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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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(1) .

(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ߪߓ߼ߦ   ࠗࡦ࠲࡯ࡀ࠶࠻࠻࡜ࡅ࠶ࠢ߇Ფᐕ೨ᐕᲧ 1.5 ୚ߩિ߮₸ߢჇᄢߒߡ޿ߊ⁁ᴫߦ߅޿ߡ㧘 㔚ሶಣℂߦࠃࠆ IP ࡞࡯࠲ߢߪォㅍ⢻ജߦᔅߕ㒢⇇߇⃻ࠇࠆ㧚߹ߚ㧘IP ࡞࡯࠲ߩ㔚ജᶖ⾌ ㊂߽ߎߩ߹߹Ⴧᄢࠍ⛯ߌࠇ߫㧘㔚ജࠦࠬ࠻߇ㆇ↪ࠦࠬ࠻ߩᄢ߈ߥᲧ㊀ࠍභ߼ࠆࠃ߁ߦߥࠆ ߣߣ߽ߦ㧘࿾⃿᷷ᥦൻࠍഥ㐳ߔࠆ⚿ᨐߣߥࠆ㧚ᄢ߈ߊߪߎߩ 2 ߟߩ໧㗴ࠍ⸃᳿ߔࠆᣇᴺߣ ߒߡ㧘శାภߩ߹߹ࡄࠤ࠶࠻ࠍಣℂߔࠆశࡄࠤ࠶࠻੤឵ᯏߩ⊓႐߇ᦼᓙߐࠇߡ޿ࠆ㧚ߔߥ ࠊߜ㧘శࡈࠔࠗࡃㅢାࠪࠬ࠹ࡓߦࠃࠅ 1 ࿁✢ߩવㅍㅦᐲ߇ 100Gbps ߦ㆐ߔࠆ⁁ᴫߦߥࠆ ߣ㧘વㅍ⢻ജߩ㜞޿శᛛⴚࠍ↪޿ߚశ੤឵ᯏ߇㧘ᰴ਎ઍߩ IP ࡞࡯࠲ࠍᜂ߁ߎߣߦߥࠆߪ ߕߢ޽ࠆߣ޿߁ᦼᓙߢ޽ࠆ㧚 శߪ㧘߽ߣ߽ߣࡆ࠶࠻ᖱႎࠍዊߐ޿ࡄࡢ࡯ߢォㅍߔࠆߎߣߪᓧᗧߢ޽ࠆ߇㧘శ⥄りࠍ೙ ᓮߔࠆߎߣߪᓧᗧߢߪߥ޿㧚శࡄࠤ࠶࠻ࠍశߩ߹߹ォㅍಣℂߔࠆࡄࠤ࠶࠻੤឵ᯏߪ㧘ᄙߊ ߩ⎇ⓥ⠪ߩᦼᓙࠍ⢛⽶޿ߘߩታ⃻ߦะߌߡ♖ജ⊛ߥ⎇ⓥ߇⛯ߌࠄࠇߡ޿ࠆ[1-2]߇㧘߹ߛ߹ ߛ㆏ߪ㆙޿ߩ߇⃻⁁ߢ޽ࠆ㧚ࡔࡕ࡝߿⺰ℂṶ▚ߥߤ㧘᣿ࠄ߆ߦశᛛⴚߢߪਇᓧᗧߥಽ㊁߇ ޽ࠆ߆ࠄߢ޽ࠆ㧚ዋߥߊߣ߽㧘⺰ℂṶ▚ߪ㔚᳇⊛ߦⴕ߁ߎߣߢ⸃᳿ࠍ࿑ࠆߩߢ޽ࠈ߁㧚  ࡔࡕ࡝ߦߪ㧘ㆃ޿శ㧔slow lights㧕ߥߤ♖ജ⊛ߦ⎇ⓥߐࠇߡ޿ࠆ߽ߩ߇޽ࠆ߇㧘ታ⃻ᕈ ߆ࠄߺࠆߣశ శࡈࠔࠗࡃㆃᑧ✢㧔optical fiber delay lines㧦FDL㧕ࠍ೑↪ߔࠆߎߣ߇ㄭ㆏ߢ ޽ࠆ㧚FDL ߪ㧘㔚᳇ RAM ߣ⇣ߥࠆᰴߩࠃ߁ߥ․ᓽࠍᜬߟ㧚 (1) RAM ߢߪછᗧߩ⫾Ⓧᤨ㑆ߣછᗧᤨೞߢߩ⺒ߺ಴ߒ߇ታⴕߢ߈ࠆ߇㧘FDL ߪߘߩ㐳ߐ ߦᲧ଀ߒߚ৻ቯ㊂ߩ⫾Ⓧᤨ㑆ߒ߆ᓧࠄࠇߥ޿㧚FDL ߢߩ⫾Ⓧᤨ㑆ߪ㧘శࡈࠔࠗࡃ㐳ߦ Ყ଀ߔࠆߩߢ㧘FDL 㐳ߩන૏ࠍㆃᑧᤨ㑆ߢ⠨߃ࠆ㧚ߎࠇࠍᤨ ᤨ㑆☸ᐲ㧔time granularity㧕 ߣ๭߱㧚 ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ *ᄢ㒋㔚᳇ㅢାᄢቇ ᖱႎㅢାᎿቇㇱ ㅢାᎿቇ⑼     !. - 1 .

