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Monotonicity and stability of periodic polling models

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(1)Monotonicity and stability of periodic polling models Christine Fricker, M.R. Jaibi. To cite this version: Christine Fricker, M.R. Jaibi. Monotonicity and stability of periodic polling models. [Research Report] RR-1690, INRIA. 1992. �inria-00076925�. HAL Id: inria-00076925 https://hal.inria.fr/inria-00076925 Submitted on 29 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..

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(177) M > U +=-/{+=CE[Y=/W\^+49249l 4J,   -/ ^ s

(178) gPCE2Cm0_CE2=4§FCE,%A2#c ^ sgZCE2=CE0_Cm24]FD27Q0.-3CE2  LO M : .O 0.CceAS2 ¶ A@ VASfE4R/ A2Y  > H#(( <

(179) ! Q) # # 3- .Z ) ' . ZT)  ^ W C P!,'.!  !,' G },.C CSF 4 ,/k0BV,.Cl 4{0.+4rAE/./149,.0_-3Cm2uFCm,  : >H5CE,€4AE/k4hCSF!2Cm0A0_-3Cm2oA2Y \^-30.+CED0 [bC /./6CFHfE4924J,AS[b-30kc ;|/1DV=VXC‚/14 - ¶ ¶ AS2YoVD0& &XY Zˆ>4J0  LO M : .O$L4ZA ^ s

(180) gPCE2Cm0_CE2=46FD=27Q0_-bCE2$> U +=4{,aAS2YeCEg l 47Q0.CE,  : 6&   &  W 4  M M &  &  W 49  L L 6&  &  W +AE/$-32Y#4JVX4J2Y#492e0$7QCmgZVCE24J2#0a/Q;9A2YWA[3[mCSF=0_+49g M AS,.4HAS[3[ ^AW s

(181) gPCE2Cm0_CE2=4}-32    M

(182) :  W 0.+4 l 4R7Q0_Cm,]-/]0_+eD/ ^ W sgZCm2CE0.CE24`-32   

(183) :  W > 5CE,PAS2ec  OL M : "O ^ s gPCE2Cm0_CE2=4 ;  : fE-bl 492L#c ¶ @ ;ƒ49jeV47Q0A0_-bCE2CSF0_+4tFD27Q0.-3Cm2 wCSFh0.+4tl 47Q0.CE, ALCSl 4‚;&-N/ ^ sgZCE2Cm0_Cm24t0_C C> -3gZ-b[A,_[3c‚;  : ;&fm-3l 492?Lec M ¶ ;ƒ-N/ ^ sgZCE2Cm0_Cm24 > 5 -32A[3[3c‚;SF,_Cmg ; .Z §-N/!AS[/kC ^ sgZCE2Cm0_Cm24 -b0!-/}-32 

(184) :!L#c]0.+4BgPCE2Cm0_CE2=-7Q-b01c CF  : FCm,  : j#4RY; AS2Yo-32  : Lec 0.+4%gZCE2Cm0_Cm2-7Q-b0kc Am/_/kDgZV0.-3CE2dCE2d0_+=4Z/k49,_s le-74WVCE[b-7Q-b4/Q> B4J27Q4  : -N/ ^ W sgZCm2CE0.CE24 > U +4%/.ASgZ4]V,.C CSF +CE[NY/ FCE, .Z}\^+4J2 3 ¶ > U +4WV,.CeY#D7Q0CSF ^AW s

(185) gZCm2CE0.CE24rCEV49,aAS0_Cm,/nL49-32=fZA[/1C ^ W sgZCm2CE0.CE24 ;0_+=4 ^AW s

(186) gPCE2Cm0_CE2=-7Q-b01c%CSF . ZFCE[b[3C\€/Q>  U +=4 7,_D7Q-NAS[HgZCE2Cm0_Cm2-7Q-b0kc)V,.CEV49,.01cwCSF6/10A0_4`V,_Ce7Q4R/_/%AS0]0_+=4uVCm[3[3-b2fd-32/k0AS2e0/ -N/^A§7CE2/k4pqD=4927Q4nCF!0_+=4{V=,_49le-3CmD/ƒ[349gPgZA 8.

(187). 1. . . K. . .

(188). . ;. .   . . -. . . *. . . . K. . I . ;.

(189). .

(190). . .   O&. *. . .

(191). . ;. . .   C . #. ;. . *. . H . .  . . -. H . . . -. -. -. . -. ;. ;. . -. -. ;. . -. . -. . -. . . . ;. . . . *.  C . &. ;. ¶. *.  . .  .

