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

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 sci- entific research documents, whether they are pub- lished 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.

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. �NNT : �. �pastel-00001340�

./ )01 2435%6 0

798;:<8;='>@?BAC8;=

"!#%$& '

( )+*-,./$0%$&1/243-5/678%$&"2 9

8:<; =">@?6ACBEDFG>HIHJBKLHJBM>ON-BOPEPA<QCBKRHJSTQ<U4V9>HIQCBKRHJSW:E:X:Y:X:E:X:E:E:X:E:X:Y:Z;#;

8: [\QCSJPQ<V]A<Q<K6BOS^D]BXSJB_INRQCPOBEHJBU4V6Sa`b_JcOBAd:X:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:Z;e

8:Gf gh>_IQi>j]ACBOST_JBUk>_JlLm9>j]ACBOS\BHnV]_JoRPOBOSpSIm6SqD]BXVrB_astou_IUk>K6POBOSv:X:E:X:Y:Z;w

x yEz1{}|&*~€78%$&‚ "3ƒ3-„*L2% }|0782„3-…"$&2O762 (r

f8:<; =">@?6ACBE†/‡ˆ†d‰‹Š8_JcOSIm]A<Hp>HJS^BHnU„cHIŒ6oRD]BOS^BŽQCSIHp>KRHJS:E:X:E:E:X:E:X:Y:

f8: ‡ˆouU4V9>_p>QCSJouKD]BOSqD8QCSJPQ<V]A<Q<K6BOS^D]BESJB_INQCPOB:E:X:Y:X:E:X:E:E:X:E:X:Y: #e

f8:Gf =">@?6ACBE‘d[’†vŠ]V]_Jouj]ACcUk>HIQClLm6BYV]_Jouj9>j]Q<A<QCSIHJB“:X:Y:X:E:X:E:E:X:E:X:Y: #w

f8:Gf8:<; [’BOSJP_IQ<V]HIQCouKƒD8m”SJ•SIHJ–U„B :X:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y: #w

f8:Gf8: ‡—>ACPm]A"KRm]U„c_IQClLm6BWD]BOSnVrB_astou_IUk>K6POBOS^DF˜‘d[’† :E:X:E:X:Y: -™

f8:Gf8:Gf =‚BED]BOSpP_IQ<V]HJBm]_\šMN#>ACBm]_JSnU„BOSIm]_JBOSqVrouK6PHIm6BA<ACBOS4:E:X:E:X:Y: #›

œ 7]{$t|&$0* x9x

ž

:<; Ÿ KLHI_JoD8m6PHIQCouK¡:Y:X:E:E:X:E:X:E:X:Y:E:X:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:¢f#f

ž : £^c¤-cK6c_p>HIQ<NRQ<HJcWD6>K6SnACBXP>uS

GI/GI/1/1 + GI

:X:E:X:Y:E:X:E:X:E: f

ž

ž :8:<; ‡ˆouU4V9>_p>QCSJouK‹>ON-BOPEAi>W?6ACB¦¥MD]cAi>Q‚V]m]_\§ :E:X:E:E:X:E:X:Y:¢f ž

ž :8: £TcOSJm]A<Hp>HTD]B’_Jc¤-cK6c_p>HIQ<NRQ<HJc:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:¢f#e

ž :Gf ‰RHp>HIQCouK]K9>_IQ<HJcWVoum]_nAi>W?6ACB

G/G/1/1 + G

:X:Y:X:E:X:E:E:X:E:X:Y:¢f#w

ž :Gf8:<; ¨^K”_JcOSIm]A<Hp>HTD]BXSIHp>HIQCouK]K9>_IQ<HJc:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:¢f#w

ž :Gf8: ©^V]V]A<QCP>HIQCouKšMAi>M?6ACB

G/G/1/1 + G

:Y:X:E:X:E:E:X:E:X:Y: ž#ž

ª yEz1{/7]{$t|t$i*ƒ/3¬«d3Lz3 ª6x

e8:<; [’Bm8+BK6P>uD8_JBU„BKLHJS+:X:E:X:E:X:Y:E:X:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:¢e ž

e8:<;#:<; ‘.K6P>uD8_JBU„BKRHXšWs­ou_IHJBXPJŒ9>_I¤-B®:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:¢e ž

e8:<;#: ‡—>uS

GI/M/1/1 + D − F IF O

^¯

EDF

^° :E:X:E:E:X:E:X:Y:¢e#w e8: ‡ˆouU4V9>_p>QCSJouKBKLHI_JBWD8QCSJPQ<V]A<Q<K6BOSTD]BXSJB_INQCPOB±:X:Y:X:E:X:E:E:X:E:X:Y:¢w#²

e8:8:<; ‡—>uSq¤-cK6c_p>A

G/M/1/1 + G

:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:¢w#² e8:8: ‡—>uS

M/M/1/1 + G

:X:Y:E:X:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:¢w#³

$t~$03L2. }$0}3L24}3L2k«/!-3L2%2% "24}3+76"2%«d9z '6ª

w8:<; ._JcA<Q<U4Q<K9>Q<_JBOS :E:Y:X:E:X:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:™#™

w8:<;#:<; =‚BOSnV]_JoPOBOSJSIm6STšMN#>ACBm]_JSqU„BOSJm]_JBOS”:E:E:X:E:X:Y:X:E:E:X:E:X:™#™

w8:<;#: =dF˜cOlRm9>HIQCouKƒD]B’HI_p>K6SIVrou_IHqQ<KLHJc¤#_p>ACBWD6>K6S

D ∞,S 0

:E:X:E:X: ›

w8: ._JoPOBOSJSIm6SqD]BOSqV]_Jo?6ACSTD]B’Ai>M?6ACBX‘d[’† :E:X:E:E:X:E:X:Y:X:E:E:X:E:X: ›#e

w8:Gf =‚BOSnV]_JoPOBOSJSIm6STD]B’HI_p>K6SIVou_IHT_JBK6ou_IUk>A<QCSJcOS :E:X:E:X:Y:X:E:E:X:E:X: ³#²

w8:ž =‚ouQCSqD]BOSq¤#_p>K6D]SqK6ouUYj]_JBOS :X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X: ³#w

w8:ž :<; ._JBU4QC–_JBE_JBK6ou_IUk>A<QCS >HIQCouK :X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X: ³#w

w8:ž : [’Bm8QC–U„BX_JBK6ou_IUk>A<QCS >HIQCouK :X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦;²#w

w8:Ge —Œ6cOou_J–U„BEPOBKRHI_p>A"A<Q<U4Q<HJB’s­ouK6PHIQCouK]K6BAn:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦;#;#;

w8:Ge8:<; £TcOSIm]A<Hp>HnV]_JcA<Q<U4Q<K9>Q<_JB‹:E:X:Y:X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦;#;

w8:Ge8: ._JBU4QC–_JBEK6ou_IUk>A<QCS >HIQCouK :Y:X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦;#;w

w8:Ge8:Gf [’Bm8QC–U„BXK6ou_IUk>A<QCS >HIQCouK :Y:X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦; -™

w8:Gw ©^V]V]A<QCP>HIQCouKƒ>m8 ?6ACBOSq‘d[’†:E:E:X:Y:X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦; #›

w8:Gw8:<; †}Q<ACB

M/M/1/1 + GI − EDF

š@PpŒ9>_I¤-BYlRm6BACPOouK6lLm6BE:Y:X:5; #³ w8:Gw8: †}Q<ACB

M/M/1/1 + GI − EDF

š@PpŒ9>_I¤-BEm]K]Q<HJc :E:E:X:Y:X:5;f ž w8:Gw8:Gf †}Q<ACB

M/M/s/s + GI − EDF

š4PpŒ9>_I¤-BElLm6BACPOouK6lRm6Bk:Y:X:5;f#w w8:Gw8:

ž

†}Q<ACB

M/M/s/s + GI − EDF

š4PpŒ9>_I¤-BXm]K]Q<HJc :E:E:X:Y:X:5;f-™

w8:™ ©^V]V]A<QCP>HIQCouKƒ>m8 ?6ACBOSq†Ÿ † "!4†/‡ˆ†d‰:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦;f-™

w8:G› ©^V]V]A<QCP>HIQCouKƒš@Ai>W?6ACB$#%!'&’Ÿ(!

∞

:X:E:X:Y:X:E:E:X:E:X:E:X:Y:E:X:E:X:¦;f#³

w8:G³ ‡ˆouU4U„BKLHp>Q<_JBOS@:E:Y:X:E:X:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦; ž ²

w8:G³8:<; &’cK6c_p>A<QCSp>HIQCouK‹>m85s­ouK6PHIQCouK6SqHJBOSIHJSqQ<K6D8QCP>HI_IQCPOBOS€:E:X:E:X:¦;

ž ²

w8:G³8: "BU4V6S^D]BXNRQCD6>¤-B :X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦; ž#ž

' )Y"!R|& /2%$&1 Rœrª

*

)+3 ‚… *L2% }|0782„}3”,+/*L9u$&3¬3R.-1}$&‚ /3 Rœ0/

1 * «}«}|t$&!-78%$&1/24/3|&73246587.7:9<;'=%>@?BAC;'=%DEF9 8ª

G :<; †6ou_IUWm]ACBOSY:X:E:X:E:Y:X:E:X:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:X:E:E:X:E:X:¦;e8;

G :¢©^V]V]A<QCP>HIQCouK6S^>m8¦?6ACBOS^>ON-BOPYPA<QCBKLHJSqQ<U4V9>HIQCBKRHJST:E:X:Y:X:E:E:X:E:X:¦;e

) )Y0H13R.-13R/!-3¬3 |&1$ 8ª1ª

e8:<;

G

m6SI•¦VB_IQCoD Vroum]_T‘d[’†BHT†Ÿ † :X:E:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:¢w-™

w8:<; [\•K9>U4QClLm6BED8m¬V]_JoRPOBOSJSJm6SnD]BOSnV]_Jo?6ACS„:X:E:E:X:Y:X:E:X:E:E:X:E:X:Y:¢›-™

w

[’>K6SnPOB\HI_p>Nu>Q<AK6oum6SncHIm6D8QCouK6SnACB’U„oD]–ACB\V]_Jouj9>j]Q<A<QCSIHJBED]B\Ai> "!# $&%')(

(*,+-"!.0/1+02345+-"!.0/Ž:

=‚BˆU„oRD]–ACB—PAi>uSJSJQClLm6BˆD]BˆAi>66789 "!. nPOouK6SJQCSIHJB BK@m]K4SI•8SIHJ–U„BˆD6>K6SACBOlLm6BA

D]BOSqSJB_IN-Bm]_JSqSJouKRHqSJoum]U4QCSTš@m]K:6m8+D]B’_JBOlRm<;HJBOSqlLm‚FQ<ACSTD]ouQ<N-BKLHTHI_p>Q<HJB_:hŸ A">Mm]K

¤#_p>K6DƒK6ouUWj]_JB„DFG>V]V]A<QCP>HIQCouK6SYD6>K6S’ACBOS’_JcOSpB>m8D]BMHJcACcOPOouU4UYm]K]QCP>HIQCouK6S>=}D6>K6S

ACBOS4_JcOSJB>m8Q<K8stou_IUk>HIQClRm6BOS>=ˆACBOSk>K9>A<•8SJBOS„D]B+HI_p>z?9P+_Joum]HIQCB_5oum U?;U„B”D6>K6S„D]B

¥¦NR_p>QCBOS4§?6ACBOSWDFG>HIHJBKLHJB=d>mUk>¤L>uSIQ<K@=.>mPQ<K6cUk>8:<:<:”Ÿ AdVrB_IU„BHMD]B4_JcVrouK6D8_JB

š5D]BOSYlLm6BOSIHIQCouK6SYD]B@HJBU4V6SED]B@HI_p>Q<HJBU„BKLH>=}D]B„SIHI_Im6PHIm]_p>HIQCouKD]B@_JcOSpB>m8‹oum€D]B

D8Q<U„BK6SIQCouK]K6BU„BKLH:

=‚BED]cN-BACouV]VrBU„BKLHXD8m”HJBU4V6Sa`b_JcOBA}BOSIH^>m&AIoum]_JDFŒLm]Q}m]K6BXV]_JcOoPOPm]V9>HIQCouKƒUk>z`

AaBm]_JB@D6>K6S’ACBOS’_JcOSJB>m8‹D]BWHJcACcOPOouU4UWm]K]QCP>HIQCouK‚:4[X>K6S’ACBMPOouKRHJBŽHJB@UWm]A<HIQ<U„cOD8Qi>

K6ouHp>U4U„BKLH>=ouK„PJŒ6B_JPJŒ6B\š\HI_p>K6SIU„BHIHI_JBnACBOS.D]ouK]K6cOBOS BK4m]K4HJBU4V6S.HI_J–OS.j]_JBŽsI:}[’BOS

:6m8 D]BWD]ouK]K6cOBOSTD]BXH •RVrBOSqV9>_as­ouQCSqHI_J–OSqD8QCBc_JBKLHJS\D]ouQ<N-BKLH’POouŒ9>j]Q<HJB_>=6AbFQ<KRHJc¤#_IQ<HJc

D]BnPOBOSD:6m8@D]ouQ<HE;HI_JB—_JBOSIVBOPHJcOBTBH.ACBOS._JcOSJB>m84lLm]Q]ACBOS.HI_p>K6SJU„BHIHJBKLH D]ouQ<N-BKRHE;HI_JB

š¦Ai>5stouQCSF:9BŽ8Q<j]ACBOSEBHX_p>V]QCD]BOS:/oum]HJB„D]ouK]K6cOB„D]ouQ<HW>ACou_JSY>N-ouQ<_Em]K6B¬¥¦D8m]_JcOB„D]B

NRQCB„§¬HI_J–OS’A<Q<U4Q<HJcOB4D6>K6S’ACB4SI•8SIHJ–U„B=rV]m]QCSJlRm6BMSpouKHI_p>Q<HJBU„BKRHYD]ouQ<H$;HI_JB@Q<K6SIHp>K8`

Hp>K6c#:

/oum]_ƒ_JBK6D8_JB€POouU4V]HJBD]B€Ai>POouKLHI_p>Q<KLHJB¡D8m HJBU4V6Sa`b_JcOBAMD6>K6S”ACBOS+_JcOSpB>m8G=

ouK D]ouQ<HƒBK]_IQCPpŒ]Q<_‹ACB€U„oD]–ACBPAi>uSJSJQClLm6BD]B€Ai>?6ACBDFG>HIHJBKLHJB¡V9>_m]K K6oum]N-B>m

V9>_p>U„–HI_JB=-ACB$IH"7&+}D]BOS._JBOlLm<;HJBOShlLm]Q]BKLHI_JBKRH D6>K6S.Ai>^?6ACB#:^K„POouK6SIQCD]–_JBTlLm‚Fm]K6B

_JBOlLm<;HJB‹BOSIH VrB_JD8m6BD]–OS lRm‚F˜BA<ACB D]cV9>uSJSJB‹POBD]cAi>QESp>K6S+>ON-ouQ<_”POouU4U„BK6POc SJouK

