ߁ߦ㧘ᤨೞ t 2 ߦ߅ߡ㧘ࡄࠤ࠶࠻ԙߪዋߥߊߣ߽ w2 㧘 0 w2 d D ߩᓙߜ ᤨ㑆߇ᔅⷐߢࠆߣ߈㧘ԙߪ㧞⇟⋡ߩ FDL ߦォㅍߐࠇࠆ㧚ߎߩߣ߈㧘FDL 㐳߇㔌ᢔ⊛ߢ ࠆߚߦ W 2. D w2 ߩⓨᦼ㑆㧔void periods㧕߇ᔅⷐߣߥࠆ㧚. ᤨೞ t 3 ߦ߅ߡߪ㧘࿑(c)ߦ␜ߔࠃ߁ߦ㧘ࡄࠤ࠶࠻Ԛߪ w3 㧘2 D w3 d 3D ߩᓙߜᤨ㑆߇ ᔅⷐߢࠆߣߥࠆߣ㧘Ԛߪ 4 ⇟⋡ߩ FDL ߦォㅍߐࠇࠆ㧚ࡄࠤ࠶࠻ԙߣߪ㧘W 3. D w3 ߩ. ⓨᦼ㑆ࠍ߽ߟ㧚ߒߚ߇ߞߡ㧘Ԙ߆ࠄԚ߹ߢߩశࡄࠤ࠶࠻ߪ㧘࿑(d)ߦ␜ߔࠃ߁ߦ㧘ࡄࠤ࠶࠻ ԙ㧘Ԛߩవ㗡ߦⓨᦼ㑆߇ߟߚ⁁ᘒߢജߐࠇࠆ㧚ߎߩⓨᦼ㑆ߪ㧘FDL ߇ᤨ㑆☸ᐲ D ࠍ නߣߔࠆ㔌ᢔ⊛ߥ୯ߢࠆߚ㧘FDL ߦォㅍߐࠇࠆࡄࠤ࠶࠻ߦᔅߕઃടߐࠇࠆ߽ߩߢ ࠆ㧚⚿ᨐ⊛ߦ㧘ࡄࠤ࠶࠻Ԙ߆ࠄԚ߹ߢߪㅪ⛯ߒߚࡄࠤ࠶࠻ߣߺߥߔߎߣ߇ߢ߈㧘ߎߩᦼ ❥ᔔᦼ㑆㧔busy period㧕ߣ߁㧚 㑆ࠍ❥ ৻⥸⊛ߦߪ㧘࿑ 4 ߦ␜ߔࠃ߁ߦ㧘⌕ߒߚࡄࠤ࠶࠻߇㧘ዋߥߊߣ߽ w ߩᓙߜᤨ㑆߇ᔅⷐ. శࡄࠤ࠶࠻ߩㅌ. i 1
(10) D. iD. 0 ❥ᔔᦼ㑆. ߬ߌߞߣ. ߬ߌߞߣ. ªwº «« D » ». W. W w. ࿑ 4 శࡃ࠶ࡈࠔߦ߅ߌࠆᓙߜⴕ. 4- - . iD. iD w.
(11) ߥߣ߈㧘ߎߩࡄࠤ࠶࠻ߪᰴߩࠃ߁ߦಣℂߐࠇࠆ㧚 (1) i 1
(12) D d w iD ߩߣ߈㧘 i 1
(13) ⇟⋡ߩ FDL ߦォㅍߐࠇࠆ㧚ߎߩߣ߈㧘వ㗡ߦߪ. . W. ªwº iD w 㧘 « » «D». iD 㧘. . 㧔2.1㧕. ߩⓨᦼ㑆 W ߇ઃടߐࠇࠆ㧚ߎߎߢ㧘 ªx º ߪ㧘 x ࠍ߃ࠆᦨዊᢛᢙࠍᗧߔࠆ㧚 (2) T. B 1
(14) D w ߩߣ߈㧘ᑄ᫈ߐࠇࠆ㧚. ⓨᦼ㑆߇ઃടߐࠇࠆಽ㧘឵ᯏߦ߆߆ࠆࡄࠤ࠶࠻⽶⩄ߪታ㓙ߩ⽶⩄ࠃࠅㆊߣߥࠆ㧚ⓨ ᦼ㑆ߪ㧘శࡃ࠶ࡈࠔ߇ⓨߩߣ߈⌕ߔࠆࡄࠤ࠶࠻ߦߪઃടߐࠇߥ߇㧘శࡃ࠶ࡈࠔߦࡄࠤ ࠶࠻߇⫾Ⓧߐࠇߡࠆߣ߈ߦߪઃടߐࠇࠆ㧚ߘߎߢ㧘ⓨᦼ㑆ࠍߚࡄࠤ࠶࠻ࠍࠄߚߦ ߁ߣޠߣߞߌ߬ޟฬ⒓ߢቯ⟵ߔࠆ㧚ߔߥࠊߜ㧘߬ߌߞߣߦߪ㧘 Ԙ శࡃ࠶ࡈࠔ߇ⓨߩߣ߈⌕ߔࠆࡄࠤ࠶࠻㧚ߎࠇࠍ㧘ೋᦼ⌕ࡄࠤ࠶࠻㧔first arrival packets㧕ߣ߮㧘ⓨᦼ㑆ࠍᜬߚߥ㧚 㕖ೋᦼ ԙ శࡃ࠶ࡈࠔߦࡄࠤ࠶࠻߇⫾Ⓧߐࠇߡࠆߣ߈⌕ߔࠆࡄࠤ࠶࠻㧚ߎࠇࠍ㧘㕖 ⌕ࡄࠤ࠶࠻㧔non-first arrival packets㧕ߣ߮㧘ⓨᦼ㑆ࠍࠎߛ㐳ߐߣߥࠆ㧚 ߩ㧞⒳㘃ߩࡄࠤ࠶࠻߇ࠆ㧚ߎࠇࠄ߬ߌߞߣ߇㓗㑆ߥߊㅪ⛯ߒߡജߐࠇࠆᦼ㑆߇㧘❥ᔔ ᦼ㑆ߣߥࠆ㧚. 㧟㧚࠻ࡅ࠶ࠢℂ⺰⸃ᨆ IP ࡄࠤ࠶࠻࠻ࡅ࠶ࠢߩ⌕ㆊ⒟ߦ߅ߡߪ㧘ᄙߊߩ࠻ࡅ࠶ࠢࠍ㓸✢ߔࠆၮᐙࡀ࠶࠻ ࡢࠢߦ߅ߡ⛔⸘⊛ߥᕈ⾰߇ࡐࠕ࠰ࡦಽᏓߦ᧤ߔࠆߎߣ߇⍮ࠄࠇߡ߅ࠅ㧘ㄭૃ⊛ߦࡑ ࡞ࠦࡈㆊ⒟ߢࠆߣߒߡ⸳⸘㗴ߦ↪ߔࠆߎߣ߇ߢ߈ࠆ㧚ࡄࠤ࠶࠻㐳ߪ 58ޔ594ޔ1518 ࡃࠗ࠻ߥߤߦࡇࠢࠍ߽ߟಽᏓߣߥߞߡ߅ࠅ㧘ਇቯᒻߢࠆ[8]㧚ߒߚ߇ߞߡ㧘M/D/1/K ࠪ ࠬ࠹ࡓ߇శࡄࠤ࠶࠻឵ᯏࡕ࠺࡞ߦ߰ߐࠊߒߣᕁࠊࠇࠆ߇㧘M/D/1/K ࠪࠬ࠹ࡓߣߩ㆑ ߪᰴߩ 2 ὐߦࠆ㧚 Ԙ 㕖ೋᦼ⌕ࡄࠤ࠶࠻ߦ㧘ⓨᦼ㑆߇ઃടߐࠇࠆߎߣ㧘 ԙ ࠪࠬ࠹ࡓኈ㊂ߦ㒢߇ࠆߩߢߪߥߊ㧘⫾Ⓧᤨ㑆ߦ㒢߇ࠆߎߣ㧚 એਅߢߪ㧘శࡄࠤ࠶࠻ᑄ᫈₸ߣᐔဋㆃᑧᤨ㑆ߩ৻⥸⸃ࠍ᳞ࠆ㧚. 3.1 ╬ଔ⽶⩄ ⌕ߔࠆశࡄࠤ࠶࠻ߪࡑ࡞ࠦࡈㆊ⒟ߦᓥ߁ߣߒ㧘ߘߩ⌕₸ࠍ O 㧘ࡄࠤ࠶࠻㐳ࠍ s 0 ߣ߅ ߊ㧚 s 0 ߩಽᏓߦߟߡᰴߩ㑐ᢙࠍቯ⟵ߔࠆ㧚 g 0 x
(15) 㧦 s 0 ߩ⏕₸ኒᐲ㑐ᢙ㧔pdf㧦probability density function㧕㧘. G0 x
(16) 㧦 s 0 ߩ⫾ⓍಽᏓ㑐ᢙ㧔CDF㧦Cumulative Distribution Function㧕㧘߹ߚߪ⏕₸. 5- - .