(3) (2) ⴣ⓭࿁ㆱߦ㑆ߦวࠊߥߌࠇ߫㧘ࡄࠤ࠶࠻ߪᑄ᫈ߐࠇࠆ㧚 శࡄࠤ࠶࠻੤឵ᯏߦ߅޿ߡࡄࠤ࠶࠻ᑄ᫈₸߿ᐔဋㆃᑧᤨ㑆ߥߤߩ⸳⸘୯ࠍ᳞߼ࠆ࠻࡜ࡅ ࠶ࠢℂ⺰ߩᄙߊߪ㧘FDL ࠍ೑↪ߒߚశࡃ࠶ࡈࠔ᭴ᚑࠍᛒߞߡ޿ࠆ㧚శࡃ࠶ࡈࠔ᭴ᚑߦߪᄙ ߊߩឭ᩺߇޽ࠆ߇㧘ㅢᏱߪᤨ㑆☸ᐲߩᢛᢙ୚ߩ FDL ࠍᢙᄙߊ↪ᗧߒߡ㧘ᤨ㑆☸ᐲߩᦨㆡ ൻࠍ࿑ࠆ߽ߩߣߥߞߡ޿ࠆ[3-6]㧚ߔߥࠊߜ㧘FDL ߩ㐳ߐಽᏓࠍߤߩࠃ߁ߦߔࠇ߫㧘ࡄࠤ࠶ ࠻ᑄ᫈₸ࠍᛥ߃ࠆߎߣ߇ߢ߈ࠆ߆‫߁޿ߣޔ‬໧㗴ߢ޽ࠆ㧚ߎࠇߦኻߒߡ㧘ㄭૃᢙ୯⸃[3-4]߿ ᢙ୯ࠪࡒࡘ࡟࡯࡚ࠪࡦ[5-6]ߥߤߐ߹ߑ߹ߥขࠅ⚵ߺ߇ߐࠇߡ޿ࠆ߇㧘޿ߕࠇ߽➅ࠅ㄰ߒ⸘ ▚ࠍᔅⷐߣߒ㧘⷗ㅢߒߩࠃ޿ℂ⺰ᑼࠍᓧߡ޿ࠆࠊߌߢߪߥ޿㧚 ᧄ⺰ᢥߪ㧘శࡄࠤ࠶࠻੤឵ᯏߩ࠻࡜ࡅ࠶ࠢℂ⺰ߦ㑐ߔࠆ޿߹߹ߢߩᚑᨐࠍ〯߹߃㧘ᢥ₂ [5]ߩ⸥ㅀߦᴪ޿ౝኈࠍℂ⸃ߒ߿ߔ޿ᒻߦᢛℂߔࠆߣߣ߽ߦ㧘ᜰᢙ㑐ᢙಽᏓࡄࠤ࠶࠻㐳ߦኻ ߔࠆᓙߜᤨ㑆ಽᏓߩㄭૃ⸃ࠍਈ߃ߡ޿ࠆ㧚ዉ಴ߒߚㄭૃ⸃ߩ♖ᐲࠍࠪࡒࡘ࡟࡯࡚ࠪࡦ⚿ᨐ ߣᲧセߒ㧘ߐࠄߦૐ⽶⩄⁁ᘒߦ߅ߌࠆశࡃ࠶ࡈࠔ᭴ᚑߩᬌ⸛ࠍⴕߞߡ޿ࠆ㧚. 㧞㧚శࡃ࠶ࡈࠔࡕ࠺࡞  శ੤឵ᯏߩ᭴ᚑߣߒߡ㧘࿑㧝. శࡃ࠶ࡈࠔ. ߦ␜ߔࠃ߁ߦ㧘ࠬ࡞࡯ࡊ࠶࠻೙. 1. 1. 2. 2. ⠨߃ࠆ㧚శࠬࠗ࠶࠴ߩታ⃻ᣇᴺ ߦߪ㧘ⓨ㑆ࠬࠗ࠶࠴[1]ߣᵄ㐳ࠬ. శࠬࠗ࠶࠴ ࡮࡮࡮. ࡮࡮࡮. 3. 㒢ߩߥ޿಴ജᓙߜⴕ೉ᒻ[7]ࠍ. 3. ࠗ࠶࠴[2]߇ឭ᩺ߐࠇߡ߅ࠅ㧘೨ ⠪ߪⓨ㑆ᄙ㊀ᣇᑼࠍᓟ⠪ߪᵄ㐳 ᄙ㊀ᣇᑼࠍ೑↪ߒߡ޿ࠆ㧚޿ߕ. N. N. ࠇ߽㧘಴ജᓙߜⴕ೉ᒻశ੤឵ᯏ ࠍታ⃻ߢ߈ࠆ㧚. ࿑㧝㧚಴ജᓙߜⴕ೉ᒻశࠬࠗ࠶࠴ߩၮᧄ᭴ᚑ.  ᓙߜⴕ೉ߣߥࠆ಴ജࡃ࠶ࡈࠔ ߦ FDL ࠍ೑↪ߔࠆ᭴ᚑࠍ㧘࿑ 㧞ߦ␜ߔ㧚㧝ߟߩ಴ജߦኻߒߡ 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ߐන૏ࠍ. 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. ⌀ⓨਛߩశㅦߢ޽ࠆ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࠶ࡈࠔ಴ജ. శࡃ࠶ࡈࠔ౉ജ (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࠶ࡈࠔ಴ജ Ԙ. ❥ᔔᦼ㑆   (d) ࡄࠤ࠶࠻Ԙ㨪Ԛߩ಴ജ. ࿑ 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 એ਄ߩㆃᑧᤨ㑆߇ᔅⷐߣߥࠆࡄࠤ࠶࠻ߪ⎕᫈. ߐࠇࠆ㧚  ࿑㧞ߩశࡃ࠶ࡈࠔߦ߅޿ߡ㧘శࡄࠤ࠶࠻߇ᵹࠇࠆ᭽ሶࠍ࿑㧟ߦ␜ߔ㧚੹㧘࿑(a)ߦ␜ߔࠃ ߁ߦ㧘Ԙ㧘ԙ㧘߅ࠃ߮Ԛߩ 3 ୘ߩశࡄࠤ࠶࠻߇㧘శࡃ࠶ࡈࠔߩ౉ജㇱߦߘࠇߙࠇᤨೞ 0 㧘. t 2 㧘߅ࠃ߮ t 3 ߦ೔⌕ߔࠆߎߣߣߔࠆ㧚߹ߚ㧘శࡄࠤ࠶࠻Ԙߪචಽߦ㐳޿ߩߢ㧘ᓟ⛯ߔࠆశ ࡄࠤ࠶࠻ԙߣԚߣߪ಴ജߢⴣ⓭ߔࠆߣߔࠆ㧚ߘߎߢߪ㧘ⴣ⓭ࠍ࿁ㆱߔࠆߚ߼ᰴߩࠃ߁ߥಣ ⟎߇ขࠄࠇࠆ㧚  ࿑(b)ߦ␜ߔࠃ߁ߦ㧘ᤨೞ t 2 ߦ߅޿ߡ㧘ࡄࠤ࠶࠻ԙߪዋߥߊߣ߽ w2 㧘 0  w2 d D ߩᓙߜ ᤨ㑆߇ᔅⷐߢ޽ࠆߣ߈㧘ԙߪ㧞⇟⋡ߩ FDL ߦォㅍߐࠇࠆ㧚ߎߩߣ߈㧘FDL 㐳߇㔌ᢔ⊛ߢ ޽ࠆߚ߼ߦ W 2. D  w2 ߩⓨᦼ㑆㧔void periods㧕߇ᔅⷐߣߥࠆ㧚.  ᤨೞ t 3 ߦ߅޿ߡߪ㧘࿑(c)ߦ␜ߔࠃ߁ߦ㧘ࡄࠤ࠶࠻Ԛߪ w3 㧘2 D  w3 d 3D ߩᓙߜᤨ㑆߇ ᔅⷐߢ޽ࠆߣߥࠆߣ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߇ߟ޿ߚ⁁ᘒߢ಴ജߐࠇࠆ㧚ߎߩⓨᦼ㑆ߪ㧘FDL ߇ᤨ㑆☸ᐲ D ࠍ න૏ߣߔࠆ㔌ᢔ⊛ߥ୯ߢ޽ࠆߚ߼㧘FDL ߦォㅍߐࠇࠆࡄࠤ࠶࠻ߦᔅߕઃടߐࠇࠆ߽ߩߢ޽ ࠆ㧚⚿ᨐ⊛ߦ㧘ࡄࠤ࠶࠻Ԙ߆ࠄԚ߹ߢߪㅪ⛯ߒߚࡄࠤ࠶࠻೉ߣߺߥߔߎߣ߇ߢ߈㧘ߎߩᦼ ❥ᔔᦼ㑆㧔busy period㧕ߣ޿߁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 ࠍ⿥߃ࠆᦨዊᢛᢙࠍᗧ๧ߔࠆ㧚 (2) T. B  1

