Perturbation analysis of an M/M/1 queue in a diffusion random environment
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Texte intégral
(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Perturbation analysis of an M/M/1 queue in a diffusion random environment Christine Fricker — Fabrice Guillemin — Philippe Robert. N° 5422 Deembre 2004. N 0249-6399. ISRN INRIA/RR--5422--FR+ENG. Thème COM. apport de recherche.
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(186)
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(191)
(192) X
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(230) -8. 22. 2
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(237) A)!
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(240) %
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(257)
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(259)
(260) :
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(262) B !
(263)
(264) 4 !
(265) !
(266) 0
(267) % !
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(269) * E$H
(270)
(271) <
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(277) H , < ,)!
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(280) (Qn (z)). !
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(282) 2 + . ÚÛÕ®Ú¯Ü.
(283)
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(288) gk|H\Ilivkn~[. ms ikXZ\ ms|H\7vtrAimsv B ¬>V\
(289) lmsoBgjpqh\7v(ikXZ\ \ n~r}ikp©mso Bf (z) = ¬ h Z X I \ k v \ ª V = e
(290) rsggjn~[*p©oZ*ikXBrAi f (z) = 1 ¬hikX~\(gj\ nZ\7o~l7\ f (z) gkr}ikpqgx«~\3g zf (z) iXZ\ v\Iln~vkv\7o~l7\f (z) v\7£qrA=ipmJ(fo (z)) ∈ C . N. n. 0. f0 (z) = 1, f1 (z) = (z + λ)/µ, µfn+1 (z) − (z + λ + µ)fn (z) + λfn−1 (z) = 0,. ) -G1. n ≥ 2.. WX~\r}HmAs\,iXZvk\I\ 5ik\Ivk[ vk\3lnZvvk\Io~l\Tv\7£qrAipmJoµp[+|Z£©p\3giX~rAi f (z) p©g r|Hms£©eCo~ms[+p©rs£zp©o Ar}vp©rsZ£\ p¦iX»h\7Jvk\I\ r}oBiX~rAi(ikXZ\|Hms£©efoZms[+pqr}£qg mJvk[ r}omsvkikXZmJsmJo~r}£&|Hms£©eCo~ms[+p©rs£Rgkefgjik\I[ z gkp©o~l\ °Br8Ar}vt g.nlmJo~hpikp©msopqg(mJCfp©msn~gk£©e%grAikpqgj«~(f\I (z)) ):gj\I\ gk¤s\Ie¶rso~ 'xgk[r}p©£:+ /VmJv*Z\irsp£qg 