Stability of Multi-server Polling Models

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(1)Stability of Multi-server Polling Models Christine Fricker, M.R. Jaïbi. To cite this version: Christine Fricker, M.R. Jaïbi. Stability of Multi-server Polling Models. [Research Report] RR-3347, INRIA. 1998. �inria-00073342�. HAL Id: inria-00073342 https://hal.inria.fr/inria-00073342 Submitted on 24 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. Stability of multi-server polling models Christine Fricker et M.R. Ja¨ıbi. N ˚ 3347 Janvier 1998 ` THEME 1. ISSN 0249-6399. apport de recherche.

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(118) | .v= 0y0 $ 3 .

(119) | .fX y0 0 ^ 3 . 8

(120) | .[s 0 c

(121) | .= 0 8 0 | .= . R V h V j c R j ` x v ]P_ h VÚx?c^V h _?erGV? SQ R ]nW©serGV0]Sc R t_ c.5]^V V V W©W©_}³xV6ÕX=D8 0 h . 3 .

(122) | .= 0t0$ 3 .

(123) | .^X 0t0y0 { U d QSR ` ] .fXq} 0 dbGV rvu]. 3. o .. . . j . i |. &. u i i 0 {. \ |. m9` rGc^serdbx +?d u 3 _tx vC]` W©Wxe h^j W X. R V h VL . 3. { g g_?x v | . X $ 0. 3 3 u. { .. |. 3 |. u. 3 3. {. . { .. j . i |. . 3 f. X^ 7^. WD0 d. .^X 0. Xc j q . . j . . u$i i 0. . i . .^X^ 7 ^ WE0. |. u i i 0. .fX^ ? ^ WE0. oDp4.oJq.

(124) XC.

(125) !"# $ % % & ' & ( % & . R w R W©serGV0] ]V0_t]^Grd#] j rptV0v _tx v . .

(126) | .fX 0t0 { U 0 h . | .

(127) | .v= 0y0 $ . c R V¨rx V _ h ]^de]fc^V W .^X 0 X QSR ]V0x v ]$c R V's hfjbjl ¦j?h { W V1 W I ¦j?h { W { . X $ 0 | o B>n` c \ : B 1¶x veV V0vU ?dV h j vewcdt U_?x v | | 0t0y0 { h .j= . .

(128) | .^X 0t0y0y0 .

(129) .^X ^ h .@d f 0 . .

(130) | .fX 0t0 U 0 h .j=4.xU 0y0 . .

(131) | .fX 0y0 { U 0 . V w _t` ]^V h w j W©serV c^Vt. .]=4.xU 0t0 {. U . :. QSR ]+dbV rv ]+c R _lcG. X. $ 0 | o . :. R w R ]G. 0 QSR V9s h^jbjl ] . 7+S Ï> . >ndr j. w _lrU]^c^_?eGrcdt biV§W$V _?x+c R V§]fc_terGcd jl _]` ]^V c jl £L` V èV0] V cc R V§£bèV ` V ]nuV§xb` WuV h V0v _?]Úx#c R V+s hfjbjlijlQSR V j?h V W { V c ^ _tx v³Vc R Vs j rrGxe]^db]^cfV0W jl £bèV èV0]©XcEdddc j ec^_tx V vLd]_lc` h _lcfxe£L` V èV0] N XcEddEdc W `1c©]wrV _ h c R _lcs j rrGxe¬]fde]^cfV0W = I R w R ] w j W©serV c^V rdA]fW$Gr_ h c j u]S]fc_terV§ _?x v j xerd+ PM.