(14) D  w ߩߣ߈㧘ᑄ᫈ߐࠇࠆ㧚.  ⓨᦼ㑆߇ઃടߐࠇࠆಽ㧘੤឵ᯏߦ߆߆ࠆࡄࠤ࠶࠻⽶⩄ߪታ㓙ߩ⽶⩄ࠃࠅㆊ೾ߣߥࠆ㧚ⓨ ᦼ㑆ߪ㧘శࡃ࠶ࡈࠔ߇ⓨߩߣ߈೔⌕ߔࠆࡄࠤ࠶࠻ߦߪઃടߐࠇߥ޿߇㧘శࡃ࠶ࡈࠔߦࡄࠤ ࠶࠻߇⫾Ⓧߐࠇߡ޿ࠆߣ߈ߦߪઃടߐࠇࠆ㧚ߘߎߢ㧘ⓨᦼ㑆ࠍ฽߼ߚࡄࠤ࠶࠻ࠍ޽ࠄߚߦ ‫߁޿ߣޠߣߞߌ߬ޟ‬ฬ⒓ߢቯ⟵ߔࠆ㧚ߔߥࠊߜ㧘߬ߌߞߣߦߪ㧘 Ԙ శࡃ࠶ࡈࠔ߇ⓨߩߣ߈೔⌕ߔࠆࡄࠤ࠶࠻㧚ߎࠇࠍ㧘ೋᦼ೔⌕ࡄࠤ࠶࠻㧔first arrival packets㧕ߣ๭߮㧘ⓨᦼ㑆ࠍᜬߚߥ޿㧚 㕖ೋᦼ ԙ శࡃ࠶ࡈࠔߦࡄࠤ࠶࠻߇⫾Ⓧߐࠇߡ޿ࠆߣ߈೔⌕ߔࠆࡄࠤ࠶࠻㧚ߎࠇࠍ㧘㕖 ೔⌕ࡄࠤ࠶࠻㧔non-first arrival packets㧕ߣ๭߮㧘ⓨᦼ㑆ࠍ฽ࠎߛ㐳ߐߣߥࠆ㧚 ߩ㧞⒳㘃ߩࡄࠤ࠶࠻߇޽ࠆ㧚ߎࠇࠄ߬ߌߞߣ߇㓗㑆ߥߊㅪ⛯ߒߡ಴ജߐࠇࠆᦼ㑆߇㧘❥ᔔ ᦼ㑆ߣߥࠆ㧚. 㧟㧚࠻࡜ࡅ࠶ࠢℂ⺰⸃ᨆ  IP ࡄࠤ࠶࠻࠻࡜ࡅ࠶ࠢߩ೔⌕ㆊ⒟ߦ߅޿ߡߪ㧘ᄙߊߩ࠻࡜ࡅ࠶ࠢࠍ㓸✢ߔࠆၮᐙࡀ࠶࠻ ࡢ࡯ࠢߦ߅޿ߡ⛔⸘⊛ߥᕈ⾰߇ࡐࠕ࠰ࡦಽᏓߦ෼᧤ߔࠆߎߣ߇⍮ࠄࠇߡ߅ࠅ㧘ㄭૃ⊛ߦࡑ ࡞ࠦࡈㆊ⒟ߢ޽ࠆߣߒߡ⸳⸘໧㗴ߦ೑↪ߔࠆߎߣ߇ߢ߈ࠆ㧚ࡄࠤ࠶࠻㐳ߪ 58‫ޔ‬594‫ޔ‬1518 ࡃࠗ࠻ߥߤߦࡇ࡯ࠢࠍ߽ߟಽᏓߣߥߞߡ߅ࠅ㧘ਇቯᒻߢ޽ࠆ[8]㧚ߒߚ߇ߞߡ㧘M/D/1/K ࠪ ࠬ࠹ࡓ߇శࡄࠤ࠶࠻੤឵ᯏࡕ࠺࡞ߦ߰ߐࠊߒ޿ߣᕁࠊࠇࠆ߇㧘M/D/1/K ࠪࠬ࠹ࡓߣߩ㆑޿ ߪᰴߩ 2 ὐߦ޽ࠆ㧚 Ԙ 㕖ೋᦼ೔⌕ࡄࠤ࠶࠻ߦ㧘ⓨᦼ㑆߇ઃടߐࠇࠆߎߣ㧘 ԙ ࠪࠬ࠹ࡓኈ㊂ߦ೙㒢߇޽ࠆߩߢߪߥߊ㧘⫾Ⓧᤨ㑆ߦ೙㒢߇޽ࠆߎߣ㧚  એਅߢߪ㧘శࡄࠤ࠶࠻ᑄ᫈₸ߣᐔဋㆃᑧᤨ㑆ߩ৻⥸⸃ࠍ᳞߼ࠆ㧚. 3.1 ╬ଔ⽶⩄  ೔⌕ߔࠆశࡄࠤ࠶࠻ߪࡑ࡞ࠦࡈㆊ⒟ߦᓥ߁ߣߒ㧘ߘߩ೔⌕₸ࠍ O 㧘ࡄࠤ࠶࠻㐳ࠍ s 0 ߣ߅ ߊ㧚 s 0 ߩಽᏓߦߟ޿ߡᰴߩ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ߦߪଐሽߒߥ޿ߣߒߡ޿ࠆ ߩߢ㧘 W ߪ >0, D @ ߩ㑆ߢဋ৻ߦಽᏓߒߡ޿ࠆߣߔࠆߎߣ߇ߢ߈ࠆ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ߔߥࠊߜࠨ࡯ࡆࠬᤨ㑆 sX ߪ㧘ࡄࠤ࠶࠻㐳 s 0 ߣⓨᦼ㑆 W ߩว⸘   sX. s0  W. (3.4). ߢ޽ࠆߩߢ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. ߢ᳞߼ࠄࠇࠆ㧚ߎߎߢ㧘 ߪ⇥ߺㄟߺⓍಽ㧔convolution integral㧕ߢ޽ࠆ㧚 㕖ೋᦼ೔⌕߬ߌߞߣߩᐔဋࠨ࡯ࡆࠬᤨ㑆 sX ߪ f.   sX. ³ xg x

