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On stochastic recursive equations and infinite server queues

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(1)On stochastic recursive equations and infinite server queues Eitan Altman. To cite this version: Eitan Altman. On stochastic recursive equations and infinite server queues. RR-5211, INRIA. 2004, pp.13. �inria-00070782�. HAL Id: inria-00070782 https://hal.inria.fr/inria-00070782 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On stochastic recursive equations and infinite server queues Eitan Altman. N° 5211 May 2004. ISSN 0249-6399. ISRN INRIA/RR--5212--FR+ENG. Thème COM. apport de recherche.

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(185) –CWAXhCSP_CT€P_NQ>AP_CD[aPRPRY~C$T\^RXhBI[\^RY~Y‘XŸPbL€hPR^_>h[VY~S)N§\YZCUN?P)NP_^RY~WAX C„}TV–^ YZ N–XhC}TVrV^,[aPR[aY~‰~\‰ CK }Š T\^/[a‰~‰  NRT&PR>f[yP YZN„hXAY‘PRC ¤ & $ 9VŠ   ±

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(187) & $ 9 $ NRWhS)

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(189) µ¶XhgOC$C$g•AAœ [aXhg’gOChXAC _ `. L. L. p. p. . (. ,. p. &. #. ,r. &. d. . . (. , . &. _ `. (. p ,. . . . L. . (. L. . . . p. L. p. . . & ? (        0  "  2 #   $ A  ( YZNSU‰~C$[\^R‰~L hXhYŽP_CŸ¤ p & $ 9 $gOWAC'P_T PR>hCcC^_r\TOgOYZSUY‘PbL;T\}  Š G’TV^RC$Ty\C^:•ˆY‘PYZNS‰‘C:[a^ }^_T\B PR>AC CœO eA‰‘YZSUY‘P›CœOeA^RC:NRNRY~T\XhNˆ›C–>h[3\C}TV^ [aXhg&}T\^ L PR>h[aP›P_>ACL SU TVY‘XfSUYZgOC,}TV^ K ( ( Š,XhY~‡|WAC$XAC$N_NˆT\}.

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(192) h^)NbP+[\Xhg/NRC$SUTVXhg–BDTVB€CX|P_N T\}ˆP_>AC NbP)[yPRC€y[a^_YZ[aA‰~C$N„[aeAe5C$[\^RY~XAr(Y‘XŸPR>hC&NbP_TMS)>f[\NQPRYZS^_C$SWA^_NRY~\CC:‡|Wh[yP_Y‘TVXhN[aXhgŒ[aeAeh‰‘Y~C$g¨PR>AC$BIPRT NRT\‰~\C(P_>ACcY~XOhXAY‘PRCŒNQC$^RVC^ ‡|WACWAC¨eA^RTVA‰~CB’ŠiFP_>AC^NbP_TMS)>f[\NQPRYZS^_C$SWA^)NQY~\C’C$‡|Wh[aPRY~T\XhN&>h[3\C'5CC$X WfNQC:gP_T&NbP_WhgOLPR>ACNQP_[\AY~‰‘Y‘PbLT\}P_>AC‡|WACWhC/WAXhgOC$^,CVCX(BDTV^RC/rVCXAC$^_[\‰eA^_T\h[\AY‘‰~YZNbP_Y~S[\N_NRWAB€eOPRY~T\XhN [\Xhg¡SUTVX|VC^_r\C$XhSUCDPRTc[cWAXAYZ‡VWhCNbP)[yP_Y‘TVXh[a^_Lc^_Cr\Y~B€C >h[\NfC$CX C:NbP)[aA‰~Y~NR>AC:gŠ(=?>hCNQY~B€eA‰~C&CUœOeA‰~Y~SYŽP CœOeA^RC:NRNRY~T\XhN+T\OP)[aY~XAC$g/B [\§\C$N+T\WA^^_C$NRWA‰ŽP)N[\eAe5C$[a‰~Y~XAr,P_T–y[a^_Y~T\WhN[aeAeA‰~YZS[yP_Y‘TVXhNTa}OPR>hC›Y~XOhXAY‘PRCNRC^_\C$^ ‡|WAC$WAC\Š "ATV^CœO[\B€eA‰‘CV•‚PR>AC$L S[aX;fC(WhNRC$g¡P_T¨^_CeA^_C$NRCX|PP_>ACf^_NQPD[\Xhg NRC$SUTVXhg¡B€T\B€CX|P)NT\},PR>AC XMWAB!fC$^