(17) ಽᏓ㑐ᢙ㧔PDF㧦probability distribution function㧕㧚 ߒߚ߇ߞߡ㧘ᐔဋࡄࠤ࠶࠻㐳 s 0 ߪᰴᑼߣߥࠅ㧘 f. s0. ³ xg x
(18) dx. 㧔3.1㧕. 0. 0. శࡄࠤ࠶࠻ߩ⽶⩄ U ߪᰴᑼߣߥࠆ㧚 . U. Os 0. 㧔3.2㧕. ᰴߦ㧘ⓨᦼ㑆 W ߩಽᏓࠍ⠨߃ࠆ㧚W ߩ⏕₸ኒᐲ㑐ᢙ㧔pdf㧕ࠍ l x
(19) 㧘⫾ⓍಽᏓ㑐ᢙ㧔CDF㧕 ࠍ L x
(20) ߣ߅ߊ㧚ࡄࠤ࠶࠻ߩ⌕ߪࡐࠕ࠰ࡦಽᏓߢ㧘ࡄࠤ࠶࠻㐳ߦߪଐሽߒߥߣߒߡࠆ ߩߢ㧘 W ߪ >0, D @ ߩ㑆ߢဋ৻ߦಽᏓߒߡࠆߣߔࠆߎߣ߇ߢ߈ࠆ㧚ߒߚ߇ߞߡ㧘 W ߩᐔဋ ୯ W ߪᰴᑼߣߥࠆ㧚 D. . W { ³ xl x
(21) dx 0. D 2. (3.3). ߐࠄߦ㧘㕖ೋᦼ⌕ࡄࠤ࠶࠻ߩ߬ߌߞߣ㐳㧘ߔߥࠊߜࠨࡆࠬᤨ㑆 sX ߪ㧘ࡄࠤ࠶࠻㐳 s 0 ߣⓨᦼ㑆 W ߩว⸘ sX. s0 W. (3.4). ߢࠆߩߢ㧘 sX ߩ⏕₸ኒᐲ㑐ᢙ㧔pdf㧕ࠍ g x
(22) 㧘⫾ⓍಽᏓ㑐ᢙ㧔CDF㧕ࠍ G x
(23) ߣ߅ߊߣ㧘 g x
(24) g 0 x
(25) l x
(26) {. f. ³ g x y
(27) l y
(28) dy 0. (3.5). f. ߢ᳞ࠄࠇࠆ㧚ߎߎߢ㧘 ߪ⇥ߺㄟߺⓍಽ㧔convolution integral㧕ߢࠆ㧚 㕖ೋᦼ⌕߬ߌߞߣߩᐔဋࠨࡆࠬᤨ㑆 sX ߪ f. sX. ³ xg x
(29) dx . (3.6). 0. ╬ଔ⽶⩄ U eq ࠍ↪ࠆߣ㧘 ࠃࠅ᳞ࠄࠇࠆ߇㧘߬ߌߞߣߩ⽶⩄ߢቯ⟵ߐࠇࠆ╬ . U eq. OS. (3.7). ࠃࠅ᳞ࠄࠇࠆ႐ว߇ࠆ㧚╬ଔ⽶⩄ߪ㧘ⓨᦼ㑆ࠍ⽶⩄ߦขࠅㄟࠎߛ߽ߩߢࠆ㧚. 6- - .
(30) 3.2 ή㒢㐳శࡃ࠶ࡈࠔ (a). ᧤᧦ઙ. ᦨೋߦ㧘 B o f ߣߒߚή㒢㐳శࡃ࠶ࡈࠔࠍ⠨߃ࠆ㧚ߎߎߢߪ㧘ή㒢ߩ⫾Ⓧᤨ㑆߇ឭଏ ߐࠇࠆߩߢ㧘ࡄࠤ࠶࠻ߩᑄ᫈ߪߥ㧚ߔߥࠊߜ㧘ࡄࠤ࠶࠻៊ᄬߩߥ⁁ᘒߢࠆ㧚ߎߩࠪ ࠬ࠹ࡓߦ߅ߌࠆᗐᓙߜᤨ㑆ಽᏓࠃࠅ㧘㒢㐳ࡃ࠶ࡈࠔߦ߅ߌࠆ⸃߇᳞ࠄࠇࠆ㧚 㧘శࡃ࠶ࡈࠔ߇❥ᔔᦼ㑆ߦߥߣ߈㧘ߔߥࠊߜⓨߩߣ߈ߩ⏕₸ࠍ Q ߣ߅ߊߣ㧘G/G/1 ࠪࠬ࠹ࡓߦ߅ߡ . U eq 1 Q. (3.8). ߇ᚑ┙ߔࠆ[9]㧚ೋᦼ⌕ࡄࠤ࠶࠻ߩᐔဋࠨࡆࠬᤨ㑆߇ s 0 ߢࠆߩߦኻߒߡ㧘㕖ೋᦼ⌕ ࡄࠤ࠶࠻ߩߘࠇߪᑼ(3.3)ࠃࠅ s 0 D 2 ߢࠆߩߢ㧘ߘࠇߙࠇࠍട㊀ᐔဋߒߡ. D· § Qs 0 1 Q
(31) ¨ s 0 ¸ 2¹ ©. S. s0 U eq. D 2. 㧔3.9㧕. ࠍᓧࠆ㧚ߎߎߢ㧘ᑼ(3.8)ࠍ↪ߚ㧚ߐࠄߦ㧘ᑼ(3.7)ߦઍߔࠆߣᰴᑼࠍᓧࠆ㧚 . U eq. U. (3.10). D U 1 2s0. D ! 0 ߢࠆ㒢ࠅ㧘 U eq !. U ߢࠆ㧚߹ߚ㧘ᤨ㑆☸ᐲ D ߇ᄢ߈ߊߥࠆߣ㧘 U eq ߪჇടߔ. ࠆ㧚ࠪࠬ࠹ࡓ߇᧤ߔࠆߚߩ᧦ઙߪ㧘 U eq 1 ߢࠅ㧘ᑼ(3.10)ࠍ↪ࠆߣ㧘ߎࠇߪ. . § ©. O ¨ s0 . D· ¸ 1 2¹. ߣߔࠆ᧦ઙߣߥࠆ㧚ታ㓙ߩ⽶⩄ U. (3.11). Os 0 ߇ 1 ࠃࠅዊߐ႐วߢ߽㧘U eq ! 1 ߣߥࠆߎߣ߇. ࠆߎߣߦᵈᗧߔࠆᔅⷐ߇ࠆ㧚ߘߎߢ㧘ᡆૃ⽶⩄ U c ࠍ . § ©. U c O ¨ s0 . D· ¸ 2¹. 㧔3.12㧕. ߣቯ⟵ߔࠆߣ㧘 U c 1 ߇㧘 U eq 1 ߣห╬ߥࠪࠬ࠹ࡓ᧤᧦ઙߢࠆ㧚. ᡆૃ⽶⩄ U c ߪ㧘ߔߴߡߩ⌕ࡄࠤ࠶࠻ߦᐔဋⓨᦼ㑆 D 2 ࠍട߃ߚ⽶⩄ߢࠆߚ㧘 . U c U eq. O. D Q 2. (3.13). 7- - .
