Contributions à l'étude des files d'attente avec clients impatients

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(1)Contributions à l’étude des files d’attente avec clients impatients Pascal Moyal. To cite this version: Pascal Moyal. Contributions à l’étude des files d’attente avec clients impatients. Mathematics [math]. Télécom ParisTech, 2005. English. �pastel-00001340�. HAL Id: pastel-00001340 https://pastel.archives-ouvertes.fr/pastel-00001340 Submitted on 21 Jul 2005. 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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(50) ; f 6+PK/ &%')( 3,+"+.8 +K+5 HK/WACBOSWPA<QCBKRHJS@SJouKLHMD]BOSY_JBV]_JcOSJBKRHp>KLHJS4D]B5D8QCBrc_JBKRHJBOS • PAi>uSJSJBOS C , C , ... BHACB\SpB_IN-Bm]_>=]SIQ1Q<A>WACBPJ6ouQ =]SJB_IHm]K PA<QCBKRHqD]BAi>YPAi>uSpSJB >Nu>KRHTm]KPA<QCBKLH\D]BAi>4PAi>uSJSJB C SJQ i < j : C [>K6SdHJoum]HJBOSdACBOSdD8QCSJPQ<V]A<Q<K6BOS D]BqSJB_INQCPOBOSdB]POBV]HJcOBq/d@BH =-m]KPA<QCBKLH lRm]Q6BOSIH D]c AI^D6>K6SACBSISJHJUBhVrBm]HN-ouQ<_BKRHI_JB_ >V]_JOS}A<m]Q8m]K@PA<QCBKRHdlLm]Q8A<m]Q]BOSIHV]_IQCou_IQ<Hp>Q<_JB#: ^KD8Q<_p> >ACou_JSlLmF m]K SJB_IN-Bm]_XSJB_IHD]BWs> OouK 3,+)H)"23 5+0%'kSIQ/Q<AVrBm]H^Q<KRHJB_I_JouU4V]_JB ACB\SpB_INRQCPOBDF m]K+PA<QCBKLHqVroum]_nPOouU4UBK6POB_TMBKSJB_INRQ<_m]K¬>m]HI_JBlLm]QrA<m]QBOSIHV]_IQCou_IQ ` Hp>Q<_JB= !,8&!<Q3,+)H)"23 5+0%'MSIQ<K6ouK: =BOS4ACouQCS4D]B V]_Jouj9>j]Q<A<Q<HJcOS5BH4ACBOS@Nu>ACBm]_JS4D]BOS4V9>_p>UHI_JBOS5lLm]QnP>_p>uPHJc_IQCSJBKRH BKLHIQC_JBUBKLH^m]K6B\?6ACBDFG>HIHJBKLHJBESJouKRHn_JcOSIm]UcOSnV9>_Ai> !,8&2O"! (*C9 +) I

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(52) 9 7 (5 ; = A!#" A/S/s/s + B − X $ $; 7 . ;. • A. E 9 E A<5. . E 9 7 D5%. * ;$ 7 E 9EFAC5 E 97 D5+ = 9 7 9C;E 7. E 5 (7 ) ;. • S • s. & . 9C; 9 ;'5 7 7:5. . . 9C;. ; 9<;';'5%'( 7. σn , n ∈ N ∗. . E. Un , n ∈ N ∗. ). , 7 -; ' +5 7 = 7 . $ $; 7 E / AC=10 ; 3 2 ' (D E E = 5 ( 4 65 5 7 -; ' ;. 7E 9 9C58E E 72 ' (D E E = 5 8 4 5 5 9 D 0 ;D : E 5 (7 = 9>=C58= 9CE D 58E ? D@A A 5 ;B5 ). • B. • X. E

(53) D 5 (D. ;. $ $; 7 . EF9. ;. D;'7. 7. ). 2 A<=10 ; < . 587 5 E 5 < 7 -; ' 5 (C. . KV9>_IHIQCPm]A<QCB_>=ACBOShD]Bm8V]_JBU4QCB_JS V9>_p>UHI_JBOS A BH S P>_p>uPHJc_IQCSp>KLHACBOSdACouQCS D]BOS SIm]Q<HJBOSnD]BOSQ<KLHJB_p>_I_IQ<N-cOBOS\BHD]BOSHJBU4V6SnD]BSJB_INQCPOBOSV]_JBK]K6BKLH^BOSJSJBKLHIQCBA<ACBUBKRHqACBOS Nu>ACBm]_JSqSJm]Q<Nu>KRHJBOSY "I*! 1+N& SIQAi>4SIm]Q<HJBXBOSIHqSJHp>HIQCouK]K9>Q<_JBYB_I¤-oD8QClLm6? B= ° • G ¯ " "! 1+N&+0!#I 3 "!#I*!. SIQBA<ACBXBOSIHqQb: Qb:D = ° • GI ¯ 2O"2 8 +ED'K/K/ ° SJQ1Ai>WSIm]Q<HJBBOSIHQb: Qb:D:.D]B\ACouQB8VouK6BKRHIQCBA<ACB ¯ SIQ A = M =ACB • M ¯ V]_JoPOBOSJSIm6SqD]BOSqBKRHI_JcOBOSTBOSIHqD]ouK6PEm]KV]_JoRPOBOSpSIm6SqD]B }ouQCSJSJouK ° = * 1+"2 +0!<+P/"5+( ° SIQAi>4SJm]Q<HJBXBOSIHqD]cHJB_IU4Q<K]QCSIHJBYBH^POouK6SIHp>KLHJB#: • D ¯.