(192) "!XJ   < ,"! 7] : Y :  G   'B

(193) ! ) # # 3 !VT,  - X]Y Z % 7 ^ W C P!,'"!  !,'  %' NPG ' )   % : # )0- + X]Y Z ) ' / X]Y ZT)  ^ W C P!,'.!  !,'  %' NPG XY Z \^+492o0.+4h-32-b0_-A[ Y#-/k0_,.-3LD0.-3Cm2

(194) Y },.C CSF 490

(195) X]Y Z L4W0.+4rY#-/k0_,_-bLD0.-3CE2`CSF -N/B-3,aAE7oAS0  > ƒ47JASD/k4dAS[3[r7QCmgZVCE24J2#0a/§CSF M Y : A,_4d2CE2=s

(196) 24Jf AS0.-3l : 4 : ;

(197) Y ^ W

(198) Y A2YuLecZ[b49gZgPA e;#FCE,AS[b[ 3KI ; :: M:

(199) YZ :  Z  : 777  :

(200) : Y : ^ W  Z  : 77 7  :

(201) M Y :

(202) M Y Z ƒc-3gZgZ4RY#-bAS0_4 -32Y#D7Q0.-3CE2 ;

(203) XY Z ^ W

(204) X ; : Y Z FCm,}A[3[ *N  3 ;mAS2Y X]Y Z -N/ ^ W s

(205) gPCE2Cm0_CE2=4 -b2 NuFCE, 3j#4Y|> U +-/ƒ-bgZV[3-b4/}0_+4 ^AW s

(206) gZCm2CE0.CE2-N7Q-30kc%CSF AS[b[ 7QCEgZVCm2492e0/ƒCSF X]Y Z A2Y?-32?VAS,_0.-7QD=[AS,rCSF ( XY Zˆ> U +4tgZCE2=CE0_Cm2-7-30kc CF€0_+4t/k49,_le-74§VCE[b-7Q-b4/{-bgZV[b-34/ 0.+492t0_+=4 ^AW s

(207) gPCE2Cm0_CE2=-7Q-b01c%CSF + XY Z  / XY Z -bA2 NuFCm2, 3 je4Y>  '   $.#% 3 -  3 '3 * 3 +=492ZAn+eDfE4€2eDg§L49,7QD/k0_CmgZ49,/AS,_4€\^A-30.-32f=;q0.+4Y#D,A0_-bCE2%CSFAh/10aASfE46Y#49V492Y/ /k0_,.CE2fm[3ctCE2t0_+=4{0kceV4{CF!0_+=4W/k49,_le-74BVCE[b-7QcP\^+-7+t-/ƒD/k4Y> ‚DVVC /k4hAWp D49D4‚; /149,.l 4Y]L#c%AnVCE[b-7Qc]CF$D2[b-3gZ-b0_4Yh0kc#V4 ) #  )79 ) !,' !/k0ASfm4n-/ 7

(208) ) H)   &0.+49,_4r-/^A2o-32 2=-30_4h2eDg%LX4J,ƒCSF 7QD/10_CmgZ49,a/ƒ\^AS-b0_-b2fZA0B0.-3gZ4 -b2 0_+=4 p D49D=4 > ^0ƒ0.+4nL49fE-b22-b2f]CSF /1D7+tA]/k0ASfm4 ;0_+=4hpqD=49D4n-N/ƒ/10.-3[b[$/_A0_D,aAS0.4YtAS2YZ0_+=4 YeD,A0_-3Cm2)CF^0.+4Z/k0AfE4%-/n2CE2=s