(193) Ta}¯SUT\XhXAC$SUPRC$g!B€T\AY~‰~C$N‚[yPš[aX€[a^_AY‘PR^)[a^_L‰‘TOS[aPRY~T\XDY‘X!PR>hCT\XACgOY~B€CXhNRY‘TVXh[a‰O[Vg —>hTMS›XACPb›TV^R§ T\} ‘m  —•MWhNRY~XAr/PR>hCC$‡|WAY~y[a‰~CXhSC,5CUPbCCX&PR>AC„[\g >ATOSXACPb›TV^R§D[aXhg&[\X Y‘XAhXAY‘PRC–NRC^_\C$^ˆ‡|WACWhCr\Y~\C$X Y~X ‘m  Š . . . . e. f. e. . <e f ge f. ÍÍ Ó. jklnmm. f.

(194) m3l.  . . ?  y, . ‘m JŠ3dŠ3™gA§\C›[\Xhg !ŠyƒDŠaƒ[VgA[arf•R™ XAC S‰~[VNRNTa}AA^)[aXfS)>AY‘XhreA^_TOSUC$N_NRC$N)• B   N A•¯m VkOŠ   )•

(195) E–нE–Š

(196) ,C$LOgOC 1³šgAYŽP_T\^ •m )m$qA•JMeA^_Y‘XArVC^  C:SP_WA^RC TaP_C$ l /³,ŠM™‰ŽP_B [aX˜• _J|PRTOS)>h[VNbP_Y~S^RC:SUWA^)NQY~\CC:‡|Wh[yP_Y‘TVXhNšY‘PR>([aeAeA‰~YZS[yP_Y‘TVXhN

(197) P_T!‡|WAC$WAC$N›YŽP_>gOC$efC$XhgOCX|P y[VS[yP_Y‘TVXhN y•  1 

(198) B  V•+m\m:l bm *oVq m O™,eA^:•hlanVnVlOŠ q /³,Šš™,‰‘PRB [aX [aXhg ™ Š

(199) T\^)gOY `b§¯• R™,eheA‰‘YZS[aPRY~T\XhN!Ta}´T\^_Ty|§VTy  N/dC$XATyy[yP_Y‘TVX;=?>ACTV^RLŒP_T T\X JMP_[yP_Y‘TVXh[a^_L J|PRTOS)>h[VNbP_Y~SŒdC:SUWA^)NQY~\C¡JMC$‡|WAC$XhSUC:N[\Xhg P_>ACY~^¨E›T\X|PR^_T\‰ •     B N  •OeAe˜Šfq —ofm$qh•fm OŠ o /³,ŠV™,‰‘PRB [\X˜•|=Š aY~B€CXAC$”/• !Š T\}B [aX˜•  „]ˆJ€‡|WACWAC:N

(200) Y‘PR> NbP)[yP_Y‘TVXh[a^_LC^_r\TOgOYZS?NQC$^RMYZSUCPRY~B€C$N [\Xhg’P_>ACe5C^R}T\^_B€[\XhSUCTa}š=E] Y~XcTy\C$^R‰~TV[Vg )•A]š^RTOSUC$C$gOY~XAr|NTa}šµb³š³ˆ³ µ¶XO}TOSUTVB’•