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(70) . . ) 0 «/*L3~«/%$ H3. 9 5 7 5 E 5 7 ;-' 5 7 +5 7.53E 7 -; ' D; 7 ;- 9 D 5 7 9 3E ;$ 7 5 D E 5 7 9 7 E E D7 C A D;- C. 2 A EDF p5 E 5 A E E 9<5. C 9 5 7 5 E 5 7 5+ h) 0 2 d7] -93L2 )+3L7]|t$t"3 0 $&z2$5 ;$ = (5%' 2 LDF 5 7 5 E 7 ;-' D; 7 ;- 9 D ?BAC587 D 58E E D 7 ;-' 5+ E E 5 A @E E 9<5 ;$ 7 5 D E @5 7 E E D7 EFA C C 9 5 7 5 E 5 7 5% d) 0 «}z*R3R~«"%$ H13 2 LDF p537 5E 7 7 ;- 9 D 5 7 9 E E 5 A E E 9<5%;$ 7 5 D E 5 7 9 7 E E D7:E A C. A AC= 7 9 ;-' D ;. . ouU4UB^K6oum6ShAbFG>N-ouK6SV]_JcOPQCSJc^D6>K6S Ai>X_JBUk>_JlRm6BqV]_JcOPOcOD]BKRHJB=8K6oum6ShKFcHIm6D8QCB_JouK6S BK D]cHp>Q<A^lRm6B+ACBOSkD8QCSJPQ<V]A<Q<K6BOS¦D]BSJB_INQCPOBOSK6ouK8`bV]_JcOBU4V]HIQ<N-BOS =qK6ouHJcOBOSQCPQ EDF BH LDF : 5 oum6S_JBNRQCBK6D8_JouK6SEAi>_I¤-BUBKLHYV9>_Ai> SIm]Q<HJB@SIm]_ACBOSV]_JouV]_IQCcHJcOSED]B4POBOS D8QCSJPQ<V]A<Q<K6BOSD]B\SJB_INQCPOBTBHSIm]_hACBOSV]_Jouj]ACUBOSlLm6B^Vro#SJBqAbFcHIm6D]B\D]BOSd?6ACBOSDFG>HIHJBKRHJB POou_I_JBOSIVrouK6D6>KLHJBOS%: . ( ) $+ . ) ' , & )).) , +%.&' ). [c?6K]QCSJSJouK6SEQCPQ}ACBOSXD8QCBc_JBKLHJBOSYNu>_IQi>j]ACBOSW>ACc>HJouQ<_JBOSEBHXV]_JoRPOBOSpSIm6SXSIHJoPJ9>uSIHIQClRm6BOS P>_p>uPHJc_IQCSp>KRH^m]K6B\?6ACBYDFG>HIHJBKLHJBW>N-BOPYPA<QCBKRHJSqQ<U4V9>HIQCBKLHJS%: AC B *283#/ 9 "!. $3,+38K3 8'/>H >mPA<QCBKLH BOSIHEACBHJBU4V6SMlRmFG>m]_p>¬ • >HIHJBK6D8_JB C >ON#>KLH4DF ;HI_JB SJB_INRWQb: B5HJBU4V6SWCVrBm]H ;HI_JB5SJm]Vc_IQCBm]_ D = >m6lLm6BA"P>uS C BOSIHqVrB_JD8m ¯ Q<AKFG>MV9>uSTV9>uSnV]m¬>HIHJBQ<K6D8_JBEACBESpB_IN-Bm]_ ° = ACB "23./ "!. T DF m]K PA<QCBKLH C BOSIH\ACBWHJBU4V6SX"B BrBOPHIQ<N-BUBKRHXV9>uSJSJc • V9>_ C D6>K6STACBj]4m BrB_Y n. n. n. n. n. a n. n. n. Tna = Wn 1{Wn <Dn } + Dn 1{Wn ≥Dn } ,. ACB1 *283#/J />H 8 + T D8mPA<QCBKLH C BOSIHACBHJBU4V6SdlLm6BV9>uSJSpBnB"BrBOPHIQ<N-BUBKRH D6>K6SnACBXSJSIHJUB ¯ j]m4BB_ SpB_INRQCPOB ° C . •. n. s n. !. n. Tns = (Wn + σn )1{Wn <Dn } + Dn 1{Wn ≥Dn } .. }oum]_YHJoum]HYSJSIHJUBkD]BM?6ACBOSYDFG>HIHJBKLHJB¦ VrB_IHJB=Ai> 3,+ 8 #N #"+0,+5 HI13 1+* π D8m PA<QCBKRH BOSIHTAi>4V]_Jouj9>j]Q<A<Q<HJc@lLm6BWPOBYPA<QCBKRH\SJouQ<H^VB_JD8m: >_B8BU4V]ACB=1Voum]_^m]K6B ?6ACBDFG>CHIHJBKRHJBkPAi>uSJSJQClLm6B¦j]m4BB_?6K]Q M/M/1/1 + B =POBHIHJB4V]_Jouj9>j]Q<A<Q<HJckSJB_p>Q<H D]ouK]K6cOB\V9>_Ai>EV]_Jouj9>j]Q<A<Q<HJcXlRmF Q<Ak>Q<H 1 + B PA<QCBKLHJSnD6>K6SACB\SISJHJUBOSYAbFBKLHI_JcOB D]B C :q[X>K6S\ACBYP>uS\DF m]K6BY?6ACBWDFG>HIHJBKRHJB@>N-BOPWPA<QCBKRHJS\Q<U4V9>HIQCBKLHJ>S =rAi>kPOouK6D8Q<HIQCouK . k. k. k.