(29) dx                    . (3.6). 0. ╬ଔ⽶⩄ U eq ࠍ↪޿ࠆߣ㧘 ࠃࠅ᳞߼ࠄࠇࠆ߇㧘߬ߌߞߣߩ⽶⩄ߢቯ⟵ߐࠇࠆ╬  . U eq. OS. (3.7). ࠃࠅ᳞߼ࠄࠇࠆ႐ว߇޽ࠆ㧚╬ଔ⽶⩄ߪ㧘ⓨᦼ㑆ࠍ⽶⩄ߦขࠅㄟࠎߛ߽ߩߢ޽ࠆ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޿⁁ᘒߢ޽ࠆ㧚ߎߩࠪ ࠬ࠹ࡓߦ߅ߌࠆ઒ᗐᓙߜᤨ㑆ಽᏓࠃࠅ㧘᦭㒢㐳ࡃ࠶ࡈࠔߦ߅ߌࠆ⸃߇᳞߼ࠄࠇࠆ㧚  ੹㧘శࡃ࠶ࡈࠔ߇❥ᔔᦼ㑆ߦߥ޿ߣ߈㧘ߔߥࠊߜⓨߩߣ߈ߩ⏕₸ࠍ Q ߣ߅ߊߣ㧘G/G/1 ࠪࠬ࠹ࡓߦ߅޿ߡ  . U eq 1  Q. (3.8). ߇ᚑ┙ߔࠆ[9]㧚ೋᦼ೔⌕ࡄࠤ࠶࠻ߩᐔဋࠨ࡯ࡆࠬᤨ㑆߇ s 0 ߢ޽ࠆߩߦኻߒߡ㧘㕖ೋᦼ೔⌕ ࡄࠤ࠶࠻ߩߘࠇߪᑼ(3.3)ࠃࠅ s 0  D 2 ߢ޽ࠆߩߢ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)ߦઍ౉ߔࠆߣᰴᑼࠍᓧࠆ㧚  . U eq. U. (3.10). D U 1 2s0.  D ! 0 ߢ޽ࠆ㒢ࠅ㧘 U eq !. U ߢ޽ࠆ㧚߹ߚ㧘ᤨ㑆☸ᐲ D ߇ᄢ߈ߊߥࠆߣ㧘 U eq ߪჇടߔ. ࠆ㧚ࠪࠬ࠹ࡓ߇෼᧤ߔࠆߚ߼ߩ᧦ઙߪ㧘 U eq  1 ߢ޽ࠅ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 ࠃࠅዊߐ޿႐วߢ߽㧘U eq ! 1 ߣߥࠆߎߣ߇޽. ࠆߎߣߦᵈᗧߔࠆᔅⷐ߇޽ࠆ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෼᧤᧦ઙߢ޽ࠆ㧚.  ᡆૃ⽶⩄ U c ߪ㧘ߔߴߡߩ೔⌕ࡄࠤ࠶࠻ߦᐔဋⓨᦼ㑆 D 2 ࠍട߃ߚ⽶⩄ߢ޽ࠆߚ߼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࠶ࡈࠔߦ߅ߌࠆ߬ߌߞߣߩ઒ᗐᓙߜᤨ㑆㧔virtual waiting time㧕ߦ㑐ߒߡ㧘 ቯᏱ⁁ᘒߦ߅ߌࠆ⏕₸ಽᏓߣߒߡᰴߩ㑐ᢙࠍቯ⟵ߔࠆ㧚. v x

(33) 㧦߬ߌߞߣߩ઒ᗐᓙߜᤨ㑆 x ߦ߅ߌࠆ⏕₸ኒᐲ㑐ᢙ㧔pdf㧕㧘߅ࠃ߮ V x

(34) 㧦߬ߌߞߣߩ઒ᗐᓙߜᤨ㑆 x ߦኻߔࠆ⫾ⓍಽᏓ㑐ᢙ㧔CDF㧕㧚 ߎߎߢ㧘 ‫ޟ‬઒ᗐ‫ޠ‬ᓙߜᤨ㑆ߣ๭߱ߩߪ㧘⃻ታߦߪߥ޿ή㒢㐳శࡃ࠶ࡈࠔࠍᗐቯߒߡ޿ࠆߚ߼ ߢ޽ࠆ㧚߹ߚ㧘߬ߌߞߣ೔⌕ࠍࡐࠕ࠰ࡦಽᏓߣߒߡ޿ࠆߚ߼㧘PASTA㧔Poisson arrival see time average㧕ߩ㑐ଥ[10]ࠃࠅ㧘೔⌕㧘ㅌ಴ߥߤߩᤨ㑆ⷐ⚛ࠍᶖ෰ߒߡቯᏱ⁁ᘒࠍ઒ቯߒ ߚ㧚ߒߚ߇ߞߡ㧘 v x

(35) ߣ V x

(36) ߪᤨ㑆ᐔဋߢ޽ࠆ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ߩߥ޿ࡄࠤ࠶࠻ߩ⏕₸ኒᐲߪࡃ࠶ࡈࠔ߇ⓨߩߣ߈೔⌕ߔࠆࡄࠤ࠶࠻ᢙߢ ޽ࠆߩߢ㧘╙ 2 ᑼ߇ᚑ┙ߔࠆ㧚  ᓙߜⴕ೉ࡕ࠺࡞ߦ߅޿ߡᓙߜᤨ㑆ಽᏓࠍ⸃ᨆ⊛ߦ᳞߼ࠆ႐วߦߪ㧘ࠨࡦࡊ࡞ᤨ㑆ߦ೔⌕ ߔࠆቴߩ᜼േࠍਤᔨߦㅊ޿㧘ߘߩ᜼േߩ⚻ㆊᤨ㑆߆ࠄᓸⓍಽᣇ⒟ᑼࠍ᳞߼ࠆᣇᴺ߇৻⥸⊛ ߢ޽ࠆ߇㧘ℂ⸃߇㔍ߒߊ߆ߟᾘ㔀ߢ޽ࠆ㧚ࠃࠅ⋥ᗵ⊛ߦ᳞߼ࠆᣇᴺߦ㧘࡟ࡌ࡞੤Ꮕᴺ㧔level crossing method㧕[11]߇޽ࠆ㧚․ߦ㧘ࡐࠕ࠰ࡦ೔⌕ㆊ⒟ߢ߆ߟ FCFS ಣℂⷙೣߩࡕ࠺࡞ߦ ߪ᦭ലߢ޽ࠆ㧚  ࿑㧡ߦ㧘࡟ࡌ࡞੤Ꮕᴺࠍℂ⸃ߔࠆߚ߼㧘⚻ㆊᤨ㑆ߦኻߔࠆ઒ᗐᓙߜᤨ㑆ᄌൻࠍ␜ߔ㧚ታ ✢ߢ␜ߒߚࠨࡦࡊ࡞᜼േࠍߺࠆߣ㧘઒ᗐᓙߜᤨ㑆ߪ߬ߌߞߣߩ೔⌕ߏߣߦု⋥ߦ਄᣹ߔࠆ ߇㧘೔⌕߇ߥ޿ߣ߈ߦߪᤨ㑆ߩ⚻ㆊߦᲧ଀ߒߡਅ㒠ߔࠆ㧚ߘߎߢ㧘છᗧߩᓙߜᤨ㑆 x ߦ⌕ ⋡ߒ㧘ߘߩ࡟ࡌ࡞ࠍ࿑ߦߪᮮὐ✢ߢ␜ߒߚ㧚᷹ⷰᤨ㑆 >0, t @ ߦ߅޿ߡ㧘ࠨࡦࡊ࡞᜼േ߇ု⋥ ਄᣹ߒߡ࡟ࡌ࡞ x ࠍ੤Ꮕߔࠆὐߩᢙࠍ N up t

(44) 㧘ਅ㒠ߒߡ੤Ꮕߔࠆὐߩᢙࠍ N down t

(45) ߣ߅ߊ ߣ㧘න૏ᤨ㑆޽ߚࠅߩ੤Ꮕὐᢙߪ‫ ޔ‬t o f ߦ߅޿ߡᰴᑼߣߥࠆߎߣ߇⸽᣿ߐࠇߡ޿ࠆ㧚. 8- - .

