Chapter 2 The Poisson Process

Transcription

1 Chapr 2 Th oisso rocss 2. Expoial ad oisso disribuios 2... Th Birh Modl I scods, a oal of popl ar bor. Sarig a ay poi i im, wha is h waiig im for h firs birh? I milliscods, a oal of lpho calls arriv a a swichig sysm. Sarig a ay poi i im, wha is h waiig im for h firs call? A paricl ravls i a sraigh li io h cosmos. Assum ha i yars, i will cofro objcs. Wha is h waiig im for h firs objc? T Wihou furhr iformaio w ca oly assum ha h objcs ar idpd, ach occurrig a a im rprsd by a uiformly disribud radom variabl ovr [, ). Thus w hav uiform i.i.d. o b dod as X,..., X.

2 2 Th waiig im for h s objc is T mi{x,..., X } // This is also calld h firs ordrd saisic. // Covioal oaio for h h ordrd saisic is X (). Th disribuio of T ivolvs boh ad. (T > x) (x/) L b vry larg. Th, T wih probabiliy, i.., lim ( T > x) for all x > L b vry larg. Th, T wih probabiliy, i.., lim ( T > x) for all x Nihr is irsig. So, w cosidr lim ( T x) > wih h irpraio ha ad grow proporioally oghr: Th birh modl wih isiy : I ui ims, hr ar birhs. Fix h raio ad l. Thus. // isiy pr scod, pr miu, pr yar, c.

3 3 Thus, amog idpd uiform r.v. ovr h irval [, ), h miimum has h followig disribuio: // Cosidr oly hos valus of such ha is igral. lim ( T x) lim( x / ) > x Th disribuio fucio is F(x) dsiy fucio is f(x) x xpoial disribuio wih ra. x, ad h. This is calld h Wih birhs i ui ims, h umbr N of birhs i ui im has h Biomial(, /) disribuio. ( ) ( + )! ( ) ( + )!( )

4 4 As, (N ) /! for. This is calld h oisso disribuio wih h dsiy, which is h limi of Biomial(, /) disribuio as. I coclusio, waiig im for h firs birh is xpoial(), umbr of birhs wihi a ui im is oisso().

5 Ma of Expoial & oisso Disribuios Thorm. oisso disribuio wih h dsiy oisso disribuio wih h ma of pr scod roof. L N b a oisso disribud radom variabl wih h dsiy pr scod. E[N ] (N ) /! / ( )! /! // shif idx: pr scod Thorm. Expoial disribuio wih h ra Expoial disribuio wih h ma of / scods

6 6 roof. L M b a xpoially disribud radom variabl wih h ra pr scod. // h oaio M suggss mmorylss i h squl x E[M ] x( d x) x x [ x( )] ( ) d x // igraio by par x [ / ] [ /] / scods

7 Mmorylss propry of xpoial disribuios Th xpoial disribuio is h coiuous aalogu of h gomric disribuio i big mmorylss. Thorm. A xpoial disribud r.v. X is mmorylss. Mor prcisly, for all a ad x, (X>a+x X>a) (X>x) // Havig waid for a ui ims, h probabiliy // of waiig for x mor uis is uchagd. roof: (X>a+x X>a) // L x. (X>a+x) / (X>a) ( x+ a) a / // l b h para. i xp. disr. (X > x) Thorm. Covrsly, vry mmorylss coiuous disribuio is a xpoial disribuio. roof. L X b a r.v. such ha (X>a+x X>a) (X>x)

8 8 for all x ad a. Equivally, or (X>a+x) (X>a)(X>x) F X (a+x) [ F X (a)] (X>x) Diffriaig w.r.. h paramr a, S a. f X (a+x) f X (a) (X>x) f X (x) f X () (X>x) Wri f X (). W hav h followig diff. q. Thus, (X>x) f X (x) x [(X>x) f X (x)] d [ (X )] d x x > x x (X>x) cosa (X>) (X>x) x for all x. Homwor. You ar waiig for Godo, who will arriv o h N h bus, whr N is Gomric(p) disribud. If h ir-arrival ims of buss ar Expoial() i.i.d., wha is h disribuio of your waiig im?