(209) -b2#0.49fE,aASL[b4 ;$- FH2CE0h-b2 2-b0_4 > U +=-/n\^-3[b[L4P/149492d0_C 4Jj7Q[bDY#4hA2#cy/10A0_-bCE2A,_ctL49+A9l#-bCE,^CF!0_+4r/kc/k0_49go> *!Cm2/1-NY#49,]2C\ A p D49D4‚;!/_A9c?p D49D=A4 @;}\^+-N7z+?-N/W/k49,_l 4YdL#cdVCE[3-N7Q-34R/6CFn[b-3gZ-b0_4Y 0kceV4WA50 ) # # 0.+4r/10aASfE4R/Q;$/.AR*c C AS06/10AfE54 @ O FCE, ¶B^ ^  C > DVVC‚/14]0_+=4]p D49D4 -N/B/.AS0.D,A0_4YoA0n0_-bgZ4 > U +492$;|0_+4vpqW D4JD4r/10A9c=/n/_A0_D,aAS0.4YuFCE,n4Jl 49,^L479ASD/14%AS0 4RAE7+%/10AfE4 4N  @/O 0_+=4B2eDg%LX4J, + X]Y C + C CSF7QD/k0_CmgZ49,/$0_+AS0^A,_4/149,.l 4YW-/}W -b2#0.49s fm,AL[34 > ƒD=0&0.+4nVCE[b[3-b2fr/1c=/10_4Jg /k0_-3[b*[ >S\HCE,.8!/ >WLX4R79ASD/k4n0_+=4hY#D,aAS0_-bCE2 / X]Y C / C CF 0_+4R/14h/k0ASfm4/ƒA,_4n-b2#0.49fE,aASL[b4 > U +=4h/14p D4927Q4/ / XY C 7QCm2/10.-30_D=0_4 DC^-32Ye49V492Y#492e0 /k4p D4927Q4R/ CF -†> -†>®Y|>,A2Y#CEg lEA,_-NASL[b4/Q>< CE,_4JCSl 4J,9;E0.+4/146/14p D49274/}AS,.4n-32Y#49V492s Ye492#0€CSF AS[b[0.+4%pqDAS2e0_-30.-34/ƒ,.49[A0_-bl 4{0.Ct0_+4rCE0_+=49,hp D49D4/€-32 0.+4%/1c=/10.49go;A2Y CF A[3[$0.+4r/1\^-30a7z+=s

(210) Cl 49,^0.-3gZ4/> U +4J,_4QFCm,_4 ; FCE,€0_+4v/149,.l#-N7Q4nCSF 0_+4hCE0.+49,€p D49D4R/Q;0_+=4 /.AS0.D,A0_-3Cm2 CFHpqD=49DP4 @%-/ƒ4p D-3lmA[3492e0ƒ0.C§0.+4h,_49V[NAE7Q4JgZ492e0&CF}0.+4r/10aASfE4R/nY#49l Cm0_4Y 0.CW-b0&Lec%AEY=Y#-30.-3CE2AS[$/k\^-30a7z+sCl 49,}0.-3gZ4/ +A9l#-b2fh0_+4/_AgZ4BV=,_CEV49,.0_-b4/!AE/ƒ0_+=4BCm,_-3fm-3s 2AS[|/1\^-b07+s

(211) Cl 49,0_-3gP4/Q> †2]0_+-N/}\^A9c ;eAW/kDLsˆ/1c=/10_4Jg \^+49,_4€0_+=4B2eDg%LX4J,CF p D49D4/ 5. ;. . C . C. *. . C. C . C. ;. ;. . . . . . . . *. C. $. E. C. . . . 5. ;. ¶. ;. . 5.

(212) N- /ƒ,_4YeD7Q4Y§Lec ¶ -/ƒCmL0A-324Y|> U +-/ƒV,.Ce7Q4Y#D=,_4hCSFH/.AS0.D,A0_-3Cm2 gtA9c%L4WAVV[b-34YZ0_C /k49l 49,aAS[|pqD=49D4/>!@ 4QF4J,_,_-b2fh0_C]0.+4n2eDg§L49,.-32fCSF0_+4hp D49D=4/Q;e\ƒ4h/kDVVC /k4{-b2Z0_+=4 FCm[3[bCS\^-b2f

(213) S> 4J 0 L4 0.+4 -b2-30.-A[ƒVCE[b[3-32=f?/kc/k0_49go>5Cm,= 

(214)

(215) 

(216) ¶

(217)  ] ; : ) # # U "!#$#&%'( 7 + C 7

(218) < 79   !    H72 ¶/

(219)  /B,_4R/1D[b0_-32=f{F,_Cmg 0.+4h/.AS0_D=,AS0.-3Cm2uCF 0.+4rpqD=49D42/  ¶

(220)  ] ;/14J,_l 4YZAm797QCm,Y#-b2f§0.C§0.+4r/_AgZ4hVXCm[3[b-32f]0aASL[b4WA2Y 0_C 0.+4r/_AgZ4h7QCm,_,_4R/1VCE2Ye-32fP/149,.l#-N7Q4€VXCm[3-N7Q-34/HAE/^-32 n>!5Cm,  ;\^+492`/10aASfmP4 3}L,_-b2f / Aole-/1-b0]0_CoCm24 CFh0_+4 ,/k0Zp D49D4R/ - *3 ^  ;Hp D49D4 - *3 -N/]/149,.l 4Y?[3-b8 4§-b2 ; AE797QCm,Y#-b2ft0_Ct0_+=4§VCE[b-7Qc Z AS0.0Am7z+4RY 0_Co/k0AfE4 3 $CE2?0.+4]CE0.+49,h+AS2Y;\^+4J2 - 43 I m;2=C p D49D=4]CSF  -//149,.l 4YuLD=0{0.+4%/14J,_l 49,nL47CEgZ4/^D2A9lmAS-b[AL[34WFCm,WA V49,.-3CeYZCSF 0.-3gZ4€Ye-/10.,_-bLD0_4RY [b-38 4€0_+4€LCED2Y / Z FCm,ƒ0_+=4

