An Alternating-Priority Server with Correlated Switchover Times

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(1)An Alternating-Priority Server with Correlated Switchover Times Eitan Altman, Robin Groenevelt. To cite this version: Eitan Altman, Robin Groenevelt. An Alternating-Priority Server with Correlated Switchover Times. RR-5368, INRIA. 2004, pp.55. �inria-00070635�. HAL Id: inria-00070635 https://hal.inria.fr/inria-00070635 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. An Alternating-Priority Server with Correlated Switchover Times Robin Groenevelt — Eitan Altman. N° 5368 Novembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5368--FR+ENG. Thème COM. apport de recherche.

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(47) kgup ¯pmrm[ocm~«¹ blmoYrwcBo~ p ¯ bgiVY[\°ÜWl®mVY[j [ tirm3#2 r3#2 Og ¹T¡ VYpqkiV¡¨[#rmeienZ©[guVYtilmnmVYlGnfgj°W½nYtuguVY[tiZ\lmti[m¹¡¨[¡ pklGoGgipodnYlmneuc¥rmeienYZ\[ ρ := ρ + ρ < 1 giVrOg ¦oGguguVYti[lf~fEnnYk[j[nY[jguVYe+[rwtik[l®wlmrw}ti[jprwtirwoTgukp[ oY©¯ nYnYook~fgup[jlmt+oeegirwª guplmormtuc ti[GpZ\[m° i = 1, 2 ª ® c (n) =IE[V V ] − IE[V ]IE[V ], n ∈ N, ª±G c (n) =IE[V V ] − IE[V ]IE[V ], n ∈ Z. 1lwgu[guVrwg pe\~f[¬o[j~ ¯ lGt Z °»·¢pguV giVYpe\klmoEm[joGgiplGopg©pqe-oYlmg©oY[Ck[jeieirw¯ tic«gil¡¨lmtiÃ ¡ pguV c (n)c :=(n)IE[V V ] − IE[Vn ∈]IE[V ¹

(48) eupoTk[ no~f[t-ehgrOgiplGorwticti[mpZ\[\pg lml®¡+e° guVT¦rOo g ] c (n) = IE[V }rmtgipknYqrwtC¹®p ¯T¯ lmt0¯ [jVrGkiV©]E−nY[jIE[V nY[§¯ guVY[+]IE[V e[CGn[o]k[W=l ¯ IE[V e¡ pgikVYlOmV[tgu]pZ\−[jIE[V e

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(50) nYoklm]IE[V titu[jrwgu[j~À] ¹C=giVY[co c(−n) rmo~ c (n) = 0 ¹ lGt n ∈ N ° ¯ giVY[ti[pqeWoYl klmtitu[jrwguplmo[gh¡¨[[jogiVY[e¡ pgikVYlOm[t§gipZ\[CeWl ¯ (0)guVY[=gh¡¨δl EnY[jnY[jej¹dguV[o c (n) = 0 ¹ lmt n ∈ Z ° NI. `. `. G. HG. ^. n,i. i. 1. 2. 2. 2 1. n,i. i. (2) i. i. 2 2. n,i. i. 2 i. (2) i. 2 i. #`. i. i. 1. /c. (2) i. i. HG/J. /. i i. 2. I. G. i. 0,i n,i. 12. 0,i. 0,1 n,2. 0,1. n,i. n,2. 12. 21. 21. 0,2 n,1. 0,2 n,1. 0,2. I. 0,2. n,1. n,1. I. −n,2 0,1. −n,2. 0,1. 12. 2 i. i. i. 12. ïðíëÀï.

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(52) . . 6. . 2. 3. 6. !"#. $. 2. %'&. 6. (%). ^. [Ckrmne[l ¯ guV[rGeueunYZ\}fgiplGo¶l ¯ giVY[#EnY[nY[Ce lG}[trOgupoY\nYo~f[jt1egirwguplmormtuc9ti[mpZ\[ ª C rwo~»ª²m kjrwo¶[ti[¡ tipggu[jo¶rGe ª>GrG IE[V V ] = v + c (n), i = 1, 2, n ∈ N, ª>w IE[V V ] = v v + c (n), n ∈ Z. ¦olGti~f[jtgil«[CehgrwYpeuVguV[¥af|+z ¹