9 Trasforms of oisso & Expo. disribuios A rasform mas a alraiv xprssio for a mahmaical objc. Exampl. For a lcric wav f(), whr im, h Fourir rasform φ(ω) iωf() d is a alraiv way o spcify h fucio: Giv h valu φ(ω) for all ω, h fucio f() is uiquly drmid. Th commo irpraio of h variabl ω is h frqucy, ad h valu φ(ω) spcifis h ampliud a h paricular frqucy ω. I shor, h o-o-o corrspodc f() a<<b φ(ω) <ω< is a rasform bw h im ad frqucy domais. Th rasform of a couuous r.v. X mas h rasform of is disribuio fucio FX() (X < ). Wh X, of w also adop h Laplac rasform: φ(ω) ωf() d

10 Dfiiio. Th Laplac rasform of a coiuous r.v. X is φx( θ) (X < M θ ) whr M θ is a idpd xpoial r.v. wih ma /θ. // φ X (θ) is h probabiliy of wiig i h rac // bw a v a im X agais a v // a h mmorylss im wih paramr θ. // Th disribuio of X is spcifid by (X<x) for all x. // A alraiv spcificaio is by φ X (θ) fo all θ. // Thi of rasform bw h x (im) domai ad // h θ (frqucy) domai. Formula. φ X (θ) E[ θx ] roof. (X < M θ ) (X < Mθ X x)dfx ( x) ( x < Mθ X x)dfx ( x) (M θ > x) df X ( x) // M θ is idp. of X x dfx ( x) θ E[ θx ]

11 Thorm. Th Laplac rasform of a xpoial() r.v. X is /(θ+). Compuaioal roof. θ X θx φx( θ) E[ ] df X( x) // F X (x) x θ x x d x / ( θ + ) Alraiv proof by h birh modl. Durig h im irval [, ], olld calls ad θ oll-fr calls arriv, rspcivly. All call arrivals ar i.i.d. W hav a rac amog h oal of (θ+) calls, of which hr ar olld calls. L. Th waiig im for h firs olld call is X M ad, for h firs oll-fr call, M θ. Th horm simply quas h Laplac rasform φ X (θ) o φx ( θ ) (X < M θ ) (probabiliy for h ovrall firs call o b olld) / ( θ + ) //qual chac for all i.i.d o wi i h rac

12 2 Dfiiio. Th Fourir rasform of a coiuous r.v. X i X is φ ( ω) E[ ω ] Similar o h Laplac rasform, h Fourir rasform of h xpoial radom variabl wih ra is foud o b φ(ω) / ( ω). From his, o ca calcula h variac of xpoial disribuio as follows: Dfiiio. Th z-rasform of a discr r.v. X is

13 3 φ ( z ) E[z X ] X z (X ) {all hads i X osss of a z-biasd coi} // z probabiliy of had {o ail i X osss of a z-biasd coi} I paricular, h z-rasform for a oisso() r.v. is z /! ( z) /! ( z ) Irpraio (will b furhr clarifid afr h discussio o oisso procss). Durig h im irval [, ], calls arriv a h swichig sysm. All call arrivals ar i.i.d. Thus is h arrival ra. If vry call has h probabiliy z of big a olld call. Th, h arrival ra is z for olld calls ad is (z) for oll-fr calls. Th probabiliy of o oll-fr calls wihi a ui im is (z).

14 Sum of Idpd oisso r.v. Thorm. L X ad Y b idpd oisso r.v. wih dsiis ad µ, rspcivly. Th X+Y is oisso(+µ). roof by h birh modl. Cosidr (+µ) uiform i.i.d. ovr h irval [, ), whr uiform i.i.d. of hm rprs arrivals of oll-fr pho calls a a lpho xchag ad h rmaiig µ rprs olld calls. As, umbr of oll-fr calls durig h irval [, ) is oisso(); umbr of olld calls durig h irval [, ) is oisso(µ); hs wo oisso r.v. ar idpd, sic oll-fr calls ar idpd of olld calls; h oal umbr of calls durig h irval [, ) is oisso wih h ma +µ. Compuaioal roof. (X+Y ) (X & Y )

15 5 (X ) (Y ) // idp. vs for fixd & µ µ! ( )! ( + µ )!!! µ ( )! ( + µ )! µ µ ( + ) ( + µ )! Alraiv roof from z-rasform. Th z-rasform for a oisso() r.v. is E[z X ] Similarly, E[z Y ] E[z X+Y ] ( + µ )(z) oisso(+µ) r.v. µ ( z) ( z). Bcaus of idpdc,, which is h z-rasform for a.