(221) [b-3gZ-b0_4Yh01ceV4 VCm[3-79c Z ; FCm[3[bCS\H4Y Leco0_+4t/k\^-30a7z+sCl 49,n0.-3gZ4 W Y Z > U CuFˆAm7Q-3[b-30A0_4]0.+4t7QCmgZVAS,.-/kCE2dCSF^0.+4/k4 /kDLsˆ/1c=/10.49gt/Q; \H4n8 4949V%0_+4R/14nV49,.-3CeY/!CSFD2A9lmAS-3[NASL-b[3-b01cvASVA,_0^CF 0_+4h/k\^-30a7z+sCl 49, 0.-3gZ4R/Q> ƒD0BFCm

(222) , ;0.+49c`AS,.4]-327Q[bDY#4Yt-32`0_+4]0.-3gZ4rY#D,.-32fP\^+-7+ 0.+4]/k49,_l 49,€-/ 2=CE0/149,.l#-b2f]-32`A]7Qc=7Q[34‚;qCm,B0.CE0aAS[ >E/k\^-30a7z+sCl 49,^0_-bgZ4 >e;q\^+=C /14rgZ4A2Z-/ƒ2C\ )9. -. -. '. . . . . . . . . . 5. . *. . L M ;. /¶. : U +=4BVCE[b[3-b2fr/1c=/10_4Jg }-/!/1-bgZ-3[NAS, 0_C A2YZA[3[CmD,ƒV,.49le-3CED/ ,.4/1D=[30/!AVV[3c]0.CW-b09> U +=4]/10aAS0.4]CSF -N/BY#4R/_7Q,.-3L4YZL#ct0.+4]/k4pqD=4927Q4 X] Y Z D(: &X] Y Z 9 ( &X] Y Z A0ƒ0.+4.>VCE[3[b-32f >{-b2/10aAS2e0/3&X Y Z \^-b0_+ 4N  3   "U ¶/

(223)  0  > †2ZVAS,.0_-N7QD[A,9;EFCE, 4RAE7+ 3_; X Y Z X§-N/BA%<oAS,.8 Cl 7+A-32oA2Yu-N/ ^AW s

(224) gPCE2Cm0_CE2=4]-32 N)\^+492`0_+4r-32-b0_-NAS[ /k0A0_4%-/B49gPV0kc

(225) +4J,_4W\H4]gZ4A2 0_+AS0rpqD=49D4/ ¶/

(226)  ]A,_4]49gPV0kctAS0h-32/10A2#0 > 4J0BD/€/1V47- FcZ0.+4W0.,AS2/1-30.-3Cm2/ Z  CF!0_+=-/^<oAS,.8 Clt7z+AS-32 > 490  O M : .O$A2Y   : 9     

(227) \ƒ4yAS[/kCo\^,_-b0_4 6FCm,]0_+4 ,a/10P7QCEgZVCm2492e0/{CF 8  L s!\^+4J2 - 43 ^ m;VD=0  O ¶ FCE, In-b2 ¶ @ s > U +=492 0.+4nFD27Q0.-3CE2/  Z  ;  Z  A2Y  Z   Y#49V492Y]CE2  CE2[bc]0_+,.CEDfm+   >@ 49V4A0_-b2fW0.+4nA,_fmDgZ492e0/}\^+-N7z+ [b4Y F,.CEg ¶-¶ s ¶ 0_C ;0_+4hCmVX4J,AS0.CE,  Z Y#4 24Y§Lec -N/ƒ/14J492Z0_C]fm-3l 4 0.+4h0_,aAS2/k-30_-bCE2/^CSF X] Y Z 0.C C  Z     U   X Y Z ; :  X Y Z   .Z      

(228) N - 43 K^  7. *. 7. ;. -. 2. ?A2 4.   . 25. +. . . . ;. . . .  . 5. . +. C . ;. ;. . '. .  * . $. . .  ). I . . <.  .  C . O.  C .  C . /*. . *. . ;. ;. . . . . . . ;. . . sH\^+492 - 43 Im;|/10aASfE4 4N  3 7QCm,_,.4/1VCE2Y/n0_CtA§V49,.-3CeY CFƒD=2A9lEA-3[NASL-b[3-30kc%CSFH0_+=4 . . . . ¶. .