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(54) . 3 . "2 8. / . 3. 48. (. UWVY[ nec}[tiplf~³¹ ¹ G[oY[jtirwgu[j~Ec knehgilmZ\[tepqeguVY[enYZ l ¯ po~f[j}[o~f[joGg-eupoYm[ Yneuc}[tiplY~Ye¹f[jrGkiV¶D¡ pguVB(Y~f)pqehgitupYnfgiplGo D Y ∼ D (1) ° +[jok[guVY[eu[jklmoY~Z\lmZ\[oEg pqe # " " # X X ª±G IE[D (Y )] = IE = IE D D (1) = d IE[Y ] + d − d IE[Y ]. c grwÃdpoY Y = N (T ) ¡¨[lGfgirmpo»ª i = 1, 2 n,i. R. n,i,k. 2. Y. 2 n,i. 2. Y. n,i,k. k=1. n,i. 2 i. n,i. 2. (2) i. 2 i. k=1. M. n,i. (2). 2 2 (T )] + (di (Nn,i (T ))] =d2i IE[Nn,i IE[Dn,i (2). − d2i )IE[Nn,i (T )]. =λ2i d2i IE[T 2 ] + di λi IE[T ].. ª G J. ïðíëÀï.

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(98) ¯ eu[tim[j~-rwgMEnY[nY[ i ~fnYtipoYgiVY[ n kcfk[Wp ¯ giVY[ti[ rwti¯ [ N odknnYehZ#gilmZ\[t[teue0[tipmo [jgu~BVY[x~fnEtunYp[joYnY [1NrwgMYginTVYe[1c¥Z\}lm[Z\tip[lfoE~gë¹l ªJd}WlGgi[poYex°0nade+pguoVTkrO[+g1gigiVYVY[1[-oE[nY¸fZ#}[j[jkgut§[je~B[jtuodmnY[CZ#~©pqe0[tgiVYl [x¯ eknnZ egulGl Z©gi[jVYti[e eu[tim[j~³¹f}[t1kcfk[m¹Yrwg+EnY[jnY[ i pe IE[L ] ª±mG λR IE[T ] =IE[T (L )] = = , 1−ρ 1−ρ ¯ lGt i = 1, 2 °MU l ~f[tipm[guVY[eu[jklGo~Z©lGZ\[oGg+oYlmgu[giVrOg IE[T ] =IE[T (L )] Mrmt [Γ ]IE[L ] =(IE[Γ ]) IE[(L ) ] +

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(100) 9. 3. . 2. C. 1. 6. i. HG. (2) i. HG. HG. a. i. i. i. i. i. HG. 2 (2) i i. i. i. 3. th. n,i. n,i. n,i. i. xepoY\[CGnTrOguplmo»ª± C¨mpm[Ce S. i. i. ∗ n,i. 2 n,i. 2 n,i. ∗ n,i. ∗ n,i. IE[(L∗n,i )2 ] = (1 − ρi )2. a. ∗ 2 n,i. 2. +. ∗ n,i. n,i. . (2). Rλi ρi (1 − ρi ) + λ2i bi (1 − ρi )2 (1 − ρ). . .. AG. .   (2) Rλi 1 − ρ2i + λ2i bi 1 2 2 λ2i IE[In,i  ]+ IE[Tn,i ]= (1 − ρi )2 1−ρ. å â®ÛhÔ>Û ÏãàÏÑØµäÎòhÝ*Î ä ÎòhÝ±Õ×Ô Ù (2) PN k=1 rk. λi [−(1−ρi )+λ2 b ] Pi i 1− N ρ k=1 k. λi. Í ÏÑà²à±ÏÑØ®Ú%ÏÑØ. . m. . i = 1, 2.. äÕ×Ô>Í ®ÐãÎ ®Ù * Ö !ðÙ ïqÝ*à²â®Õ