16 6 Thorm. L X ad Y b idpd oisso radom variabls wih dsiis ad µ, rspcivly. Wri p /(+µ). Th, udr h codiio of X+Y, h disribuio of X is Biomial(, p). Tha is, (X X +Y ) p ( p ) roof by oisso modl. L h arrivals of oll-fr lpho calls wih µ olld calls all b uiform i.i.d. ovr h im irval [, ). Assum ha of h oal (+µ) calls ur ou o fall i h irval [, ). Th probabiliy for xacly i h collcio of o b oll-fr is ( ) ( + ) µ ( µ ) ( µ + + ) [( + µ ) ][( + µ ) ] [( + µ ) + ] jus li drawig cards o by o from a dc of (+µ). As, h idpd radom variabls X ad Y rprs h umbrs of oll-fr ad olld calls, rspcivly, ovr h im irval [, ) ad hc h abov probabiliy covrgs o (X X+Y). Tha is, (X X+Y) ( ) ( + ) µ ( µ ) ( µ + + ) lim µ µ µ [( + ) ] [( + ) ] [( + ) + ] µ ( + µ) biomial probabiliy wih paramrs ad /(+µ)p Compuaioal roof.

17 7 (X X+ Y ) (X & Y ) (X+ Y ) (X ) (Y ) (X+ Y ) // wo idp. vs for fixd & µ [ /!][ / ( )!] ( + µ ) ( + µ ) /! + µ µ + µ

18 Spliig a oisso radom variabl Th prcdig horm mrgs idpd oisso radom variabls. Th followig horm is h ivrs. I splis a oisso radom variabl. Thorm. Assum h oal umbr of fauls is oisso wih dsiy ad hr is a idpd probabiliy p for ach faul o lad o a bradow. Th, () h umbr of bradows is oisso(p) disribud (2) h umbr of rcovrabl fauls is oisso(p) disribud, ad (3) hs wo oisso radom variabls ar idpd. Compuaioal roof of (). L N rprs h oal umbr of fauls ad X h umbr of bradows. W wa o show ha X is oisso(p) disribud. (X) (X N ) (N ) p p ( ) /! ( p) ( p) /!( )! // codiioig o N

19 9 [( p) /!] ( p) / ( )! [( p) /!] ( p) / m! m p [( p) /!] m // wriig m- p ( p) /! Compuaioal roof of (3). L X rprs h umbr of bradows, Y h umbr of rcovrabl fauls, ad N X+Y. W shall show ha X ad Y ar idpd. (X ad Yj) (X ad N+j) (X N+j) (N+j) + j j j p ( p) + / ( + j)! p j ( p [( p) /!] [( p) (X) (Yj) ) / j!] Iuiiv proof of (, 2) by h birh modl. lac fauls o h im irval [, ] by uiform i.i.d. Th umbr of fauls i h irval [, ] is Biomial(, /). // Biomial(, /) oisso() wh For ay sigl faul,

20 2 (I is a bradow & falls wihi h irval [, ]) (bradow) (wihi irval [, ]) // idp. v p / p/ Morovr, his probabiliy for ach faul is idpd of ha of vry ohr faul. Thus, h disribuio of h umbr of bradows i [, ] is Biomial(, p/). Similarly, h disribuio of rcovrabl fauls i [, ] is Biomial(, (p)/). Rcovrabl Bradow Now l. Th umbr of fauls, brasows, ad rcovrabls i h irval [, ] bcoms oisso(), oisso(p), ad oisso((p)), rspcivly. Iuiio bhid (3) by h birh modl. Th aformiod disribuios Biomial(, p/) ad Biomial(, (p)/) ar gaivly corrlad o ach ohr bcaus h four rcagls shar h oal pool of. Bu, as, h corrlaio bw h wo iy rcagls o h lf bcoms gligibl. // Imagi sow flurris,, fall dow o h big // plai of ara. Th umbr of flurris gahrig o // a ui-ara roof is idpd of ha o aohr.

