So Aedx Te og Ia o a Mxe o Ode-Sea a Maae-Reale Sye Aa V Iye ad Ava Ja Teoe 4: e ad q be e obably geeag o o e eady-ae be o ode ee e ye by a avg H ode ad a M ode eevely Te ad q Wee ad be e ee oo o e ollowg b o : Te aaee v a eae o vaably o Reale H dead oe wee < < v Poo o Teoe 4: Fo oo See Ka97 o o a led aoa leadg o e exa exeo gve e eoe ee below Te oey a exloed aaly a bewee wo oeve Reale H ode aval e ye Maova T N be o e ode e aae ye a ee by e ode aval o Reale H e: P N x N- x N- x N- x N-4 x4 PN x N- x ad eeoe be o ode e ye a ee by a Reale H ode aval a ebedded Maov a ee: Tao obable o e bedded Maov a PN N Seady-ae obably a a H aval ee be ye Pobably Geeag Fo PGF o eady-ae dbo {} Alo dee γ Tao obably o o M/M/ qee w gve e obably o be ye a e gve a ee wee be ye a e Γ Pobably Geeag Fo PGF o γ Γ alae Tao o Γ e Γ d Had e e bewee wo obevao e bedded Maov a bee oa T wold be eqal o γt Hee T a ado vaable w a a Hyeexoeal dbo w F A e e ad eeoe γ da γ Te eady-ae obable wll ay aa-kologoov eqao
da γ da da Γ γ ee da e d a e ST o d a da e d e a α β wee α e ST o da Teeoe d e a Γ Ug e olex ovolo Teoe w ay a e alae Tao o e od o e wo o e olex ovolo o e ao o e wo o we a we : d d a α Γ π β Γ π ~ ~ wee ~ ay ove be ad Wle e ao obably o o a M/M/ ye volve oded eel o ad a a olaed o e alae Tao o e PGF o e ao obable oaavely eay o exe ad gve a : Γ wee λ λ λ λ 4 λ λ e aval ae ad e eve ae o e M/M/ ye Teeoe: d α π wee We ow α Ug ay Iegal Fola w ay a a loed a a eloe e ole ad e egao ae a lowe ee e o d π we a we Og e eveg algeba: { } { } H H N wee H H / / / ad gve e aaee o e wo aval oee ad e eve oe H ad H ae oa We oe a o e ae o e Hyeexoeal dbo w balaed ea e exeo e deoao a be led a:
wee a eae o vaably o Reale H dead oe wee < To olve o H ad H we eed wo boday odo Fo e odo by deo o a PGF e be e ee oo o e N H H Fo e eod odo we oe a a a eal oo ad o ee e aalyy o e ego < we ave N H H H H Ug ee wo boday odo o olve o wo ow H ad H lggg e vale ba ad lyg ee a ee: q Seady-ae obably a a Reale M ode aval wll ee ode e ye T alo e e-aveage o e be o ode e ye e e be ye a ee a a ado o e q q e g o dbo {q} A eady-ae e ae o o ae eqal o e ae e ae eeoe: ν λq q ν λq q q q Teoe 4: Te ead-e o Reale M a a Hyeexoeal dbo o degee ad eade o Reale H a a Geealed Hyeexoeal dbo o degee Moeove e wo dbo de oly e weg a ae aged o e ae wo ooe xoeal dbo A e A e ad e e F Wee ad q A F A ad A A Poo o Teoe 4: Te lead-e exeeed by a eale e ae a e eady-ae e-ye e by a ode o a eale e aae qeg ye Fog o Reale M a ala avg ode o M wll ee a oal o oe ode e ye w obably q w g q ad a wll deee e lead-e exeeed by ala ode Te alae ao gve a: I q I aval ee oal ye Gve a a aval ee ye e wag e o aval wll be e eve e o oe a -old ovolo o e xoeal eve e dbo Addg ow eve e e e--ye o aval wll be old ovolo: I q q q
Slyg o Teoe 4 we e aal-ao exao o q a o a eay o ve: q q Wee: I Teeoe: q Ae oe lao e above ao a be veed a ollowg: q e e e Te ex e al o obag e el o ow a e e above exeo eo q / Now we a we e o be--ye ee by a M aval a: q ad e d o lead-e a be exeed a: e e Fe o ove a e above dbo a Hyeexoeal dbo we eed o ow a > > To aol we develo a odeg bewee dee oo o ad ye aaee Te odeg < < < < < < M ad o ove we < eall a < < < < Nex evalae e o a dee vale o A < > / > / < < Fally a I lea a oe e eo-le a ee eal ad d o: bewee ad oo e bewee ax / ad / oo ad e ae / oo Se >/ ad we ow a / eeoe </ Pg ee all ogee we ge e deed odeg Noe a alo ove a a ee eal ad d oo ad eeoe we a e e ada ola o oba loed-o exeo o e ee oo o Plggg odeg exeo o above ove a > > Te ae e we aed o o Reale H yeld e o be--ye ee by a H aval: A A ad e d o Reale H lead-e: 4