(132) . o. :j. . X. R V h V { g | g']c R VCc j c_lrc h _q¥+w+r j _?v¬_tx v { g | : gj$g9ugc R V¨V aesV w cfV0v ]^Sc^w Rb;j ptV h c^W$VsV h pb]fcnx']` ]fde]^cfV W= V h Vt LiVP]` s s j ]^VSc R _tcßc R V s j rGrxe³]^db]^cfV0W]ßx j cI]^c^_t rGV? t2 V?c R _lciw j xuveGc^ j x ]ßpb j r_lcfV0v _?x v~veV cfV h W$xeV³ R w R £bèV0èV ]Ú_ h V³]^c^_t rGV? - £L` V èV]Ú` x ]^c^_?erGV c R V$£L` V èVrGV0xetc R w j xLptV h tV0] c j 7"SGc R s j ]fcfptV§s h^j _?erGcdt QSR Vs h V pb j ` ]_?x _lrde]f]] Rej P]c R _lcc R V+£bèV0èV ]$XcddEdc _ h V©]fW³èrc^_?xeV j ` ]^rGd~]^c^_?erGV j?h ` xu] c_terVt _tw w jLh vexec j X jLh ^ X h V ]suV0wc^Gp?V rdt 1¶xc R V]^V w j x vw _t]^Vt uc R VW$V0_txwdew rGVc^W$V w j xLptV h ?V ]ic j 7" Va w r` v xe+_?xLdA]fc_terVV R _=pb j?h xCc R V³]^db]^cfV0W _?x vAV ptV h dA£bèV0èV§]P` x ]^c^_?erGV x'c R V³s h V wV0vexe+]fV0x ]^Vt QSR V h V ¦jLh V? iV_t]]^` W$V§c R _l` c X c j V x ]` h V§c R V³xLcfV h _terGcd jl c R V w dewrVc^W$V ] uV w _t` ]^Vc R V£bèV ` V ] R XcEdddc W w j xLc h èc^V ¦jLh xLcfV h _terV pb]fcSv ` h _lc^ j x ]S_tx vAiV ¦j w ` ] j xCc R V³V R _=pb j?hnjl c R V³£bèV èV0] X cdEddc W x_` xu]fc_terVs j rGrxe+]fde]^cfV0 W >nV w _?` ]^V jt c R Vxb` W³V h xe jl c R V£bèV ` V ] F.^XEO 0 dbV rv ] . V c. o | | : |j |. |P^ . o : j . o d ^ ddd ^ :j . 3. .^X 0. W _qa b ;^ ?^ W $ Xc 3 : 3 j 3 3 X e d { © QSR VP]fV c]ßx j c7V W©secd? tc R ` ] ] ex Gc^Vt QSR VSr j w _lr ]^c^_?erGcd]ßveV c^V h W$xeV0v©?d MÚèV ` V ]nXcEddEdc w j W©s j ]^V_©]^c^_?erGV§s j rrGxe]` ]^db]^cfV0W R rV ¦jLh _ £bèV0èV_ß]S` x ]^c^_?erGV?>nd¨veV xecf j x jl 2. o. ÌE.

(133) XDO. & &

(134) . s j r rxe+]` ]^de]fc^V W ] ]^c^_t rGV? QSR V³s hfjbjl ] c R 0V x_lr] j _?]fV0v j xCc R V t_ wc c R t_ c 5. ]^V V@6 XD=D8 0 ¦j?h n uV w _t` ]^V. 1 1 i 1 043 ]Px j x veV w h V0_t]^x u g,. o g . :g j g. 3 . 3 3 3. u o j X $ . . . 3 h 3 S g. 3 3 3. : u j. 0 ^. . X. R w R ]PV £bèp=_trGV0xLcc j g $. . 3 1 V &1 1 [. . X : g9ug. . . 3 i 1 1 <1. o. | . . . : |j. . . | . |. . . . : 3 . 3u 3j 3. 0. U. _?x v¨tptV ]nc R _lc ¦j?h c R V

(135) g ]Pr_ h ?VV0x j è R j xC_©]fV c jl s j ]fcfptVs h^j u_terGcdt g. 0 { g `g. 0 $ g. 0 a 7"Nc R V h V g ]nc R V§r j _tvC_lcSc^W$V U©_txu v g . 0 ]Sc R V§r j _?vA_ h^h Gp? V vüV ciV V x,U©_tx v _tcP£bèV èV n . . l %,+SP 6ÕX8!#"!$&%(')*,+.-&"/0.12312 54 &x" 4 H,, !"`! I &: S¡Lc j w R _?]fc^w ghfj w V ]]fV0]_?x v¨c R V h - s serw _lc^ j x ] CU f. X ?l| 0 es sU$=Cl{76=O=e 6 =9889:";5<=$ ! 4 H 5> K% % !"G! I :& - x xU - s ser2 gih^j _?UG U| f. X0tt| 0 . susU |uX |76b|?{ X?. F. 5. F. 6 {8?@5-&12$A"$&%BC,8"!$&%12DE0.",F!; HG & I ! I 9&: ;9vx ! 9! 9 9! a% &t 9v[9! >m9_lc R

(136) sV h eZPV ] XEO.fX0t X 0 s s |U?I6L|?|Ou 6 |8JKL

(137) "!M2 5> !F x ( ON && & %&& !! )H % x K% % !" ! I 9&: %! M &tE BC (D[ % - x xU - s ser2 ghfj _t G C.fX0tC 0 es sUu|?I6u}t}e 6 C98JPKL