21 2 Exampl of machi rpair shop A shop cosiss of M machis ad has a sigl rpairma. L h sa b h umbr of machis ha ar dow. Th sa spac is {,,, M}. Assum ha. h amou of im a machi rus bfor braig dow is Expoial() ad 2. h amou of im i as h rpairma o fix ay bro machi is Expoial(µ). Th probabiliy of ay of hs idpd vs o occur firs is proporioal o h ra. Upo h occurrc of a v, h waiig im for vryhig mmorylss is rwd. Th followig diagram shows h ras for rasiios. µ µ, ( M ),,,, M

22 22 L,, 2, b h saioary probabiliis of h Marov chai. Bcaus of h balacd flow bw sas ad, ( ) ( ) [ ] M M M M µ µ µ // sam as igvcor of rasiio marix bu br ( ) ( ) [ ] ( ) M M M M M M,,,,!! µ µ Now w ivo h boudary codiio. ( ) ( ) + M M M M!! µ I h log ru, h avrag umbr of dowd machis is ( ) ( ) µ ρ ρ ρ + whr,!!!! M M M M M M M

23 Mhod of Radomizaio A difficul qusio. Fix m. u balls radomly (uiform ad idpd disribuio) io m larg boxs. Calcula G() {vry box gs occupid by a las o of h balls} Radomizaio soluio. Disribu oisso() may balls radomly (uiform ad idpd disribuio) io m boxs, ach box idpdly rcivs oisso(/m) balls by h spliig of a oisso radom variabl. Hc (Box is occupid wh oal umbr of balls is oisso()) (Box is occupid wh his box coais oisso(/m) /m Thus, may balls) (all m boxs occupid wh hr ar oisso() balls) ( /m ) m // idpd & idical boxs L N b a oisso() disribud r.v. Th abov probabiliy, hrafr dod as φ(), rlas o h objciv fucio G() as follow. ( /m ) m φ() (N ) (all m boxs occupid wh hr ar balls)

24 24 (all m boxs occupid wh hr ar balls)! G()! Thus, h fucio φ() i h coiuous variabl ad h squc G(), G(2),, G(), i h discr idx uiquly drmi ach ohr by φ( )! G( ) Mahmaically, h fucio φ() is a rasform of h squc of G(), G(2),, G(), // Thi of as h discr im ad as h frqucy. W hav compud h fucio φ(), so w wa o ivr h rasform o g G(). Sic h righ-had sid is i h form of a sris xpasio wih rspc o, w shall rwri h soluio of φ() ( /m ) m i h sam form. φ() ( /m ) m m m / m m m ( ) // biomial xpasio of ( ) m ( / m)! m m ( ) // Taylor xpasio m m ( / m) ( )! m // irchag Σ ordr

25 25 This givs aohr sris xpasio of φ() wih rspc o. Th compariso of corrspodig cofficis i h wo xpasios yilds h o clos-form soluio: G() m m m ) ( m

26 Th oisso rocss A gric modl of pur birh: T T 2 T 3 T 4 S S 2 S 3 S 4 N() arrival cou Th procss ca b dscribd by ay of h followig: Th irarrival ims T, T 2, T 3, // discr idx // coiuous valu S Σ j T j ad T S S Th arrival ims S, S, S 2, S 3, // discr idx // coiuous valu N() S Th arrival cous N() i [, ], // coiuous-im // discr valu

27 Fiv quival dfiiios of oisso procss Dfiiio. Th oisso procss wih h isiy is a pur-birh procss. Th fiv dfiiios blow will spcify his pur-birh procss by imposig propris o: Dfiiio-A: Arrival ims ar ordrd saisics i h limiig modl. Dfiiio-B: Arrival cous ar oisso; codiioal arrival ims ar ordrd saisics Dfiiio-C: Irarrival ims ar xpoial i.i.d. Dfiiio-D: Icrms i arrival cou ar idpd; ar saioary oisso. Dfiiio-E: Icrms i arrival cou ar idpd; add wih probabiliy ovr im. Dfiiio-A (Arrival ims ar ordrd saisics i h limiig modl). Cosidr birhs ha ar uiform i.i.d. ovr h irval [, ). L. Th, h arrival im S has h sam disribuio as h h smalls amog hs uiform i.i.d. // h smalls amog i.i.d is calld h h ordrd saisic

28 28 Dfiiio-B (Arrival cous ar oisso; codiioal arrival ims ar ordrd saisics). Assum ha N() is oisso() for >. // Corrlaio bw N( ) & N( 2 ) o b spcifid blow For all >, h disribuio of S, S 2,, S udr h codiio ha N() ar h sam as h ordrd saisics of uiform i.i.d. ovr [, ].! f s,, s N( ), < s < s2 < < s ( ) // (s, s 2,, s ) rags ovr h -dim riagl, // which has h volum /(!). // Wri h uiform i.i.d. as U, U 2,, U. // Th f ( u,, u N( ) ), < u j for all j, // i.., (u, u 2,, u ) ragig ovr -dim squar. Dfiiio-C (Irarrival ims ar xpoial i.i.d.). Wh T, T 2, T 3, ar i.i.d. xpoially disribud wih ma /, his is h oisso procss wih h isiy.