(46) ਄᣹࡟ࡌ࡞੤Ꮕὐ. ਅ㒠࡟ࡌ࡞੤Ꮕὐ. ઒ᗐᓙߜᤨ㑆. x. ࠨࡦࡊ࡞᜼േ 0 ᤨ㑆 t . 0. ࿑ 5 ⚻ㆊᤨ㑆ߦኻߔࠆ઒ᗐᓙߜᤨ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ᒰߚࠅߩ਄᣹੤Ꮕὐᢙߣਅ㒠੤Ꮕὐᢙߪ㧘 t o f ߦ߅޿ߡ╬ߒ޿㧘 (2) ߎߩᤨ㑆ᒰߚࠅߩ੤Ꮕὐᢙߪ㧘࡟ࡌ࡞ x ߦ߅ߌࠆ⏕₸ኒᐲ㑐ᢙߦ╬ߒ޿㧚  ᑼ㧔3.16㧕ࠍ↪޿ߡ㧘઒ᗐᓙߜᤨ㑆ಽᏓ㑐ᢙߦ㑐ߔࠆᣇ⒟ᑼࠍ᳞߼ࠆ㧚੹㧘࿑ 3 ߦ␜ߔ ࡄࠤ࠶࠻Ԙ㧘ԙ㧘߅ࠃ߮Ԛߩ೔⌕ࠍ઒ᗐᓙߜᤨ㑆ߢ޽ࠄࠊߔߣ㧘࿑ 6 ߣߥࠆ㧚ߎߎߢ㧘਄ ᣹੤Ꮕὐߩߺࠍ⠨߃ࠆ㧚  ࡄࠤ࠶࠻Ԙߪೋᦼ೔⌕ࡄࠤ࠶࠻ߢ޽ࠆߩߢ㧘࡟ࡌ࡞ 0 ߆ࠄု⋥ߦ┙ߜ਄߇ࠆ㧚ߎߩ਄᣹ ߇࡟ࡌ࡞ x ࠍ੤Ꮕߔࠆߦߪ㧘ࡄࠤ࠶࠻㐳߇ x એ਄ߢ޽ࠆᔅⷐ߇޽ࠆ㧚න૏ᤨ㑆ߦ O ୘೔⌕ ߔࠆࡄࠤ࠶࠻ߩ㐳ߐ CDF ߇ G0 x

(50) ߢ޽ࠆߩߢ㧘੤Ꮕߔࠆᤨ㑆ഀวߪ O >1  G0 x

(51) @ ߣߥࠆ㧚 ৻ᣇ㧘ᤨ㑆 t 2 ߢ೔⌕ߔࠆࡄࠤ࠶࠻ԙߪ㕖ೋᦼ೔⌕ࡄࠤ࠶࠻ߢ޽ࠆߩߢ㧘ߘߩ߬ߌߞߣ㐳 CDF ߪ G x

(52) ߢ޽ࠆ㧚 t 2 ߦ߅ߌࠆ઒ᗐᓙߜᤨ㑆ࠍ [ ߣߔࠆߣ㧘ࡄࠤ࠶࠻ԙߩ೔⌕ߦࠃࠅ࡟ ࡌ࡞ x ࠍ੤Ꮕߔࠆᤨ㑆ഀวߪ O >1  G x  [

(53) @ ߣߥࠆ㧚࡟ࡌ࡞ 0 ߢ޽ࠆሽ࿷⏕₸ߪ Q 㧘࡟ࡌ ࡞ [ ߢ޽ࠆሽ࿷⏕₸ߪ v [

(54) ߢ޽ࠆߩߢ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▸࿐ߦ޽ࠆ [ ో૕ߦട㊀ᐔဋߒߚ㧚 ࡜ ࡊ࡜ࠬ-ࠬ࠴ࡘ࡞࠴ࠚࠬᄌ឵  ᑼ(3.17㧕ߪ pdf ߦኻߔࠆⓍಽᣇ⒟ᑼߢ޽ࠅ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ߢ᳞߼ࠆ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ᢙߩ࡜ࡊ࡜ࠬᄌ឵ࠍ␜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)ࠍ࡜ࡊ࡜ࠬㅒᄌ឵ߔࠆߣ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ߢ޽ࠆ T. B  1

(75) D ࠍ⿥߃. ࠆߣ㧘߬ߌߞߣߪᑄ᫈ߐࠇࠆ㧚޽ࠄߚ߼ߡ㧘߬ߌߞߣࠍಽ㘃ߔࠆߣ㧘ᰴߩ 3 ⒳㘃ߣߥࠆ㧚 (i) ೋᦼ೔⌕ࡄࠤ࠶࠻㧧ࡃ࠶ࡈࠔߪⓨߢ޽ࠆߩߢ x. 0 ߢ޽ࠅ㧘ㅢㆊࡄࠤ࠶࠻ߢ޽ࠆ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 ߘࠇࠄߣ඙೎ߔࠆ㧚ᑄ᫈ߐࠇߚ߬ߌߞߣߪࡃ࠶ࡈࠔౝߦሽ࿷ߒߥ޿ߎߣ߇㧘ή㒢㐳ࡃ࠶ࡈ ࠔࡕ࠺࡞ߣߩ㆑޿ߢ޽ࠆ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࠺࡞ߢߩ઒ᗐᓙߜᤨ㑆 x ࠍᄌᢙߣߔࠆ⏕₸ኒᐲ㑐ᢙ pdf㧚v x

(78) ߣߩ ㆑޿ߪ㧘ᑄ᫈ߐࠇߚ߬ߌߞߣ߇ሽ࿷ߔࠆߩߢ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฽߹ߥ޿ᐔဋㆃᑧᤨ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ⓨߢ޽ࠆ⏕₸㧚ߔߥࠊߜ㧘. QT. VT 0

(85) 㧚. 㧔3.21㧕. ᑄ᫈ߐࠇߚ߬ߌߞߣ߇޽ࠆߣ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⒟ᑼߢ޽ࠆߚ߼㧘 >0, T @ ߦ޽ࠆ x ߦኻߒߡߪ㧘 vT x

(91) ߣ v x

(92) ⸃ߩᒻߪหߓߢ޽ࠅ㧘⋧੕ߦᲧ଀㑐ଥߦ޽ࠆ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)ߦઍ౉ߒߡ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)ߩഀวߢ޽ࠆ㧚ߒߚ߇ߞߡ㧘߬ߌߞߣಽ 㘃(ii)ߩഀว㧘ߔߥࠊߜㅢㆊ߬ߌߞߣߩ߁ߜ㕖ೋᦼ೔⌕ࡄࠤ࠶࠻ߢ޽ࠆ⏕₸ߪ [1  Q V T