(32) ߣߥࠅ㧘㕖ೋᦼ⌕ࡄࠤ࠶࠻ߦᐔဋⓨᦼ㑆 D 2 ࠍઃടߒߚಽ㧘 U eq ࠃࠅᄢ߈ߥ୯ߣߥࠆ㧚 (b). ᗐᓙߜᤨ㑆ಽᏓ. ή㒢㐳శࡃ࠶ࡈࠔߦ߅ߌࠆ߬ߌߞߣߩᗐᓙߜᤨ㑆㧔virtual waiting time㧕ߦ㑐ߒߡ㧘 ቯᏱ⁁ᘒߦ߅ߌࠆ⏕₸ಽᏓߣߒߡᰴߩ㑐ᢙࠍቯ⟵ߔࠆ㧚. v x
(33) 㧦߬ߌߞߣߩᗐᓙߜᤨ㑆 x ߦ߅ߌࠆ⏕₸ኒᐲ㑐ᢙ㧔pdf㧕㧘߅ࠃ߮ V x
(34) 㧦߬ߌߞߣߩᗐᓙߜᤨ㑆 x ߦኻߔࠆ⫾ⓍಽᏓ㑐ᢙ㧔CDF㧕㧚 ߎߎߢ㧘 ޟᗐޠᓙߜᤨ㑆ߣ߱ߩߪ㧘ታߦߪߥή㒢㐳శࡃ࠶ࡈࠔࠍᗐቯߒߡࠆߚ ߢࠆ㧚߹ߚ㧘߬ߌߞߣ⌕ࠍࡐࠕ࠰ࡦಽᏓߣߒߡࠆߚ㧘PASTA㧔Poisson arrival see time average㧕ߩ㑐ଥ[10]ࠃࠅ㧘⌕㧘ㅌߥߤߩᤨ㑆ⷐ⚛ࠍᶖߒߡቯᏱ⁁ᘒࠍቯߒ ߚ㧚ߒߚ߇ߞߡ㧘 v x
(35) ߣ V x
(36) ߪᤨ㑆ᐔဋߢࠆ㧚ⷙᩰൻ᧦ઙࠃࠅ V f
(37). f. Q ³ v [
(38) d[. 1 ߮ࠃ߅ޔ. 0. x. V x
(39) V 0
(40) ³ v [
(41) d[. 㧔3.14㧕. 0. ࠃࠅ㧘ᰴᑼࠍᓧࠆ㧚. V 0
(42) Q 㧘߅ࠃ߮ v 0
(43) OQ. (3.15). ߎߎߢ㧘ᓙߜᤨ㑆ߩߥࡄࠤ࠶࠻ߩ⏕₸ኒᐲߪࡃ࠶ࡈࠔ߇ⓨߩߣ߈⌕ߔࠆࡄࠤ࠶࠻ᢙߢ ࠆߩߢ㧘╙ 2 ᑼ߇ᚑ┙ߔࠆ㧚 ᓙߜⴕࡕ࠺࡞ߦ߅ߡᓙߜᤨ㑆ಽᏓࠍ⸃ᨆ⊛ߦ᳞ࠆ႐วߦߪ㧘ࠨࡦࡊ࡞ᤨ㑆ߦ⌕ ߔࠆቴߩേࠍਤᔨߦㅊ㧘ߘߩേߩ⚻ㆊᤨ㑆߆ࠄᓸⓍಽᣇ⒟ᑼࠍ᳞ࠆᣇᴺ߇৻⥸⊛ ߢࠆ߇㧘ℂ⸃߇㔍ߒߊ߆ߟᾘ㔀ߢࠆ㧚ࠃࠅ⋥ᗵ⊛ߦ᳞ࠆᣇᴺߦ㧘ࡌ࡞Ꮕᴺ㧔level crossing method㧕[11]߇ࠆ㧚․ߦ㧘ࡐࠕ࠰ࡦ⌕ㆊ⒟ߢ߆ߟ FCFS ಣℂⷙೣߩࡕ࠺࡞ߦ ߪലߢࠆ㧚 ࿑㧡ߦ㧘ࡌ࡞Ꮕᴺࠍℂ⸃ߔࠆߚ㧘⚻ㆊᤨ㑆ߦኻߔࠆᗐᓙߜᤨ㑆ᄌൻࠍ␜ߔ㧚ታ ✢ߢ␜ߒߚࠨࡦࡊ࡞േࠍߺࠆߣ㧘ᗐᓙߜᤨ㑆ߪ߬ߌߞߣߩ⌕ߏߣߦု⋥ߦߔࠆ ߇㧘⌕߇ߥߣ߈ߦߪᤨ㑆ߩ⚻ㆊߦᲧߒߡਅ㒠ߔࠆ㧚ߘߎߢ㧘છᗧߩᓙߜᤨ㑆 x ߦ⌕ ⋡ߒ㧘ߘߩࡌ࡞ࠍ࿑ߦߪᮮὐ✢ߢ␜ߒߚ㧚᷹ⷰᤨ㑆 >0, t @ ߦ߅ߡ㧘ࠨࡦࡊ࡞േ߇ု⋥ ߒߡࡌ࡞ x ࠍᏅߔࠆὐߩᢙࠍ N up t
(44) 㧘ਅ㒠ߒߡᏅߔࠆὐߩᢙࠍ N down t
(45) ߣ߅ߊ ߣ㧘නᤨ㑆ߚࠅߩᏅὐᢙߪ ޔt o f ߦ߅ߡᰴᑼߣߥࠆߎߣ߇⸽ߐࠇߡࠆ㧚. 8- - .
(46) ࡌ࡞Ꮕὐ. ਅ㒠ࡌ࡞Ꮕὐ. ᗐᓙߜᤨ㑆. x. ࠨࡦࡊ࡞േ 0 ᤨ㑆 t . 0. ࿑ 5 ⚻ㆊᤨ㑆ߦኻߔࠆᗐᓙߜᤨ㑆㧔ࡌ࡞Ꮕᴺ㧕. . lim t of. N up t
(47) t. lim t of. N down t
(48) t. v x
(49). 㧔3.16㧕. ߔߥࠊߜ㧘 (1) ᤨ㑆ᒰߚࠅߩᏅὐᢙߣਅ㒠Ꮕὐᢙߪ㧘 t o f ߦ߅ߡ╬ߒ㧘 (2) ߎߩᤨ㑆ᒰߚࠅߩᏅὐᢙߪ㧘ࡌ࡞ x ߦ߅ߌࠆ⏕₸ኒᐲ㑐ᢙߦ╬ߒ㧚 ᑼ㧔3.16㧕ࠍ↪ߡ㧘ᗐᓙߜᤨ㑆ಽᏓ㑐ᢙߦ㑐ߔࠆᣇ⒟ᑼࠍ᳞ࠆ㧚㧘࿑ 3 ߦ␜ߔ ࡄࠤ࠶࠻Ԙ㧘ԙ㧘߅ࠃ߮Ԛߩ⌕ࠍᗐᓙߜᤨ㑆ߢࠄࠊߔߣ㧘࿑ 6 ߣߥࠆ㧚ߎߎߢ㧘 Ꮕὐߩߺࠍ⠨߃ࠆ㧚 ࡄࠤ࠶࠻Ԙߪೋᦼ⌕ࡄࠤ࠶࠻ߢࠆߩߢ㧘ࡌ࡞ 0 ߆ࠄု⋥ߦ┙ߜ߇ࠆ㧚ߎߩ ߇ࡌ࡞ x ࠍᏅߔࠆߦߪ㧘ࡄࠤ࠶࠻㐳߇ x એߢࠆᔅⷐ߇ࠆ㧚නᤨ㑆ߦ O ⌕ ߔࠆࡄࠤ࠶࠻ߩ㐳ߐ CDF ߇ G0 x
(50) ߢࠆߩߢ㧘Ꮕߔࠆᤨ㑆ഀวߪ O >1 G0 x
(51) @ ߣߥࠆ㧚 ৻ᣇ㧘ᤨ㑆 t 2 ߢ⌕ߔࠆࡄࠤ࠶࠻ԙߪ㕖ೋᦼ⌕ࡄࠤ࠶࠻ߢࠆߩߢ㧘ߘߩ߬ߌߞߣ㐳 CDF ߪ G x
(52) ߢࠆ㧚 t 2 ߦ߅ߌࠆᗐᓙߜᤨ㑆ࠍ [ ߣߔࠆߣ㧘ࡄࠤ࠶࠻ԙߩ⌕ߦࠃࠅ ࡌ࡞ x ࠍᏅߔࠆᤨ㑆ഀวߪ O >1 G x [
(53) @ ߣߥࠆ㧚ࡌ࡞ 0 ߢࠆሽ⏕₸ߪ Q 㧘ࡌ ࡞ [ ߢࠆሽ⏕₸ߪ v [
(54) ߢࠆߩߢ㧘ട㊀ᐔဋࠍߣࠆߣᑼ㧔3.16㧕ߪ x. v x
(55). O >1 G0 x
(56) @Q O ³ >1 G x [
(57) @v [
(58) d[. 㧔3.17㧕. 0. ߣ᳞߹ࠆ㧚ߎߎߢ㧘ฝㄝ╙ 2 㗄ߪ㧘 0 [ d x ߩ▸࿐ߦࠆ [ ోߦട㊀ᐔဋߒߚ㧚 ࡊࠬ-ࠬ࠴ࡘ࡞࠴ࠚࠬᄌ឵ ᑼ(3.17㧕ߪ pdf ߦኻߔࠆⓍಽᣇ⒟ᑼߢࠅ㧘ߎࠇࠍ. 9- - .
(59) ᗐᓙߜᤨ㑆. S3 S2 . x. x [. [ s1 0. 0. t 2 t3. ᤨ㑆 t . ࿑ 6 ᗐᓙߜᤨ㑆ߩផ⒖ 㧔Laplace-Stieltjes transform㧦LST㧕ߢ᳞ࠆ㧚ቯᢙ㗄ࠍ㒰ߔࠆߚ㧘 x ߢᓸಽߒߡ㧘 . dv x
(60) dx. x. OQg 0 x
(61) Ov x
(62) O ³ g x [
(63) v [
(64) d[. 㧔3.18㧕. 0. ࠍᓧ㧘ߐࠄߦ LST ࠍታⴕߒߡ . Tv * T
(65) v 0
(66) OQg 0 * T
(67) Ov * T
(68) Og * T
(69) v * T
(70). ࠍᓧࠆ㧚ߎߎߢ㧘㧖ߪฦ㑐ᢙߩࡊࠬᄌ឵ࠍ␜ߔ㧚 v 0
(71). v. *. (3.19). OQ ࠍ↪ࠆߣ㧘ᦨ⚳⊛ߦ. *
(72) T
(73) OQ[1 g 0 * T ] T O[1 g T
(74) ]. 㧔3.20㧕. ߣߥࠆ㧚ᑼ(3.20)ࠍࡊࠬㅒᄌ឵ߔࠆߣ㧘ᗐᓙߜᤨ㑆 x ߦኻߔࠆ pdf ታ㑐ᢙࠍᓧࠆߎ ߣ߇ߢ߈ࠆ㧚. 3.3 㒢㐳శࡃ࠶ࡈࠔ (a) 㑐ᢙߣߘߩቯ⟵ 㒢㐳శࡃ࠶ࡈࠔߢߪ㧘ᗐᓙߜᤨ㑆 x ߇ᦨᄢ⸵ኈㆃᑧᤨ㑆ߢࠆ T. B 1
(75) D ࠍ߃. ࠆߣ㧘߬ߌߞߣߪᑄ᫈ߐࠇࠆ㧚ࠄߚߡ㧘߬ߌߞߣࠍಽ㘃ߔࠆߣ㧘ᰴߩ 3 ⒳㘃ߣߥࠆ㧚 (i) ೋᦼ⌕ࡄࠤ࠶࠻㧧ࡃ࠶ࡈࠔߪⓨߢࠆߩߢ x. 0 ߢࠅ㧘ㅢㆊࡄࠤ࠶࠻ߢࠆ㧘 (ii) 㕖ೋᦼ⌕ࡄࠤ࠶࠻ߢ߆ߟㅢㆊࡄࠤ࠶࠻㧘ߔߥࠊߜ 0 x d T 㧘 (iii) 㕖ೋᦼ⌕ࡄࠤ࠶࠻ߢ߆ߟᑄ᫈ߐࠇࠆࡄࠤ࠶࠻㧘 T x 㧚 㒢㐳శࡃ࠶ࡈࠔߦ߅ߌࠆ㑐ᢙࠍ㧘એਅߩࠃ߁ߦ㧘ਅઃ T ࠍᷝ߃ߡή㒢㐳శࡃ࠶ࡈࠔߩ ߘࠇࠄߣߔࠆ㧚ᑄ᫈ߐࠇߚ߬ߌߞߣߪࡃ࠶ࡈࠔౝߦሽߒߥߎߣ߇㧘ή㒢㐳ࡃ࠶ࡈ ࠔࡕ࠺࡞ߣߩ㆑ߢࠆ㧚. - 10 .
