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On the Rates of Convergence of Erlang's Model

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(1)On the Rates of Convergence of Erlang’s Model Christine Fricker, Philippe Robert, Danielle Tibi. To cite this version: Christine Fricker, Philippe Robert, Danielle Tibi. On the Rates of Convergence of Erlang’s Model. [Research Report] RR-3368, INRIA. 1998. �inria-00073321�. HAL Id: inria-00073321 https://hal.inria.fr/inria-00073321 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. On the rates of convergence of Erlang’s model Christine Fricker, Philippe Robert, Danielle Tibi. No 3368 F´evrier 1998 ` THEME 2. N 0249-6399. apport de recherche.

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(51) K=e Bfp e |

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(53) r K=eBƒp e | =feB. e ]w |

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(57) &®± 5Ã^YhwRTV—°beacacb9¬_`^Td³_ h

(58) ¬_ca`a>r:VmnowV—°mnafhˆb]gKbe^now_l~¢VKuhwRTV®}Tuˆb¢gV0oˆoD= _C£ Ve£

(59) hwR•j9hioˆj9hw_lo nV0ohwRnV —°beaca`b9¬_c^Td&~¢_`¯:V0uwV0^fhˆ_cjea¤V0s;mnjkhw_cbe^. . [. Rd.

(60) . ` d 8 =?AB^C mon 8 = ` ABfp. ¬_ hˆRTbemTh

(61) je^;q mT}n}•V0uOr:bemn^n~Tjkuˆqe´. 8 =?ABB. e^g hi9j r q r m s = ` AB&u

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(70) ^  =A&u

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(72) BÅ´n_C£ Ve£  =?A&uzBDC p:8 = B=A&u

(73) z9B:C p$3• = ƒ=A&u

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(76)  + &R  (0 %'&  

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(197)    !"$# &%' (*) ,+-/.0%. hˆRnj9h”jkuˆuw_c‡eV§jeuwV!deuˆVKVK^ je^n~œhwRnVjeuwuˆ_`‡9jeacojeuwV”hˆRTVFowjeUV§—°beuYr:bkhˆR }Tuˆb¢gV,owowV0o0£ 5P—ij5dfuwV0VK^ rnjkaca®_co ^nbkhYuˆVyxyV,gÅhwV,~ —°beuAhˆRTV!}Tuˆb¢gV,owowV0o”‰§je^n~ W¢´hwRTV0^ª_ h uwV,o"_l~¢VK^•gV”hw_cU”V§_c^ hwRTV'mTuw^ ¬_ca`ar•V!de_c‡eVK^ r>q hˆRTV§oˆjkU”V&‡9jkacmTV&V ¢}:be^TV0^fhˆ_cjea`acq ~T_co"hwuˆ_`rnm¢hwV,~©—°bfur:bkhwRª}nuwb¢gV,owowV0o0£-5P—{jFdeuˆVKV0^ rnjea`aJ_coAuˆVyxyV,gÅhˆV0~ _c^ ‰ rTm¢h]^Tbkh _`^œW¢´•hwRTV0^©be^nVYbk—6hˆRTVuwV,~Frnjkacaco³_c^ ‰_loi}•jk_c^fhˆV0~©_`^5dfuwV0VK^5je^n~š_`hˆo]uwV,o"_l~¢mnjea±uˆV0ow_c~TVK^ngKV hˆ_`U”VRnjeoihwRnVoˆjkU”r VY‡9jea`mTV”jeoi—°beu³r hwRnV ^TV0¬r deuˆVKV0^5rnjea`aO_`^ W¢£ {behw_lgV hwRnjkh]¬_`hwR©hwRT_lo gbfmT}Tac_`^Td'¬V R•j1‡e,V > 8 =AB > 8 =AB³jk^n~ > 8 =AB > 8 =ABi_`—Jjk^n~šbf^Tacq_`—vhˆRTVKuˆV jeuwVYo"hw_caca