(120) ] ߢ޽ࠆ㧚ߎߩ㕖ೋᦼ೔⌕ࡄࠤ࠶࠻ߦߪ㧘FDL ߦ౉ജߔࠆ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㧘ᰴߦࠃ߁ߦߒߡ᳞߼ࠆߎߣ߇ߢ߈ ࠆ㧚ಽ㘃(i) ߬ߌߞߣߩᐔဋ㐳ߪ s 0 ߢ޽ࠅ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⒟ߢ޽ࠆߩߢ㧘PASTA ߩ㑐ଥߦࠃࠅ㧘ࡄࠤ࠶࠻ᑄ᫈₸ߪࡄ ࠤ࠶࠻߇೔⌕ߒߚߣ߈ߩ㐽Ⴇ⏕₸ߦ╬ߒ޿㧚㐽Ⴇߪ x   PB. T ߩߣ߈⿠߈ࠆߩߢ㧘. 1  VT T

(128). 㧔3.32㧕. ߇ࡄࠤ࠶࠻ᑄ᫈₸ߩၮᧄᑼߢ޽ࠆ㧚  VT T

(129) ࠍ᳞߼ࠆ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 ߦߟ޿ߡ.   QT. Q Q  OS T V T

(132). 㧔3.34㧕. ࠍᓧߡ㧘ᑼ(3.34)ࠍᑼ(3.33)㧘(3.26)ߦઍ౉ߔࠆߣ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 ߣ޿߁᦭㒢㐳శࡃ࠶ࡈࠔߦߡቯ⟵ߐࠇߚᄌᢙ߇฽߹ࠇߡ޿ࠆ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࠶ࡈࠔߢߩᄌᢙߩߺߢ⴫ߐࠇࠆᑼ߇ᓧࠄࠇߚ㧚ߒߚ߇ߞߡ㧘ή㒢㐳శࡃ࠶ࡈࠔ ࡕ࠺࡞ߢߩ⸃ࠍ᳞߼ࠇ߫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 ߢ޽ࠆߩߢ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৻⥸ᑼߢ޽ࠆ㧚  M/G/1 ࠪࠬ࠹ࡓߦ߅޿ߡ♽ౝቴᢙ߇ K એ਄ߣߥࠆ⏕₸ࠍ E K ߣ߅ߊߣ㧘M/G/1/K ࠪࠬ࠹ ࡓߩᑄ᫈₸ߪᰴᑼߢ⴫ߐࠇࠆ[12]㧚   PB. 1  U

(152) E K. 㧔3.41㧕. 1  UE K. M/M/1/K ࠪࠬ࠹ࡓߩ଀ࠍઃ㍳ߦ␜ߔ㧚 K એౝߣ޿߁♽ౝቴᢙ೙㒢߇㧘శࡃ࠶ࡈࠔߩ႐ว ߪ T એਅߣ޿߁ㆃᑧᤨ㑆೙㒢ߦᄌࠊࠅ㧘E K ࠍ [1  V T

(153) ] ߦ⟎߈឵߃ߚᒻ߇ᑼ(3.40)ߢ޽ࠆ㧚 ߐࠄߦ㧘⽶⩄ࠍᡆૃ⽶⩄ߦ⟎߈឵߃ࠆߣ㧘ㄭૃߩߥ޿ᑼ(3.39)ߣߥࠆ㧚ߒߚ߇ߞߡ㧘ᡆૃ ⽶⩄߇ታല⊛ߥ⽶⩄ߢ޽ࠆߣ⹺⼂ߢ߈ࠆ㧚. 㧠㧚ᜰᢙ㑐ᢙಽᏓࡄࠤ࠶࠻㐳ߢߩㄭૃ⸃ᴺ  ᑼ(3.38)ࠍ↪޿ߡᑄ᫈₸ࠍ⸘▚ߔࠆߦߪ㧘V T

(154) ߩ୯߇ᔅⷐߢ޽ࠆ㧚ߔߥࠊߜή㒢㐳ࡃ࠶ ࡈࠔࡕ࠺࡞ߦ߅ߌࠆ઒ᗐᓙߜᤨ㑆ಽᏓࠍ㧘ᓙߜᤨ㑆 x ߦ㑐ߔࠆ㑐ᢙߩᒻߢ᳞߼ࠆᔅⷐ߇޽ ࠆ㧚ߎߩ㑐ᢙߪᑼ(3.20)ߩ v. *. T

(155) ࠍ࡜ࡊ࡜ࠬㅒᄌ឵ߒߡ᳞߼ࠄࠇࠆ߇㧘ታ㓙ߩߣߎࠈ࡜ࡊ࡜. ࠬㅒᄌ឵ߢ᳞߼ࠄࠇࠆ㑐ᢙᒻߪ߈ࠊ߼ߡ㒢ࠄࠇߡ޿ࠆ㧚ⶄ㔀ߥ㑐ᢙࠍ࡜ࡊ࡜ࠬㅒᄌ឵ߔࠆ ߎߣߪ৻⥸ߦ࿎㔍ߢ޽ࠅ㧘߆ߟᢙ୯⊛ߦ⸘▚ߔࠆߎߣ߽ߢ߈ߥ޿㧚ߎߩߚ߼㧘ᑼ(3.20)ࠍ ↪޿ߡ⸘▚ߒߚ଀ࠍ⪺⠪ߩ⍮ࠆ㒢ࠅߥߊ㧘ታ㓙ߩᢙ୯⸘▚ߢߪㅢᏱߎߩᑼࠍ↪޿ߕߦᢙ୯ ࠪࡒࡘ࡟࡯࡚ࠪࡦࠍⴕߞߡ޿ࠆ㧚ߎࠇߢߪ㧘ᑼ(3.20)ࠍ᳞߼ߚᗧ๧߇ߥ޿㧚  ⃻ታߩࠪࠬ࠹ࡓߪߔߴߡ᦭㒢ࡃ࠶ࡈࠔࠪࠬ࠹ࡓߢ޽ࠆ߇㧘ߘߩ⸃ᨆߦ޽ߚࠅ㧘ή㒢ࡃ࠶ ࡈࠔࡕ࠺࡞ߩ߅ߌࠆ⸃ࠍ᳞߼ߡ߆ࠄᑼ(3.41)ࠍ↪޿ߡ᦭㒢ࡃ࠶ࡈࠔࠪࠬ࠹ࡓߦㆡ↪ߔࠆߣ ޿߁ᚻᴺߪ৻⥸⊛ߢ޽ࠅ㧘ࠃߊⴕࠊࠇߡ޿ࠆ㧚ߒ߆ߒߥ߇ࠄ㧘ߎߩ႐ว㧘ᓙߜᤨ㑆ಽᏓߩ 㑐ᢙᒻࠍ᳞߼ߥߌࠇ߫ߥࠄߥ޿ߣ޿߁࿎㔍ߐ߇޽ࠆ㧚  એਅߢߪ㧘శࡄࠤ࠶࠻㐳߇ᜰᢙ㑐ᢙಽᏓߔࠆ႐วߢߩㄭૃ⸃ࠍ᳞߼㧘ߘߩ♖ᐲࠍᢙ୯ࠪ ࡒࡘ࡟࡯࡚ࠪࡦߦࠃࠆ⚿ᨐߣᲧセߔࠆ㧚ࠨ࡯ࡆࠬᤨ㑆߇ᜰᢙ㑐ᢙಽᏓߔࠆࠪࠬ࠹ࡓߪᓙߜ ⴕ೉ࡕ࠺࡞ߢߪ M/M/1/K ߦ⋧ᒰߒ㧘ዉ಴߇ᦨ߽◲නߥࡕ࠺࡞ߢ޽ࠆ߇㧘ᑄ᫈₸ߦ㑐ߒߡ ᣿␜⊛ߥ⴫⃻߇ᓧࠄࠇߡ޿ߥ޿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৻ಽᏓߣ઒ቯߔࠆ㧚ฦ⏕₸ኒᐲ㑐ᢙߣߘߩ࡜ࡊ࡜ࠬᄌ឵ߪ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) ߪන૏ࠬ࠹࠶ࡊ㑐ᢙߢ޽ࠆ㧚ᑼ(3.5)ߦ߅ߌࠆ⇥ߺㄟߺⓍಽߪ㧘࡜ࡊ࡜ࠬᄌ឵ 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). ߢ޽ࠆ㧚ᑼ(4.1)㧘(4.2)ࠍᑼ(3.20)ߦઍ౉ߔࠆߣᰴᑼߣߥࠆ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)ࠍ࡜ࡊ࡜ࠬㅒᄌ឵ߔࠇ߫ㆃᑧᤨ㑆ߦ㑐ߔࠆ⏕₸ኒᐲ㑐ᢙࠍᓧࠆߎߣ߇ߢ߈ࠆ㧚  ߒ߆ߒߥ߇ࠄ㧘ᑼ(4.4)ߩ᣿␜⊛ߥㅒᄌ឵ࠍ᳞߼ࠆߎߣߪ࿎㔍ߢ޽ࠆߩߢ㧘 DT  1 ߣߒ  . 1 DT 1  e  DT | 1  2 DT.