(138) "!M

(139) "$&%Q*RS

(140) 8T1VU$ 9! 9!$( &x" !! 4H ! H9& % v k K% % !" ! I :& (D[ % 1 lal<l Qh _?x ] - ` c j W©_tc ß j xLc h^j r2 |UV.fX0tC 0 susU X ?t76>X lU | 6 O8PKWLOX)$ Y> !VD C ) HP !" I D C % x &( & Q V w R UZ V s j?h cP> ¡ ZOUeC ui¤TÏ1 X0tuO 6 }8Z,L[F!;"R<q % v ! I 9:& I D C@!! P ! & !9! P49! ( !! P 9!! ! v[9! ÚèV0èV xeA¡Lde]fc^V W©] QSR V jLh d+_?x v - s s rGw _tcf j x ] =Cf. X0tb} 0 es s X\6b|t|u. 5. F. F. E. /. /. F. 6 8S]YL[F&^2F&M_<`"!$&%#')aJb#McDDMd";a<l $%!(H%! v[9! H9& &x !" & v !" e &t I ! [9! 9v[9! - x xU gih^j _?UG ==7.^X tt| 0 s sU O?UI6u}9U=e. oDp4.oJq.

(141)

(142) !"# $ % % & ' & ( % & 6 8P*. Xq}. 2 $ 4H 9 !" 2I DC P Rc V³{l| c R - x xb` _tr - rGrV h c j x9 j x V h V0x wVx# j W j xLcfwV rGr j 1rrGx j ] j wc§X ?Ou bsusU$==UI6 ==?}e % &t 9v[9! &tD C 57?" ! & 9 P & &[9!V9! 9v[9!' 9 !" ßs h V s h x?c UX ?L}b 6 XtX8]* R<=<

(143) "!$&% K

(144) "R< $ 4Ha !" I JC C % v ( ' &(D 9 ! 9 P ! ! &t%x !" - x xU - s ser2 gih^j u_tU O.^X ? O 0 s sU X?EX O76>X {b}b 6 XD=98]@] ,M_+R12 "$&% ')OJ2"!Mc0M2 9! 9 9! 9! )H &[ G 9 !" Ú` V èV xeA¡Lde]^cfV0W$] QSR V j?h d+_tx v - s serw _lcf j xu] UX C@.^X ?l| 0 es sU$ = Xt\X 6 =l{? 6 X {8 $ )HRR 9 !" I DC7 &:(E[! C Q V w R tZPV s j?h c ==?}l 1 ;Z 1 - m#_=d~X ?l|u 6 X |8

(145) 5 KP1.!3!Md"%!M_<

(146) "!$&% b * Î"!$&!X)<=RM2 & &[2H& 9!JC & % v ? ! P V 3ÚxeptV h ]^Gcd? ?¡LV sec X ? X? [!9 v&[% ' Q V0w R qZPV s j?h cIS¡ ;Q Z X Ub}bX? g ` h vuèa 6 DX C9889

(147) a[D12 l &tD)H')H 9! x[! H9&' ! !"& !" %!(9H%! P v[9! 1 l<lal Qh _?x ] - èc j W©_lc > j xLc h^j r2 b|u X .fX0t O 0 es s u U?I 6buDX = 6 EX O8ZC8 "D\UR<=-&1Vß I 9&: 9! 9 ! 9 - veplx - s s r5 gih^j _?U =C.fX0t?{ 0 susU X | UI6X=} C 6 X=} 8

(148) 58"< <=F&DM1 $ )H'!!P@ 9&:9(D[9! 9 !" I ÚèV èV x +¡bdb]^cfV0W©] QSR V jLh d - s ser2G =e X .fX0t C 0 s s OL} 6b C 6 X 82':U0!`"!$&%`

(149) 5*&!D U2"!& > !7DC P &x" 4H&t! &t JC 9 &[ DC H%! x[! !"@)H !LK% % !"7! I &: ghfj erGV0Wd g V h V vu_tw R $ 1¶x ¦j?h W©_lc^]^5 =t7 .fX0t = 0 susUuI{ 6 =Ou . . < <C!@5-&12,$H""!$&%B[H"!D1VR<=M M. >x g hfj w V V v xeL] jl j j ^ h j W` xew _tcf xU > xLc rU_tx v~ j W$suècfxeu em 6 XEU8]* R<=<)"!$&%9C [H"!D1VR<=M M2 C F. 2. ! ! !$&t9%v !". ÌE.

(150) Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unité de recherche INRIA Rhône-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN Unité de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unité de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. ´ Editeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) .

(151) ISSN 0249-6399.

(152)