HI_p>Q<HJBU„BKLH:q‘.KƒHJB_IU„BOSTQCD]c>A<QCSJHJBOS>=9ouKƒV9>_IACBYD]BJ67K/LM *!. &%'N(O(*,+-"!.0/L+P2Q

345+-"!.0/Ž: =‚BOS5PA<QCBKRHJS ouKLH¦m]K6B”V9>HIQCBK6POBVoum]_¦BKLHI_JB_+BK SpB_INRQCPOB=q>m D]BAišD]B

Ai>ulLm6BA<ACB4Q<ACSXPpŒ6ouQCSIQCSJSJBKRHWD]B4lRm]Q<HIHJB_EAi>5?6ACB#:5‡ˆB4U„oD]–ACB„> Q<K]Q<HIQi>ACBU„BKRHWcHJc„POouK8`

SIHI_Im]Q<H4Voum]_5D]cOP_IQ<_JB+ACBOS4_JcOSJB>m8 HJcACcV]Œ6ouK]QClRm6BOSko&R ACBOS„PA<QCBKRHJS„U4QCS„BK >HIHJBKLHJB

_p>uPOP_JoRPpŒ9>QCBKLHk>m¡joum]H@DFm]K POB_IHp>Q<K¡HJBU4V6S@SIQˆACBm]_4>V]VBAˆK‚F˜cHp>Q<H@V9>uSWV]_IQCS@BK

POouU4V]HJB#: ^KPOouU4V]_JBK6D€lRm‚FQ<AhBOSIHXVrB_IHIQ<K6BKLH>=/V]A<m6SXAi>_I¤-BU„BKLH>=}Vroum]_ED]cOP_IQ<_JB4ACBOS

_JcOSJB>m8 D8Q<HJSM¥MHJBU4V6Sa`b_JcOBA}§-:

^K_JBV]_JcOSpBKLHJB m6SIm6BA<ACBU„BKRH4ACB5POouU4Vrou_IHJBU„BKLHkD]BOSW?6ACBOS@DFG>HIHJBKLHJB V9>_@D]BOS

V]_JoRPOBOSJSJm6S SIHJoPJŒ9>uSJHIQClLm6BOS—lRm]Q9ACBOS—D]cOP_IQ<N-BKLHqD]B^Uk>K]QC–_JB’BŽŒ9>m6SIHIQ<N-B#: ^K PJŒ6B_JPpŒ6B

›

>ACou_JSEš Sp>N-ouQ<_Yš¦lLm6BA<ACB@POouK6D8Q<HIQCouKPOBOS’V]_JoPOBOSJSIm6SX>HIHJBQ<¤#K6BKLHWACBM_Jc¤#Q<U„B4SIHp>HIQCouK8`

K9>Q<_JBXBH—ACBm]_JSqPOouU4Vrou_IHJBU„BKLHJS\>uSI•RU4V]HJouHIQClRm6BOS—VrB_IU„BHIHJBKLHTSJoum]N-BKRHqD]BXP>ACPm]ACB_

ACBOS5¥5VrB_as­ou_IUk>K6POBOSM§D]BOSESI•8SIHJ–U„BOS

¯

HJBU4V6SYD]B4_JcVouK6SpB=V]_Jouj9>j]Q<A<Q<HJckD]B4VrB_IHJB

Vroum]_TD]BOS—?6ACBOSTš@P>V9>uPQ<HJcYA<Q<U4Q<HJcOB#:<:<:

° :

[X>K6S¬ACB P>uS+V9>_IHIQCPm]A<QCB_D]B‹Ai>?6ACB€DFG>HIHJBKLHJB¡>ON-BOP€PA<QCBKRHJS¬Q<U4V9>HIQCBKRHJS>=XACB

U„oD]–ACB V]_Jouj9>j]Q<A<QCSIHJBPOouU4Vou_IHJB m]K V9>_p>U„–HI_JB€SIm]V]V]ACcU„BKLHp>Q<_JB>ON-BOPACB D]cAi>Q

D]BOS^PA<QCBKLHJS:n=">„V]_JBU4QC–_JBYlRm6BOSIHIQCouKƒSJm]_TPOBEK6oum]N-B>mƒU„oRD]–ACBWBOSIHTD]BYS >ON-ouQ<_^SJQ‚ACBOS

V]_JoPOBOSJSIm6S4lRm]QnD]cOP_IQ<N-BKLH„Œ9>j]Q<HIm6BA<ACBU„BKRH5ACBOSM?6ACBOS4DFG>HIHJBKRHJB”PAi>uSJSIQClRm6BOS@•SJouKRH

>uD6>V]HJcOS@BH@SIQ<K6ouK@=D]BkACBOSW_JBV]_JcOSJBKLHJB_4D8QCBc_JBU4U„BKLH:%^K¡PpŒ6B_JPJŒ6B_p>‹BK6SIm]Q<HJB š

_JcVrouK6D8_JBY>m8”lLm6BOSJHIQCouK6SqSIm]Q<N#>KLHJBOSYŠ

•

27]{}$t|t$0* Š/šTlLm6BA<ACB POouK6D8Q<HIQCouKMSJm]_"SJBOS‚V9>_p>U„–HI_JBOS}Ai>n?6ACBhDFG>HIHJBKLHJB—>HIHJBQ<KLHa`

BA<ACBXm]K”_Jc¤#Q<U„BESIHp>HIQCouK]K9>Q<_JB

•

!L76|0!R /|M}3L2 «d3R 9u~€76/!-3L2 ŠkPOouU4U„BKRH„P>ACPm]ACB_4ACBOSMD8QCBc_JBKLHJS„Q<K6D8QCP>z`

HJBm]_JSˆD]BOSˆVrB_astou_IUk>K6POBOS—D8m+SI•8SIHJ–U„B=8>m5V]_JBU4QCB_—_p>K]¤@D]BOSJlLm6BACS =LAi>YV]_Jouj8`

>j]Q<A<Q<HJcYD]B’VrB_IHJBEDFm]K¬PA<QCBKLH^BK+stouK6PHIQCouKƒD]BOSnV9>_p>U„–HI_JBOS^D8m”SJ•SIHJ–U„B

•

$t~€3"2%$&//3R~3R ŠPOouU4U„BKLH U4QCBm8kD8Q<U„BK6SIQCouK]K6B_hAi>\?6ACB

¯

K6ouUYj]_JB\D]B

SJB_IN-Bm]_JS>=]NQ<HJBOSJSJBXDF˜BŽ]cOPm]HIQCouK

°

Vroum]_qD8Q<U4Q<KLm6B_\POBHIHJBXVB_IHJB

•

^{! +/1$0… /3¡|07} $&2!L$&«|&$t/3 /323L.H"$&!-3 Š’SIm]Q<N#>KLHElRm6BA<ACB4D8QCSJPQ<V]A<Q<K6B4SJB_INQ<_

ACBOSEPA<QCBKLHJSWVoum]_Y>U„cA<QCou_JB_WPOBOSEVrB_astou_IUk>K6POBOSEBHWBK V9>_IHIQCPm]A<QCB_>="Vroum]_YBK

VB_JD8_JBXACBEU„ouQ<K6SnVo#SpSIQ<j]ACB

! #"$"%&('*)

[X>K6SˆAi>WA<Q<HIHJc_p>HIm]_JB=8ACBOS _JcOSJm]A<Hp>HJSnSJouKLHnD]cAJšEK6ouUWj]_JBm8 D6>K6S—ACBOS—P>uS—o&R+ouK >WV]m

_p>V]V]_JoPJŒ6B_nACB\U„oD]–ACB\D]B\Ai>Y?6ACB\DFG>HIHJBKLHJBY>N-BOP\PA<QCBKRHJSnQ<U4V9>HIQCBKLHJS^šWPOBA<m]QD]B\Ai>

?6ACBXDFG>HIHJBKRHJBXPAi>uSJSIQClRm6B

¯S >K6SnD]cAi>QCS

°

:.‘dSJSJBKRHIQCBA<ACBU„BKLH>=POBAi>WK‚F˜BOSIH—Vo#SJSJQ<j]ACB\lRm6B

ACou_JSJlRm6BˆAi>\D8QCSJPQ<V]A<Q<K6B—D]BˆSpB_INRQCPOB—BOSIHM3,+)"2+-+"!..+)H0/ 3,+)"2+-+ / 1+"%>+

¯

†/‡ˆ†d‰

°

:'#‹>QCS

m]K6B^¤#_p>K6D]B^lRm9>KLHIQ<HJc’D]B^U„oRD]–ACBOS =LV9>_IU4QACBOSJlRm6BACShHJoum6S—POBm8ko&R¦Ai>ED8QCSJPQ<V]A<Q<K6B’D]B

SJB_INQCPOB+D]cVrBK6D¡D8m D]cAi>QqD]BOS4PA<QCBKRHJS>= K6B VBm]H ;HI_JB+cHIm6D8QCc V9>_„POBOS@U„cHIŒ6oD]BOS

PAi>uSJSJQClLm6BOS:[X>K6SˆPOBOS P>uS>=RACBOS V]_JoRPOBOSJSJm6S D]cOP_IQ<N#>KLHˆŒ9>j]Q<HIm6BA<ACBU„BKLHnACBOSˆSI•8SIHJ–U„BOS

¯

POouK]¤-BOSIHIQCouK¡D]B4Ai>5?6ACB„DFG>HIHJBKRHJB=}PpŒ9>_I¤-BkD]B@HI_p>Nu>Q<AˆD]BOSESJB_IN-Bm]_JS

°

K6B4S >HIQCSastouKRH

V]A<m6S^DF˜cOlLm9>HIQCouK6STUk>_32-oNQCBK]K6BOS^BHnACBOST>_I¤#m]U„BKRHJSTŒ9>j]Q<HIm6BACS^SJouKLHTQ<K9>uD6>V]HJcOS:

‡ˆouU4U„B¦ouKN#>+ACB„N-ouQ<_>=}POB_IHp>Q<K6BOS@D]B5POBOSWD8QCSJPQ<V]A<Q<K6BOSMD]BkSpB_INRQCPOBkV]_JcOSJBKLHJBKRH

POBVrBK6D6>KLH^m]K+Q<KLHJc_);H^P_Im6PQi>A/D6>K6SnAi>MV]_p>HIQClLm6B#:ˆ‡ˆouU4U„BKLH^D]cOP_IQ<_JBYD]BXUk>K]QC–_JB

Sp>HIQCSIs­>QCSp>KRHJBMACBOS’U„oRD]–ACBOSXPOou_I_JBOSIVrouK6D6>KLHJS“=/>kA<Q<HIHJc_p>HIm]_JBkBOSIH6Aam6SJlRm‚FQCPQd>uSJSJB-4

_JcOD8m]Q<HJBWSIm]_nACBXSIm&AIBH:

=hF˜ouj4AaBOPHIQsED]B+POB HI_p>ON#>Q<ATBOSIH„D]ouK6P+D]B¬POouKLHI_IQ<j]m6B_¦šAi>‹POouK6SJHI_Im6PHIQCouK“DFm]K6B

HIŒ6cOou_IQCBˆSIVrcOPQ?9lLm6Bˆ>m8X?6ACBOS}DFG>HIHJBKRHJBn>ON-BOPˆPA<QCBKLHJS}Q<U4V9>HIQCBKLHJS>=#BKMV]_JouVro#Sp>KLHD]BOS

D]BOSJP_IQ<V]HIQCouK6Shou_IQ<¤#Q<K9>ACBOSd>uD6>V]Hp>j]ACBOSh>m@V]A<m6S/¤#_p>K6DMK6ouUWj]_JBnVo#SpSIQ<j]ACB D]B U„oD]–ACBOS:

5 ouHI_JBEV]_JcOoPOPm]V9>HIQCouK BOSIH^D]ouK6P@>m]Hp>KLH\HIŒ6cOou_IQClLm6BMlRm‚FG>V]V]A<QCP>HIQ<N-B¦ŠˆK6oum6S\P>ACPm8`

ACouK6SqD6>K6STPOB_IHp>Q<K6SqP>uSnV]_JcOPQCS>=8ACBOSnVrB_as­ou_IUk>K6POBOSnV9>_qD]BOSnHJBOPJŒ]K]QClRm6BOSLI7698 ( =6BH

HuPJŒ6ouK6SdV9>_—>Q<A<ACBm]_JShD]BqD]ouK]K6B_ˆD]BOS.U„cHIŒ6oD]BOShDF˜cHIm6D]Bq¤-cK6c_p>ACBOS =>uD6>V]HJcOBOSˆ>m8

V9>_IHIQCPm]Ai>_IQ<HJcOS^D]B’Ai>M?6ACBXDFG>HIHJBKRHJBM>N-BOPEPA<QCBKLHJSTQ<U4V9>HIQCBKLHJS:

& '

=‚B PJŒ9>V]Q<HI_JB BOSIH@D]cN-ouA<m“šƒAi>¬D]BOSJP_IQ<V]HIQCouK D8mU„oD]–ACB D]BOSM?6ACBOSMDFG>HIHJBKRHJB

>ON-BOPTPA<QCBKRHJSdQ<U4V9>HIQCBKRHJS:

5

oum6Sd•MQ<KLHI_JoD8m]QCSJouK6ShACBOShV]_IQ<K6PQ<V9>ACBOS D]cŽ?6K]Q<HIQCouK6SˆBHdK6o`

Hp>HIQCouK6S\lRm6BWK6oum6S^m]HIQ<A<QCSpouK6S’HJoum]H’>m‹ACouK]¤ D]BWPOBHIHJBMHIŒ6–OSJB#:’‘.K‹V9>_IHIQCPm]A<QCB_>=rK6oum6S

D]cŽ?6K]QCSJSJouK6S—ACBOS—V]_JoPOBOSJSIm6SnSIHJoPJŒ9>uSIHIQClRm6BOSnD]cOP_IQ<N#>KLHqACB’SI•8SIHJ–U„B’BHnACBOSnD8QCSJPQ<V]A<Q<K6BOS

D]BXSJB_INRQCPOBOSTlLm6B’K6oum6S^cHIm6D8QCouK6S:

=‚BqPpŒ9>V]Q<HI_JB^f’BOSIHdPOouK6Sp>uP_Jc\š\Ai>’V]_Jouj]ACcUk>HIQClRm6BTDF˜BK6SJBUWj]ACBTBH >m8@_JcOSIm]A<Hp>HJS

BHTU„cHIŒ6oD]BOS\BŽ8QCSIHp>KRHJS^SIm]_^Ai>kSJHp>j]Q<A<Q<HJcWBH^ACBOSTVrB_astou_IUk>K6POBOS’D]BOST?6ACBOS^DFG>HIHJBKRHJB

>ON-BOPXPA<QCBKLHJSTQ<U4V9>HIQCBKLHJS: 5 oum6SnPOouU4U„BKOouK6STV9>_n_p>V]VBACB_TACBOS—_JcOSIm]A<Hp>HJSqD]BXSIHp>z`

j]Q<A<Q<HJc=‚D]B@POouK]¤-BOSIHIQCouKš5AbF˜cHp>HESIHp>HIQCouK]K9>Q<_JB=‚D]B@HJBU4V6SXDFG>HIHJBKLHJB„Voum]_XD]BOS\?6ACBOS

DFG>HIHJBKLHJBhD8Q<_IQ<¤-cOBOS"V9>_‚Ai>nD8QCSJPQ<V]A<Q<K6Bh†/‡ˆ†d‰XBH_p>V]VrBACouK6S‚ACBOSs­ou_IUYm]ACBOS}>K9>A<•RHIQClRm6BOS