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(108) 9. 3. H8. 6. :B<. >. A. /. -. 9. $ 2. d$. ". !$% '. "2 * 9. Tc. "% $. . h h. 6. i. ). \ (. H). 2. . !$ (. & :6 9*6 f&E. -. >. `. &E. IE[Wq,i ] = i = 1, 2. 1. -. HW. 9/2 658:9. ) (%. .. 2. ). . _ (. 2 02. . . ))(. ). . 8. )P)% $ ( . (. 3. ª± m. (2) 2 (1 − ρ)IE[In,i ] λ i bi + , 2(1 − ρi ) 2R(1 − ρi ). c^. ). (2). (2). (2). λ 1 b1 λ1 ρ22 b1 +λ2 (1 − ρ1 )2 b2 + 2(1−ρ1 ) 2(1−ρ1 )(1−ρ)(1−ρ+2ρ1 ρ2 ) 2 ∆ ρ2 (1−ρ) 2 (1−ρ1 )R + − δ2 +C ψ, + 2(1−ρ) 2 1−ρ1. IE[Wq,1 ] =. NZ%. . ψ :=. 2 2. "2. &. 9 8 6 3 48 6. Z] . ! (. . 8. 1−ρ1 R(1−ρ+2ρ ρ ) 1 2 !")! $% P. 3. hP d$ . . . (f . 6. 2. C. 6. Z(. %. =$. . %&. 8. !. (f%). + 2 $ 2. . . ) (%!). 658. . #"% $. . 6. . h h. ). i. . )=Z. 658. 658. 658. 6. $ ). 8. ") $!] . "% $. '. 658. T. . . . ª±wE $!. h h . 6 658 9 6. . . (. 24353 2. 46. 3. N)] !$% . ¦o¶guVY[nYoTklmtitu[jrwgu[C~kjrmeu[\ª C = 0¨giVYpqe pe poBklGtuti[jeu}lmo~f[jok[¡ p¤giV Y¹ ¯ lmtiZ#nYr\ª>° ®m ²° ¡ VYÉÀ[jtuÉ [ giVY½Y[-lGtMe[jrtumk[jnTt+ehgimlmldZ\¯ [Ce [tMlmrmoBtutiwprmdkjprOoYguplmrOoBg{ErGnYe1[enYld[ lGoBi ª rmie =giVY1,[-2EnYgu[VYnY[1[#epcfe1e[gu[Z\Z*}fghcG[j°+VUWrCmVY[C[#e0trmrweMorw~fo lGZ&_ *O% rwtiprm YGn[ [¯ nYlG[t giEVYnY[ [jnYn[+lmo O¡ rmVYkjrO[gigupVYlm[jo t{ lGtMtulGoYZ lmg¨gurVY[1kn¬Tetigielmg§Z\E[nYt¨[jrwnYti[xtuppme¨[j[e0¸Y¡ rmk×VYgi[oc IguVY[x°{eu[]¨timlm[otM~fppgie{plGYoYnpeuoc©lmgutMVYlG[o ¡§OrwrmpkjguprOoYgiplGgio©pZ\}Yti[lfp~fo9nTgikVY[j[e IE[Ls,i ] =λi IE[Wq,i ] + ρi ,. . i = 1, 2.. I. HG/J. `. G. th. P`. G. n,i. ²ª G ¯ }Yne yguVYg[¨rwmgiGpZ\[j~B[§koYn[je[jgu~flG[jZ©~[jgut+lgieV[jturOmg[MrmgituVYti[Wpmk[jne+e~fgulGnZ©tupoY[jtie"r9guVOrwrGg

(109) krmrwtugutiplmpmo¥[j~VrGe [ ¯ gilmlti[{¡§VrwppZ g ®lmVt1[tguVpo[#guOVrG[¨krwwgurmpkjlmrO¯ o¶gupgilml9o"¬°"oYUWpqVYeV [ [¸d}[Ck×gu[C~ti[Z rwpoYpoY©OrGkrOgiplGo9gipZ\[pqe rmo~guV[[¸d}[Ck×gi[j~oEnZ#[t+l knegulGZ\[t ] giVrOg1rwtitupm[C~9[ ¯ lGtu[giVY[girwGm[C~knegulGZ©IE[I[jt pe ]/2IE[I °MUWVYpqe Z\[jrwoeWguVrwg λ IE[I ]/2IE[I ] IE[I ] ª>mE (1 + λ b ) . IE[W |vac] = 2IE[I ] å å â®Û0Û¦ñCâ®áCÛ{Ô±ÐãÛhÎà²à±à±Ý ÏÑÕ×Ý±ØÛÔ>à²Í£âCÕ

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(111) Î Í Ï ñ ®áÖwÛ¦Ýðæ³ÛhÛhØ ÎØ®é Ù IE[Wq,i ] =. IE[Dn,i ] IE[In,i ] IE[Wq,i |vac] + IE[Wq,i |busy]. IE[Cn,i ] IE[Cn,i ]. J. T`. 2 n,i. i. q,i. n,i 2 n,i. 2 n,i. n,i. Hk. ··· +. ë ë&Ø. ihjklm. 2 2 [1−ρ1 −2ρ2 (1−ρ)]δ2 +(1−ρ1 )2 δ1 2R(1−ρ+2ρ1 ρ2 ). n,i. i i. δ1. δ2.