(201) ,TVXA!r –T\XArf• Gc[a^)S)>’l\n\naofŠ k /³,ŠA™,‰‘PRB [aX˜•A]Š –T\XhNQP_[\X|PRT\e5T\Wh‰‘T|N• Š  Y~W˜• RJMP_[ahY‘‰~YŽPbLV•AG’T\XAT\PRTVXAY~SYŽPbL [\Xhg&µ¶XMy[a^_Y~[\X|P·–Wh[\XVP_Y P_Y‘C:NY~X'ƒ„C$XAC^)[a‰˜]TV‰‘‰~Y‘Xhr JOLMNQPRC$B N •   % •heAe˜Šhq|k bk#M•5m Vl "Š+´?[VSSUC$‰‘‰~Yš[aXfgŸ]Š+´^  B [aWhg˜E •  % ! y•

(202) JMeA^_Y‘XArVC^:•˜NQC:SUTVXhg¨C:gOY‘PRY~T\X˜• l\n\nVqAŠ  /™Š+™Š´›TV^RTyM§VTy5+•   )# " y#• aTV>AXŒ‹ Y‘‰~C%L $ JMTVXhN•ˆm o PR^)[aXhNR‰Z[yPRC:g&}^_T\B dWhN_NQYZ[aX Š /™Š™Š˜´T\^_TyM§\Ty[\Xhg¨J¯Š˜ƒDŠ "AT|NRN$• RJ|P_TOS)>h[\NQPRYZS[\‰‘‰~L^_C$SWA^)NQY~\CNRC$‡|WAC$XhSUC:N–[\Xhg’P_>ACY~^r\CXhC^)[a‰ Y~”$[aPRY~T\XhN y&•  <   # % • • ,TfŠm\•OeAe+Š˜m m\•¯m VlOŠ /™Š˜´^_[\XhgMP$• R=?>AC!NQPRTOS)>h[VNbP_Y~SC$‡|Wh[aPRY~T\X 9 ,-  .8

(203) 9 3 4 'Y‘PR>¡NbP)[yPRY~T\Xf[a^_L(ST|%C &SUY~CX|P_N )•    9  B N y• 

(204) TV‰ÂŠ˜m •¯m Š ~m$n ™Š+´^_[\XhgMP:•]Š "A^)[aXh§\CX¡[aXfgŸ´„Š  Y~NRC§¯•  9 ! )# % •™§y[\gOC$BDY~C 

(205) C^_‰Z[arh• ´C$^R‰~Y‘X+•˜m VlOŠ ~m\m ]Š+ƒ„‰Z[\N_NQC$^RB [\Xc[aXhg !Š !(Š 'ˆ[\Th• _J|PRTOS)>h[VNbP_Y~S\C$SUPRTV^–gO*Y )5C$^RC$XhSUCDC$‡|Wh[aPRY~T\XhNYŽP_>ŒNQP_[aPRY~T\Xh[\^RL STMC &SUY~CX|P_N ,• +

(206) + 9'  %~• 

(207) T\‰˜q|lO•OeAe kAm •hm |kOŠ ~m:l dŠMƒ„^_TMCXAC$\C‰‘P›[\Xhg ³ŠO™,‰‘PRB [aX+-• R™,X[\‰ŽP_C^_Xh[yP_Y‘Xhr/eA^RY~T\^_Y‘PbLDNRC^_\C$^

(208) YŽP_>SUTV^R^_C‰Z[yP_C$g€[aXhg S^RT|NRN ST\^_^RC$‰~[aPRC:g&y[\S$[yPRY~T\XfN •MB€[\XMWhNRS^RY~eOP:Š ~m$q ™.Š \C$[\X bG'[a^_Y‘C[\Xhg ]ŠVdTVfC$^QP:•

(209) ).