(76) vT x
(77) 㧦⌕߬ߌߞߣో㧔⸥ಽ㘃ߢ(I)㧘(ii)㧘߅ࠃ߮(iii)㧕ߦኻߒߡቯ⟵ߐࠇ㧘ή㒢 㐳ࡃ࠶ࡈࠔࡕ࠺࡞ߢߩᗐᓙߜᤨ㑆 x ࠍᄌᢙߣߔࠆ⏕₸ኒᐲ㑐ᢙ pdf㧚v x
(78) ߣߩ ㆑ߪ㧘ᑄ᫈ߐࠇߚ߬ߌߞߣ߇ሽߔࠆߩߢ㧘ࡃ࠶ࡈࠔౝߦ߅ߌࠆታ㓙ߩ߬ߌ. ߞߣኒᐲߪࠃࠅዊߐ㧚ߎߩߚ㧘ⓨߣߥࠆ⏕₸ QT ߇ Q ࠃࠅᄢ߈ߊߥࠆὐߢ ࠆ㧚 VT x
(79) 㧦 vT x
(80) ߩ⫾ⓍಽᏓ㑐ᢙ CDF㧘 WT x
(81) 㧦ㅢㆊ߬ߌߞߣ㧔ߔߥࠊߜ(I)ߣ(ii)㧕ߩߺࠍኻ⽎ߣߒߚ㧘ᗐᓙߜᤨ㑆 x ߦ㑐ߔࠆ ⫾ⓍಽᏓ㑐ᢙ CDF㧚ߎߩቯ⟵ߦࠃࠅ㧘ᰴᑼ߇ᚑ┙ߔࠆ㧚 WT x
(82). VT x
(83) VT T
(84). (3.20). WT 㧦ㅢㆊ߬ߌߞߣߩߺࠍኻ⽎ߣߒߚᐔဋㆃᑧᤨ㑆㧘. wT 㧦ㅢㆊߒߚታࡄࠤ࠶࠻ߩߺࠍኻ⽎ߣߒ㧘ⓨᦼ㑆ࠍ߹ߥᐔဋㆃᑧᤨ㑆㧘. S T 㧦ㅢㆊ߬ߌߞߣߩߺࠍኻ⽎ߣߒߚᐔဋ߬ߌߞߣ㐳㧘 PB 㧦⌕ోࡄࠤ࠶࠻ߦኻߔࠆ㐽Ⴇ⏕₸㧘߅ࠃ߮ࡄࠤ࠶࠻ᑄ᫈₸㧘៊ᄬ₸㧘 QT 㧦ࡃ࠶ࡈࠔ߇ⓨߢࠆ⏕₸㧚ߔߥࠊߜ㧘. QT. VT 0
(85) 㧚. 㧔3.21㧕. ᑄ᫈ߐࠇߚ߬ߌߞߣ߇ࠆߣ㧘ⓨߩഀวߪჇടߔࠆ㧚⽶⩄ߪߘߩಽᷫዋߔࠆߩߢ㧘 ᑼ(3.7)㧘(3.8)ߦઍࠊࠅߦᰴᑼ߇ᚑ┙ߔࠆ㧚 1 QT. 1 PB
(86) OST. 㧔3.22㧕. (b) ᐔဋㆃᑧᤨ㑆ߣᐔဋ߬ߌߞߣ㐳 x d T ߢ⌕ߔࠆ߬ߌߞߣߦኻߒߡߪ㧘ࡌ࡞Ꮕᴺࠍ↪ߔࠆߣ㧘ᑼ(3.17)ߣห᭽ߦ x. vT x
(87). O >1 G0 x
(88) @QT O ³ >1 G x [
(89) @vT [
(90) d[. 㧔3.23㧕. 0. ߇ᚑࠅ┙ߟ㧚ᑼ(3.17)ߣหߓᣇ⒟ᑼߢࠆߚ㧘 >0, T @ ߦࠆ x ߦኻߒߡߪ㧘 vT x
(91) ߣ v x
(92) ⸃ߩᒻߪหߓߢࠅ㧘⋧ߦᲧ㑐ଥߦࠆ㧚ߘߎߢ㧘ᰴᑼߣ߅ߊ㧚. - 11 .
(93) vT x
(94). Dv x
(95) 㧘 D ! 0 㧘߅ࠃ߮ 0 d x d T. 㧔3.24㧕. ᑼ(3.24)ࠍᑼ(3.23)ߦઍߒߡ㧘ᑼ(3.17)ߣᲧセߔࠆߣ㧘 . D. VT 0
(96) V 0
(97). QT Q. 㧔3.25㧕. ߣߥࠆ㧚ᑼ(3.25)ࠃࠅ㧘ᗐᓙߜᤨ㑆ಽᏓߦ㑐ߔࠆᰴߩ㑐ଥᑼࠍᓧࠆ㧚. QT v x
(98) 㧘 VT x
(99) Q. vT x
(100). QT V x
(101) 㧘 x d T Q. 㧔3.26㧕. ᑼ(3.20㧕㧘(3.26)ࠍ↪ࠆߣ㧘ㅢㆊ߬ߌߞߣߦኻߔࠆㆃᑧᤨ㑆 CDF ߪ㧘. V x
(102) 㧘xdT V T
(103). WT x
(104). 㧔3.27㧕. ߣߥࠅ㧘ߘߩᐔဋㆃᑧᤨ㑆ߪᰴᑼߣߥࠆ㧚 T. WT. T. T V x
(105) dV x
(106) dx ª V x
(107) º x x ³0 dx V T
(108) «¬ V T
(109) »¼ ³0 V T
(110) dx 0. T. ³ xdW T x
(111) 0. V x
(112) dx V T
(113) 0. T. T ³. V x
(114) dx V T
(115) 0. T. T ³. 㧔3.28㧕. ㅢㆊ߬ߌߞߣߩ߁ߜ㧘ᓙߜᤨ㑆߇ 0 ߣߥࠆ⏕₸ߪ㧘ᑼ(3.27)ࠃࠅ WT 0
(116). V 0
(117) V T
(118). Q V T
(119). 㧔3.29㧕. ߣߥࠆ㧚ߎࠇߪ㧘߬ߌߞߣಽ㘃(I)ߣ(ii)ߦኻߔࠆ(i)ߩഀวߢࠆ㧚ߒߚ߇ߞߡ㧘߬ߌߞߣಽ 㘃(ii)ߩഀว㧘ߔߥࠊߜㅢㆊ߬ߌߞߣߩ߁ߜ㕖ೋᦼ⌕ࡄࠤ࠶࠻ߢࠆ⏕₸ߪ [1 Q V T
(120) ] ߢࠆ㧚ߎߩ㕖ೋᦼ⌕ࡄࠤ࠶࠻ߦߪ㧘FDL ߦജߔࠆ㓙㧘ᐔဋ D 2 ߩⓨᦼ㑆߇ടࠊࠆ ߚ㧘ㅢㆊߔࠆታࡄࠤ࠶࠻ߩߺࠍߺߚߣ߈ߩᐔဋㆃᑧᤨ㑆ߣߒߡᰴᑼࠍᓧࠆ㧚. V x
(121) Dª Q º dx «1 V T
(122) 2 ¬ V T
(123) »¼ 0. T. wT. T ³. 㧔3.30㧕. ㅢㆊ߬ߌߞߣߩߺࠍኻ⽎ߣߒߚᐔဋ߬ߌߞߣ㐳 S T ߪ㧘ᰴߦࠃ߁ߦߒߡ᳞ࠆߎߣ߇ߢ߈ ࠆ㧚ಽ㘃(i) ߬ߌߞߣߩᐔဋ㐳ߪ s 0 ߢࠅ㧘ಽ㘃(ii) ߬ߌߞߣߩߘࠇߪ㧔 s 0 D 2 㧕ߢ ࠆߩߢ㧘ߘࠇߙࠇߩሽ⏕₸ߢട㊀ᐔဋࠍߣࠇ߫㧘ᰴᑼࠍᓧࠆ㧚. - 12 .