A e A e I ae A > A > ad ee a egave g bewee wo xoeal ooe a a a Geealed Hyeexoeal dbo Valdao o Teoe 4 Rel: : I e ae v e we Hyeexoeal dbo degeeae o a xoeal dbo e lead-e dbo deved above ae eded o a a M/M/ ye Poo o : A v we ee a ad eeoe o e odeg bewee oo oved Teoe 4 aed o dee elag oo o a b eqao o oee eay o ow a w ove T lead o A Teeoe A ad bo dbo ollae o e ae xoeal dbo w e ae λ ν w oe w e M/M/ ye el : I e ae e we e Poo aval daea e lead-e dbo deved above ae eded o a a H/M/ ye Poo o : A we ee a ad eeoe A A ad e lead-e dbo o Reale H ollae o a xoeal dbo w ae Ug e adad aaly o a GI/M/ ye we ge exaly e ae el o a H/M/ ye eo 4: eo ad deo o oe oa ode ed ae ae a ollowg: ode wo ado vaable: X w dbo o F ad Y w dbo o G e: I F G e X ad o be alle a Y e oa ode ee X Y I X Y e X ad o be alle a Y e eag ovex ode ee X x Y I X x Y ad X Y e X ad o be alle a Y e ovex ode ee X x Y 4 eve ode aleady deed I e ollowg we ovde a oal do abo ow e vao oa ode ae oeed o e oal exeed veoy o e X ad Y eee wo lead-e dead ado vaable ad ode a eqaly o e o φ X φ Y I e ga o F alway above e ga o G e X Y ad e eqaly e o all eag o φ T oly dae a Y lage a X ad doe o oe abo e vaably Teeoe o e o eabl e o doae I e ga o F ee alway above e ga o G o oe oly oe o below e X x Y ad e eqaly e o all eag ovex o Te evo odeg le ad aga odeg o e o ow e o doae Howeve o e ae baeo level baode ae a eag ovex o o e lead-e dead ad bo ee ode a be ed o ow e exeed baode doae I e ga o F oe e ga o G oly oe o below ad e ea ae eqal e X x Y ad e eqaly old o all ovex o T dae a Y oe vaable a X ad e gve e ae bae-o level e veoy o ae a ovex o o e leade dead ode a be ed o ow e exeed veoy o doae 4 I we e ga o F o g by ay dae ad e ed ga o F ee eve oe 5
e ga o G o oe oly oe o below e 6 X d Y Idee o e e o e ado vaable odeg oae e vaably Te oal o doae a ll be ow ee Teoe 49 beae e veoy o doe o ave ay ooe a deed o e abole e o e lead-e dead eo o g-age o S : e be a eal o deed o a odeed e U o e eal le e S be e be o g age o Ta : S S wee e e exeded ove all e < < < U abay b e ad S x x x e be o g age o e eqee x x x eo e beg daded Teoe 44: Fo o HM/M/ odel e lead-e exeeed by Reale H oe deve a e lead-e exeeed by Reale M e ee Poo o Teoe 44: Te eeay ad e odo o deve odeg exeed e o g-age o a: S F F o all w e g eqee beg - e ae o eqaly a ay g-ed veo o Reale M lead-e d old o e Reale H lead-e d a o oe o below Fo a gve le g- e d o o eae a ado vaable w d gve a F F xe e d deee o a ollowg: F F { A e { e } e { A e e } e d } { A e A e Now we ow a by oo o F F ad e deee o above wll be egave We alo ow a A > A > > > aed o e odeg above we a alo ow A Te eod e e d deee o alway egave ode wo ae ae wee A > e ad ae wee A e We ow a e g-age o ae e eqed odo bo ae I ae e e e d deee o ove We ow a a e d deee o egave A eae ee wll alway ea egave o wll o e eo le o e e ay a Te a { A e Alo e > e > Teeoe } { } e A e e { } { } A e e > A e e > ad e d deee o alway ove o all > Pg ogee S F F ad ae o eqaly e g eqee - Nex ode e ae wee A e I ae d deee o alway egave S F F oollay 45: x S S }
Poo o oollay 45: ode e ae we g- e oo o Teoe 44 Se A all o ae Addoally we oe a a e d deee o eo Pg e age ae ove a e d deee o alway ove o all > Ta F F o w ove Te by deo o oaally lage ode ode e ae we g- A oved Teoe 44 S F F Howeve S F F a o be e beae a wold ea > w o e Teeoe S F F a e d o oe e d o exaly oe o below w ove x y deo o oaally ovex ode e el b ove S S Pooo 46: V ad V Poo o Pooo 46: Te Reale M lead-e Hyeexoeal ad well-ow a oee o vaao wll be geae a Reale H lead-e Geealed Hyeexoeal w o gaaeed o gve a oee o vaao geae a To ove V < wll be e o ow a > a A A A > A a A A > A A xadg e le ad de Ug