29 29 Dfiiio-D (Icrms i arrival cou ar saioary oisso ad idpd). A pur birh procss {N()} is calld h oisso procss wih h isiy if h followig ar ru. (Saioary oisso icrm) Th icrm ovr ay im duraio of lgh is oisso() disribud. // (N(s+)N(s) ) ( ) /! for all s, (Idpd icrm) Icrms ovr disjoi irvals ar idpd. Dfiiio-E (Icrms i arrival cou ar idpd; add wih probabiliy ovr ifiisimal im). A pur birh procss {N()} is calld h oisso procss wih h isiy if h followig ar ru. Icrms ovr disjoi irvals ar idpd. Th icrm ovr ay im duraio of lgh saisfis h followig: (N(+ )N() ) + o( ) (N(+ )N() 2) o( )

30 3 (N(+ ) N()) + o( ) W shall xami a xampl of applicaio bfor provig h quivalc of hs fiv dfiiios.

31 Exampl of oisso procss i Aloha CSMA ac graio is by a oisso procss wih isiy /sc. Wh a pac is grad, h rasmissio is ampd immdia, which lass p scods. Durig h vulrabl priod, h rasmissio of a pac may collid wih aohr. I ha cas, h rasmissio fails ad h pac is discardd. I is possibl ha a similar pac may b amog hos grad lar o for r-amp. Do G avrag chal raffic (i.., umbr of ampd pac rasmissios pr priod of p scods) p probabiliy of succssful rasmissio (i.., wh o collisio durig h vulrabl priod) S G hroughpu of h chal (i.., avrag umbr of succssful rasmissio pr priod of p scods) Th hroughpu S is o a moooic fucio of h raffic lvl G. W ar irsd i h raffic lvl ha yilds h maximum hroughpu. Cosidr pur Aloha ad slod Aloha sparaly.

32 32 ur Aloha (duraio of h vulrabl priod 2p) ( ) ( ) ! max S wh G G dg ds G S G G G G p

33 33 Slod Aloha (duraio of h vulrabl priod p) S max G S G ds dg G G + G ( ) wh G.368 G

34 oisso icrm roof of Dfiiio-E (ifiisimal icrm) Dfiiio-D (oisso icrm) Abbrvia ( ) { N ( ) }. W wa o prov ha Dfiiio-E ( ) ( ) for! W shall prov his formula by iducio o. Wri h for brviy. Firs, w calcula ().

35 35 Nx, w show how o obai () from h iducio hypohss ha ( ) )! ( ) (. ( ) ( ) { } ( ) ( ) ( ) { } ( ) { } ( ) ( ) { } ( ) { } ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) h h h h o h h h o h h o h N N h N N N h N N h N h // ) ( ) ) ( lim( lim //idpd icrm, '

36 36 ( ) c +! ) ( Th boudary codiio () c. ( )! ) ( Homwor. rov ha Dfiiio-D Dfiiio-E. ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) // idpd icrms,,, 2 h o h h N h N N N h N N N h N N h N h ( ) ( ) ( ) ( ) ( ) ( ) ( ) )! ( ) ( )! ( ) ( ' )' ( ' + +

37 Irarrival im disribuios roof of Dfiiios-D ad -E (icrm i arrival cou) Dfiiio-C (i.i.d. irarrival ims) Agai, abbrvia ( ) { N ( ) }. (T > ) () // oisso icrm ovr h irval (, ) (T 2 > T s) ( birh i (s, s+] T s) ( birh i (s, s+] birh i (, s]) ( birh i (s, s+]) // idp. icr. ovr 3 disjoi irvals // saioary oisso icrm This is ru for all s. Hc (T 2 > ) ad T 2 is idpd of T. (T 3 > T r, T 2 s) ( birh i (s, s+] T r, T 2 s) ( birh i (s, s+] birh i (, r]; birh i (r, s]) ( birh i (s, s+])

38 This is ru for all r ad s. Hc (T 3 > ) ad T 3 is idpd of T ad T 2. Rpaig h sam argums, w sablish h i.i.d. disribuios of T, T 2, T 3,, T, 38

39 Arrival im disribuios Thorm. Evry amog Dfiiios-B, -C, -D, ad -E implis ha S has h gamma disribuio wih paramrs ad. Tha is, S has h probabiliy dsiy fucio f() () / ()! roof from Dfiiios-B ad -D. N() is oisso(). Thus, {S } {N() } // N() S ( ) j j j! f S () d d {S } ( ) j j j d d! ( ) j j j d d! ( ) ( ) j j j j j! )! ( ( ) ( ) j j j!! () / ()!