(112) C. "2 8. G. 3 3. / . 48. (. ygirmmm[C~¿knegilmZ\[t9giVrOg¶rmtutipm[Ce\¡ V[o¢guVY[Ëe[jtum[jt9peYnTec¾VrGe gul ¡§rmp¤g ¯ lmtguVY[knYtitu[joGg rmk ntutiehpgimlm[CZ\~[rwt§gpo¶ª>rmeuo[~¶tiEepqgukp[+"girwl#ti[¬oYpoTpqeu¨Vgu}YVY[nEe¨nYgu[jVYnY[[[¸f}[ ¯ [Clmk×tigu[x[C~ giVYgu[pZ\girm[moYG[[j[C~~fk[jn~9egugul©lGZ\eu[[tit+m[+~fpqgu~³VY°M[ UWLVYpqeWGknTpGeh[jgielmZ\[teMguVTrOg ª± C b + b IE[L |busy]. IE[W |busy] = 2b ¯ [te0po\guVY[xEnY[nY[G¹m¬tehg{ti° [jÀprmgpgi[W[ guV Crw gMÀgiguVY[[+qeW[¸fn}e§[jgik×VgirO[jg ~\gu¡§VYrm[p¤gi[p¸foY}[jgukpguZ\[C~ [ lU ¯ lr\lGfkngirmegupo\lG¯ Z©guVY[j[1tWoEponYZ-guVT[[jt{eucdl egi[kZ%nehgipqlme Z\IE[W odnYZ#[t+l knehgilmZ\[tej¹ L ¹rwgWguV[Gn[nY] [=ibªJp+oBeuIE[W [tudpqk[]rmo~pogiVY[EnY[nY[®§pe q,i. (2) i. q,i. i. AG. i. q,i. G. s,i. i. q,i. s,i. ^xogiVY[lwguV[t Vrmo~³¹. IE[Ls,i ] := λi IE[Ws,i ] = ρi + λi IE[Wq,i ].. IE[Ls,i ] =. IE[In,i ] IE[Dn,i ] IE[Ls,i |busy] + IE[Ls,i |vac] IE[Cn,i ] IE[Cn,i ]. =ρi (1 + IE[Lq,i |busy]) + (1 − ρi ). ]¨lGZ#YpopoY©giVY[jeu[rGehgWgh¡¨l\[CEnrOgiplGoe§Gpm[Ce 1 IE[Lq,i |busy] = bi. 2 λi IE[In,i ] . 2IE[In,i ]. 2 (1 − ρi )IE[In,i ] IE[Wq,i ] − 2IE[In,i ]. s0nYggupoY gulGm[giVY[t [jEnrwguplmoeª± G¦Áª±Gm{Gpm[Ce. !. ª>GG. .. J. 2 IE[In,i ] IE[Wq,i ] =(1 − ρi ) (1 + λi bi ) + ρi 2IE[In,i ]. (2) 2 (1 − ρi )IE[In,i ] bi + IE[Wq,i ] − 2bi 2IE[In,i ]. !. >ª mE UWVY[giVY[lGtu[jZ ¯ lmlO¡+e dc}YnYmGpo\po¥giVY[wrwnY[je l ¯ ] rwo~ IE[I ] ° +lmgupqk[guVTrOg1olwg1lmok[ Vr®my+[oB¡¨[rwrGgueu[jeutunYoZ\rwgu[jpm~[gu}YVti[ldeul ¡ ¯ ppqgie kVYml®pmm[[jo¶t§pgioBpZ\y+[j}Ye+}rm[joo~~fpgu¸BV[]xY° nIE[I ec}[jtuplf~Ye§gul T[nYoklmtitu[jrwgu[C~ =. (2) 2 IE[In,i ] λ i bi + . 2(1 − ρi ) 2IE[In,i ]. n,i. . 2 n,i.

(113) . "[glmguVËGn[nY[je[-pq~f[oEgu¯ pqkrwcB~Ypegutipnfgu[C~³¹³p±° [m° λ := λ ¹ c(j) := c (j) ¹ b := b ¹ v := v ¹ ¹Wrwo~ ρˆ := ρ ¹ lmt i = 1, 2 °¢UWVY[ehgrwYpp¤ghc klmo~Yp¤giplGoSpqe©giVY[o ρˆ = λd < 1/2 ° ¦o δ := δ 2. 2 i. i. i. (2). (2) i. I. i. ïðíëÀï. i.

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