1ªWXZ\ mJvjiXZmsJmso~rs£pixe [+\IrJgjnZv\ m}§iXZ\Igk\{|Hms£©eCo~ms[+p©rs£©gFp©gR|Zvk\3lpqgj\I£e(ikXZ\gj|H\Ilikvtr}£~[+\3rsgknZvk\msHiXZ\zmJ|B\Ivr}ikmJv gkp©o~l\ B p©g gk\7£¯ ¢rs8wxmJpoCiIª B Wm»h\i\7v[+poZ\ ¬ ¨\TlmJ[*|~nhik\,iXZ\T£p©[+p¦ipo~8rs£n~\=rsg ik\7oBZg+ikmp©oh«~o~p¦ixe¸m}`ikXZ\TvtrA ipm f (z)/f (z) ¬R dψ(x) XZ\7v\ f (z) ¬ n = 0, 1, 2, . . . r}v\iXZ\
(291) rJgkgkmhlpqrAni\|BmJ£efoZmJ[+p©rs£©gI¬> XZpqltXOgkr}ikpqgxe v\Il7nZvvk\Io~l\`vk\I£©r}ikp©mso ) -G1¨ p¦iX iXZ\ p©oZp¦ip©rs£lmJo~hpikp©mso~g r}o~ f (z) = 1/µ. f (z) = 0 cfikvtr}p©sXCijmJvkr}vtlmJ[+|Znhir}ikp©mso~gefp©\7£qiX~rAi3¬hmsv z ∈/ [−(√λ + √µ) , −(√λ − √µ) ] ¬ n. n. ∗ n. ∗ n. n. 0. 1. 2. X~\7v\. 1 fn (z) = Z+ − Z −. ". λ+z−µ+ 2µ. p. δ(z). !. n Z+. z+λ+µ± Z± = 2µ. −λ − z + µ + 2µ. +. p. 2. 2. 2. fn∗ (z) =. iX~rAi. Z. X~\7v\. 0. dψ(x). X~rsg¨r(gknZ|Z|HmsvkiVp©o~l7£n~Z\Ip©o. dψ(x) = χ(z), z−x. p©gikX~\*l7msoCikp©ofnZ\I,vtrslikp©msoT XZmCgj\ nikX¶r}|Z|Zvm8ºhp[rsoJip©g χ(z) J m v p©gJpJ\7o
(292) fe χ(z) z∈ / (−∞, 0] 'Îi{pqg\IrJgjp©£©e ltXZ\Ilt¤J\IikX~r}i msv. ¬. −∞. fn∗ (z)/fn (z). fn∗ (z) . n→∞ fn (z). χ(z) = lim. z > 0 Z + > Z− > 0 χ(z) =. ÕÕ ÑK~ê Lóó. fn∗ (z). n 1 n Z+ − Z − . µ(Z+ − Z− ). cCip\I£¦i:wx\Ig¨iXZ\7mJvke4+ /®gxitrAik\3gFiX~rAi¨ikX~\z[+\IrJgjnZv\ . δ(z). r}o~ iXZ\7o
(293) msv. 2 λ+z−µ+. !. n Z−. #. ,. δ(z). pikX =√(z + λ + µ) −√ 4λµ ªz_=msv\7mAs\IvI¬ZikXZ\*rsggjmhlpqrAi\I
(294) |Hms£©eCo~ms[+p©rs£©g mJv z δ(z) √ √ ∈ / [−( λ + µ) , −( λ − µ) ] 2. p. p. δ(z). .. z>0. r}v\ sp©s\7oTfe. . (−∞, 0]. rso~. ªWX~\nZo~likp©mso. ¬ )- 1.
(295) -I . 22. 2
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(297) : . W X~\nZo~likp©mso%msoTiXZ\.vp©sXCiX~rso~=gkp©Z\(m}VÅ n~rAipmJo ) - 1{lIr}o=H\+r}o~rs£eCikpql7rs££©e,lmsoCipofnZ\3,ikm ikXZ\ l7ms[+|Z£©\º|Z£qr}o~\Z\7|ZvpJ\I m}®iXZ\gj\Is[+\7oCi √λ + √µ) , −(√λ − √µ) ] rso~iXZ\msvpJpoª&_=msv\ |~vk\3lpqgj\I£eJ¬BiXZ\.nZo~lipmJo χ(z) X~rJgr nZoZp nZ[−( \*|BmJ£\+rAi r}oB=pig`vk\3gjpqhnZ\.p©g`\ nBr}£ikm ª WX~\\7p©s\IoCJ\Ilikmsv{rJgkgkmhlpqrAi\I