40 4 roof from Dfiiio-C. S Σ j T j, h sum of xpoial i.i.d. I is wll ow ha h sum of xpoial() i.i.d. has h gamma(, ) disribuio. roof from Dfiiio-E. ( < S < + ) {N() ad birh i (, + )} + o( ) {N() } { birh i (, + )} + o( ) // idpd icrm [ () / ()!] [ + o( )] + o( ) // Df-E Df-D (oisso icrm) f S () lim ( < S < + )/ () / ()! // {N() }

41 Codiioal arrival ims Thorm 5.2. Dfiiio-D -C (i.i.d. irarrival ims) Dfiiio-B (oisso arrival cous; codiioal arrival ims ar ordrd saisics.) roof. L < x < < x <. L x, x / x,..., x / x 2 all b ifiisimally small. f S,,S (x,, x N()) lim {x x S <x + x,, x S <x + x, " " x S <x + x N()} / x x 2... x lim {x " x" S <x + x,, x S <x + x, x S <x + x, N()} / {N()} x x 2... x lim {x x S <x + x,, x S <x + x, " " x S <x + x, T + >x } / {N()} x x 2... x // blow w rasla vs abou S j io T j for all j lim {x x S <x + x,, x S <x + x, " " x x T <x x + x, T + >x } / {N()} x x 2... x // S T +x lim {x x T <x + x, x 2 x T 2 x 2 x + x 2,, " " x x T <x x + x, T + >x } / {N()} x x 2... x

42 42 lim {x x T <x + x, x 2 x T 2 x 2 x + x 2,, " " x x T <x x + x } {T + >x } / {N()} x x 2... x // idpdc f T,,T (x, x 2 x,, x x ) {T + >x } / {N()} x (x 2 x ) (x x (x ) / [ () /!] // irarrival ims ar xpoial i.i.d.! /, < x < < x < This says ha, udr h codiio of N(), h uordrd arrival ims {S,, S } ar h uiform i.i.d. ovr h irval (, ].

43 43 A Opimizaio Exampl. () Ims arriv a a procssig pla i accordac o a oisso procss wih isiy. A a fixd im, all ims ar dispachd from h sysm. Wha is h xpcd valu of h oal wai W of all ims? (2) Th ims ar o b dispachd from h sysm i wo bachs, firs a a irmdia im, τ (, ) ad h a h im. Choos τ so as o miimiz h oal xpcd wai of all ims. Aswr o (). Codiioig o N(), E[ W ] ( ) j E[ W N( ) j] j j! Udr h codiio of N(), h uordrd arrival ims (S, S 2,, S ) ar uiform i.i.d. ovr h irval (, ] by Thorm 5.2. Hc h codiioal r.v. [W N() ] has h sam disribuio as h sum of uiform i.i.d. ovr h irval (, ]. Thus, E[W N() ] /2. Hc, E[ W ] ( ) j E[ W N( ) j] j j! ( j j / 2) (/2) E[N()] (/2) 2 /2 ( ) Aswr o (2). Miimiz τ 2 /2 + (τ) 2 /2 by diffriaio w.r.. τ. Th opimal valu of τ is /2. j! j

44 44 Exampl of lcroic cour. Elcrical pulss arriv a a cour i accordac o a oisso procss wih isiy. L S, S 2, b arrival ims, ad l A, A 2, b hir rspciv iiial ampliuds, which ar i.i.d. wih a uspcifid disribuio. Through xpoial auaio, h ampliud of a puls j bcoms A j αx a h ag x. Thus, h oal ampliud a im is N ( ) A() Aj j α ( S W wa o calcula E[A()]. Codiioig o N(), ( ) E[ A( )] E[ A( ) N( ) ]! Udr h codiio of N(), h uordrd arrival ims (S, S 2,, S ) ar uiform i.i.d. ovr h irval (, ] by Thorm 5.2. Hc h codiioal r.v. [A() N() ] has h sam disribuio as Aj j uiform i.i.d. ovr (, ]. Thus, E[A() N() ] E[ Aj j α ( Y ) j α ( Y ) j ) α ( Y ) j, whr Y j ar ] // uordrd A j EA E[ ] // uordrd A From h uiform disribuio of Y, α α ( Y E[ ) α ( y) dy ] α