(198) owbeU”VuˆV0~5rnjkacaco³_c^ ‰e£ †nmTuwhwRTV0uwU”beuˆV ¬RTVKo^ > =AB C> =ABÅ´nhwRTV0^FhwRTV hy¬®b&}Tuˆb¢gV0oˆowV0ojkuˆV]hwRTV owjeUVjk—ÖhwV0 u AÅ£J²RTV0^FhwRTV mnuw^~¢b>V0o®^Tbkh{gKbe^;hˆje_`^8 jk^>q&uwV,~§rn8 jea`aC´>hwRnV³}Tuˆb¢gV,owo®_lo®jea`uˆV0jf~¢q&j9hV0s;mT_cac_`rTuˆ_cmToU G¢uwV0UV0UYr:VKu®hwR•j9hhwRTV o"hˆjkhw_cbe^njeuwq}Tuˆb¢gV0oˆo{_lo³r•VKhy¬V0VK^šhˆRTV0owVYhy¬®b§}Tuˆb¢gV,owowV0o0£³QRTVYhw_cU”V ~8 jk—ÖhwVKu]¬RT_lgR5^Tb'uwV,~šrnjkacaO_co }nuwV,o"V0^fh_lo{gV0u"hjk_c^Tacq!o"U&jea`acVKuhˆRnjk^hwRnVAU”j ¢_`U mTUbk—6 _`^•~¢VK}:VK^n~TVK^;h{V ¢}:be^TV0^fhˆ_cjea`acq'~T_co"hwuˆ_`rnm¢hwV,~ ujk^•~¢beU ‡9jkuˆ_ljkrTacV0o = s B®¬_`hwRF}njeuˆjeU”VhwV0u ‰e´¢RTV0^ngV M,. MH. ;. ;. ;. <-=. 8 q r a c &h l ] 8 ‡AB , <  w= + ! s $ r ac ! &9h l -<   =z9B H J pˆ<   = T ih B h , T ih u A v]wu 7. P— ¿_lo6hwRnV³~¢_loyhˆuw_crTm¢hˆ_`bf^§bk—  8s ej ^n~ }•jkujkU”VhˆVKuz±´  VY¼®jkU  o_c^TV,sfm•jkac_ hyqf´no"V0V P´Tfd _`‡fV0o ÷ 8 _c^F}njkuwhw_lgmnacjeu0´ 5 , 8. %$. . 7.  < . RnVK^ngKVe´. 8 q r ac 9& h l ‡w ! s$. =. J~¢VK^nbkhwV,o6hˆRTVit

(199) be_loˆo"bf^§~¢_co"hwuˆ_crTm¢hw_cbe^&¬_`hwR V"‰ X.   B . p]=aEpjT i 8. 8 q r a c & h l ]w ! s $.

(200) . . ,. T. ih u. ÷ T‚aLB i9h Ái—OgbemnuˆowVe´fhwRTV]aljeo"h®_c^TV0s;mnjea`_`hyq&gKje^§r•V }Tuˆb9‡eV,~”¬_ hˆR!V0a`V0UV0^;hˆjkuˆq”U”V0jk^•oK´;hwRnVihˆVKuˆU _`^!hwRTV]U”_c~T~Ta`V _lo]o"_cU”}Ta`qFhˆRTV}Tuwbfrnjkrn_`ac_ hyq'hwR•j9h]hwRTV U”j ¢_`U mTU–bk—J V ¢}•bf^TVK^;hˆ_cjea±ujk^n~¢bfU ‡1jeuw_ljkrna`V,o{V TgKVKV0~no$AÅ£ QRnV]_c^TV,sfm•jkac_ hyq7V"‰ X®¬_`aca¤r•VAm•o"V,~'_c^hwRTVowV0s;mTV0a.£JÁ³mTu}nuwbf}•b;o"_`hw_cbe^'_co}nuwb9‡fV0~¤£ W+-    QRTV³owV0gKbe^n~§acjeuwdfV0o"hJVK_cdeV0^;‡9jea`mTV³gbe^>‡fVKuˆdeV0o6hw† b pa³¬RTVK^ RUTAÀ´¢owVK V Hc/‰ IJC´¢owb jeo_c^ hˆRTV o"mT}:VKuw¡Ãguˆ_ hˆ_cg0jkaIgKjeowVibfmTu®r•bfmT^n~7V"‰0ƒX6_lo®gKbeuˆuwV,gÅhjeo®jA—°mT^nghw_cbe^bk*— AÅ£ ibkhw_lgV]RTb9¬®VK‡fVKuvhˆRnj9h®hwRn_co r:bemn^n~§df_`‡fV0o    jfo®je^!mT8 }n}•V0u®r:bemn^n~&—°bfuhˆRTV³hw_cUV]hˆb uˆV0jfgR§V0s;mT_ca`_crTuˆ_`mnU£