BŽQCSIHp>KRHJBOSˆVoum]_ ACBTP>ACPm]ArD]BTAi>EV]_Jouj9>j]Q<A<Q<HJc\D]B^VrB_IHJB^BHˆD8m5HJBU4V6SˆDFG>HIHJBKLHJB’D]BOS

PA<QCBKLHJS:

‘.K6SIm]Q<HJB=K6oum6S/U„BHIHJouK6S/AbFG>uPOPOBKRH}SJm]_"Ai>qPOouU4V9>_p>QCSpouKMD]BOS/VB_as­ou_IUk>K6POBOS}BKLHI_JB

ACBOSED8QCBrc_JBKRHJBOSYD8QCSJPQ<V]A<Q<K6BOSWD]B„SJB_INQCPOBOSXBHYBK V9>_IHIQCPm]A<QCB_WSJm]_XAbF˜ouV]HIQ<Uk>A<Q<HJc5D]B@Ai>

D8QCSJPQ<V]A<Q<K6BE‘d[’†

¯

+"+K/* N94+0! + +K/"

° :

‘.K8?6K@=u>V]_J–OS/>ON-ouQ<_"VouQ<KRHJc.AbFQ<U4Vo#SpSIQ<j]Q<A<Q<HJcdDFG>V]V]A<QClRm6B_"Ai>ˆU„cHIŒ6oD]ouACou¤#QCB >uD6>V8`

HJcOBY>m8 ?6ACBOST†/‡ˆ†d‰+D6>K6STACBYP>uSqo&RƒAi>4D8QCSJPQ<V]A<Q<K6BWBOSIHT‘d[’† =6K6oum6S^D]cOP_IQ<N-ouK6STm]K6B

U„cHIŒ6oRD]B4ou_IQ<¤#Q<K9>ACB„BŽ8QCSIHp>KRHYD6>K6S’Ai>¦A<Q<HIHJc_p>HIm]_JB„Vroum]_ED]cOP_IQ<_JB@AbF˜cN-ouA<m]HIQCouKDFm]K

SI•SJHJ–U„BXSJoum]U4QCSTš@‘d[’† =]Uk>QCSqSp>K6SqVB_IHJBED]BXPA<QCBKRHJS:

"oum6STACBOSqPJŒ9>V]Q<HI_JBOS\SIm]Q<N#>KLHJSqSJouKRHTD]cN-ouA<m6S^šMK6o#STPOouKLHI_IQ<j]m]HIQCouK6S:

[X>K6ShACB\PJŒ9>V]Q<HI_JB ž =RK6oum6SˆPOouU4V]ACcHJouK6SˆACBOSˆ_JcOSIm]A<Hp>HJSˆD]B^SJHp>j]Q<A<Q<HJc^BŽ8QCSIHp>KLHJS—BH

D]cOP_IQ<HJS^D6>K6STACBYPJŒ9>V]Q<HI_JB4f@Vroum]_Tm]K6BWD8QCSJPQ<V]A<Q<K6BMD]BESJB_INQCPOBYlRm6BACPOouK6lLm6BWBH^D6>K6S

m]K€P>uD8_JB4V]A<m6SE¤-cK6c_p>AdVroum]_XACBOSXACouQCSWD]BOSXV9>_p>U„–HI_JBOSYD8mSJ•SIHJ–U„B#: 5 oum6SXU„BHa`

HJouK6S.BK@oum]HI_JBnBK4cNQCD]BK6POB—m]K6B—U„cHIŒ6oRD]BnB_I¤-oD8QClRm6Bq>uD6>V]HJcOBnD8m@P>ACPm]A6D]B >A<U =

Voum]_—D]cHJB_IU4Q<K6B_TšYlRm6BA<ACB’POouK6D8Q<HIQCouK ACB\SJ•SIHJ–U„B’>HIHJBQ<KLH—ACB\_Jc¤#Q<U„B’SIHp>HIQCouK]K9>Q<_JB#:

[X>K6S’ACBMPpŒ9>V]Q<HI_JB„e4=rK6oum6S’K6oum6S’Q<KLHJc_JBOSJSpouK6SY>m P>ACPm]AD]BMAi>5V]_Jouj9>j]Q<A<Q<HJc„D]B

VB_IHJB BH”š¡Ai>POouU4V9>_p>QCSJouK BKRHI_JB ACBOS+D8QCSJPQ<V]A<Q<K6BOS”D]B SpB_INRQCPOB D]B POB‹VrouQ<KLH”D]B

NRm6B#: 5 oum6SYD]ouK]K6ouK6S@D]Bm8BK6P>uD8_JBU„BKRHJS@Voum]_WAi>”V]_Jouj9>j]Q<A<Q<HJc D]BkVB_IHJB=D6>K6S

ACBMP>uS6#%!'&$!;4škstou_IHJB@PJŒ9>_I¤-B„>N-BOP4D]cAi>QCSXlLm6BACPOouK6lRm6BOSXBHXD6>K6S’ACB@P>uS & ! # !;

>ON-BOP D]cAi>QCS”D]cHJB_IU4Q<K]QCSIHJBOS%:

5

oum6S+D]ouK]K6ouK6S”BK6SIm]Q<HJB‹m]K6BK6oum]N-BA<ACB V]_JBm]N-B D]B

AbF˜ouV]HIQ<Uk>A<Q<HJc”D]B Ai>‹D8QCSJPQ<V]A<Q<K6B”‘d[’†Vroum]_4Ai>‹POouU4V9>_p>QCSJouK“SIHJoPJŒ9>uSIHIQClRm6B+SIm]_4Ai>

VB_IHJB5D]BOSMPA<QCBKRHJS>=U„ouKLHI_JouK6S@AbF˜ouV]HIQ<Uk>A<Q<HJc DF˜‘d[’† BKHJB_IU„B¦D]BkV]_Jouj9>j]Q<A<Q<HJc D]B

VB_IHJB=6BH^D]ouK]K6ouK6Sqm]K”Uk> AIou_p>KLH\D8m”¤L>Q<K¬DF˜‘d[’†Vroum]_qAi>@V]_Jouj9>j]Q<A<Q<HJcYD]BEVB_IHJB#:

;²

5 oum6SdV]_Joum]N-ouK6S V9>_ˆ>Q<A<ACBm]_JSˆlLm6BqAi>XD8QCSJPQ<V]A<Q<K6B\D]BqSJB_INQCPOBT=‚[’†

¯

9 K/" N94+P!G

6+ + /*

°

BOSJHnAi>”¥WU„ouQ<K6SqjrouK]K6BW§¦D8QCSJPQ<V]A<Q<K6BED]BESpB_INRQCPOBXSIm]Q<N#>KLH^POBOSqP_IQ<HJ–_JBOS:

[X>K6S—ACB’PJŒ9>V]Q<HI_JBWw4=RK6oum6SnD]ouK]K6ouK6Sqm]K6BXD]BOSJP_IQ<V]HIQCouK+K6oum]N-BA<ACBXD]B\Ai>E?6ACBE>N-BOP

PA<QCBKRHJS’Q<U4V9>HIQCBKLHJS’_Jc¤#QCBMV9>_\‘h[\† BKƒQ<KRHI_JoRD8m]QCS >KLH’m]KƒV]_JoPOBOSJSIm6S^Uk>_32-oNQCBKš

N#>ACBm]_JSqU„BOSIm]_JBOSTVouK6PHIm6BA<ACBOSMŠ.ACBXV]_JoPOBOSJSIm6STD]BOS 3,+38 ,/"=6lRm]Q‚¤L>_JD]BEBK¬U„cU„ouQ<_JB

ACBOS/D]cAi>QCS‚_JcOSIQCD8m6BACS}D]B.HJoum6S‚ACBOS/PA<QCBKLHJS/V]_JcOSJBKLHJS/D6>K6S"ACBhSI•SJHJ–U„B=%oumWD]c AJš—VrB_JD8m6S:

5 oum6S’D]ouK]K6ouK6S\m]K6BYA<Q<U4Q<HJBL:6m]QCD]BMBH^m]KHIŒ6cOou_J–U„B@POBKLHI_p>AA<Q<U4Q<HJBMVoum]_\HJoum6S

ACBOSTV]_JoRPOBOSJSJm6S#‹>_32-oNQCBK6S’š4Nu>ACBm]_JS^U„BOSIm]_JBOSqVrouK6PHIm6BA<ACBOS\V]_JcOSJBKLHp>KRH^Ai>„U?;U„B

D8•K9>U4QClLm6B lRm6B¦ACBkV]_JoPOBOSJSIm6S@D]BOS@V]_Jo?6ACS

¯

ACBOS3,+38>( K/K/ / .+N&! /-3 8 +"

° : 5 oum6S

BKD]cOD8m]QCSJouK6S\AbFG>V]V]_JoO8Q<Uk>HIQCouK‹D]BOS^V]_JoPOBOSJSIm6STD]BYVB_as­ou_IUk>K6POBOS’D]BEAi>4?6ACBY‘d[’†

¯

V]_JoPOBOSJSIm6S’D]B@POouK]¤-BOSIHIQCouK@=rV]_JoPOBOSJSIm6S’D]BOS\VB_IHJBOS’BHXD]BOS’SJB_INRQCPOBOS

°

V9>_XD]BOS\V]_Jo`

POBOSJSJm6SdPOouKLHIQ<KRm6S BHdD]cHJB_IU4Q<K]QCSIHJBOS =LBHdAbF˜BOSIHIQ<Uk>HIQCouK5D]BOSD:6m6PHIm9>HIQCouK6Sˆš’POBOSdA<Q<U4Q<HJBOS

V9>_TD]BOSqV]_JoRPOBOSJSJm6SqD]BXD8QCBm6SJQCouK‚:

‘dK¦oum]HI_JB=RK6oum6Sˆ>uD6>V]HJouK6SˆPOBOS _JcOSIm]A<Hp>HJSˆšEAi>’?6ACB^DFG>HIHJBKLHJB’†Ÿ † >ON-BOP^D]cAi>QCS

D]cHJB_IU4Q<K]QCSIHJBOS =1lRm]QVBm]H;HI_JBYD]cOP_IQ<HJBEV9>_TACBXU?;U„BEV]_JoPOBOSJSIm6S>=1>Q<K6SIQ"lRm‚FG>m”V]_Jo`

POBOSJSJm6SqD]BOSqSJB_INQCPOBOSn_JcOSIQCD8m6BACS^D]B’Ai>M?6ACB?IH"7&+G39+:

5

oum6SˆPOouK6PA<m6ouK6SnBHˆD]ouK]K6ouK6SˆACBOS VB_JSIVrBOPHIQ<N-BOS—D]BTPOBOSˆHI_p>ON#>m85BK¦PJŒ9>V]Q<HI_JBY™:

"

5 oum6S/Q<KLHI_JoD8m]QCSJouK6SQCPQACBOS/D]cŽ?6K]Q<HIQCouK6S.lRm6BhK6oum6S>A<ACouK6S}m]HIQ<A<QCSJB_HJoum]H>mMACouK]¤’D]B POB

HI_p>ON#>Q<Ab: 5 oum6SnD]cŽ?6K]QCSpSJouK6S—HJoum]HqDFG>jou_JD+BK V9>_IHIQCB 8:<;=]šMV9>_IHIQ<_qD]B\Ai>@D]cŽ?6K]Q<HIQCouK

¤-cK6c_p>ACB„DFm]K6B@?6ACB„DFG>HIHJBKRHJB=/Ai> 7 "!. &%')( (*,+-"!.0/ +02345+-"!.0/Ž:

5

oum6S

D]cŽ?6K]QCSJSJouK6S BK V9>_IHIQCB 8:Ai>D8QCSpPQ<V]A<Q<K6BD]B‹SJB_INQCPOB‘d[’† =ˆlRm]Q6AIoum6B_p>m]K _uACB

POBKLHI_p>AV9>_‚Ai>qSIm]Q<HJB#: 5 oum6S"cN-olLm6ouK6S/BK8?6KWBKYV9>_IHIQCB 8:GfnACBOS/D8QCBrc_JBKRHJBOS/N#>_IQi>j]ACBOS

BHXV]_JoRPOBOSJSJm6S I 3 1+-8 +K2?&!#(KK/"=lRm]Q.VB_IU„BHIHI_JouKRHMD]B„D]cOP_IQ<_JB„ACB„POouU4Vou_IHJBU„BKRH

D]BOSqD8QCBrc_JBKRHJSqU„oD]–ACBOSTcHIm6D8QCcOS:

!"$#% &"&'( !)*(+-,./( !)

5

oum6S@_p>V]VrBACouK6S„QCPQnAi>‹D]cŽ?6K]Q<HIQCouK ¤-cK6c_p>ACB”D8m U„oRD]–ACB”V]_Jouj9>j]Q<A<QCSIHJB”D]B+Ai>ƒ?6ACB

DFG>HIHJBKLHJB=uBKXQ<KLHI_JoD8m]QCSp>KLH}ACBOSV]_JoPOBOSJSIm6S‚D]Bj9>uSpBd>m88lRm6BA<ACBOS‚K6oum6Srs­B_JouK6S_JcŽs­c_JBK8`

POB’HJoum]H^>m”ACouK]¤„D]BEPOBHqHI_p>Nu>Q<Ab:

0

$&|&3ƒ21<78O3R3

‰ouQCBKLHXD]BOS’PA<QCBKLHJSXlRm]Q/BKLHI_JBKRHED6>K6S\m]KƒUk>¤L>uSIQ<K€š5D]BOS\Q<K6SIHp>KLHJSX>ACc>HJouQ<_JBOS>=BK

D]BUk>K6D6>KLH—m]K¦SpB_INRQCPOB^D]BTD8m]_JcOB’>ACc>HJouQ<_JB#: ‡ˆBOS PA<QCBKRHJSq>HIHJBK6D]BKLHnD6>K6S m]K6Bq?6ACB

DFG>HIHJBKLHJBqlRm6BhAbFm]K„D]BOS.SpB_IN-Bm]_JS.D8mMUk>¤L>uSJQ<K„SJouQ<HA<Q<j]_JB—Voum]_Vroum]N-ouQ<_D;HI_JBnSpB_INRQCS:

^KSJB„V]Ai>uPOBkSIm]_Em]K¡BOSIV9>uPOB4V]_Jouj9>j]Q<A<QCSJc

Ω, F , P

^: ^K¡>V]VrBA<ACB

C 1 , C 2 , ...