(298) p¦iXTikX~\(\7p©s\Iof8rs£n~\ 0 zp©g{=ikXZ\.0 s\3likmJv{ XZpqltX nikXl7ms [+|BmJoZ\7oC(1i{pq−g ρρ) ª ):WXZpqg s\3likmJv l£©\Ir}v£©eB\I£mJoZJg¨im L (N) ª 1 °~vkmJ[ u&\Ivkvmsoh xcCip\I£¦i:wx\Igp©ofs\IvgkpmJomsv[(nZ£qrZ¬>gk\7\ gj¤J\7er}oB 'xgj[rsp£ + /rso~»³{\IoZvp©l7p + /ά>ikXZ\ l7msoCikp©ofnZmsnBg¨gk|H\Ilivkn~[ m}ikXZ\ [+\3rsgknZvk\ dψ(x) pqgsp©s\7o
(299) fe 2. 2. n. 2 ρ. . 1 dψ(x) = lim (χ(x − iε) − χ(x + iε)). ε→0 dx 2iπ. 'Îipqg\3rsgkp£©eOltXZ\3lt¤s\I»ikX~r}iikX~\=r}HmAs\
(300) £p©[+p¦i pqgoZmJo¸ofnZ£©£{mJoZ£eOmsv. √ √ 2 √ µ) , −( λ − µ)2 ). rso~¹¬Zp©o ikX~r}izl7rsgk\s¬. x. p©o ikXZ\Tp©oCik\IvkAr}£. √ (−( λ +. √ s ρ (x + λ + µ)2 dx. dψ(x) = − 1− πx 4λµ. 'Îi,pqg ¨msvkikX. oZm}ipoZ¸ikXBrAi pqg J\7ve l£©mJgk\=ikm¸ikXBrAi
(301) ikXZ\lmJvkv\Igk|Hmso~hp©oZOgk|H\Ilivrs£[+\3rsgknZvk\ rJgkgkmhlpqrAi\I
(302) p¦iX%]VXZ\7fehgkXZ\7dψ(x)
(303) |BmJ£efoZmJ[+p©rs£©gIª '¢o,ârsli3¬~iXZ\(|Hms£©efoZms[+pqr}£qg nZo~Z\7vl7mso~gkpqh\7vtrAipmJo XZ\Ivk\ Zp¦¥§\7v`vms[ ]VXZ\ICehgkXZ\7,|Hms£©efoZms[+pqr}£qgzmJoZ£e
(304) iXZvkmJnZsXikXZ\*p©oZp¦ip©rs£>lmsoBhp¦ipmJo~g )âgk\7\]VXZp©X~r}vtr +Ì0/&msv rso \7ºhX~r}n~gjikp©s\`ivk\3rAik[+\IoJi msl£qrsggkp©lIr}£§msvkikXZmJsmJo~r}£§|Hms£©eCo~ms[+p©rs£©g 1ª 'Îi{pqg \IrJgjp©£©eltXZ\Ilt¤J\I ikX~r}i Z. √ √ −( λ− µ)2 √ √ −( λ+ µ)2. √ √ Z 1 − x2 2 λµρ 1 √ dx dψ(x) = π −1 λ − 2 λµx + µ Z ∞ p 2ρ X n/2 1 = ρ Un (x) 1 − x2 dx = ρ, π n=0 −1. X~\7v\ U (x) ¬ n = 0, 1, 2, . . . r}v\ikX~\]VXZ\IfefgkXZ\IT|BmJ£efoZmJ[*pqr}£qgm}FiXZ\gj\3lmsoBT¤fpo~®¬® XZpqltX%r}v\ mJvjiXZmso~msv[+rs£B pikX,v\Igk|B\3li im.iXZ\ ¨\7p©sXCi [*\3rsgknZv\ w(x)dx pikX n. 'Îi mJ££©mA{gVikX~r}i ikXZ\im}itr}£¹[rsggm}. w(x) = dψ(x) Z. p. p©g. 1 − x2 1(−1,1) (x).. 0. dψ(x) = 1.. WX~\msvkikX~mssmJo~r}£©pixeOm} iXZ\%gk\ nZ\Io~l\3g (Q (z)) msv z ∈ [−(√λ + √µ) , −(√λ − √µ) ] r}oB Å nBrAikp©msBo ) -0? 1rsvk\ikXZ\Ivk\7msv\\Igjir}~£pqgjX~\I¹ª 'ÎiRpqg>¨msvkikX+oZmsikp©oZikX~r}iFikX~\{|BmJpoCiRgk|B\3livknZ[ m}HikX~\{ms|H\7vtrAimsv lmJoCir}p©o~g>mJoZ£©e(mso~\ |BmJpoCiRr}oB iX~rAi iXZ\lmJoCikp©oCn~msn~ggk|B\3likvnZ[p©g¨iXZ\ poCi\7v8rs£ (−(√λ + √µ) , −(B √λ − √µ) ) ª −∞. 2. n. 2. 2. 2. ÚÛÕ®Ú¯Ü.
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