(229) /k49,_l 4J,\^-b0_+PY#D,A0_-bCE2 / XY Z > U +492%CE2[bcW24J\ AS,.,_-blEA[/gyARchCe797D,}0_Cz>K YASV0.-32f 0.+4rAS,.fEDgZ4J2#0a/Q;#\H4hfE490  Z         Z      

(230)    

(231) N - 43 I  †0r-/{4RAE/1cw0_Co/k494Z0.+AS0]0.+4ZV,.C CSFhCFn[b49gZgPA 49je0_492Y=/{0_C Z 0.Co4/10aASL[b-/1+w-30/ ^AW s

(232) gPCE2Cm0_CE2=-7Q-b01c‚> U +=4o/1c=/10.49gt/ §/.AS0_-N/ Fc A)Y#CmgZ-32AS27Q4tV,.CEV49,.01c AE/§\H4o/k+AS[b[nY#4JgZCE2/k0_,aAS0_4t-b2 0.+4]249je0€[349gZgyA#> ƒc   ;|\ƒ4%Y#492=CE0_4]0.+4  ,a/10h7QCmgZVCE24J2#0a/BCSF A§l 470_CE, +ARle-32= f I 67QCEgZVCm2492e0/Q> H#(( ?

(233) ! E) #$#   + !  %' 

(234) 

(235)    -  ,!P%' ) H7  %'  57] ' 7] ") %  : Y  :  ^ W :  Y :   ' XY Z ^AW X] Y Z 

(236) !  ) #$# 4N  3 G },.C CSF  0P-/Z492CmDfE+ 0_C 7QCmgZVAS,. 4 AS2Y  > -327Q4  gyARcL4d7QCm2/1-NY#49,_4RY Am/hAu/kDLsˆ/1c=/10.49g CF  ; :9;0. +4%AE/./149,.0_-bCE2dCSF^0.+4][34JgZgZA{FCm[3[bCS\€/€L#c`0_,A2/1-b0_-bl#-b01c‚> ƒ4R79ASD/k4B0.+4h/k4pqD=4927Q4 X]L Y Z CF}0.+4  ,a/10^7CEgZVCE2=492#0a/}CSF XY Z$-N/&2Cm0^A]<oA,_8 Cl 7+A-32$;\ƒ4r2494Y`/1CE gP4W7JAS[7D[A0_-3Cm2/Q> 4WV,.Ce7Q494Y Lect-32YeD7Q0_-bCE2$> 490 ,/k20 3 ¶ A2Yo/1D=VVC /14 : L Y : ^ W :  Y : > 4J0   O M : .OL4 ^ s

(237) gZCm2CE0.CE24 .Z ^A2Y Z  A,_4h0_+=492 ^ sgZCE2Cm0_Cm24 > DV=VXC‚/14]2C\ X]L Y : ^ W X] Y : > s +=492 - ¶ Q^ m;$/.A9c - ¶ ¶ ;  :    fE-bl 492oLec ¶ Y#49V492Y/nCm2[3coCm2   A2Y 7CE-327Q-Y#4\^-30_+ :  n> U +#D/Q; 

(238) P X]Y :   :    U   XL Y M  

(239) P X]Y :   :       :     

(240) P X]Y :   Y Y.   .   :    

(241) P

(242)   X]L Y :    .   :    

(243) P X] Y :    ^ U    X] Y M 0.+4 -324RpqDA[3-b01c ,.4/1D=[30_-b2f F,_Cmg 0_+=4 ^ s

(244) gPCE2Cm0_CE2=-7Q-b01c CF .Z oA2YF,_Cmg 0_+A0 XL Y : ^ W X Y : > 5. *. . . . ;. . . . . . H. 9. . *. . !. . ;. . ;. L.  . . .  . . . ;. ;. ;. ;. ;.  . ;. . . . ;. . . ;.  . ¶@. . . . . . ;. I . . . ;. :. . ;.

(245) s + 492 - ¶ I m;m/_A9c - ¶ 0.+AS0 (2X]Y : ( : &0X]Y : U   XL  Y M   . . *.  ;     Y#4JVX4J2Y/Cm2[3crCE2    L >}@^49gZ49g]L49, ( L 6&0X]: Y : +49,.4 > U +=492$; 

(246)  XY :    :     

(247)  XY :    :       L    :       

(248) P X]Y   : Y Y. L Y         :      

(249) P

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