(124) ST. s0. D ·ª Q º Q § ¨ s0 ¸ «1 2 ¹ ¬ V T
(125) »¼ V T
(126) ©. s0 . Dª Q º 1 « 2 ¬ V T
(127) »¼. 㧔3.31㧕. (c) ࡄࠤ࠶࠻ᑄ᫈₸ ࡄࠤ࠶࠻⌕ߪࡐࠕ࠰ࡦㆊ⒟ߢࠆߩߢ㧘PASTA ߩ㑐ଥߦࠃࠅ㧘ࡄࠤ࠶࠻ᑄ᫈₸ߪࡄ ࠤ࠶࠻߇⌕ߒߚߣ߈ߩ㐽Ⴇ⏕₸ߦ╬ߒ㧚㐽Ⴇߪ x PB. T ߩߣ߈߈ࠆߩߢ㧘. 1 VT T
(128). 㧔3.32㧕. ߇ࡄࠤ࠶࠻ᑄ᫈₸ߩၮᧄᑼߢࠆ㧚 VT T
(129) ࠍ᳞ࠆ㧚ᑼ(3.22)ߣ(3.32)㧘ߐࠄߦᑼ(3.26㧕ࠃࠅ VT T
(130). 1 QT OS T. QT V T
(131) Q. 㧔3.33㧕. ߆ࠄ Q T ߦߟߡ. QT. Q Q OS T V T
(132). 㧔3.34㧕. ࠍᓧߡ㧘ᑼ(3.34)ࠍᑼ(3.33)㧘(3.26)ߦઍߔࠆߣ㧘ᰴᑼࠍᓧࠆ㧚. V x
(133) 㧘 VT T
(134) Q OS T V T
(135). VT x
(136). PB. 1. V T
(137) Q OS T V T
(138). V T
(139) Q OS T V T
(140). 㧔3.35㧕. 㧔3.36㧕. ᑼ(3.36)ߢߪ㧘S T ߣ߁㒢㐳శࡃ࠶ࡈࠔߦߡቯ⟵ߐࠇߚᄌᢙ߇߹ࠇߡࠆ㧚ߘߎߢ㧘 ᑼ(3.31)㧘߅ࠃ߮ᑼ(3.2)㧘(3.7)㨪(3.9)ࠍ↪ߡᄌᒻߔࠆߣ Q OS T V T
(141). D· § 1 O ¨ s 0 ¸>1 V T
(142) @ 2¹ ©. 㧔3.37㧕. ࠍᓧ㧘ᑼ(3.36)ߪ⚿ዪᰴᑼߣߥࠆ㧚. PB. ª D ·º § «1 O ¨ s 0 2 ¸»>1 V T
(143) @ © ¹¼ ¬ D· § 1 O ¨ s 0 ¸>1 V T
(144) @ 2¹ ©. 㧔3.38㧕. ή㒢㐳శࡃ࠶ࡈࠔߢߩᄌᢙߩߺߢߐࠇࠆᑼ߇ᓧࠄࠇߚ㧚ߒߚ߇ߞߡ㧘ή㒢㐳శࡃ࠶ࡈࠔ ࡕ࠺࡞ߢߩ⸃ࠍ᳞ࠇ߫㧘㒢㐳శࡃ࠶ࡈࠔߢߩ⸃ࠍᓧࠆߎߣ߇ߢ߈ࠆ㧚. - 13 .
(145) (d) ⠨ኤ ᑼ(3.38)ߪ㧘ᑼ(3.12)ߩᡆૃ⽶⩄ U c ࠍ↪ࠆߣ PB. 1 U c
(146) >1 V T
(147) @ 1 U c>1 V T
(148) @. 㧔3.39㧕. ߣ߅ߌࠆ㧚 D o 0 ߩߣ߈㧘 U c o PB. U ߢࠆߩߢ㧘ᑼ(3.39)ߪ. 1 U
(149) >1 V T
(150) @ 1 U >1 V T
(151) @. 㧔3.40㧕. ߣߥࠆ㧚ᑼ(3.40)ߪ㧘M/G/1/K ࠪࠬ࠹ࡓߩᑄ᫈₸ߦ㑐ߔࠆ৻⥸ᑼߢࠆ㧚 M/G/1 ࠪࠬ࠹ࡓߦ߅ߡ♽ౝቴᢙ߇ K એߣߥࠆ⏕₸ࠍ E K ߣ߅ߊߣ㧘M/G/1/K ࠪࠬ࠹ ࡓߩᑄ᫈₸ߪᰴᑼߢߐࠇࠆ[12]㧚 PB. 1 U
(152) E K. 㧔3.41㧕. 1 UE K. M/M/1/K ࠪࠬ࠹ࡓߩࠍઃ㍳ߦ␜ߔ㧚 K એౝߣ߁♽ౝቴᢙ㒢߇㧘శࡃ࠶ࡈࠔߩ႐ว ߪ T એਅߣ߁ㆃᑧᤨ㑆㒢ߦᄌࠊࠅ㧘E K ࠍ [1 V T
(153) ] ߦ⟎߈឵߃ߚᒻ߇ᑼ(3.40)ߢࠆ㧚 ߐࠄߦ㧘⽶⩄ࠍᡆૃ⽶⩄ߦ⟎߈឵߃ࠆߣ㧘ㄭૃߩߥᑼ(3.39)ߣߥࠆ㧚ߒߚ߇ߞߡ㧘ᡆૃ ⽶⩄߇ታല⊛ߥ⽶⩄ߢࠆߣ⼂ߢ߈ࠆ㧚. 㧠㧚ᜰᢙ㑐ᢙಽᏓࡄࠤ࠶࠻㐳ߢߩㄭૃ⸃ᴺ ᑼ(3.38)ࠍ↪ߡᑄ᫈₸ࠍ⸘▚ߔࠆߦߪ㧘V T
(154) ߩ୯߇ᔅⷐߢࠆ㧚ߔߥࠊߜή㒢㐳ࡃ࠶ ࡈࠔࡕ࠺࡞ߦ߅ߌࠆᗐᓙߜᤨ㑆ಽᏓࠍ㧘ᓙߜᤨ㑆 x ߦ㑐ߔࠆ㑐ᢙߩᒻߢ᳞ࠆᔅⷐ߇ ࠆ㧚ߎߩ㑐ᢙߪᑼ(3.20)ߩ v. *. T
(155) ࠍࡊࠬㅒᄌ឵ߒߡ᳞ࠄࠇࠆ߇㧘ታ㓙ߩߣߎࠈࡊ. ࠬㅒᄌ឵ߢ᳞ࠄࠇࠆ㑐ᢙᒻߪ߈ࠊߡ㒢ࠄࠇߡࠆ㧚ⶄ㔀ߥ㑐ᢙࠍࡊࠬㅒᄌ឵ߔࠆ ߎߣߪ৻⥸ߦ࿎㔍ߢࠅ㧘߆ߟᢙ୯⊛ߦ⸘▚ߔࠆߎߣ߽ߢ߈ߥ㧚ߎߩߚ㧘ᑼ(3.20)ࠍ ↪ߡ⸘▚ߒߚࠍ⪺⠪ߩ⍮ࠆ㒢ࠅߥߊ㧘ታ㓙ߩᢙ୯⸘▚ߢߪㅢᏱߎߩᑼࠍ↪ߕߦᢙ୯ ࠪࡒࡘ࡚ࠪࡦࠍⴕߞߡࠆ㧚ߎࠇߢߪ㧘ᑼ(3.20)ࠍ᳞ߚᗧ߇ߥ㧚 ታߩࠪࠬ࠹ࡓߪߔߴߡ㒢ࡃ࠶ࡈࠔࠪࠬ࠹ࡓߢࠆ߇㧘ߘߩ⸃ᨆߦߚࠅ㧘ή㒢ࡃ࠶ ࡈࠔࡕ࠺࡞ߩ߅ߌࠆ⸃ࠍ᳞ߡ߆ࠄᑼ(3.41)ࠍ↪ߡ㒢ࡃ࠶ࡈࠔࠪࠬ࠹ࡓߦㆡ↪ߔࠆߣ ߁ᚻᴺߪ৻⥸⊛ߢࠅ㧘ࠃߊⴕࠊࠇߡࠆ㧚ߒ߆ߒߥ߇ࠄ㧘ߎߩ႐ว㧘ᓙߜᤨ㑆ಽᏓߩ 㑐ᢙᒻࠍ᳞ߥߌࠇ߫ߥࠄߥߣ߁࿎㔍ߐ߇ࠆ㧚 એਅߢߪ㧘శࡄࠤ࠶࠻㐳߇ᜰᢙ㑐ᢙಽᏓߔࠆ႐วߢߩㄭૃ⸃ࠍ᳞㧘ߘߩ♖ᐲࠍᢙ୯ࠪ ࡒࡘ࡚ࠪࡦߦࠃࠆ⚿ᨐߣᲧセߔࠆ㧚ࠨࡆࠬᤨ㑆߇ᜰᢙ㑐ᢙಽᏓߔࠆࠪࠬ࠹ࡓߪᓙߜ ⴕࡕ࠺࡞ߢߪ M/M/1/K ߦ⋧ᒰߒ㧘ዉ߇ᦨ߽◲නߥࡕ࠺࡞ߢࠆ߇㧘ᑄ᫈₸ߦ㑐ߒߡ ␜⊛ߥ߇ᓧࠄࠇߡߥ㧚. - 14 .