45 45 Thus, E[A() N() ] EA E[ A( )] α α ( ) E[ A( ) N( ) ]! α EA α α EA α α EA α EA α ( )! as

46 Combiig ad spliig of oisso procsss Thorm. Combiig wo idpd oisso procsss wih isiis ad 2 rsuls i a oisso procss wih isiy + 2. roof. Th combid procss ihris h propry of idpd icrms. Combiig wo idpd oisso r.v. wih dsiis ad 2 rsuls i a oisso r.v. wih dsiy ( + 2 ). Hc h icrm ovr a irval of ay lgh is oisso. Dfiiio-D ow prvails. Thorm. If vry arrival i a oisso procss wih isiy is idpdly of yp- wih probabiliy p (ad yp-2 wih probabiliy p), h h wo yps of arrivals form wo idpd oisso procsss wih rspciv isiis p ad (p). N(), -p p N (), p N 2 (), 2 (-p) roof. rviously, w hav s h spli of a oisso r.v. io wo idpd os. Thus, icrms of yp- ad yp-2 ovr a irval of lgh ar oisso(p) ad

47 47 oisso((p)), rspcivly. Thus h wo procsss {N ()} > ad {N 2 ()} > hav saioary oisso icrm. Morovr, for a fixd >, N N m N N m () { ( ) ( ) } { ( ) } { ( ) }, 2 2 I ordr o show ha {N ()} > is a oisso procss by Dfiiio-D, w d o show idpd icrm (Homwor blow). I ordr o show ha h wo oisso procsss rsulig from h spli ar idpd of ach ohr, w d idpdc bw h r.v. N (s) ad N 2 () for all s ad (Homwor 2 blow). Homwor. rov idpd icrm. Homwor 2. rov his idpdc usig sam () ad idpd icrm.

48 48 Iuiio bhid h Thorm. Divid h irval (, ] io qual subirvals. L so ha a subirval has o arrival a yp- arrival a yp-2 arrival mulipl arrivals wih probabiliy / + o(/) wih probabiliy p/ + o(/) wih probabiliy (p)/ + o(/) wih probabiliy o(/) Fix a. Spli h oisso r.v. N() io h wo oisso r.v. N () ad N 2 () wih rspciv dsiis p ad (p). To s ha N () ad N 2 () ar idpd, w assum, for isac, ha N (). Th, of h subirvals, will coai yp- arrivals, ad hc h probabiliy for ach of h ohr o coai a yp-2 is (yp-2 o yp-) ((p)/) / (p/) (p)/ Thus, giv ha N (), h disribuio of N 2 () is Biomial(, (p)/) oisso((p)) // o logr ivolv roposiio 5.3. Suppos N() is a oisso procss wih isiy. A v a im s is classifid as yp-i wih probabiliy i ( s), i,, umbr ( ), whr i ( s) i. Th, h N i of yp-i vs occurrig up o im, i,,, ar idpd oisso r.v. wih ma

49 49 ( s) ds ( ) E[ Ni ( )] i i whr i i ( s) ds is h probabiliy for a v i h irval (, ) o b yp-i. To simplify h oaio, w shall prov oly h spcial cas wh 2. roposiio 5.3 (Spcial cas wih jus yp- & yp-2). I a oisso procss wih isiy, if a arrival a im s i is idpdly of yp- wih probabiliy p(s) ad yp-2 wih probabiliy p(s), h h wo yps of arrivals durig h irval (, ) ar idpd oisso r.v. wih rspciv dsiis p( s) ds ad [ p( s)] ds Th prcdig Thorm Corollary 5.3 p(s) p p s s roof. Th proof is similar o ha i spliig a oisso r.v. io wo.

50 5 {m yp- arrivals ad yp-2 arrivals i (, )} (m+ arrivals i (, )) (m ar yp- ad ar yp-2 m+ arrivals i (, )) m + [ () m+ /(m+)!] p m q, // Each of h m+ arrrivals idpdly has h // uiform probabiliy p o b of yp-, whr // p [ p( s) / ] ds by codiioig o h uiform // arrival im ad qp. [ p (p) m /m!] [ q (q) /!] This o oly sablishs h dsird oisso disribuios wih dsiis p also provs hir idpdc. p( s) ds ad q [ p( s)] ds bu

51 5 Trmiology. For h us i lar xampls, w ow iroduc som basic rmiology of quuig. (s) M/M/s quu, whr s Irarrival im is Mmorylss Srvic im is Mmorylss Numbr of srvrs Assumpios: M: Ir-arrival im ar Expoial() i.i.d. M: Srvic ims ar Expoial(µ) i.i.d. All hs ar idpd.