ACBOSˆPA<QCBKRHJSqD6>K6S—AbF˜ou_JD8_JBXD]B\ACBm]_JSq>_I_IQ<N-cOBOS>=]BHnouK K6ouHJB’Voum]_—HJoum]H

n

⁼

T n

AbFQ<K6SJHp>KLH

DFG>_I_IQ<N-cOBOSTD]B

C n

^=9BH

σ n

=8Ai>„D8m]_JcOBED8m¬SJB_INQCPOBXD]BUk>K6D]cXV9>_

C n

^:

^K”SIm]V]Vo#SpB\lRm6B\V‚:˜S:ACBOST>_I_IQ<N-cOBOSqSpouKLHqSIQ<U4V]ACBOS

¯

m]K”SpBm]A‚PA<QCBKLHqBKLHI_JBWšMAi>WstouQCS

° =

Ai>SJm]Q<HJB D]BOS Q<K6SIHp>KLHJS”DFG>_I_IQ<N-cOBOS

T n , n ∈ N ∗

BOSIH”D]ouK6Pm]K6B SIm]Q<HJB D]BNu>_IQ`

>j]ACBOS >ACc>HJouQ<_JBOSˆV‚:˜S:SIHI_IQCPHJBU„BKLH—P_JouQCSJSp>KRHJB#:^K„K6ouHJB

(N t , t ≥ 0)

=-ACBTV]_JoRPOBOSJSJm6S SIHJoRPpŒ9>uSIHIQClRm6BqlLm]Q6POouU4V]HJB=8š’PJŒ9>ulRm6BqQ<K6SJHp>KLH

t

=-ACB—K6ouUYj]_JB^D]BqPA<QCBKLHJSˆBKLHI_JcOS D6>K6S ACBXSI•SJHJ–U„B A m6SJlRm‚FGš

t

:.©^m]HI_JBU„BKLH\D8Q<H>=]Vroum]_nHJoum]H

t

⁼

N t := X

n∈ N ∗

1 {T n ≤t} .

;

^K”K6ouHJB_p>4V9>_^>Q<A<ACBm]_JS

U 1 := T 1

BHqVroum]_nHJoum]H

n ≥ 1

^=8Ai>

(n + 1)

`bQC–U„B?+0!. 1+N +"Q +"+P%&H)

U n+1 := T n+1 − T n .

^K¦D8Q<H >ACou_JS lRm6B

(N t , t ≥ 0)

^{BOSJHhACB 3,+38>( "/ / /$IK/F} +1+"+P%'H K/"=

U n , n ∈ N ∗ ,

BOSJHdAi>

/ +5 $I"/F+0!. 1+N +1+K+0%'H)K/EBH

σ n , n ∈ N ∗

BOSIHˆAi>MSIm]Q<HJB\D]BOSF *283#/1I / 1+"%>+( K/:^K SIm]V]Vro#SJBXV9>_\>Q<A<ACBm]_JSTlRm6BXACBOSqSIm]Q<HJBOSqD]BOSTQ<KLHJB_p>_I_IQ<N-cOBOS’BHTD]BOSnHJBU4V6S^D]BESJB_INQCPOBOS

SJouKRHnSIHp>HIQCouK]K9>Q<_JBOSqBH—ouK¦K6ouHJBX>ACou_JS

U

=m]K6B\N#>_IQi>j]ACB’>ACc>HJouQ<_JB

¯

Nr:G>8:

°

¤-cK6c_IQClRm6B

>•L>KLHTVroum]_qACouQAi>MACouQ"D]BOSqQ<KLHJB_p>_I_IQ<N-cOBOS\BH

σ

=8m]K6BXN:G>8:d>•L>KLH^Voum]_nACouQ‚Ai>MACouQ/D]BOS HJBU4V6S^D]BXSJB_INRQCPOBOS%:^K”SIm]V]Vro#SJBXD]BXV]A<m6SqlRm6BXPOBOSnNr:G>8:dSJouKLHTQ<KLHJc¤#_p>j]ACBOSMŠ

E [U ] < ∞

^BH

E [σ] < ∞ ,

ouK”K6ouHJBYBK”oum]HI_JB

λ := (E [U ]) −1

^BH

µ := (E [σ]) −1

BHTouK”D]cŽ?6K]Q<HTAi> (16 +"I

ρ

D8mƒSI•SJHJ–U„B’V9>_EŠ

ρ := λ

µ = E [σ]

E [U ] .

="> "!# POou_I_JBOSIVrouK6D6>KLHJB BOSIH+D]ouK6PAbF˜ouj4AIBH¦V]_Jouj9>j]Q<A<QCSJHJB BKLHIQC–_JBU„BKLH

D]cHJB_IU4Q<K6c’V9>_nPOB\V]_JoRPOBOSpSIm6S—D]BOSˆBKRHI_JcOBOS>=8V9>_nPOBHIHJB’SIm]Q<HJB’D]BOS HJBU4V6SnD]B’SJB_INQCPOB=

BH@V9>_MACBOSWV]_JouV]_IQCcHJcOSkD8m ¥”Uk>¤L>uSIQ<K“§#: 5 ouU4U„cU„BKLH>= POBOS@V]_JouV]_IQCcHJcOS4SJouKLH@ACB

K6ouUWj]_JBMD]BWSJB_IN-Bm]_JS>=1Ai>kP>V9>uPQ<HJcMD]BWAi>@?6ACBMDFG>HIHJBKLHJB

¯

oum$# &%1+

°

=P#F˜BOSIH\škD8Q<_JB=

ACBWK6ouUYj]_JB4D]BMPA<QCBKLHJSXlRm]Q/VrBm]N-BKLH1;HI_JBMU4QCS’BKƒU?;U„BMHJBU4V6S’BK >HIHJBKLHJB=‚BH’Sp>

&+P/*( +3#+0! I$/ 1+"%>+( *=M+'0('ACBYP_IQ<HJ–_JBYSIm]Q<N#>KLHqACBOlRm6BA‚ACBOS^SJB_IN-Bm]_JSTPpŒ6ouQCSIQCSJSJBKRHTACB

PA<QCBKRHˆV]_IQCou_IQ<Hp>Q<_JBXšWSJB_INQ<_ ACou_JSJlRm‚FQ<ACSˆSJouKLHnD8QCSIVrouK]Q<j]ACBOS—BH lRm‚FQ<A1•5>YD]BOS—PA<QCBKLHJSnBK

>HIHJBKRHJB#:h=‚BOS^D8QCSJPQ<V]A<Q<K6BOSTD]BXSJB_INQCPOB’ACBOSqV]A<m6Sqm6SIm6BA<ACBOSqSJouKRHYŠ

•

^{*) ,+ ¯ 6++ /*G( &2O0/ 6++K/"#/ -+K%') °} SIQLouK@SJB_IH"HJoum&AIoum]_JS"ACBhV]A<m6S}>K6PQCBK@PA<QCBKLH

>_I_IQ<N-c#:^‘.KV9>_IHIQCPm]A<QCB_>=SIQ/Q<A/K‚F•¬>5lLm‚Fm]KSJB_IN-Bm]_>=Ai>kD8QCSpPQ<V]A<Q<K6BY_JBNQCBKLHEš

- /.

¯

6++K/"D+0! / 6++ /* 8 4

°

=V]m]QCSJlRm‚FG>ACou_JSˆm]K¦PA<QCBKRHnBKLHI_JcE>ON#>KLH—m]K+>m]HI_JB

SJou_IHns­ou_JPOcU„BKLH\>Nu>KRHqA<m]Q"SJoum6ST†/‡ˆ†d‰ =

•

^{) ,+ ¯} '/*$()&2O0/ 6+ + /*F/ 1+"%'N

°

SIQnouKSpB_IHMHJoum&AIoum]_JS4ACB D]B_IK]QCB_„PA<QCBKLH

>_I_IQ<N-c#:d‘.K+V9>_IHIQCPm]A<QCB_>=6SIQQ<Ar•5>Wm]K+SJB_IN-Bm]_>=8Ai>MD8QCSJPQ<V]A<Q<K6B’_JBNRQCBKRHTš - 0.

¯

'/*6+0! / 6++K/"8

° =

•

^{1*2*3 4.65 SIQ/ouK} SJB_IH\ACBOS’PA<QCBKLHJSE>mŒ9>uSp>_JD‹SIm]Q<Nu>KRH’m]KƒHIQ<_p>¤-BMm]K]Qs­ou_IU„B SIm]_qAi>MVrouV]m]Ai>HIQCouKƒD]B’Ai>M?6ACBXDFG>HIHJBKRHJB=

•

^{7 + ¯87 + 8>(KK/ / 8 +9+ 6} +"+P!"

°

SIQ"m]K ¥4SJB_IN-Bm]_E§+SpB_IH^SIQ<UWm]A<Hp>K6cU„BKLHXHJoum6S^ACBOS

PA<QCBKLHJSˆV]_JcOSJBKLHJS—D6>K6S ACBTSI•8SIHJ–U„B=šXm]K6B^NRQ<HJBOSJSpBqD8Q<NRQCSpcOBqV9>_hACBTK6ouUYj]_JB’D]B

PA<QCBKLHJS\SJB_INRQCSqš„PJŒ9>ulRm6BXQ<K6SIHp>KLH>=

•

^{6+PK/ &%')(} 3,+"+.8 +K+5 HK/ƒŠWACBOSWPA<QCBKRHJS@SJouKLHMD]BOSY_JBV]_JcOSJBKRHp>KLHJS4D]B5D8QCBrc_JBKRHJBOS PAi>uSJSJBOS

C 1 , C 2 , ...

BH—ACB\SpB_IN-Bm]_>=]SIQ1Q<A‚>WACB’PJŒ6ouQ =]SJB_IHˆm]K PA<QCBKRHqD]B’Ai>YPAi>uSpSJB

C i

>Nu>KRHTm]K”PA<QCBKLH\D]B’Ai>4PAi>uSJSJB

C j

SJQ

i < j

^:

[’>K6SdHJoum]HJBOSdACBOSdD8QCSJPQ<V]A<Q<K6BOS D]BqSJB_INQCPOBOSdBŽ]POBV]HJcOBq†/‡ˆ†d‰@BH ‰ =-m]K„PA<QCBKLH lRm]Q6BOSIH

D]c AIš^D6>K6SACB—SI•SJHJ–U„BhVrBm]HN-ouQ<_BKRHI_JB_ >V]_J–OS}A<m]Q8m]K@PA<QCBKRHdlLm]Q8A<m]Q]BOSIHV]_IQCou_IQ<Hp>Q<_JB#:

^KD8Q<_p> >ACou_JS’lLm‚Fm]K SJB_IN-Bm]_XSJB_IH’D]BWs­> OouK 3,+)H)"23 5+0%'kSIQ/Q<AVrBm]H^Q<KRHJB_I_JouU4V]_JB

ACB\SpB_INRQCPOB’DFm]K+PA<QCBKLHqVroum]_nPOouU4U„BK6POB_TšMBK”SJB_INRQ<_ˆm]K¬>m]HI_JB’lLm]QrA<m]Q‚BOSIHˆV]_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>U„–HI_JBOS5lLm]QnP>_p>uPHJc_IQCSJBKRH

BKLHIQC–_JBU„BKLH^m]K6B\?6ACB’DFG>HIHJBKLHJBESJouKRHn_JcOSIm]U„cOSnV9>_—Ai> !,8&2O"! (*C9 +) I"! 9&0=

K6ouHp>HIQCouK m]K]Q<N-B_JSJBA<ACBU„BKLH5_JBOPOouK]KLm6B”D6>K6S@Ai>ƒHIŒ6cOou_IQCB+D]BOSM?6ACBOS4DFG>HIHJBKLHJB¬lLm6BA`

POouK6lLm6BOSEŠ

)+3-,./$0%$&1“(tt

E

9

7

(5 ; =

9C; 9 ;'5 7

9C; E D 5

(D E

A/S/s/s + B − X

^A!#"

• A

^{; $ ;$ 7 E9 EA<5 E9 7 D5% & 7:5 ;} 9<;';'5%'(

7

U n , n ∈ N ∗

⁾

• S

^{; * ;$ 7 E9EFAC5 E97 D5+}

σ n , n ∈ N ∗

^{7, = 7 7 ;-' 5+ 7}

= 9

7

9C;E

7. E5

(7

)

• s

^{; $ ;$ 7 E/ AC=10 ; 32 ' (D E E = 5 ( 4 565 7 ;-' D;'7}

)

• B

^{7E9 9C58E E72 ' (D E E = 5 8 4 5 5 D90 D;: ; 2 A<=10 ; <}

E5

(7 = 9>=C58= 9CE D 58E

?

D@A A

5 ;B5

)

• X

^{; $ ;$ 7 EF9 587 5 E5 < 7 ;-' 5 (C}

‘.K„V9>_IHIQCPm]A<QCB_>=ACBOShD]Bm8„V]_JBU4QCB_JS V9>_p>U„–HI_JBOS

A

^BH

S

P>_p>uPHJc_IQCSp>KLHˆACBOSdACouQCS D]BOS SIm]Q<HJBOSnD]BOSˆQ<KLHJB_p>_I_IQ<N-cOBOS\BH—D]BOS—HJBU4V6SnD]B’SJB_INQCPOBOS—V]_JBK]K6BKLH^BOSJSJBKLHIQCBA<ACBU„BKRHqACBOS

Nu>ACBm]_JSqSJm]Q<Nu>KRHJBOSYŠ

• G

^{¯"I*! 1+N&°} SIQAi>4SIm]Q<HJBXBOSIHqSJHp>HIQCouK]K9>Q<_JBYB_I¤-oD8QClLm6B?=

• GI

^{¯" "!} 1+N&+0!#I 3 "!#I*!.