(156) 1 s0. 0. 1 D. 0. 0 ࡄࠤ࠶࠻㐳 x. D. 0 ⓨᦼ㑆㐳 x. (a)ࡄࠤ࠶࠻㐳ಽᏓ. (b)ⓨᦼ㑆㐳ಽᏓ. ࿑ 7 ࡄࠤ࠶࠻㐳ߣⓨᦼ㑆㐳ߩ⏕₸ኒᐲಽᏓ. 4.1 ㄭૃᑼ ࿑ 7 ߦ␜ߔࠃ߁ߦ㧘శࡄࠤ࠶࠻㐳ߪᐔဋ୯ s 0 ߩᜰᢙ㑐ᢙಽᏓ㧘ⓨᦼ㑆ߪ >0, D @ ߩ㑆ߢߩ ဋ৻ಽᏓߣቯߔࠆ㧚ฦ⏕₸ኒᐲ㑐ᢙߣߘߩࡊࠬᄌ឵ߪ㧘ᰴᑼߣߥࠆ㧚. g 0 x
(157). 1. 1 x s0 * e 㧘 g 0 T
(158) s0. s 0T 1. 1 >u x
(159) u x D
(160) @㧘 l * T
(161) D. l x
(162). (4.1). 㧘 . 1 1 e DT DT.
(163). (4.2㧕. ߎߎߢ㧘 u x
(164) ߪනࠬ࠹࠶ࡊ㑐ᢙߢࠆ㧚ᑼ(3.5)ߦ߅ߌࠆ⇥ߺㄟߺⓍಽߪ㧘ࡊࠬᄌ឵ ߢߪනߥࠆⓍߣߥࠆߚ㧘. g * T
(165). g 0 T
(166) l * T
(167) *. 1. 1 1 e DT s 0T 1 DT.
(168). (4.3). ߢࠆ㧚ᑼ(4.1)㧘(4.2)ࠍᑼ(3.20)ߦઍߔࠆߣᰴᑼߣߥࠆ㧚. ª. º » ¬ s 0T 1¼ ª º 1 1 T O «1 1 e DT » ¬ s 0T 1 DT ¼. OQ «1 . v. *. T
(169). 1. (4.4).
(170). ᑼ(4.4)ࠍࡊࠬㅒᄌ឵ߔࠇ߫ㆃᑧᤨ㑆ߦ㑐ߔࠆ⏕₸ኒᐲ㑐ᢙࠍᓧࠆߎߣ߇ߢ߈ࠆ㧚 ߒ߆ߒߥ߇ࠄ㧘ᑼ(4.4)ߩ␜⊛ߥㅒᄌ឵ࠍ᳞ࠆߎߣߪ࿎㔍ߢࠆߩߢ㧘 DT 1 ߣߒ . 1 DT 1 e DT | 1 2 DT.
(171). (4.5). - 15 .
(172) ߢㄭૃߒߡ㧘 DT
(173) એߩߴ߈ਸ਼ࠍήⷞߔࠆ㧚ᑼ(4.4)ߪ߈ࠊߡ◲නൻߐࠇߡ 2. v. *. T
(174). OQ. (4.6). ª D ·º § «1 O ¨ s 0 2 ¸» © ¹¼ ¬. 1 T s0. ߣߥࠅ㧘ߘߩㅒᄌ឵ߪᰴᑼߣߥࠆ㧚 v x
(175). OQe >1O s. 0 D. 2
(176) @x s0. OQe 1 U c
(177) x s. (4.7). . 0. ᑼ(4.5)ߦ߅ߡㄭૃߔࠆߚߩ᧦ઙߢࠆ DT 1 ߪ㧘ᑼ(4.7)ࠃࠅ 1 U c
(178) D s 0 1 ߣ ห╬ߢࠆ㧚ߎߩ᧦ઙߪ㧘 D ߇ s 0 ߦኻߒߡᭂߡዊߐ߆㧘⽶⩄ U c ߇ 1 ߦㄭߊ㊀႐ว ߦ⋧ᒰߔࠆ㧚ߒߚ߇ߞߡ㧘ᑄ᫈₸߇ᄢ߈႐วߩㄭૃߢࠆ㧚 CDF ߪ㧘ᑼ(3.14)ࠃࠅ᳞ࠆߎߣ߇ߢ߈ࠆߩߢ㧘ᑼ(4.7)ࠍઍߒߡ x. V x
(179). V 0
(180) ³ OQe 1 U
(181) [ s0 d[ c. Q OQ. 0. . 1. U D 1 U 2s0. e 1 U c
(182) x s0. s0 c e 1 U
(183) [ 1 Uc. >. @. s0 x 0. 1 U eq e 1 U c
(184) x s0. (4.8). ߣ㧘ᑼ(3.10)ߩ╬ଔ⽶⩄㧘ᑼ(3.12)ߩᡆૃ⽶⩄ࠍ↪ࠆߣ߈ࠊߡ◲නߥᑼߣߥࠆ㧚 㧝㧚⽶⩄ߦኻߔࠆᦨᄢ☸ᐲ ⽶⩄. ᦨᄢ FDL ☸ᐲ. U. Dmax s 0. 0.01. 198. 0.1. 18. 0.2. 8. 0.3. 4.7. 0.4. 3. 0.5. 2. 0.6. 1.3. 0.7. 0.86. 0.8. 0.5. 0.9. 0.22. 0.99. 0.02. ᑼ(4.8)ߪ ޔD o 0 ߦ߅ߡ U eq o. U ޔU c o U ߢ. ࠆߩߢ㧘 V x
(185). 1 Ue 1 U
(186) x s0. (4.9). ߣߥࠆ㧚ᑼ(4.9)ߪ㧘M/M/1 ࠪࠬ࠹ࡓߦ߅ߌࠆቴߩᓙߜᤨ 㑆ಽᏓ㑐ᢙߣหᒻߢࠆ[13]㧚ߒߚ߇ߞߡ㧘ᑼ(4.8)߆ࠄ㧘 Ყଥᢙߢߪ╬ଔ⽶⩄߇㧘ᜰᢙଥᢙߢߪᡆૃ⽶⩄߇ታല ⊛ߥ⽶⩄ߣߥߞߡࠆߎߣ߇ℂ⸃ߐࠇࠆ㧚 ᑼ(4.8)ࠍᑼ(3.39)ߦઍߒߡ㧘ᑄ᫈₸ߩㄭૃᑼࠍᓧࠆ㧚 PB. 1 U c
(187) U eq e 1 U c
(188) T s 1 U cU eq e 1 U c
(189) T. s0. 0. (4.10). 4.2 ᢙ୯ ᑼ(4.10)ߦࠃࠅᑄ᫈₸ߩ⸘▚ࠍߔࠆ႐ว㧘 U eq 1 ߩ. - 16 .
(190) ᧤᧦ઙࠍ⏕ߔࠆᔅⷐ߇ࠆ㧚ᑼ (3.11)ߦࠃࠅ㧘FDL ☸ᐲ D ߦߪᰴᑼߩࠃ߁ߥᦨᄢ㒢 ߇ࠆ㧚 . · D §1 D 2¨¨ 1¸¸ { max s0 s0 ¹ ©U. (4.11). ⽶⩄ߦኻߔࠆᦨᄢ☸ᐲ Dmax s 0 ࠍ㧘 1 ߦ␜ߔ㧚ߚߣ߃߫㧘⽶⩄ 0.5 ߢߪ 2ޔ0.8 ߢߪ 0.5 ߇㒢୯ߣߥࠆ㧚 ᐔဋࡄࠤ࠶࠻㐳 s 0 ࠍ 1㧘ߔߥࠊߜනᤨ㑆㐳ߣߒߚߣ߈㧘 D ߦኻߔࠆᑄ᫈₸ PB ߩ⸘▚ ⚿ᨐࠍ࿑ 8 ߦ␜ߔ㧚ታ✢ߪᑼ(4.10)ߦࠃࠆ⸘▚୯㧘ὐߪᢥ₂[5]ߩ࿑ࠃࠅ⺒ߺขߞߚࠪࡒࡘ ࡚ࠪࡦ⸘▚୯ߢࠆ㧚⽶⩄ߪᲧセߢ߈ࠆ U 0.8 ߣߒߚ㧚ㄭૃᑼߦࠃࠆ⸘▚ߪࠪࡒࡘ ࡚ࠪࡦ⸘▚⚿ᨐߣࠃ৻⥌ࠍߺߖߚ㧚⺋Ꮕߪ B. 512 ߩ႐วߢ D ߇ᄢ߈ߣᄢ߈ߊ㧘. ᦨᄢߢ߽ 20㧑ߢࠆ㧚߹ߚ㧘 D ߦኻߔࠆᑄ᫈₸ߩะࠍᛠីߔࠆߎߣ߇ߢ߈ߡࠆ㧚 ฦᢥ₂ߦߡࠄ߆ߦߐࠇߡࠆࠃ߁ߦ[3-6]㧘 (1) D ߇ዊߐߣ㧘ᦨᄢㆃᑧᤨ㑆 T. B 1
(191) D ߇ዊߐߊߥࠆߚ㧘ᑄ᫈₸ߪჇടߔࠆ㧚. (2) D ߇ᄢ߈ߣⓨᦼ㑆߇Ⴧ߃㧘╬ଔ⽶⩄߇Ⴧടߔࠆߚᑄ᫈₸ߪჇടߔࠆ㧚 (3) ߒߚ߇ߞߡ㧘D ߩ୯ߦߪᑄ᫈₸ࠍᦨዊߦߔࠆᦨㆡ୯߇ሽߒ㧘ᦨᄢ☸ᐲ Dmax s 0 ߩ 1/2 ઃㄭߦߘߩᦨㆡ୯߇ሽߔࠆ㧚. 㪝㪛㪣 ☸ᐲ㩷 㪛 㪇. 㪇㪅㪇㪌 㪇㪅㪈. 㪇㪅㪈㪌 㪇㪅㪉. 㪇㪅㪉㪌 㪇㪅㪊. 㪈㪅㪇. 㪇㪅㪊㪌 㪇㪅㪋. U. 㪇㪅㪋㪌. 0 .8. 㪈㪇㪄㪈 ᑄ᫈₸㩷 㪧㪙. 㪈㪇㪄㪉 㪈㪇㪄㪊. 㪙㪔㪈㪉㪍 㪙㪔㪉㪌㪍. 㪈㪇㪄㪋 㪈㪇㪄㪌 㪈㪇㪄㪍. 㪙㪔㪌㪈㪉. 㪈㪇㪄㪎 ࿑ 8. FDL ☸ᐲ D ߦኻߔࠆࡄࠤ࠶࠻ᑄ᫈₸㧔 U. 0.8 㧕. 㧔ታ✢ߪㄭૃᑼ⸘▚㧘ὐߪࠪࡒࡘ࡚ࠪࡦ⸘▚⚿ᨐ[5]㧕. - 17 .