(210) U• 0/+1  2 #• ,TfŠ˜m blO•MeAe˜Š+m:1l !_m:q •hm aofŠ ~mo &

(211) Š /[aB€C$gh[[aXf3g '!4Š ¯>h[aXArf5•  XAYZ‡VWhCXAC:NRNTa}fPR>AC?NQTV‰‘WAPRY~T\X}TV^‚T\eOP_Y‘B [a‰ONQP_[aPRYZSˆ^_T\WOP_Y‘XhrY~X!TVefC$X ´,E?G'] ‡|WAC$WACY~XAr XACPb›TV^R§ON )• # 6   # N #• ,TfŠ˜m$!n _m3lO•5m\m !_m:q\nh•fm VkAŠ e f. * |=B\  b. ,D =; ; US  b

(212) DNC0B Mk ;

(213) . iD S. lDrb  

(214) ;

(215). -_. D =; 

(216)

(217)  

(218) . `. e f. .

(219) DPC. th   Dh

(220). 

(221)  =B. _ i`. g). e f. uS . |=; 

(222) ]DPC. otgt. T US. ,Dg? g?  vJ . e f. g. . e f. ,   rb. e f. Ji

(223) . 0 

(224) sDNC. e if. %

(225) KJ. tcxD =. _. a BD S

(226) ? .

(227) 8 8. ,   rb. ` B UDKJ. ,   rb` B UDKJ. R`. e f. R?  . S |=; 

(228) d . B.  =

(229). . e f. S m=; 

(230) d . e. ytwt. T US. Dg? g?  vJ. f. R  D  KJ. e f. e ytwt. R D'=B i

(231)  = _D'S

(232). *. Dr?. $. e f e. f. ,CB 3 . e. `. h

(233) T c-? B. D %DPC B -tRD =; 

(234) ;

(235) D 3

(236) UB  rb? . 8. f. B. lD. tc . _D S\.  rb  . ҁÐ͘ÒNM.

(237) m:q.  

(238)     

(239). ~m:k /G Š ^_WAX|PR”([aXhg ™Š‚G'[\§\Ty,NR§MY•G’TOgOC$‰‘Y~XArcMYZgOCTcPR^)[ &S&WhNRY~XArŸG¨ \ƒ 3Y~XOhXAY‘PbLŸY~XAeAWAPDeA^_T! SC$N_NQC:%N *™ ST\B€eA^_T\B€YZNQCfCPb›C$CX G'[a^_§\TyMYZ[aX[\Xhg  d B€TOgOC‰ZN•  ) + 9     B.  . +

(240)  y•OeAe˜9Š aq\!q yo • 

(241) T\‰—Š˜m • ,ThŠ5kO• \WAXAC m Š ~m dŠ  TyLMXAC:N• Q=?>hCcNbP)[aAY~‰‘Y‘PbL;T\}[ ‡|WAC$WAC'Y‘PR> XAT\#X —Y~XhgOC$efC$XhgOCX|P&Y~X|PRC^ ¶[a^_^RY~y[a‰[aXhg NRC^_MY~SC P_Y‘B€C:N a)•    %  N'    #• ,ThŠfqA•OeAe˜Šho  ¶kVlanh•hm lAŠ ~m  !ŠAG’Y~T\^)[aXhgOY5[aXhg³ŠO™,‰‘PRB [aX˜ • :E›T\XAXAC:SP_Y‘MY‘PbL!Y~X™g 

(242) ,TOS CUPbT\^_§MN *[€·–WACWhCY~XAr=?>AC$T\^_CUP YZS[\‰+™,eheA^RT|[\S)> a•  ! 

(243) f•˜E[aBh^RYZgOr\CV• €•fG'[\^_S)>’lanVnaohŠ e f. .  

(244) d . e. lD.  c=  Dh

(245) :. |<. f. D'=. s?. TBW. D =. e if . ,D =; ; US  b

(246) 7DNC. ÍÍ Ó. jklnmm. . tcoA-DUVn

(247) UB DKt. D  . D. >. = S.

(248) Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France). Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France).   

(249).   . ISSN 0249-6399.

(250)

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