°

SIQ‚BA<ACBXBOSIHqQb:Qb:˜D =

• M

^{¯ 2O"2 8 +ED'K/K/ °} SJQ1Ai>WSIm]Q<HJB’BOSIH—Qb:Qb:˜D:.D]B\ACouQBŽ8VouK6BKRHIQCBA<ACB

¯

SIQ

A = M

^=ACB

V]_JoPOBOSJSIm6SqD]BOSqBKRHI_JcOBOSTBOSIHqD]ouK6PEm]K”V]_JoRPOBOSpSIm6SqD]B }ouQCSJSJouK

° =

• D

^{¯ * 1+"2} +0!<+P/"5+(

°

SIQAi>4SJm]Q<HJBXBOSIHqD]cHJB_IU4Q<K]QCSIHJBYBH^POouK6SIHp>KLHJB#:

;

‘.K+¤-cK6c_p>A-=]SJQ>m6Pm]K6BX>UWj]Q<¤#m­HJcXK‚F˜BOSIHnVro#SJSIQ<j]ACB=8ouK K6ouHJB_p> :!:!S>=m]K6BT?6ACBE>N-BOP

s

SJB_IN-Bm]_JS =.D]Bkj]m4BB_MD]B P>V9>uPQ<HJc¦Q<K8?6K]QCB=._Jc¤#QCB5V9>_@Ai>¬D8QCSJPQ<V]A<Q<K6B †/‡ˆ†d‰: >_

BŽ]BU4V]ACB=]Ai>Y?6ACB

M/M/1

BOSIH—Ai>W?6ACBXDFG>HIHJBKLHJBEo&R+ACBOSnPA<QCBKRHJSTBKLHI_JBKRHTSIm]Q<N#>KLHqm]K V]_JoPOBOSJSIm6SnD]B6}ouQCSJSJouK@=]BK+D]BUk>K6D6>KLH^m]K SpB_INRQCPOB’D]B\ACouQBŽVrouK6BKLHIQCBA<ACB=9>N-BOPXm]K

SJB_IN-Bm]_>=6m]K”j]m4BrB_qD]BEHp>Q<A<ACBXQ<K8?6K]QCBEBHqHI_p>Q<HJcOBEBKƒ†}Ÿ † M:

|&3ƒ7 H13L!¬!L|t$0325$t~€«}78%$&3R2

}oum]_4U„oRD]cA<QCSpB_WACB5HI_p>K6Sas­B_IH@D]B¦D]ouK]K6cOBOS4BK¡HJBU4V6Sa`b_JcOBA-=houKD]ouQ<H„> Aaoum]HJB_Mm]K6B

POouKRHI_p>Q<KLHJB5>m€SJ•SIHJ–U„B@BK SIVrcOPQ?1>KLHYlRm6BMACBOS’HuPJŒ6BOSESpouKLHXVrB_JD8m6BOSESIQACBMHJBU4V6S

lRm‚F˜BA<ACBOSV9>uSJSJBKRHD6>K6SACB—SI•8SIHJ–U„BˆBOSIH}V]A<m6S¤#_p>K6DMlRm‚Fm]K4D]cAi>Q6lLm]QACBm]_.BOSIHd>A<ACoum6c#:

="B’U„oRD]–ACBXlRm6B\K6oum6SnPOouK6SIQCD]c_JouK6SqBOSIHnD]ouK6P\m]K6B\?6ACBXDFG>HIHJBKLHJBEo&R ACBOSnPA<QCBKRHJS

ouKRH„m]K IH"7&+

¯

oum345+-"!#( °

Vroum]_?;HI_JB”SJB_INQCS>= >m D]BAiš‹D8m6lRm6BAqQ<ACS„SJou_IHJBKRHkD8m

Uk>¤L>uSIQ<KSp>K6SE>ON-ouQ<_McHJc4Sp>HIQCSasb>Q<HJS:@©Tm]HI_JBU„BKRHYD8Q<H>=‚ACB

n

`bQC–U„B„PA<QCBKRH

C n

^BKRHI_JB

D6>K6S\Ai>„?6ACBWDFG>HIHJBKRHJB„> BrBOPHJcMD8mD]cAi>Q

D n

=1lRm]Q}BOSJHTm]K6BWNu>_IQi>j]ACB4>ACc>HJouQ<_JBMV‚:˜S:

SIHI_IQCPHJBU„BKRH”Vro#SIQ<HIQ<N-B#: ‰RQ’Q<A\K‚FG>V9>uS V]m >HIHJBQ<K6D8_JB‹ACBSJB_IN-Bm]_¬š€Ai>?6K D]B‹Sp>

V9>HIQCBK6POB

¯

+8'0(' šAbFQ<K6SIHp>KRH

T n + D n

^° =dQ<AqBOSJH@cA<Q<U4Q<K6c¬D8m SI•SJHJ–U„B#:¡="BOS4D]cAi>QCS SJB_JouKRHTD]ouK6PEPOouK6SIQCD]c_JcOSH"+P2+0!#8&++N"/YBH / IH # />1+"%>+5(K:

>_n>Q<A<ACBm]_JS>=LAi>YSIm]Q<HJB

D n , n ∈ N ∗

SJB_p>EHJoum&Aaoum]_JSˆSIm]V]Vro#SJcOBqSIHp>HIQCouK]K9>Q<_JB’D]B ACouQ‚¤-cK6c_IQClRm6BXAi>MACouQ"D]B

D

^: ^K”SIm]V]Vro#SJB_p>4D]B’V]A<m6S^lLm6B

E [D] < ∞

BHTouK+K6ouHJB_p>

Γ := (E [D]) −1 .

>_¦>Q<A<ACBm]_JS>=nSJQnACB”PA<QCBKLH

C n

BOSIHkHJoum&Aaoum]_JS5D6>K6S„ACB”SI•8SIHJ–U„B¬š AbFQ<K6SIHp>KLH

t

^=—ouK

>V]VrBA<ACB_p>’SJouK IH"7&+ +)HK/K+ <"

t DR n (t)

^ACB HJBU4V6S_JBOSJHp>KLHhš

t

>Nu>KRH.AbF˜BŽV]Q<_p>HIQCouK D]BESpouK”D]cAi>Qb:^Kƒ>4>ACou_JSYŠ

DR n (t) = D n − (t − T n ).

^K„VrBm]HhD]cOP_IQ<_JBTAi> F9 "!. &%')($(*+"!#0/8+P23495+"!#0/^>ON-BOPqAi>XK6ouU„BK6PAi>HIm]_JB

SIm]Q<N#>KLHJB=6Q<KLHI_JoD8m]Q<HJBEV9>_ G >_I_JB_

¯ G >_pe-™

° Š

)+3L,}$0%$0 (&i(

,.|&3“ 178O33v7BH13R!“!L|t$&3R2¡$t~€«}78%$032 7 9C; 9 ;'57 1 9<; E9

A

&

A/S/s/s + D − X

)

A!7"

• A

)

S

s

^{; * ;$ 7}

) A<=%= 9 7 E9 AC= EF9(D; 9CE8E

)

; 7

?

(5%' =

E9 EFAC5

D ; A8

7 7.D 7

7:9<;';'5%'(

7 )

7.

= 7

7 ;-' 5 7 E/ A<=10 ; & 7 ;-' D;'7

)

• D

^{; * ;$ 7 E9EA<5 ;'5 D < 7 E9<5 7 7 E5 (7}

) E58=%5

9

AC58; 7 .D7 D 9CD 80 D@ D 7 ;-' 5+ )

• X

^{7 EF9 587 5 E5 < 7 ;-' 5 (C}

3~€7]‚ "3ƒ(tt ‰RQnACB+POouKRHI_p>Q<_JB”K‚F˜BOSIH4V9>uS4V]_JcOPQCSpc=ˆouK SJm]V]Vo#SJB_p>HJoum&AIoum]_JS

lLm6B’Ai>@D8QCSJPQ<V]A<Q<K6BWD]BXSJB_INQCPOBEBOSIHnK6ouK8`bV]_JcOBU4V]HIQ<N-B#:

=">„D]cŽ?6K]Q<HIQCouKV]_JcOPOcOD]BKLHJBMQ<K6D8QClRm6BWlRm6BXAbF˜ouK‹SIm]V]Vro#SJBXHJoum&AIoum]_JS^lRm6BEACBYj]m4BrB_

BOSIHqD]B’Hp>Q<A<ACBXQ<K8?6K]QCB#:h="BXV9>_p>U„–HI_JBW>ACc>HJouQ<_JB

D

^¯ACBED]cAi>Q

°

_JBU4V]Ai>uPOBM>ACou_JSqD6>K6S

Ai>K6ouU„BK6PAi>HIm]_JB¬D]BXBK6D6>A<ATPAi>uSJSJQClLm6B ACB V9>_p>U„–HI_JB

B

^¯Ai>‹Hp>Q<A<ACB+D8m j]m4BB_

°

POouU4U„Bqsb>uPHJBm]_ A<Q<U4Q<Hp>KRH Ai>WP>V9>uPQ<HJc\D8m¦SI•8SIHJ–U„B#:=">X?6ACB^DFG>HIHJBKRHJB’>N-BOP^PA<QCBKRHJS

Q<U4V9>HIQCBKLHJSYBOSJH’D]ouK6P„m]KU„oD]–ACB„š¦VrB_IHJB=">m U?;U„B@HIQ<HI_JB4V9>_XBŽ8BU4V]ACBklLm‚Fm]K6B

?6ACB¬š‹j]m4BB_4?6K]Q

G/G/s/s + B

^¯o&R m]K“PA<QCBKRH5BOSIHkVB_JD8m“SJQqQ<ATBKLHI_JBƒD6>K6S„ACB SI•SJHJ–U„BE>ACou_JSqlLm6B’ACBEj]m4BrB_qBOSIH^D]c AIšMV]ACBQ<K

° :

ŸArBOSIH—V9>_q>Q<A<ACBm]_JSTšYK6ouHJB_qlLm6B’D6>K6S—Ai>WA<Q<HIHJc_p>HIm]_JB=]ACBOS—D]cAi>QCSnVrBm]N-BKLHT>m6SJSIQ

K6B^V9>uS8;HI_JBXcA<Q<U4Q<K9>HJouQ<_JBOS

¯

>m6lRm6BA‚P>uS—Q<ACS—K6BXSJB_IN-BKLHqlRm‚FGš@D]ouK]K6B_qD]BOS—V]_IQCou_IQ<HJcOS

>m8¡PA<QCBKLHJS

°

:¡¨^K PA<QCBKLH„_JBOSJHJB >ACou_JSkD6>K6S@ACB+SI•SJHJ–U„B >V]_J–OS4BŽ8V]Q<_p>HIQCouK“D]B+Sp>

V9>HIQCBK6POB=]U?;U„BESIQQ<A‚K‚FG>MV9>uS\>HIHJBQ<KLH^ACBXSJB_IN-Bm]_:

‘.K8?6K@=ACBOSXD]cAi>QCSXVrBm]N-BKLHM>m6SJSIQ ;HI_JB@cA<Q<U4Q<K9>HJouQ<_JBOS>=‚Uk>QCS / C!

/ 1+"%>+( 5ŠdD6>K6SqPOBEP>uSnm]K”PA<QCBKRH^BOSIHqcA<Q<U4Q<K6cYSIQQ<AK‚FG>@V9>uS$ 1+"2+P!GHESJouK”SpB_INRQCPOBEš

AbFQ<K6SIHp>KLH

T n + D n

^: #>QCShK6oum6SdK6oum6S A<Q<U4Q<HJB_JouK6S D6>K6S POBqHI_p>Nu>Q<Ar>m84P>uShV]_JcOPQCSJcOS D6>K6SnAi>4D]cŽ?6K]Q<HIQCouK 8:<;#:

() &'( ,(. ) . )# (&'"$+-,)

=‚BOSED8QCSJPQ<V]A<Q<K6BOSED]B4SpB_INRQCPOBOSXU„BKRHIQCouK]K6cOBOSWD6>K6SEAi>¦V9>_IHIQCB@V]_JcOPOcOD]BKLHJBkSpouKLHEj]QCBK

S]_q>V]V]A<QCP>j]ACBYšMm]K6B\?6ACBXDFG>HIHJBKLHJBY>N-BOPEPA<QCBKLHJSTQ<U4V9>HIQCBKLHJS: =">@D8•K9>U4QClLm6BXD]B

Ai>W?6ACBEDFG>HIHJBKLHJBMBOSIHT>ACou_JSTPOBA<ACBYDFm]K6B’?6ACBEDFG>HIHJBKLHJBMPAi>uSJSIQClRm6B=8U„oD8Q?9cOBEV9>_nACBOS

cA<Q<U4Q<K9>HIQCouK6STD]BEPA<QCBKRHJS>=6lRm]Q‚K6BXSJouKLHTV]A<m6STš4SJB_INQ<_:

‡ˆBOPQdcHp>KLH>=>m6Pm]K6B„DF˜BKLHI_JBkBA<ACBOSXK6B@HIQCBKLH 3,+"+.8 +K+TPOouU4V]HJBkD]BOSED]cAi>QCSED]BOS

PA<QCBKLHJS:

5

oum6SnV]_JcOSpBKLHJouK6SqQCPQ/D]Bm8+D8QCSpPQ<V]A<Q<K6BOSTD]BESJB_INQCPOBOS>=]D]cŽ?6K]QCBOS^D6>K6SqACBXP>uS

o&R„ACBOShPA<QCBKRHJSˆSpouKLHˆ> BBOPHJcOShD]B^D]cAi>QCShBHˆlLm]QAIoum6B_JouKLHˆm]K„_uACB^POBKLHI_p>ArD6>K6S POBHIHJB

cHIm6D]B„ŠACBOSqD8QCSJPQ<V]A<Q<K6BOS

EDF

^BH

LDF

^:

)+3-,./$0%$&1“(i(?t

C

9

587 5 E5

7

;-' 5

. D

4

E

9

9'

E5

(7 58=

9

(5

(7

7

5+

) 0 2

9<;'E5 7

9 E5

58;'7 5 A

;$

=

(5%'

2 A

EDF

^{5"7 5 9 D 4 7 ;-' 5}

) E 7 ;-' D; 7 ;- E

E5

A E

<

E9<5$;$ 7 5 D E3

5

7 9

7 E ED7 .A<D;- C

E

7 ;- 9CEFAC;'7. E5 ,D 7 D 3.A<= E(5A 9 7 A 7 ;-' 5+ 8C

;w

C 9 5 7 5 E5 7 ;-' 5 7 5+ ) 0 «/*L3~€«/%$H3 2 A

EDF p

⁵

7.53E 7 ;-' D; 7 ;- 9 D 5 7 9 3E E5 A E E9<5

;$ 7 5 D

E 5

7 9

7 E ED7 ACD;-

C

C 9 57 5 E5 7 5+ h) 0 2 d7] -93L2 )+3L7]|t$t"3 0 $&z2$5 A

;$

=

(5%'

2

LDF

^{5 7 5 E 7 ;-' D; 7 ;- 9 D ?BAC587 D 58E AC=}

E

D 7

;-' 5+

E E5

A

@E

E9<5 ;$ 7 5 D E

@5

7 9

7 E ED7 EFA C

C 9 5 7 5 E5

7

5%

d)

0

«}z*R3R~«"%$H13

2

LDF p

^{537 5E 7 ;-' D ;}

7 ;- 9 D 5 7 9 E E5 A E E9<5%;$ 7 5 D E

5 7 9 7 E ED7:EA C

‡ˆouU4U„B^K6oum6ShAbFG>N-ouK6SˆV]_JcOPQCSJc^D6>K6S Ai>X_JBUk>_JlRm6BqV]_JcOPOcOD]BKRHJB=8K6oum6ShK‚F˜cHIm6D8QCB_JouK6S

BK D]cHp>Q<A^lRm6B+ACBOSkD8QCSJPQ<V]A<Q<K6BOS¦D]B”SJB_INQCPOBOS„K6ouK8`bV]_JcOBU4V]HIQ<N-BOS =qK6ouHJcOBOS„QCPQ

EDF

BH

LDF

^{: 5} oum6S’_JBNRQCBK6D8_JouK6SEAi>_I¤-BU„BKLHYV9>_’Ai> SIm]Q<HJB@SIm]_’ACBOS’V]_JouV]_IQCcHJcOSED]B4POBOS D8QCSJPQ<V]A<Q<K6BOS—D]B\SJB_INQCPOBTBH—SIm]_hACBOSˆV]_Jouj]AC–U„BOS—lLm6B^Vro#SJBqAbF˜cHIm6D]B\D]BOSd?6ACBOS—DFG>HIHJBKRHJB

POou_I_JBOSIVrouK6D6>KLHJBOS%:

( ) $+ . ) ' , & )).) ,

+%.&' )

[’cŽ?6K]QCSJSJouK6SEQCPQ}ACBOSXD8QCBc_JBKLHJBOSYNu>_IQi>j]ACBOSW>ACc>HJouQ<_JBOSEBHXV]_JoRPOBOSpSIm6SXSIHJoPJŒ9>uSIHIQClRm6BOS

P>_p>uPHJc_IQCSp>KRH^m]K6B\?6ACBYDFG>HIHJBKLHJBW>N-BOPYPA<QCBKRHJSqQ<U4V9>HIQCBKLHJS%:

•

^{ACB *283#/} 9 "!. $3,+38K3 8'/>H

W n

>mPA<QCBKLH

C n

BOSIHEACB„HJBU4V6SMlRm‚FG>m]_p>¬š

>HIHJBK6D8_JB

C n

>ON#>KLH4DF;HI_JB SJB_INRQb: ‡ˆB5HJBU4V6SWVrBm]H ;HI_JB5SJm]Vc_IQCBm]_„š

D n

⁼

>m6lLm6BA"P>uS

C n

BOSIHqVrB_JD8m

¯

Q<A‚K‚FG>MV9>uSTV9>uSnV]m¬>HIHJBQ<K6D8_JBEACBESpB_IN-Bm]_

° =

•

ACB "23./ "!.