(192) 1 㧘U. D ߦኻߔࠆㅢㆊࡄࠤ࠶࠻ߩᐔဋㆃᑧᤨ㑆 wT ߩ⸘▚୯ࠍ㧘࿑ 9 ߦ␜ߔ㧚s 0. 0.8. ߣߒ㧘ታ✢ߪᑼ(4.8) 㧘(3.30)ࠍ↪ߚㄭૃ⸘▚୯ߢࠅ㧘ὐߪᢥ₂[5]ߩ࿑ࠃࠅ⺒ߺขߞߚ ࠪࡒࡘ࡚ࠪࡦ⸘▚୯ߢࠆ㧚ߎߩ႐วߪ㧘3%એਅߩ⺋Ꮕߢ৻⥌ߒߚ㧚D ߇ᄢ߈ߣ㧘 ╬ଔ⽶⩄߇Ⴧടߔࠆߩߢᐔဋㆃᑧᤨ㑆ߪჇടߔࠆ㧚. 㪐㪇. U. 㪏㪇. 0 .8. ᐔဋㆃᑧᤨ㑆㩷 䌷㪫. 㪎㪇 㪍㪇. 㪙㪔㪌㪈㪉. 㪌㪇 㪋㪇. 㪙㪔㪉㪌㪍. 㪊㪇 㪉㪇. 㪙㪔㪈㪉㪏. 㪈㪇 㪇 㪇. 㪇㪅㪇㪌. 㪇㪅㪈. 㪇㪅㪈㪌. 㪇㪅㪉. 㪇㪅㪉㪌. 㪇㪅㪊. 㪇㪅㪊㪌. 㪇㪅㪋. 㪇㪅㪋㪌. ࿑ 9. FDL ☸ᐲ D ߦኻߔࠆㅢㆊࡄࠤ࠶࠻ߩᐔဋㆃᑧᤨ㑆㧔 U. 0.8 㧕. 㪝㪛㪣 ☸ᐲ㩷 㪛. 㧔ታ✢ߪㄭૃᑼ⸘▚㧘ὐߪࠪࡒࡘ࡚ࠪࡦ⸘▚⚿ᨐ[5]㧕 ᑼ(4.10)ߪ♖ᐲߩ㜞ㄭૃᑼߢࠆߎߣ߇ࠄ߆ߦߐࠇߚߩߢ㧘⸳⸘᧦ઙߩᛠីߦ↪ ߔࠆߎߣߣߔࠆ㧚ߚߣ߃߫㧘࿑ 8㧘9 ߦ߅ߡ㧘 B 㧩128㨪512 ߣߒߡࠆ߇㧘శࠬࠗ࠶࠴ ߩታᕈࠍ⠨ᘦߔࠆߣㆊߦᄢ߈ߥ୯ߢࠆ㧚߹ߚ㧘ታ㓙ߩࡀ࠶࠻ࡢࠢ⸳⸘ߢߪ㧘0.8 ߣ߁㜞⽶⩄⁁ᘒࠍၮᧄߣߖߕߦ㧘ᦨᄢߢ߽⽶⩄ 0.5 ߦ㒢ߒߡࠆ㧚ߘߎߢ㧘ዊߐߥ B ୯ࠍ߽ߟశࡃ࠶ࡈࠔߩᕈ⢻ࠍ⺞ߴߡߺࠆ㧚 ࿑ 10 ߪ㧘 B 㧩32 ߣߒߚߣ߈ D ߦኻߔࠆᑄ᫈₸ PB ࠍ␜ߔ㧚⽶⩄ U ࠍ 0.3㨪0.7 ߣߒߚ㧚 ⽶⩄ߦࠃࠅᑄ᫈₸ߪᄢ߈ߊᷫዋߔࠆ㧚⽶⩄ 0.5 ߢߪ D =1 ઃㄭߦߡ㧘0.01㧑એਅߩᑄ᫈₸߇ ታߐࠇࠆ㧚⽶⩄ࠍ 0.5 ߣߒ㧘 B ࠍࡄࡔ࠲ߣߒߚ႐วߩᑄ᫈₸ PB ࠍ㧘࿑ 11 ߦ␜ߔ㧚. B 㧩4 ߩߣ߈ D 㧩1.4㧘 B 㧩64 ߩߣ߈ D 㧩1.2 ߦߡᑄ᫈₸ߪᦨዊߣߥࠆ߇㧘ᄢ߈ߊߪᄌൻ ߒߡߥ㧚ᑄ᫈₸߇ 0.01㧑એਅߣߥࠆߩߪ B ߇ 32 એߩߣ߈ߢࠅ㧘⽶⩄㒢ࠍ⺖ߒ ߡ߽శࡃ࠶ࡈࠔߦߪࠆ⒟ᐲߩⷙᮨ߇ᔅⷐߥߎߣ߇ࠊ߆ࠆ㧚࿑ 12 ߪ U =0.5 ߣߒߚߣ߈㧘D ߦኻߔࠆᐔဋㆃᑧᤨ㑆 wT ࠍ␜ߔ㧚 D 㧩1.2㨪1.4 ߢࠇ߫㧘ᐔဋㆃᑧᤨ㑆ߦᄢ߈ߥჇടߪ. - 18 .
(193) ߥ㧚. 㪈. 㪝㪛㪣 ☸ᐲ㩷 㪛. 㪇. 㪉㪅㪇. 㪈㪅㪇. U. 㪈㪇㪄㪈. 0.7. U. 㪄㪉. 㪈㪇. 㪈㪇㪄㪊. 㪊㪅㪇. B. 0.6. U. 32. 0.5. 㪈㪇. 㪈㪇㪄㪌. U. 0.4. 㪄㪍. 㪈㪇. 㪈㪇㪄㪎 㪈㪇㪄㪏 㪈㪇㪄㪐 㪈㪇㪄㪈㪇. U. 0.3. 㪈㪇㪄㪈㪈 㪈㪇㪄㪈㪉 FDL ☸ᐲ D ߦኻߔࠆࡄࠤ࠶࠻ᑄ᫈₸㧔 B. ࿑ 10.. 32 㧕. 㪝㪛㪣 ☸ᐲ㩷 㪛 㪇. 㪇㪅㪉. 㪇㪅㪋. 㪇㪅㪍. 㪇㪅㪏. 㪈㪅㪇. 㪈㪅㪉. 㪈㪅㪋. 㪈㪅㪍 㪈㪅㪏. 㪈㪅㪇. U. 0.5. FDL ☸ᐲ D ߦኻߔࠆࡄࠤ࠶࠻ᑄ᫈₸㧔 U. 0.5 㧕. 㪙㪔㪋 㪙㪔㪏. 㪈㪇㪄㪈. ᑄ᫈₸㩷 㪧㪙. ᑄ᫈₸㩷 㪧㪙. 㪄㪋. 㪈㪇㪄㪉. 㪙㪔㪈㪍. 㪈㪇. 㪄㪊. 㪙㪔㪊㪉 㪄㪋. 㪈㪇. 㪈㪇㪄㪌 㪈㪇㪄㪍. 㪙㪔㪍㪋. 㪈㪇㪄㪎 㪈㪇㪄㪏 ࿑ 11.. - 19 . 㪉㪅㪇.