(171). (4.5). - 15 .

(172) ߢㄭૃߒߡ㧘 DT

(173) એ਄ߩߴ߈ਸ਼ࠍήⷞߔࠆ㧚ᑼ(4.4)ߪ߈ࠊ߼ߡ◲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)ߦ߅޿ߡㄭૃߔࠆߚ߼ߩ᧦ઙߢ޽ࠆ DT  1 ߪ㧘ᑼ(4.7)ࠃࠅ 1  U c

(178) D s 0  1 ߣ ห╬ߢ޽ࠆ㧚ߎߩ᧦ઙߪ㧘 D ߇ s 0 ߦኻߒߡᭂ߼ߡዊߐ޿߆㧘⽶⩄ U c ߇ 1 ߦㄭߊ㊀޿႐ว ߦ⋧ᒰߔࠆ㧚ߒߚ߇ߞߡ㧘ᑄ᫈₸߇ᄢ߈޿႐วߩㄭૃߢ޽ࠆ㧚 CDF ߪ㧘ᑼ(3.14)ࠃࠅ᳞߼ࠆߎߣ߇ߢ߈ࠆߩߢ㧘ᑼ(4.7)ࠍઍ౉ߒߡ 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↪޿ࠆߣ߈ࠊ߼ߡ◲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 ߦ߅޿ߡ 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ᢙߣหᒻߢ޽ࠆ[13]㧚ߒߚ߇ߞߡ㧘ᑼ(4.8)߆ࠄ㧘 Ყ଀ଥᢙߢߪ╬ଔ⽶⩄߇㧘ᜰᢙଥᢙߢߪᡆૃ⽶⩄߇ታല ⊛ߥ⽶⩄ߣߥߞߡ޿ࠆߎߣ߇ℂ⸃ߐࠇࠆ㧚  ᑼ(4.8)ࠍᑼ(3.39)ߦઍ౉ߒߡ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⏕⹺ߔࠆᔅⷐ߇޽ࠆ㧚ᑼ (3.11)ߦࠃࠅ㧘FDL ☸ᐲ D ߦߪᰴᑼߩࠃ߁ߥᦨᄢ೙㒢 ߇޽ࠆ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ߔߥࠊߜන૏ᤨ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⸘▚୯ߢ޽ࠆ㧚⽶⩄ߪᲧセߢ߈ࠆ U 0.8 ߣߒߚ㧚ㄭૃᑼߦࠃࠆ⸘▚ߪࠪࡒࡘ ࡟࡯࡚ࠪࡦ⸘▚⚿ᨐߣࠃ޿৻⥌ࠍߺߖߚ㧚⺋Ꮕߪ B. 512 ߩ႐วߢ D ߇ᄢ߈޿ߣᄢ߈ߊ㧘. ᦨᄢߢ߽ 20㧑ߢ޽ࠆ㧚߹ߚ㧘 D ߦኻߔࠆᑄ᫈₸ߩ௑ะࠍᛠីߔࠆߎߣ߇ߢ߈ߡ޿ࠆ㧚  ฦᢥ₂ߦߡ᣿ࠄ߆ߦߐࠇߡ޿ࠆࠃ߁ߦ[3-6]㧘 (1) D ߇ዊߐ޿ߣ㧘ᦨᄢㆃᑧᤨ㑆 T. B  1