(71) %; D]B^VrB_IHJBXBOSIHnV]A<m6SnPOouU4V]ACB]B BOSIHnD]ouK]K6cOBXV9>_nAi>MV]_Jouj9>j]Q<A<Q<HJcElRm6B C SJBN-ouQ<H V]_JouVo#SJB_nm]K¬HJBU4V6STDFG>HIHJBKLHJBWπSIm]Vrc_IQCBm]_T4SJouKD]cAi>Qh BOSIHnVrB_JD8m ] = P [W > D ] . π = P [C ¯ 8:<; ° [\c?6K]QCSpSJouK6SnUk>Q<KLHJBK9>KRHÂCBOS 3,+ 8 ( K/ / I/ I 3 1+-8 +K2?&!#(KK/5 ACBO3,+ 8 ( K/ / I/ ( 8&! "IK/"5+ 8&! BOSIH@ACB¦V]_JoRPOBOSJSJm6S4POouU4V]Hp>KLHACB • K6ouUYj]_JBWD]BXPA<QCBKLHJS\D6>K6SnACBESI8(XSIHJ,UtBX≥40)Pp9>ulLm6BXQ<K6SJHp>KLH t = ACB 3,+38>( K/K/ /8 *!#( J # &%1+ (Q , t ≥ 0) BOSJH"ACB.V]_JoPOBOSJSIm6S/lLm]QLPOouU4V]HJB • ACBK6ouUYj]_JBWD]BEPA<QCBKLHJS^D6>K6SqACBj]4m BB_^MAbF Q<K6SIHp>KRH t :.©^Q<K6SI-Q =8Vroum]_nHJoum]H t k. k. k. k. k. k. . . t. . . . . . t. Qt = (Xt − s)+ ,. •. Voum]_.HJoum]H n ≥ 1 =RouK4K6ouHJBnV9>_ T˜ =-ACB n`bQCUBqQ<K6SIHp>KLHhD]BnSpou_IHIQCBnD8mkSJB_IN-Bm]_>= P#FBOSIHa` z` D8Q<_JB =1AbF Q<K6SIHp>KLH\D]BXAi> n`bQCUBX?6KD]BESJB_INQCPOB#: =B 3,+ 8 ( K/ / I/?IK/6! / BOSIHd>ACou_JS}ACBV]_JoRPOBOSpSIm6S}VouK6PHIm6BA ^HI_p> AaBOPHJouQ<_JBOSV:S%: I / 1+"%>+( K/ PuD8Ai¤ ° POouU4(SV]Hp>, KRt H≥ACBn0)K6ouUYj]_JB\D]BTSJB_INQCPOBOSdHJB_IU4Q<K6cOS Aam6SJlL¯ mFG t :©Tm]HI_JBUBKRH D8Q<H>=]Vroum]_nHJoum]H n. . t. t. St =. X. 1Tñ ≤t ,. Voum]_nHJoum]H n ≥ 1 =]ouK+K6ouHJBV9>_ σ˜ =Ai>@D8m]_JcOBXD8m n`bQCUBXSJB_INQCPOB+N"!

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(73) I"/ IH 34 +"0/WD8mSISJHJUB (D , t ≥ 0) BOSIHqACBXV]_JoPOBOSJSIm6SqPOouU4V8` Hp>KLHWACBOSXQ<K6SIHp>KRHJSYD]BSJou_IHIQCBOSWD8mSI8SIHJUB ¯ SJB_INQCPOBOSYoumVrB_IHJBOS ° Aam6SJlRmFG t : ©^m]HI_JBUBKLH^D8Q<H>=6Voum]_nHJoum]H t . t. Dt := St + Pt .. RQTouK_JcOou_JD]ouK]K6BACBOSkD]cAi>QCS5_JcOSIQCD8m6BACSkD]BOS PA<QCBKRHJS¦V]_JcOSJBKLHJS¦D6>K6SACB¬j]m4BrB_ t V9>_Mou_JD8_JBP_JouQCSJSp>KRH ¯ Vroum]_YACBOSESJB_INQ<_YBK QEDF =}D]ouK6P ° =}ouKK6ouHJBPOBOSWD]cAi>QCS _JcOSIQCD8m6BACS t. R1 (t) < R2 (t) < .... < RQt (t).

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