52 52 Exampl of ifii-srvr sysm (M/G/ ). Cusomrs arriv a a srvic saio accordig o a oisso procss wih isiy. Upo arrival h cusomr is immdialy srvd by o of h ifiily may srvrs. Th srvic im has h disribuio fucio G. Q: Wha is h disribuio of h umbr of cusomrs ha hav compld srvic by a fixd im? Wha is h disribuio of h umbr of cusomrs ha ar big srvicd a im? To apply roposiio 5.3, w classify cusomrs as yp- or yp-2 dpdig whhr h srvic is compld by im. Thus, a cusomr arrivig a im s < is of h yp- or yp-2 wih rspciv probabiliis G(s) or G(s). // No ha G(s) wh <s Now apply roposiio 5.3 wih p(s) G(s).

53 53 Exampl of highway cours. Cars r a uidircioal highway of h lgh d i accordac wih a oisso procss wih isiy. Evry car idpdly ravls a a cosa spd X, which is radom wih h disribuio fucio F X. Wh a fasr car cours a slowr o, i passs i wih o im big los. If you r h highway, wha spd should you choos i ordr o miimiz h xpcd umbr of cours, i.., passig ohrs or big passd? Soluio. Your r h highway a im, choos h fixd spd x, ad xi a im d / x. Ohrs r h highway accordig o a oisso procss wih isiy. A car rig h road a im, chooss a spd X accordig o h disribuio F X, ad xis a + d/x. Wri T d/x. To apply roposiio 5.3, l a v ma ha a car ohr ha yours rs h road. A v a im is said o b of yp- if ha car will cour your car, i.., if < ad +T > or > ad +T < Thus w ar irsd i h disribuio fucio F T, which rlas o F X as follows. F T Thus, T ( ) { } X d FX d, // whr F {A v a im is of yp-} X F X

54 54 d { + T > } F T ( ) F if - X < d { } ( ) - + T < F T FX if > From roposiio 5.3, h oal umbr of yp- vs is oisso wih dsiy E [ N ] {a v a im is of yp-} d F F T T ( ) d + F ( ) ( y) dy + F ( y) dy // y To miimiz E[ N ], w d o choos h cosa spd x, or, quival, o choos h ravl im d / x. d E[ N] [ FT ( ) + FT ( )] [ 2FT ( ) ] d Thus, d E[ N] d F F T F X ( ) F T ( ) 2 ( d ) F ( d / ) / X 2 ( x) F ( x) X X 2 Tha is, h opimal spd x is h mdia spd udr h disribuio G. T T d

55 55 Exampl of racig h HIV ifcios. Suppos ha idividuals corac HIV virus i accordac wih a oisso procss wih a uow isiy. Th icubaio priod, i.., h duraio from h im wh a idividual bcoms HIV ifcd uil AIDS sympoms appar, has a ow disribuio fucio G ad is idpd for vry prso. L ( ) N h umbr of idividuals who hav show sympoms of h disas by im. N 2 ( ) h umbr of HIV-posiiv idividuals who hav o y show ay sympoms by im. Q: Suppos a valu of N () is obsrvd. Ca w us i o sima N 2 ()? Rcall from roposiio 5.3 ha E E [ N ( ) ] G( s) ds G( y) [ N ( ) ] G( s) ds G( y) 2 L h obsrvd valu of N () b. Th soluio is i wo sps: Firs, us his obsrvd valu o sima h isiy of h poiso procss i coracig h HIV virus. E [ N ( ) ] G( y) G ( y) dy dy dy dy // firs - ordr sima

56 56 Th, us h sima of o fid E[N 2 ()]. E [ N 2 ( ) ] G( y) dy G G ( y) ( y) Spcial isac. Suppos G is xpoial wih ma /µ yars ad ha 22 idividuals hav xhibid AIDS sympoms durig h firs 6 yars of h pidmic. Th, µ y G y E ( ) [ N ( ) ] ( y) µ y dy µ y ( y) dy ( ) dy dy µ ( )/ µ µ ( )/ µ 2 G G Wih µ., 6, ad 22, 22 E N ( ).6 6 dy dy.6 [ ] ( ) ( )