(191) D ߇ዊߐߊߥࠆߚ߼㧘ᑄ᫈₸ߪჇടߔࠆ㧚. (2) D ߇ᄢ߈޿ߣⓨᦼ㑆߇Ⴧ߃㧘╬ଔ⽶⩄߇Ⴧടߔࠆߚ߼ᑄ᫈₸ߪჇടߔࠆ㧚 (3) ߒߚ߇ߞߡ㧘D ߩ୯ߦߪᑄ᫈₸ࠍᦨዊߦߔࠆᦨㆡ୯߇ሽ࿷ߒ㧘ᦨᄢ☸ᐲ Dmax s 0 ߩ 1/2 ઃㄭߦߘߩᦨㆡ୯߇ሽ࿷ߔࠆ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ὐߪࠪࡒࡘ࡟࡯࡚ࠪࡦ⸘▚⚿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⸘▚୯ߢ޽ࠅ㧘ὐߪᢥ₂[5]ߩ࿑ࠃࠅ⺒ߺขߞߚ ࠪࡒࡘ࡟࡯࡚ࠪࡦ⸘▚୯ߢ޽ࠆ㧚ߎߩ႐วߪ㧘3%એਅߩ⺋Ꮕߢ৻⥌ߒߚ㧚D ߇ᄢ߈޿ߣ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ὐߪࠪࡒࡘ࡟࡯࡚ࠪࡦ⸘▚⚿ᨐ[5]㧕  ᑼ(4.10)ߪ♖ᐲߩ㜞޿ㄭૃᑼߢ޽ࠆߎߣ߇᣿ࠄ߆ߦߐࠇߚߩߢ㧘⸳⸘᧦ઙߩᛠីߦ೑↪ ߔࠆߎߣߣߔࠆ㧚ߚߣ߃߫㧘࿑ 8㧘9 ߦ߅޿ߡ㧘 B 㧩128㨪512 ߣߒߡ޿ࠆ߇㧘శࠬࠗ࠶࠴ ߩታ⃻ᕈࠍ⠨ᘦߔࠆߣㆊ೾ߦᄢ߈ߥ୯ߢ޽ࠆ㧚߹ߚ㧘ታ㓙ߩࡀ࠶࠻ࡢ࡯ࠢ⸳⸘ߢߪ㧘0.8 ߣ޿߁㜞⽶⩄⁁ᘒࠍၮᧄߣߖߕߦ㧘ᦨᄢߢ߽⽶⩄ 0.5 ߦ೙㒢ߒߡ޿ࠆ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 ታ⃻ߐࠇࠆ㧚⽶⩄ࠍ 0.5 ߣߒ㧘 B ࠍࡄ࡜ࡔ࡯࠲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 ߒߡ޿ߥ޿㧚ᑄ᫈₸߇ 0.01㧑એਅߣߥࠆߩߪ B ߇ 32 એ਄ߩߣ߈ߢ޽ࠅ㧘⽶⩄೙㒢ࠍ⺖ߒ ߡ߽శࡃ࠶ࡈࠔߦߪ޽ࠆ⒟ᐲߩⷙᮨ߇ᔅⷐߥߎߣ߇ࠊ߆ࠆ㧚࿑ 12 ߪ U =0.5 ߣߒߚߣ߈㧘D ߦኻߔࠆᐔဋㆃᑧᤨ㑆 wT ࠍ␜ߔ㧚 D 㧩1.2㨪1.4 ߢ޽ࠇ߫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ᢙಽᏓߦ޽ࠆߣߔࠆߣ㧘ㆃᑧᤨ㑆೙㒢એ਄ߣߥࠆࡄࠤ࠶࠻ߦኻߒߡ 㐳޿࠹࡯࡞ߩಽᏓ߇ሽ࿷ߔࠆ㧚IP ࡀ࠶࠻ࡢ࡯ࠢߢߪࡄࠤ࠶࠻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₸߇ࠪࡒࡘ࡟࡯࡚ࠪࡦ⸘▚⚿ᨐߣߒߡ⸥タߐࠇߡ޿ࠆ߇㧘ᜰ ᢙ㑐ᢙಽᏓߣᲧセߒߡ㧘߅ߩ߅ߩ 1/15㧘1/300 ⒟ᐲᑄ᫈₸ߣߥߞߡ޿ࠆ㧚ߒߚ߇ߞߡ㧘ታ ⸳⸘߳ߩㆡᔕࠍ⠨߃ࠆߣ㧘ᜰᢙ㑐ᢙಽᏓߩߺߢߪਇ⿷ߢ㧘ဋ৻ಽᏓߣ࿕ቯ㐳㧘ߐࠄߦછᗧ ಽᏓߩࡄࠤ࠶࠻㐳ࠍᗐቯߒߚ࠻࡜ࡅ࠶ࠢℂ⺰߇ᔅⷐߢ޽ࠆ㧚  ߒ߆ߒߥ߇ࠄ㧘ታ⸳⸘ߦὶὐࠍᒰߡߚ႐ว㧘ࠪࡒࡘ࡟࡯࡚ࠪࡦ⸘▚ߪᾘ㔀ߔ߉ߡ৻⥸ߩ ೑↪ߦߪਇะ߈ߢ޽ࠆߣ⠨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ಽᏓࠍ᳞߼ࠆ࠻ ࡜ࡅ࠶ࠢℂ⺰ࠍ⠨ኤߒߚ㧚ℂ⺰ߩኻ⽎ߣߒߚశࡃ࠶ࡈࠔ᭴ᚑߦ߅޿ߡ㧘FDL ᤨ㑆☸ᐲߩᦨ ㆡൻࠍ࿑ࠆߎߣ߇ߘߩਥߥ⋡⊛ߢ޽ࠆ㧚ᓧࠄࠇߚ⚿ᨐߪ㧘ᰴߩߣ߅ࠅߢ޽ࠆ㧚 (1). ᜰᢙ㑐ᢙಽᏓࡄࠤ࠶࠻㐳ߦኻߒߡ㧘ᓙߜᤨ㑆ಽᏓߣᑄ᫈₸ࠍ᳞߼ࠆㄭૃᑼࠍ᳞߼ߚ㧚 ㄭૃᑼߦ߅޿ߡᡆૃ⽶⩄ࠍቯ⟵ߒ㧘ዉ౉ߔࠆߎߣ߇ᑼߩ৻⥸ൻߦ᦭ലߢ޽ࠆߎߣࠍ᣿ ࠄ߆ߦߒߚ㧚. (2). ㄭૃᑼߦࠃࠆ⸘▚୯ߪႎ๔ߐࠇߡ޿ࠆࠪࡒࡘ࡟࡯࡚ࠪࡦ⚿ᨐߣ㧘ᑄ᫈₸ߢ 20㧑એਅ㧘 ᐔဋㆃᑧᤨ㑆ߢ 3㧑એਅߩᏅߢ޽ࠅ㧘ࠃ޿৻⥌ࠍߺߖߚ㧚. (3). ⸳⸘଀ߣߒߡ㧘⽶⩄ 0.5 ߦ߅ߌࠆᦨㆡൻࠍ᳞߼㧘0.01㧑એਅߩࡄࠤ࠶࠻ᑄ᫈₸ߣߔࠆ. - 20 .

(195) ߚ߼ߦߪ㧘ᤨ㑆☸ᐲࠍ 1.2㨪1.4㧘FDL ࠍ 32 ᧄએ਄ߣߔࠆᔅⷐ߇޽ࠆߎߣࠍ᣿ࠄ߆ߦ ߒߚ㧚  ታ⸳⸘ߦ೑↪ߔࠆߚ߼ߦߪ㧘ࡄࠤ࠶࠻㐳ߦߟ޿ߡဋ৻ಽᏓ߿࿕ቯ㐳㧘છᗧಽᏓߣߒߚℂ ⺰ᑼ߇ᔅⷐߢ޽ࠅ㧘੹ᓟߩ⺖㗴ߣߒߚ㧚 ઃ㍳ M/M/1/K ࠪࠬ࠹ࡓߦ߅ߌࠆᑄ᫈₸  ๭㊂ࠍ a ߣ߅ߊߣ߈㧘M/M/1/K ࠪࠬ࠹ࡓߩᑄ᫈₸ߪᰴᑼߢ޽ࠆ[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 એ਄ߣߥࠆ⏕₸ 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)ߪᰴᑼߢ⴫ߐࠇࠆ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߿ޟ‬ᖱႎ੤឵Ꮏቇ‫ޠ‬᫪ർ಴ 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߿ޟ‬ᖱႎ੤឵Ꮏቇ‫ޠ‬᫪ർ಴ 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࠶࠻ࡢ࡯ࠢ⸳⸘ 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߿ޟ‬ᖱႎ੤឵Ꮏቇ‫ޠ‬᫪ർ಴ 㧔2009 ᐕ㧕㧘p.112㧘5.4 ▵㧘ᑼ(5.38)㧚. - 22 .

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