(201) ²œVAo"Rnjea`aIowVKV]_`^'hwRTV ^nV >h{}Tuˆbe}:bfow_ hˆ_`bf^§hwR•j9h    _cohˆRTVgbfuwuˆV0gh®bfuˆ~TVKubk—

(202) U&jkde^n_ hˆmn~¢Ve£ TAÀ ºJ^º Ž ÷ º   @E  3 a  K  #  "!K9 = ƒ=÷ BB  !K    8 ƒ=÷ BDC a  ]w.

(203)  ` 8 T ƒ. . =  7 B C 8   <-= 8 ‡AB. . <. 7. ,. =. ,. =. +. UT. /. 0. Y Y. [. ]. -3. 01%2. 43. !#. 8 0. 01%2. !#. ÔÔ ÝfŠ0ìÅì&‹Åë. H. ;. L. :. EF C. w. . LlH. . L. ]w. +.

(204) W‰I . (  & 9 ?0

(205) ^  % W & ! ?% %'. 

(206) /  †nuwbfU hˆRTV'g0jeowV 3 C ae´

(207) ¬V!Ä;^nb9¬ hˆRnj9h hwRTV nuoyhY}•jkuwhYbk—{bfmTuY}Tuˆbe}:bfow_`hw_cbe^ _coYhwuˆmT V GJhwRTV n} uwb¢gV,owo³¬RT_lgR o"hˆjeu"ho{—°uˆbeU I!^TVKV,~To j9h a`V,jeo"h³hˆRTVYhˆ_`U”V    8 hˆb'uˆV0jfgRFhwRTV”dfb;b¢~šuˆVKdf_`bf^©jkuˆbemn^n~ 3•–—°bfuV0s;mT_ca`_crTuˆ_`mTUF´¢behwRTV0uw¬_lo"VihwRTV }Tuˆb¢gV,owo_cojkh~T_co"hˆje^ngV‰e£6z{aca¤¬®V³Rnj1‡fV{hˆb }Tuˆb9‡eV³_cohˆRnj9hhwRn_co hˆ_`U”V§_loA_c^n~¢V0V0~ªowm §g_cVK^;hhwbšuˆV0jfgR V,s;mT_`ac_crTuw_cmTUF£FQRTV!~¢_`¯:V0uwV0^ngV&¬_`hwR hˆRTV!g0jeowV†3 CSa&_loAhˆRnj9h hˆRTV}Tuwb¢gKV0oˆo]g0jk^©o"hˆjeu"hAjkr:b9‡eV 3n ´±j9hA—°bfu]V TjkU”}TacVe£QRTr V”RT_`h"hˆ_`^Td'hw_cU”V”bk:— 3• o"hˆjeu"hˆ_`^nd§—°uˆbeU «_loacV0oˆoAV0jfo"q5hwb5Rnje^n~¢acV”hwRnje^ hwRTV!RT_ hwhw_c^Tdšhˆ_`U”V&—°uˆbeU Ihw7b 3•ª£F² V§owRnjea`amno"V”hˆRTV&—°bfa`acb9¬_`^nd }nuwb¢gV,~¢mTuˆV Y!gKbe^no"hwuˆmngÅh”j©gKbemT}Tac_c^Td©be—{hy¬b }Tuˆb>gKV0oˆo"V,†o =?> =AB

(208) B&uL=> =AB

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