T n a

DFm]K PA<QCBKLH

C n

BOSIH\ACBWHJBU4V6SXB"BrBOPHIQ<N-BU„BKRHXV9>uSJSJc V9>_

C n

D6>K6STACB’j]m4BrB_YŠ

T n a = W n 1 {W n <D n } + D n 1 {W n ≥D n } ,

•

ACB1 *283#/J />H 8 +

T n s

D8m„PA<QCBKLH

C n

BOSIHACB—HJBU4V6SdlLm6BˆV9>uSJSpBnB"BrBOPHIQ<N-BU„BKRH

C n

D6>K6SnACBXSJ•SIHJ–U„B

¯

j]m4BB_! SpB_INRQCPOB

° Š

T n s = (W n + σ n )1 {W n <D n } + D n 1 {W n ≥D n } .

}oum]_YHJoum]HYSJ•SIHJ–U„BkD]BM?6ACBOSYDFG>HIHJBKLHJB¦š VrB_IHJB=‚Ai> 3,+ 8 #N #"+0,+5 HI13 1+*

π k

^D8m

PA<QCBKRH

C k

BOSIHTAi>4V]_Jouj9>j]Q<A<Q<HJc@lLm6BWPOBYPA<QCBKRH\SJouQ<H^VB_JD8m‚: >_’BŽ8BU4V]ACB=1Voum]_^m]K6B

?6ACB„DFG>HIHJBKRHJBkPAi>uSJSJQClLm6B„š¦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<HJcXlRm‚FQ<A•k>Q<H

1 + B

PA<QCBKLHJSnD6>K6S—ACB\SI•SJHJ–U„BOSˆšYAbF˜BKLHI_JcOB D]B

C k

:q[X>K6S\ACBYP>uS\DFm]K6BY?6ACBWDFG>HIHJBKRHJB@>N-BOPWPA<QCBKRHJS\Q<U4V9>HIQCBKLHJS>=rAi>kPOouK6D8Q<HIQCouK

D]B^VrB_IHJBXBOSIHnV]A<m6SnPOouU4V]ACBŽ]B„Š

π k

BOSIHnD]ouK]K6cOBXV9>_nAi>MV]_Jouj9>j]Q<A<Q<HJcElRm6B

C k

SJB’N-ouQ<H V]_JouVo#SJB_nm]K¬HJBU4V6STDFG>HIHJBKLHJBWSIm]Vrc_IQCBm]_Tš4SJouK”D]cAi>QhŠ

π k = P [C k

BOSIHnVrB_JD8m

] = P [W k > D k ] .

^{¯8:<; °}

[\cŽ?6K]QCSpSJouK6SnUk>Q<KLHJBK9>KRH^ACBOS 3,+ 8 ( K/ / I/ I 3 1+-8 +K2?&!#(KK/5Š

•

^{ACBO3,+ 8 ( K/ / I/ (} 8&!"IK/"5+ 8&!

(X t , t ≥ 0)

BOSIH@ACB¦V]_JoRPOBOSJSJm6S4POouU4V]Hp>KLH„ACB K6ouUYj]_JBWD]BXPA<QCBKLHJS\D6>K6SnACBESI•8SIHJ–U„BXš4PpŒ9>ulLm6BXQ<K6SJHp>KLH

t

⁼

•

^{ACB 3,+38>( K/K/ /8 *!#( J # &%1+}

(Q t , t ≥ 0)

BOSJH"ACB.V]_JoPOBOSJSIm6S/lLm]QLPOouU4V]HJB ACB’K6ouUYj]_JBWD]BEPA<QCBKLHJS^D6>K6SqACB’j]m4BB_^šMAbFQ<K6SIHp>KRH

t

:.©^Q<K6SIQ-=8Vroum]_nHJoum]H

t

^Š

Q t = (X t − s) + ,

•

Voum]_.HJoum]H

n ≥ 1

=RouK4K6ouHJBnV9>_

T ˜ n

^=-ACB

n

`bQC–U„BqQ<K6SIHp>KLHhD]BnSpou_IHIQCBnD8mkSJB_IN-Bm]_>=

P#F˜BOSIHa` šz` D8Q<_JB=1AbFQ<K6SIHp>KLH\D]BXAi>

n

`bQC–U„BX?6KƒD]BESJB_INQCPOB#: =‚B 3,+ 8 ( K/ / I/?IK/6! / I / 1+"%>+( K/

(S t , t ≥ 0)

BOSIHd>ACou_JS}ACBˆV]_JoRPOBOSpSIm6S}VouK6PHIm6BA

¯

š^HI_p> AaBOPHJouQ<_JBOSV‚:˜S%:

PšuD8Aiš¤

°

POouU4V]Hp>KRH—ACBnK6ouUYj]_JB\D]BTSJB_INQCPOBOSdHJB_IU4Q<K6cOSAam6SJlLm‚FGš

t

:©Tm]HI_JBU„BKRH D8Q<H>=]Vroum]_nHJoum]H

t

^Š

S t = X

n∈ N ∗

1 T ˜

n ≤t ,

•

Voum]_nHJoum]H

n ≥ 1

=]ouK+K6ouHJB’V9>_

˜

σ n

=Ai>@D8m]_JcOBXD8m

n

`bQC–U„BXSJB_INQCPOB+N"! 34 + 1/>1+"%' 9+"=+'0('dAi>@D8m]_JcOBED8m¬SJB_INQCPOBXlLm]Q‚SpB’HJB_IU4Q<K6BWš

T ˜ n

⁼

•

Voum]_MHJoum]H

n ≥ 1

=douKK6ouHJB5V9>_

T ˇ n

^=ACB

n

`bQC–U„B5Q<K6SIHp>KRH@o&R¡m]KPA<QCBKLH4BOSIH VrB_JD8m‚:.=‚B3,+38>( K/K/ / IK/ 3 1+* K/

(P t , t ≥ 0)

BOSJHn>ACou_JSqACB^V]_JoPOBOSJSIm6SnPšuD8Aiš¤

POouU4V]Hp>KLH^ACBXK6ouUYj]_JBWD]B’VB_IHJBOS A m6SJlRm‚FGš

t

^: }oum]_THJoum]H

t

^Š

P t = X

n∈ N ∗

1 T ˇ n ≤t ,

•

ACBF3,+38>( K/K/ /I"/ IH 34 +"0/WD8mƒSI•SJHJ–U„B

(D t , t ≥ 0)

BOSIHqACBXV]_JoPOBOSJSIm6SqPOouU4V8`

Hp>KLHWACBOSXQ<K6SIHp>KRHJSYD]B„SJou_IHIQCBOSWD8mSI•8SIHJ–U„B

¯

SJB_INQCPOBOSYoum€VrB_IHJBOS

°

Aam6SJlRm‚FGš

t

^:

©^m]HI_JBU„BKLH^D8Q<H>=6Voum]_nHJoum]H

t

^Š

D t := S t + P t .

‰RQTouK“_JcOou_JD]ouK]K6BƒACBOSkD]cAi>QCS5_JcOSIQCD8m6BACSkD]BOS

Q t

PA<QCBKRHJS¦V]_JcOSJBKLHJS¦D6>K6S„ACB¬j]m4BrB_

š

t

V9>_Mou_JD8_JB„P_JouQCSJSp>KRH

¯

Vroum]_YACBOSESJB_INQ<_YBK

EDF

^{=}D]ouK6P °} =}ouKK6ouHJB„POBOSWD]cAi>QCS _JcOSIQCD8m6BACSŠ

R 1 (t) < R 2 (t) < .... < R Q t (t).

! "#$&%'()#** +#,-#. %+/ .012+43+/5+

X

) +

678+8**

(X t , t ≥ 0)

;›

R 1 (t)

^BOSIH D]ouK6P^ACB 3# / 3 *5+5IH"7&++NHK/"+5 " )+0! /*-&!.

t

:©vm]KkQ<K6SJHp>KLH—D]Bn?6K D]B SJB_INQCPOB

T ˜ n

=rACB4SJB_IN-Bm]_ESJB_IHED]ouK6P4BK

EDF

ACB4PA<QCBKLHED]B4D]cAi>Q._JcOSJQCD8m6BA

R 1 ( ˜ T n )

^:

^K¡K6ouHJB_p>¬c¤L>ACBU„BKRH4ACB IH*C&+F+0!<+05+&6 (*+*!.63,+K+8 +"+0-++) N+0! /"-&!#

t

^V9>_

D(t)

^: ‰Rm]V]Vro#SJouK6SnlRm6B’Ai>W?6ACBXBOSIH—HI_p>Q<HJcOBEBK”‘d[’†—:9‰ouQ<H

n 0

=8ACB\KLm]U„c_JokD8m+PA<QCBKRH V]_IQCou_IQ<Hp>Q<_JBEšEAbFQ<K6SIHp>KLH

t

D6>K6SˆAbF˜ou_JD8_JB’D]BOSˆ>_I_IQ<N-cOBOS

¯

POB\PA<QCBKLHnBOSIH—D]ouK6P

C n 0

^{° :^K}

>4>ACou_JSWŠ

R 1 (t) = DR n 0 (t)

^BH

D(t) = D n 0 = R 1 (t) + t − T n 0 .

^K”K6ouHJB_p>„c¤L>ACBU„BKLH

Z t := R Q t (t),

ACBL3# / " + &! IH*C&+ +)HK/"+5 <"

t

^DFm]K€PA<QCBKRHWD6>K6SXACB@j]m4BB_Yš

t

^:M© PJŒ9>ulRm6BM?6K D]BMSJB_INQCPOB

T ˜ n

=ouKƒSJB_IH’D]ouK6PYBK

LDF

ACBYPA<QCBKRHXD]ouKLH\ACBWD]cAi>Q/_JcOSIQCD8m6BAš

T ˜ n

^BOSIH

Z T ˜ n

:=‚B V]_JoPOBOSJSIm6S

Z t , t ∈ R +

BOSJH@D]ouK6P”ACB?3 + 8>(KK/ / I/ K/ H"C+/ +)HK/K+ <"/ 2?&+02? #:

^K>5Nm lLm6B=rHJoum]HE>m ACouK]¤+D]B4POBMHI_p>ON#>Q<A-=HJoum6S’ACBOS’V]_JoRPOBOSpSIm6SXBH’ACBOSXSIm]Q<HJBOS

D]Bƒj9>uSJBOS D8m U„oD]–ACB

¯

BKLHI_JcOBOS>=^SJB_INRQCPOBOS =qD]cAi>QCS

°

SJB_JouKLH”>m U„ouQ<K6S¦POouK6SJQCD]c_JcOS

SIHp>HIQCouK]K9>Q<_JBOS%:h‡ˆBOPQ1cHp>KLH>=_IQCBK5K6BqD8Q<HhVroum]_hAbFQ<K6SIHp>KRH lRm6BqACBqSJ•SIHJ–U„BqA<m]Q`bU?;U„B=

BH¦V9>_a`bAišU?;U„B¬ACBOS5Nu>_IQi>j]ACBOS+D]B¬VB_as­ou_IUk>K6POBOS+POouU4U„BƒACBOS5HJBU4V6S DFG>HIHJBKRHJB

BHED]B4SJc AIoum]_XBHXACBOS’V]_JoRPOBOSpSIm6SXD]B@VB_as­ou_IUk>K6POBOS

X

^BH

Q

=SJouQCBKLHWD]BOS’V]_JoRPOBOSJSJm6S SIHp>HIQCouK]K9>Q<_JBOS%:

=/> 8! I /"- #"+0+0 HD]B¬Ai>‹?6ACB¬DFG>HIHJBKRHJBƒBOSJHkAbF˜BK6SJBUWj]ACBƒD]BOSkNu>ACBm]_JS¦D]BOS

V9>_p>U„–HI_JBOS@D]B4Ai>¦?6ACB5DFG>HIHJBKLHJB

¯

ACouQCSWD]BOSM>_I_IQ<N-cOBOS>=}D]BOSYD]BUk>K6D]BOS@D]B„SJB_INQCPOBOS

BH4D]BOS4D]cAi>QCS

°

HJBA<ACBOS4lRm6B¦ACB+SI•SJHJ–U„BkV]m]QCSJSpB >HIHJBQ<K6D8_JB”ACB5_Jc¤#Q<U„B+SJHp>HIQCouK]K9>Q<_JB

¯

BHMD6>K6SYPOB„P>uS>=/Q<AdAbFG>HIHJBQ<K6D8_p>”stou_JPOcU„BKRHWm]K Aaoum]_

° : ‘.K€V9>_IHIQCPm]A<QCB_>=Ai>+ACouQ D8m

V]_JoPOBOSJSIm6S

X

D]BNQCBKLH”SJHp>HIQCouK]K9>Q<_JB=\BH+m]K PA<QCBKRH”lLm]QEBKRHI_JB D6>K6S+ACB‹SI•8SIHJ–U„B HI_Joum]N-BYHJoum&Aaoum]_JS^ACBESI•8SIHJ–U„BYš@AbF˜cHp>H\SIHp>HIQCouK]K9>Q<_JB=Sp>K6SqVrB_IHIm]_IjB_^POBH^cHp>H:ˆ=">

%'-+ /"+.8&! /"-95+.8&!.!#&++N’D]B

X

BOSJH—>ACou_JS—Ai>MACouQD]B\V]_Jouj9>j]Q<A<Q<HJcESIm]_

N

HJBA<ACBXlLm6B\Vroum]_

HJoum]H

t

š5V9>_IHIQ<_ED]B@AbFG>HIHJBQ<KRHJB„D]B@AbF˜cHp>HESIHp>HIQCouK]K9>Q<_JB=

X t

SpouQ<H’D8QCSJHI_IQ<j]m6c4SIm]Q<Nu>KRH POBHIHJBXACouQb:

^K”K6ouHJB_p>4=1š@AbF˜cHp>HTSIHp>HIQCouK]K9>Q<_JB„Š

• X ∞

=]Ai>MN-B_JSIQCouK¬SIHp>HIQCouK]K9>Q<_JBWD]B

X t

• Q ∞

=8Ai>@N-B_JSIQCouKƒSIHp>HIQCouK]K9>Q<_JBED]B

Q t

⁼

• T a

=8Ai>MN-B_JSIQCouK¬SIHp>HIQCouK]K9>Q<_JBWD8m”HJBU4V6SqDFG>HIHJBKLHJB=

• T s

=8Ai>@N-B_JSIQCouK”SIHp>HIQCouK]K9>Q<_JBWD8m”HJBU4V6SqD]BESJc AIoum]_:

• W

=8Ai>@N-B_JSJQCouK¬SIHp>HIQCouK]K9>Q<_JBED8m”HJBU4V6STDFG>HIHJBKLHJBWV]_JouVo#SJc#:

^K >V]VrBA<ACB 3 8+P!. I ( 8&! /*.+ ( 5+ 8&! DFm]K6Bƒ?6ACB‹DFG>HIHJBKLHJB SIHp>j]ACB=nm]KvVrouQ<KLH+D]B

_JBK6oum]N-BA<ACBU„BKRHTVroum]_ˆSJouK V]_JoPOBOSJSIm6S—D]B’POouK]¤-BOSIHIQCouK@=]P#F˜BOSIHqšWD8Q<_JB=8m]K¦Q<K6SJHp>KLHqo&R

m]KPA<QCBKRHkBKLHI_JB+D6>K6S@m]K SJ•SIHJ–U„B¦NQCD]B#: ^K“>V]VrBA<ACB¦V9>_k>Q<A<ACBm]_JS # / D ( D9(**=

ACBYHJBU4V6S’lRm]Q}SF˜cOPOoum]ACB@BKRHI_JBMD]Bm8ƒVouQ<KRHJS\D]BMPOouK6SIHI_Im6PHIQCouK€SIm6POPOBOSpSIQstS%:T¨^K # / D

( D9(*7WSJBXD]cOPOouU4Vro#SJBEBK

# I/ED ( D9(*

=

^{# /} D13<1+K+ 8>

+

^{+5 7 3 1+"+.8}

,

o&R¦Ai> # I/ED 3 1+"+.8 +BOSIHnACB\HJBU4V6SnV9>uSJSJcXBKRHI_JB’ACB’VouQ<KRHqD]BXPOouK6SIHI_Im6PHIQCouK¬BHnACB’V]_JBŽ`

U4QCB_YQ<K6SIHp>KRHWD]B„NQCD6>¤-BkBHYAbFC+5413 1+"+.8 BOSIHYACB„HJBU4V6SWV9>uSJSJc„BKRHI_JBkPOBHYQ<K6SJHp>KLH

D]BWNQCD6>¤-B@BHXACBMVouQ<KRHED]BMPOouK6SJHI_Im6PHIQCouKSIm]Q<N#>KLH: >_ED]cŽ?6K]Q<HIQCouKD]BMAi>¦SIHp>j]Q<A<Q<HJc

DFm]K6B¦?6ACB+DFG>HIHJBKLHJB=ˆm]K6B ?6ACB+DFG>HIHJBKLHJB¬SIHp>j]ACB¬>ƒV‚:˜S:¡m]K6B Q<K8?6K]Q<HJc”D]B VrouQ<KLHJS

D]B\POouK6SJHI_Im6PHIQCouK@=6BHnACBOSnD8m]_JcOBOSqD]BOS# / D 3<1+K+8>'/YSJm6POPOBOSJSIQ<N-BOSˆs­ou_IU„BKLHnm]K6BXSIm]Q<HJB

Qb:Qb:˜D:7=6DF˜BOSIVrc_p>K6POB’?6K]QCB#:

© SIm]V]Vro#SJB_ lRm6BTACB^SI•8SIHJ–U„B\>HIHJBQ<¤#K6B^m]KLAaoum]_ˆAbF˜cHp>HˆSJHp>HIQCouK]K9>Q<_JB=]ouK5D]cŽ?6K]Q<H

m]K”Q<U4Vou_IHp>KRH^Q<K6D8QCP>HJBm]_’D]BEVrB_as­ou_IUk>K6POBOSTVroum]_Tm]K6B’?6ACBWDFG>HIHJBKLHJB4š@VrB_IHJB=6Ai>

V]_Jouj9>j]Q<A<Q<HJcYD]B’VrB_IHJBYšMAbF˜cHp>HTSIHp>HIQCouK]K9>Q<_JB=1oum3,+38 #N #"+0+0 HI 3 -+" :

)+3-,./$0%$&1“(Cxt

ACD ; ACD@ E 9 ;- ) E9«/9{}7]{$&|t$0*@/3@«d3R%3 7 A

9<; "

π := P h

D E5 (; 9 9 7 E 7 >7 =

E

9 7 9(58A

9<5 ; /

7

; D

i .

^{¯8: °}

/oum]_}m]K6B.?6ACB DFG>HIHJBKLHJBn>ON-BOPˆPA<QCBKLHJS}Q<U4V9>HIQCBKLHJS =

π

BOSIH"D]ouK]K6cOB V9>_/Ai>qV]_Jouj9>j]Q<A<Q<HJc

š\AbF˜cHp>HhSIHp>HIQCouK]K9>Q<_JBTlRm‚Fm]KkPA<QCBKRHhD]ouQ<N-BT>HIHJBK6D8_JBqV]A<m6ShACouK]¤#HJBU4V6ShlRm6BnSJouKkD]cAi>Q

Voum]_J;HI_JBESpB_INRQdŠ

π = lim

n→∞ π n = lim

n→∞ P [W n > D n ]

= lim

n→∞ P [W n > D] = P [W > D] .

^{¯8:Gf °}

#²

"

.¬

5 oum6S BŽVro#SJouK6S QCPQ9ACBOSdV]_IQ<K6PQ<V9>m85_JcOSIm]A<Hp>HJS—BŽQCSIHp>KRHJS SJm]_hACBTSJm&AaBHhD]BOSh?6ACBOS—>ON-BOP

PA<QCBKLHJSqQ<U4V9>HIQCBKRHJS: 5 oum6SnV]_JcOSpBKLHJouK6STBKƒoum]HI_JBXACBOSTD8Q kPm]A<HJcOSTU„cHIŒ6oRD]ouACou¤#QClRm6BOS

SJoum]ACBN-cOBOSYV9>_MPOB„U„oD]–ACB=/POouU4U„B5m]K6B„Q<KRHI_JoRD8m6PHIQCouK š K6o#SMPOouKLHI_IQ<j]m]HIQCouK6S =.lLm]Q

SIm]Q<NR_JouKRHTPOBXPJŒ9>V]Q<HI_JB#:

=">WA<Q<HIHJc_p>HIm]_JB=9>jrouK6D6>KLHJBEBK+POB’lLm]QPOouK6POB_IK6B’ACB’D]ouUk>Q<K6B’¤-cK6c_p>A‚D]BOSˆ?6ACBOS

DFG>HIHJBKLHJB=}BOSIHEBK _Jc>A<Q<HJck>uSJSJB-44_JcOD8m]Q<HJBkD]–OSXACou_JSElRm6B@AbF˜ouK€Vo#SpB4D]BOSXA<Q<U4Q<HJBOSESIm]_

ACBkHJBU4V6S4DFG>HIHJBKRHJB+D]BOSMPA<QCBKRHJS: ^KD8QCSIHIQ<K]¤#m6B_p>QCPQ HI_JouQCS@H •RVrBOS@DFG>V]V]_JoRPpŒ6BOS>=

>m88lRm6BA<ACBOS’POou_I_JBOSIVouK6D8_JouKRHEACBOS\HI_JouQCS’V9>_IHIQCBOSXD]B@POBWPpŒ9>V]Q<HI_JB„DF˜cHp>HXD]BWAbFG>_IHXD8m

V]_Jouj]AC–U„B#:

=">V]_JBU4QC–_JB‹POouK6SIQCSIHJB š>uD6>V]HJB_¦Ai>HIŒ6cOou_IQCB‹PAi>uSJSIQClRm6B‹D]BOS5?6ACBOS¦DFG>HIHJBKRHJB

†/‡ˆ†d‰ >m P>uS5o&R ACBOSkPA<QCBKRHJS5SJouKRH„Q<U4V9>HIQCBKLHJS%: ^K“PJŒ6B_JPpŒ6BƒD6>K6S5POB”P>uD8_JBƒš

Sp>ON-ouQ<_„šƒlLm6BA<ACBOS4POouK6D8Q<HIQCouK6S@ACBOSMV]_JoPOBOSJSIm6S@lLm]Q—SJouKRHMACBOSMU„cHI_IQClRm6BOS@Œ9>j]Q<HIm6BA<ACBOS

DF˜cHIm6D]BXD8m”SI•SJHJ–U„B

¯

POouK]¤-BOSIHIQCouK@=9PJŒ9>_I¤-BED]B’HI_p>Nu>Q<Ab:<:<:

°

_JBOSJHJBKLHqB_I¤-oD8QClLm6BOSqBHqš

BK€D]cOD8m]Q<_JB4ACBOSEP>_p>uPHJc_IQCSIHIQClRm6BOSYD8m€SI•8SIHJ–U„B„š¦AbF˜cHp>HYSJHp>HIQCouK]K9>Q<_JB#:„=">¦HIŒ6cOou_IQCB

BOSIHM>uSJSpB-4„_Jouj]m6SIHJBŠMBK¡SIm]V]Vro#Sp>KLH4>mU„ouQ<K6SMlRm6B„ACBOSWQ<KLHJB_p>_I_IQ<N-cOBOS>=.ACBOSWHJBU4V6S

D]B5SJB_INQCPOB¦BHMACBOS@D]cAi>QCSWs­ou_IU„BKLH„D]BOS@SIm]Q<HJBOSMQb:Qb:˜D:7=douKVBm]H4>uD6>V]HJB_4POB_IHp>Q<K6BOS

D]BOSJP_IQ<V]HIQCouK6SYD]B4Ai> ?6ACB„DFG>HIHJBKLHJB¦PAi>uSJSIQClRm6B„†/‡ˆ†d‰‹BHYD]ouK]K6B_MD]BOSXs­ou_IUYm]ACBOSWBŽR`

V]A<QCPQ<HJBOSqVroum]_nAi>@SIHp>j]Q<A<Q<HJc=6Ai>MV]_Jouj9>j]Q<A<Q<HJcWD]BXVB_IHJBXoum¬ACB’HJBU4V6STDFG>HIHJBKLHJB#:

=">ƒD]Bm8QC–U„B >V]V]_JoRPpŒ6BkVro#SJBkACBkV]_Jouj]AC–U„B¦D]BOSE?6ACBOS@DFG>HIHJBKRHJB >N-BOP5PA<QCBKRHJS

Q<U4V9>HIQCBKLHJSkBKHJB_IU„B+DF˜ou_JD]ouK]K9>K6POBU„BKRHŠ¦lLm6BA<ACB+D8QCSpPQ<V]A<Q<K6B+D]B¦SpB_INRQCPOB+D]ouQ<Ha`

ouK>V]V]A<QClLm6B_YVoum]_EVrB_JD8_JB4ACB4U„ouQ<K6SWD]B„PA<QCBKLHJSWVro#SJSIQ<j]ACB ©^V]V9>_p>CHEQCPQdAbFQCD]cOB

lLm‚Fm]K6B\>m]HI_JBTD8QCSJPQ<V]A<Q<K6B^D]BTSJB_INQCPOBqlRm6BT†/‡ˆ†d‰ =LlRm]Q6V]_JBK]K6B\BKkPOouU4V]HJB^ACBOShD]cAi>QCS

D]BOSqPA<QCBKLHJS>=6VBm]HqHI_p>Q<HJB_TAi>@D]BUk>K6D]BYD]BXUk>K]QC–_JBXV]A<m6SqVB_as­ou_IUk>KLHJB#:

Ÿ A]cU„B_I¤-BnD]BOSHI_p>Nu>m8@_Jc>A<QCSJcOS.SIm]_ACB—SIm&AIBHlRm6BˆAi>\D8QCSJPQ<V]A<Q<K6BnD]BnSJB_INQCPOBˆ‘h[\†

BOSIHYouV]HIQ<Uk>ACB5š”V]A<m6SIQCBm]_JSEHIQ<HI_JBOS%:+‡ˆBOPQˆcHp>KLH>=POBOSE_JcOSJm]A<Hp>HJSWSJouKRHYVroum]_EAbFŒ6Bm]_JB=

SJouQ<HkVo#SJcOSkBK“HJB_IU„BOS„HJouHp>ACBU„BKRH¦D]cHJB_IU4Q<K]QCSJHJBOS

¯

ouK POouK]K9>CHkACB”SJPOcK9>_IQCoD]BOS

_JBOlRm<;HJBOS>=}ouKou_JD]ouK]K9>K6POBkACBkHI_p>Q<HJBU„BKLH@D]BOSEHuPpŒ6BOSYVroum]_YD]ouK]K6B_YAi>+U„BQ<A<ACBm]_JB

_JcVrouK6SJB

°

=SJouQ<HTV]m]_JBU„BKRHEBŽVrc_IQ<U„BKLHp>m8ŠˆQ<A>V]V9>_p>CH’lRm6BEACBOS’>_I¤#m]U„BKLHJS’N#>A`

>j]ACBOS@Vroum]_@AbF˜cHIm6D]B V]_Jouj9>j]Q<A<QCSIHJB+DFm]K6B5?6ACB >N-BOP+PA<QCBKLHJS4Q<U4V9>HIQCBKRHJS4¤-c_JcOB¦V9>_

†/‡ˆ†d‰¦K6BYSJouKLHqV9>uS^>uD6>V]Hp>j]ACBOS^š@m]K”SI•8SIHJ–U„BX‘d[’†V9>_TBŽ]BU4V]ACB#:

5 oum6S”V]_JcOSpBKLHJouK6SD]ouK6P m]K6B€HI_JouQCSIQC–U„B¡>V]V]_JoRPpŒ6B=EPOBA<ACBD]B€Ai>

1

N& M+P2O

< *+P! " 6 38 +ED

¯£

°4¯

[’=/‰8²8; °

ŠEBA<ACB4V]_JouVro#SJB4m]K V]_JoPOBOSJSIm6SWSIHJoRPpŒ9>uSa`

HIQClRm6B+ou_IQ<¤#Q<K9>AqVroum]_„D]cOP_IQ<_JB+m]K6B¦?6ACB+DFG>HIHJBKLHJB>ON-BOP”D]cAi>QCS4K6ouK cA<Q<U4Q<K9>HJouQ<_JBOS

_Jc¤#QCBXV9>_^‘d[’†ˆ:

‡ˆB’PpŒ9>V]Q<HI_JBXSF˜ou_I¤L>K]QCSJB’POouU4U„B’SIm]Q<H\ŠBK V9>_IHIQCBXf8:<;=K6oum6S V9>uSpSJouK6S—BK¦_JBNm6B

ACBOS5_JcOSIm]A<Hp>HJS BŽQCSJHp>KLHJS¦SIm]_kACBƒV]_Jouj]AC–U„BƒD]BƒAi>‹?6ACB†/‡ˆ†d‰ >ON-BOPPA<QCBKLHJS¦Q<U4V9>z`

HIQCBKRHJS: ‘.KV9>_IHIQCB+f8:4=K6oum6SWV]_JcOSpBKLHJouK6S4AbF˜cHp>H4D]B5AbFG>_IH@SJm]_WAi>ƒlLm6BOSJHIQCouK¡D]B¦Ai>

POouU4V9>_p>QCSJouK¡D]BOS’VB_as­ou_IUk>K6POBOSYBKRHI_JB4ACBOSED8QCBc_JBKLHJBOSWD8QCSJPQ<V]A<Q<K6BOSED]B„SJB_INQCPOB4BH

BKƒV9>_IHIQCPm]A<QCB_\AbF˜ouV]HIQ<Uk>A<Q<HJcMD]BEAbFG>A<¤-ou_IQ<HIŒ]U„B@‘d[’†—:9‘dK¬V9>_IHIQCB@f8:Gf4=6K6oum6S^V]_JcOSJBK8`

HJouK6SWAi>+V]_Jouj]ACcUk>HIQClRm6B D]B„AbF˜cHIm6D]B„V]_Jouj9>j]Q<A<QCSJHJB¦DFm]K6B„?6ACBkDFG>HIHJBKRHJB5_Jc¤#QCBkV9>_

‘d[’†BH@Ai>‹POouU4V]ACBŽQ<HJc+D]B+POB¦U„oD]–ACB#: 5 oum6S@N-o%•-ouK6S„BK SJBOPHIQCouK“f8:Gf8:ACBOS@_JcŽ`

SIm]A<Hp>HJSXBŽ8Vrc_IQ<U„BKLHp>m8€BŽQCSJHp>KLHESIm]_’ACB@P>ACPm]AdD]BOS’VB_as­ou_IUk>K6POBOSEDF˜‘d[’†vBHEBK

SJBOPHIQCouK f8:Gf8:Gf4=LK6oum6S cN-oRlRm6ouK6Sdj]_IQC–N-BU„BKRH Ai>EU„cHIŒ6oRD]B^ou_IQ<¤#Q<K9>ACBTD]BTAi>X£ “BH

D]ouK]K6ouK6S\AbFQ<KLHJc_);H’HIŒ6cOou_IQClRm6BWlRm6BYPOBHIHJBWU„cHIŒ6oD]BYVrBm]H^V]_JcOSpBKLHJB_’D6>K6S^ACBWP>uD8_JB

D]BXK6ouHI_JBEV]_Jouj]AC–U„B#:

-. ). )' + ' ) () %!)