(194) 㪍㪇. U. 0 .5. ᐔဋㆃᑧᤨ㑆㩷 㪮㪫. 㪌㪇. 㪙㪔㪍㪋. 㪋㪇 㪊㪇. 㪙㪔㪊㪉. 㪉㪇. 㪙㪔㪈㪍. 㪈㪇 㪙㪔㪏 㪇 㪇. 㪇㪅㪉. 㪇㪅㪋. 㪇㪅㪍. 㪇㪅㪏. 㪈㪅㪇. 㪈㪅㪉. 㪈㪅㪋. 㪈㪅㪍. 㪈㪅㪏. 㪉㪅㪇. 㪝㪛㪣 ☸ᐲ㩷 㪛 ࿑ 12.. FDL ☸ᐲ D ߦኻߔࠆࡄࠤ࠶࠻ᑄ᫈₸㧔 U. 0.5 㧕. 4.3 ⠨ኤ ࡄࠤ࠶࠻㐳߇ᜰᢙ㑐ᢙಽᏓߦࠆߣߔࠆߣ㧘ㆃᑧᤨ㑆㒢એߣߥࠆࡄࠤ࠶࠻ߦኻߒߡ 㐳࠹࡞ߩಽᏓ߇ሽߔࠆ㧚IP ࡀ࠶࠻ࡢࠢߢߪࡄࠤ࠶࠻㐳ߦߪ 1518 ࡃࠗ࠻ߩ㒢߇ ࠆߚ㐳࠹࡞ߪߥߊ㧘ᜰᢙ㑐ᢙಽᏓߣߪᦨᖡ୯⸳⸘ߣߥࠆ㧚ᢥ₂[5]ߦߪ㧘ဋ৻ಽᏓ ߣ࿕ቯ㐳ߩ႐วߦ߅ߌࠆᑄ᫈₸߇ࠪࡒࡘ࡚ࠪࡦ⸘▚⚿ᨐߣߒߡ⸥タߐࠇߡࠆ߇㧘ᜰ ᢙ㑐ᢙಽᏓߣᲧセߒߡ㧘߅ߩ߅ߩ 1/15㧘1/300 ⒟ᐲᑄ᫈₸ߣߥߞߡࠆ㧚ߒߚ߇ߞߡ㧘ታ ⸳⸘߳ߩㆡᔕࠍ⠨߃ࠆߣ㧘ᜰᢙ㑐ᢙಽᏓߩߺߢߪਇ⿷ߢ㧘ဋ৻ಽᏓߣ࿕ቯ㐳㧘ߐࠄߦછᗧ ಽᏓߩࡄࠤ࠶࠻㐳ࠍᗐቯߒߚ࠻ࡅ࠶ࠢℂ⺰߇ᔅⷐߢࠆ㧚 ߒ߆ߒߥ߇ࠄ㧘ታ⸳⸘ߦὶὐࠍᒰߡߚ႐ว㧘ࠪࡒࡘ࡚ࠪࡦ⸘▚ߪᾘ㔀ߔ߉ߡ৻⥸ߩ ↪ߦߪਇะ߈ߢࠆߣ⠨߃ࠄࠇࠆߩߢ㧘છᗧಽᏓߩࡄࠤ࠶࠻㐳ߦኻߒߡ♖ᐲߩ㜞ㄭૃ ⸘▚ᑼ߇ᦸ߹ࠇࠆ㧚ᓟߩ⺖㗴ߣߒߚ㧚. 5.. ߹ߣ. శࡄࠤ࠶࠻឵ᯏ⸳⸘㗴ࠍ⸃ߔࠆߚ㧘ࡄࠤ࠶࠻ᑄ᫈₸ߣㆃᑧᤨ㑆ಽᏓࠍ᳞ࠆ࠻ ࡅ࠶ࠢℂ⺰ࠍ⠨ኤߒߚ㧚ℂ⺰ߩኻ⽎ߣߒߚశࡃ࠶ࡈࠔ᭴ᚑߦ߅ߡ㧘FDL ᤨ㑆☸ᐲߩᦨ ㆡൻࠍ࿑ࠆߎߣ߇ߘߩਥߥ⋡⊛ߢࠆ㧚ᓧࠄࠇߚ⚿ᨐߪ㧘ᰴߩߣ߅ࠅߢࠆ㧚 (1). ᜰᢙ㑐ᢙಽᏓࡄࠤ࠶࠻㐳ߦኻߒߡ㧘ᓙߜᤨ㑆ಽᏓߣᑄ᫈₸ࠍ᳞ࠆㄭૃᑼࠍ᳞ߚ㧚 ㄭૃᑼߦ߅ߡᡆૃ⽶⩄ࠍቯ⟵ߒ㧘ዉߔࠆߎߣ߇ᑼߩ৻⥸ൻߦലߢࠆߎߣࠍ ࠄ߆ߦߒߚ㧚. (2). ㄭૃᑼߦࠃࠆ⸘▚୯ߪႎ๔ߐࠇߡࠆࠪࡒࡘ࡚ࠪࡦ⚿ᨐߣ㧘ᑄ᫈₸ߢ 20㧑એਅ㧘 ᐔဋㆃᑧᤨ㑆ߢ 3㧑એਅߩᏅߢࠅ㧘ࠃ৻⥌ࠍߺߖߚ㧚. (3). ⸳⸘ߣߒߡ㧘⽶⩄ 0.5 ߦ߅ߌࠆᦨㆡൻࠍ᳞㧘0.01㧑એਅߩࡄࠤ࠶࠻ᑄ᫈₸ߣߔࠆ. - 20 .
(195) ߚߦߪ㧘ᤨ㑆☸ᐲࠍ 1.2㨪1.4㧘FDL ࠍ 32 ᧄએߣߔࠆᔅⷐ߇ࠆߎߣࠍࠄ߆ߦ ߒߚ㧚 ታ⸳⸘ߦ↪ߔࠆߚߦߪ㧘ࡄࠤ࠶࠻㐳ߦߟߡဋ৻ಽᏓ߿࿕ቯ㐳㧘છᗧಽᏓߣߒߚℂ ⺰ᑼ߇ᔅⷐߢࠅ㧘ᓟߩ⺖㗴ߣߒߚ㧚 ઃ㍳ M/M/1/K ࠪࠬ࠹ࡓߦ߅ߌࠆᑄ᫈₸ ㊂ࠍ a ߣ߅ߊߣ߈㧘M/M/1/K ࠪࠬ࠹ࡓߩᑄ᫈₸ߪᰴᑼߢࠆ[13]㧚 . B. 1 a
(196) a K 1 a. 㧔ઃ 1㧕. K 1. ࠪࠬ࠹ࡓౝቴᢙ r ߣߥࠆ⏕₸ࠍ Pr ߣ߅ߊߣ㧘ߘߩ⏕₸߇ K એߣߥࠆ⏕₸ E K ߪ f. EK. ¦ Pr. r K 1. f. r. 0. B.
(197). a K 1 1 a a 2 P0. aK . 㧔ઃ 2㧕. r K 1. ߣߥࠆ㧚ߎߎߢ㧘 P0. . ¦ a
(198) P. 1 1 a
(199) ࠍ↪ߚ㧚ߒߚ߇ߞߡ㧘ᑼ(ઃ 1)ߪᰴᑼߢߐࠇࠆ㧚. 1 a
(200) E K 1 aE. 㧔ઃ 3㧕. K. ෳ⠨ᢥ₂ [1] P. Gambili et al., “Transparent Optical Packet Switching: Network Architecture and Demonstrators in the KEOPS Project,” IEEE J. of Selected Area in Commun., Vol. 16, No. 7, pp. 1245-1259,1998. [2] D. K. Hunter et al., “WASPNET: A Wavelength Switched Packet Network,” IEEE Commun. Mag., Vol. 37, No. 3, pp. 120-129, 1999. [3] F. Callegati, “Optical Buffers for Variable Length Packets,” IEEE Commun. Lett., Vol. 4, No. 9, pp. 292-294, 2000. [4] Xiaohua Ma, “Modeling and Design of WDM Optical Buffers in Asynchronous and Variable-Length Optical Packets Switches,” Optical Commun., No. 269, pp. 53-63, 2007. [5] Jianming Liu et al., “Blocking and Delay Analysis of Single Wavelength Optical Buffer with General Packet Size Distribution,” J. Lightwave Technol., Vol. 27, No. 8, pp. 955-966, 2009. [6] H. E. Kankaya and N. Akar, “Exact Analysis of Single-Wavelength Optical Buffers with Feedback Markov Fluid Queues,” J. Opt. Commun. Netw., Vol. 1, No. 6, pp. 530-542, 2009. [7] ᜕⪺ߒߐ߿ޟᖱႎ឵Ꮏቇޠർ 㧔2009 ᐕ㧕㧘p.61㧘3.3.2 ▵㧚 [8] Fei Xue et al., “Design and Experimental Demonstration of a Variable-Length. - 21 .
(201) Optical Packet Pouting System With Unified Contention Resolution,” J. Lightwave Technol., Vol. 22, No. 11, pp. 2570-2581, 2004. [9] ᜕⪺ߒߐ߿ޟᖱႎ឵Ꮏቇޠർ 㧔2009 ᐕ㧕㧘p.105㧘5.3 ▵㧘ᑼ(5.16)㧚 [10] ห㧘p.93㧘4.5.2 ▵㧚 [11] Percy H. Brill, “A Brief Outline of the Level Crossing Method in Stochastic Models,” CORS Bulletin Vol. 34, No. 4, pp. 1-8, 2000. [12] Ṛᩮື㧘દ⮮ᄢテ㧘የ┨ᴦ㇢⪺ޟጤᵄ⻠ᐳࠗࡦ࠲ࡀ࠶࠻ 5 ࡀ࠶࠻ࡢࠢ⸳⸘ ℂ⺰ޠጤᵄᦠᐫ㧔2001 ᐕ㧕㧘p.58㧘ᑼ(2.42)㧚 [13] ߚߣ߃߫㧘ห㧘p.66㧘ᑼ(2.51)㧚 [14] ᜕⪺ߒߐ߿ޟᖱႎ឵Ꮏቇޠർ 㧔2009 ᐕ㧕㧘p.112㧘5.4 ▵㧘ᑼ(5.38)㧚. - 22 .
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