A A ad eaagg e odo ede o > a w e > Teoe 47: Te obably a o o e wo eale oadg ode ae: g o g α α A A o α α α A α A 7 o Poo o Teoe 47: To ow e dboal o o e le law alably o ye we ee o e ea ad Naaao 995 Teoe ad oe a e o ye ae e odo aed ee e dbo o e be--ye lead-e dead o a la e ae a e dbo o e be o aval a ado e--ye o e eqlb aval oe o a la I e ae o Reale M we e e Kelo ad Sev988 el o e Poo oe Te g o e lead-e dead gve a G F λ λ wee e alae ao o e oo-e Iveg e g g I e ae o e Reale H we e a el o Jewell 96 a elae e alae o e o e be o aval eqlb oe e g o e alae o e eaval e dey α o ae a ollowg: F
α e g ν o α ν α o g d d A e d d A e d d A obeved e Teoe 4 e o o e lead-e a obao o e exoeal e w ae oble o ly g wo ay eed o ve e ao Pooo 48: I e dead ee ae e ae e υ λ e oadg ode o e Reale H ad Reale M ae odeed a eag ovex ee a Poo o Pooo 48: e M eee e be o Hyeexoeal eewal dead aval w ey ν e ad le M eee e be o Poo dead aval w ey λ e Te a Teoe 47 e ae a M ad e ae a M Now o ow x wll be e o ow a a x M ad b M x T aoled ollowg wo e a Te Hyeexoeal dbo beg a xe o xoeal a FR eeag Fale Rae dbo w le e NWU New Woe a Ued xeao oey ee Ro996 Ug e el o Ro996 eo 96 we ow M x M Noe a v odeg e ae a e x odeg y deo o x ode M x M P M > y y P M > y y Teeoe y P > y P M > y df y y P M > y df y x P M > y b We ow o Saed ad Saa994 a o > M M y deo o oaally lage ode M M P M > y P M > y P M > y a eag o o We alo ow o oollay 45 a w by deo ea a o ay eag o φ φ φ Teeoe: P M > y P M > y P M > y P M y > y P obg a ad b ove x oeexale ae Pooo 48: aaee: v τ 9 5 ewboyale 4 x > y M el: S 5 ad S 4 Teeoe eve og e S S o e w ea a e oge ode oaally lage ode o e oadg ode o e Howeve a we owed oal o doae old 7 67 6 5 Teoe 49: I e dead ey ν o Reale H ad e dead ey λ o Reale M ae odeed a ν λ e e oal exeed veoy o ae odeed a Poo o Teoe 49: M Fo lay aee ode ee agay bae-o veoy ye: Reale w lead-e lead-e dead oadg ode ad oal exeed veoy o We ove e al el e ollowg o e x 8
9 a Reale M exeee a Poo dead oe w ey λ ad exeee a lead-e e Reale exeee a Poo dead oe w ey λ ν ad a lead-e ae a Reale M We ow below a b e Reale exeee a Poo dead oe w ey ν ae a Reale ad a leade We ow below a e Reale exeee e ae Poo dead oe w ey ν a Reale ad a leade We ow below a d Reale H exeee a dead oe a Hyeexoeal eewal w ey ν ad a leade We ow below a obg e above el Poo o a a: We ow o Teoe 47 a Reale M lead-e dead oadg ode gve by g o T a alo be we a ob w ob w wee x a dee ado vaable w Geoe x x ad ea eqal dbo Slaly ob w ob w Gve λ ν we ow a w e odo o d y deo o deve ode x ee Mlle ad Soya6 Slaly x Se ovex ode loed de xe ee Saed ad Saa994 Teoe A6 b we a we x Gve a ado vaable X ad a oa le e ovex exeed veoy o o be X b X X X Alo le X X aeve a X Now we a we by oey o ovex ode by deo o X
b wee e la le ay alo beg w g Poo o a b: Te lead-e dead oadg ode o Reale eeeed by Te lead-e dead o Reale wll be Ρ ν wee Ρ x eee Poo ado vaable w ea x Nex oe a by Teoe A9 o Saed ad Saa994 we a we ν x Ρ ν y Teoe A6d o el 5A o e ae eeee we a we ν x Ρ ν Ta x Te ae aaly a a a lead o Poo o a : We ow o oollay 45 a x T Reale ad Reale exeee e ae Poo ead w ey ν ad Reale lead-e alle e ee o ovex ode a e lead-e o Reale Teeoe x ad T ollow o Sog994 Poo o a d: odoed o a e lead-e ealao l ad ag a e e Reale H lead-e dead l a e dbo o eqlb Hyeexoeal eewal aval e l ad Reale lead-e dead l a e dbo o Poo aval e l bo w ae ey ν Te eexoeal dbo beg a xe o exoeal a FR dbo w le e NWU oey ee Ro996 Teeoe g e el o Ro996 eo 96 Teoe 964 we ow l x l Noe a v odeg e ae a e x odeg w oled w e a a e ea ae eqal le x odeg Se ovex ode loed de xe ee Saed ad Saa994 Teoe A6 b we a we x Te ae aaly a a a lead o