[’BV]m]QCS¦ACouK]¤#HJBU4V6S

¯

‘._IAi>K]¤<=

‘d_IAb;%™

°

=nouKvSp>Q<H¦cHIm6D8QCB_ >N-BOPƒD]BOS5V]_JoRPOBOSpSIm6S

Uk>_32-o%NRQCBK6S SIQ<U4V]ACBOS =EAbF˜cHp>H‹SIHp>HIQCouK]K9>Q<_JBDFm]K6B€?6ACB¡DFG>HIHJBKLHJB ¥ PAi>uSpSIQClLm6B §

M/M/s/s + B

¤-c_JcOB„V9>_@†/‡ˆ†d‰ o&R

B

BOSIHYAi>”Hp>Q<A<ACB=/A<Q<U4Q<HJcOB¦BHWD]cHJB_IU4Q<K]QCSIHJB=

D8m j]m4BB_:±‡ˆBOS”U„oD]–ACBOS”SJouKRH”ACBOS+U„oD]–ACBOS¬šVrB_IHJBOS+ACBOS+V]A<m6S”SJQ<U4V]ACBOS: ^K

D]cOP_IQ<HqV9>_asb>Q<HJBU„BKLH^ACBXPOouU4Vou_IHJBU„BKRH^D8m”V]_JoPOBOSJSIm6SnD]BXPOouK]¤-BOSJHIQCouK

(X t , t ≥ 0)

BH—ouK¦BK+D]cOD8m]Q<H—lLm6B^ACB\SI•SJHJ–U„B\>HIHJBQ<KRHˆAi>WSIHp>HIQCouK]K9>_IQ<HJcXSp>K6SˆPOouK6D8Q<HIQCouK”SIm]_ˆSJBOS

V9>_p>U„–HI_JBOS:}>_T>Q<A<ACBm]_JS =8Ai>YV]_JouV]_IQCcHJc 1©X‰ "© >ON-BOP’AbF˜B_I¤-oD8QCPQ<HJcXD]B

X

^VrB_IU„BH

D]B¦D]ouK]K6B_WAi>”V]_Jouj9>j]Q<A<Q<HJc D]BkVrB_IHJB5š”AbF˜cHp>HMSIHp>HIQCouK]K9>Q<_JB¦V9>_WAi>”V]_Jouj9>j]Q<A<Q<HJc+š

AbF˜cHp>HTSJHp>HIQCouK]K9>Q<_JBYlRm‚FQ<A• >Q<H

s + B

PA<QCBKRHJS^D6>K6SqACBXSI•SJHJ–U„B#:

="B”D]cAi>Q

¯

oum 3 5+*!#( *=—SIm]Q<N#>KLH5ACBOSk>m]HJBm]_JS

° > BBOPHJc¬>m8 PA<QCBKLHJS¦>V]V9>_p>CH

POouU4U„BXm]K6BEA<Q<U4Q<Hp>HIQCouK

¯

Uk>QCSqD]BXHp>Q<A<ACBE>ACc>HJouQ<_JB

°

D]B’Ai>@P>V9>uPQ<HJcYD8m”SI•8SIHJ–U„B#:

G >_I_JB_

¯ G >_pe-™

°

>4=ACB@V]_JBU4QCB_>="Spoum]ACBN-c@ACBWV]_Jouj]AC–U„B„Voum]_Xm]K6BM?6ACB4DFG>HIHJBKRHJB

M/M/s/s+ D − F CF S

Š"VBm]Ha` ouK5>uD6>V]HJB_hACB _JcOSJm]A<Hp>HdV]_JcOPOcOD]BKLH Voum]_.HI_Joum]N-B_

Ai>V]_Jouj9>j]Q<A<Q<HJc‹D]B¬VrB_IHJBƒDFm]KvPA<QCBKLH ¢=">_JcVrouK6SJBƒBOSJHkK6ouK@=qV]m]QCSplLm‚FG>ACou_JS¦ACB

V]_JoPOBOSJSIm6SWD]B5POouK]¤-BOSIHIQCouK

X

K‚F˜BOSIHYV9>uSYUk>_32-oNQCBK‚: G >_I_JB_MPOouK6SJHI_Im]Q<HM>ACou_JSMm]K U„oD]–ACB I 698>(MVroum]_TACBN-B_XPOBHIHJBWQ<K6D]cHJB_IU4Q<K9>HIQCouK‚:Y‡ˆBEsb>QCSp>KLH>=Q<A}BŽŒ]Q<jrBYAi>kACouQ

SIHp>HIQCouK]K9>Q<_JBWD8m”K6ouUYj]_JBMD]BEPA<QCBKRHJS^D6>K6SnACBYSI•SJHJ–U„BEBHTD]ouK]K6B

π

Ai>@V]_Jouj9>j]Q<A<Q<HJc

D]B’VB_IHJB„Š

π = 1

s! ρ s e (λ−sµ)d P [X ∞ = 0] ,

^{¯f8:<; °}

o&R

d

BOSJH„Ai>‹N#>ACBm]_5D]BOSkD]cAi>QCS5Q<K]Q<HIQi>m8: &’K6BOD]BK 2-oBH ^oN#>ACBK 2-o

¯ & Ew#› °

ouKLHMVouQ<KRHJc¦BHMPOou_I_IQ<¤-c D]BOSMB_I_JBm]_JS@D6>K6SMAi>+V]_JBm]N-B D]B

¯

f8:<;

°

=HJoum]HMBK¡N#>A<QCD6>KLH

Ai>+s­ou_IUYm]ACB#:#m]_32-BNRQCP

¯

#m]_ ™; °

>BK6SIm]Q<HJB cAi>_I¤#QqPOB¦_JcOSIm]A<Hp>H4BH4D]ouK]K6c5Ai>ƒACouQ

D8m5K6ouUWj]_JB\D]B\PA<QCBKLHJSqšYAbF˜cHp>H—SIHp>HIQCouK]K9>Q<_JB\ACou_JSJlLm6B^ACBOS—D]cAi>QCS—SJouKLHˆACBTU4Q<K]Q<UWm]U

DFm]K6B\POouK6SJHp>KLHJBEBHnDFm]K6B\ACouQ‚BŽVrouK6BKLHIQCBA<ACB#: }>_T>Q<A<ACBm]_JS>=]†}Q<K6PpŒ¬>WD]ouK]K6c’Ai>WACouQ

D8m HJBU4V6SYDFG>HIHJBKLHJB5D6>K6SEm]K6BM?6ACB

GI/M/1/1 + D − F IF O

^¯ †/Q<K9w#² ° :„[’BOS

POouK6SIQCD]c_p>HIQCouK6SXD8m¬U?;U„BMH •VrBYouKLH’U„BK6c@>mP>ACPm]AD]BWAi>„V]_Jouj9>j]Q<A<Q<HJc@D]BWVB_IHJB

DFm]K6BY?6ACB

M/G/1/1 + D − F IF O

^{¯To 2‹BH —QA U„S>= [ q›#e#> ° BHXDF}m]K6BM?6ACB

M/G/1/1 + D − F IF O

>ON-BOP5D]cAi>QCS Aam6SJlLm‚FGš+Ai>¦?6KD8mSpB_INRQCPOB

¯

[ q›#ej ° :

‡ Q<HJouK6STBK6POou_JB

‡ˆouŒ9w#› =

&\Œ6o-w#f /BH

/> 26™ ž "Vroum]_TDFG>m]HI_JBOSq_JcOSIm]A<Hp>HJS^SIm]_qD]BOS

U„oRD]–ACBOSqD]BYPOB’H •VB#:

/oum]_\Q<K]Q<HIQCB_\Ai>4¤-cK6c_p>A<QCSp>HIQCouK‹D]BYPOBOS^_JcOSIm]A<Hp>HJS’š4m]K6BX?6ACB

G/G/1/1 + G − F IF O

^{= =‚=/™uf} .D]ouK]K6BWm]K6BWjrou_IK6BMSIm]Vc_IQCBm]_JBMŒ6Bm]_IQCSIHIQClRm6BWVoum]_\Ai>5V]_Jouj9>j]Q<A<Q<HJc D]B’VB_IHJB„Š

π ≤ P[σ > U ].

^{¯f8: °}

/oum]_^m]K6B’?6ACBEDFG>HIHJBKRHJBWPAi>uSJSJQClLm6B

G/G/1

_Jc¤#QCBXV9>_T†Ÿ † =@=]ACBED]BOSJP_IQ<V]HJBm]_

DF˜cHp>H@BOSJHMAi>SIm]Q<HJB¦D]BOS@HJBU4V6S4DFG>HIHJBKLHJB”V]_JouVo#SJcOS4>m8PA<QCBKLHJS

¯

oum8 + 8>9

°

(W n , n ∈ N )

:.=dF˜cOlRm9>HIQCouKƒD]BX=Q<K6D8ACB•”D]ouK]K6BEVoum]_nHJoum]H

n

^Š

W n+1 = [W n − U n+1 + σ n ] + .

^{¯f8:Gf °}

=dF˜cHIm6D]B D]BkAi>¬SIHp>j]Q<A<Q<HJc D]BkAi>+?6ACB

G/G/1

^D8m6B š¬=‚o%•RK6BOS

¯

=‚o•]w °

m]HIQ<A<QCSJB¦POB

V]_JoRPOBOSJSJm6S—_JBUk>_JlLm9>j]ACB#: =‚o•K6BOSqV]_Joum]N-BWlLm‚FGšMAi>4POouK6D8Q<HIQCouK

E [U ] > E [σ] ,

^{¯f8:ž °}

ACBTSI•8SIHJ–U„B^>HIHJBQ<KRH—m]K5_Jc¤#Q<U„B\SIHp>HIQCouK]K9>Q<_JB=]BH D]cOP_IQ<HˆACB^_Jc¤#Q<U„B^SIHp>HIQCouK]K9>Q<_JB’D8m

SI•SJHJ–U„BE>ON-BOP

W

=8Ai>MACouQ"SIHp>HIQCouK]K9>Q<_JBWD8m8 + 8>I#:

£TBN-BK6ouK6S >m“HJBU4V6Sa`b_JcOBAb: G >uPOPOBA<A<Q’BH \cj]m]HJB_IK6B

¯ G \›8; °

BH G >uPOPOBA<A<Q-=

G

o%•-B_}BH\cj]m]HJB_IK6B

¯ GG

’›

ž °

U„ouKLHI_JBKRHdlLm6B ACB8 + 8>9MDFm]K6Bh?6ACB DFG>HIHJBKRHJB

GI/GI/1/1 + GI − F IF O

BOSIHhm]K6B\PJŒ9>CK6B’D]B#>_32-o%N5šEV9>_IHIQ<_—DFm]K6B\ouj6SJB_INu>z`

HIQCouKƒSIQ<U4V]ACB¦ŠhACB8 + 8 9BOSIHTACBYU?;U„BYlRm6BYPOBA<m]QDFm]KSI•8SIHJ–U„BEo&RƒACBOS^PA<QCBKRHJS>=

lLm]Q.POouK]K9>QCSJSJBKRHWš5ACBm]_Y>_I_IQ<N-cOB@Ai>¦Nu>ACBm]_XD8m8 + 8>9

¯

6 +N ( /"8&2O1+K/

°

=‚K6B

_JBKLHI_JBKLHXB"BBOPHIQ<N-BU„BKLHXD6>K6S\ACBYSI•8SIHJ–U„BWlLm6BWSIQ"ACB8 + 8>IƒBOSJHTV]A<m6S^VBHIQ<H\lLm6B

ACBm]_’D]cAi>Q

¯

>m6lRm6BA.P>uS^Q<ACS’SJB_JouKRH’stou_JPOcU„BKRHESJB_INQCS

°

:’¨TK6BW?6ACB4>ON-BOP 6 +)?( I/"Q

8&2O1+ /„K‚F˜BK6D8m]_JBkD]ouK6P4V9>uSED]B4VrB_IHJB„D]B4PA<QCBKLHJSY_JBKLHI_JcOSMD6>K6SXACB„SI•8SIHJ–U„B+Š’m]K

PA<QCBKLHXVrBm]H\Sp>N-ouQ<_ED]–OS\ACBMD]cj]m]HES%FQ<A/Vroum]_I_p> >HIHJBQ<K6D8_JB@ACBMSpB_INRQCPOB@oum‹K6ouK€>ON#>KLH

AbF˜BŽV]Q<_p>HIQCouK D]BkSpouK¡D]cAi>Q-=.BHMK6B„_JBKRHI_JB5V9>uS@SIQ >”K‚F˜BOSJHYV9>uSMACB5P>uS%:¬=‚B8 + Q

8>I‹D]BWAi>k?6ACB„>N-BOP 6 +)O( /"82?-+ / BHED]ouK6P=POBA<m]QdD]BMAi>5?6ACB4>ON-BOP !#6 +)

( /"82?-+ / °

N-c_IQ?9B\>ACou_JSˆm]K6B\cOlLm9>HIQCouK¦Uk>_32-oNQCBK]K6BXD8mkU?;U„B\H •RVrBTlRm6B

¯

f8:Gf

° Š

W n+1 = [W n + σ n