Poisson Arrival Process

Transcription

1 1 Poisso Arrival Procss Arrivals occur i) i a mmorylss mar ii) [ o arrival durig Δ ] = λδ + ( Δ ) P o [ o arrival durig Δ ] = 1 λδ + ( Δ ) P o P j arrivals durig Δ = o Δ for j = 2,3, ( ) o Δ whr lim = Δ Δ C 1 C 2 C λ is rfrrd o as h arrival ra. 1 2 Thorm. Poisso ad Epoial Disribuio 1 ( ) L K do h umbr of arrivals durig h im irval,, ad l do h ir-arrival im: whr =. Th w claim h = = 1, 2, ( λ) λ 1) K has a Poisso disribuio wih P[ K = ] = =,1, 2,! λ 2) is poially disribud wih f ( ) = λ Proof par 1 W will firs prov ha h umbr of arrivals is Poisso. For oaioal covic, wri P ( ) = P[ K = ]. Proof is by iducio.

2 2 For =, P P ( +Δ ) = o arrival durig (, +Δ ) Usig h Bays rul, o arrival durig (, ) P +Δ = P ( +Δ ) Po arrival durig, o arrival durig, Usig h mmorylss propry, ( +Δ ) Po arrival durig, o arrival durig, o arrival durig (, ) [ o arrival durig ay priod] λ o( ) = P +Δ = P Δ = 1 Δ + Δ W hav show λ P ( +Δ ) = Po arrival durig, 1 Δ + o Δ ( λ ) = P ()1 Δ + o Δ 1 Dividig boh sids q. 1 by Δ, ( Δ) P( +Δ) P( ) o = λ P() + P() Δ Δ Taig h limi Δ, dp () + λ P () =. ( 2) d Th gral soluio of q. 2 is λ P ( ) = c for som cosa c. As a auiliary codiio, w rquir ha P () = 1, which lads o c= 1. Fially w hav P () = = λ ( λ)! λ, which is Poisso

3 3 For h gral, = 1,2, ( +Δ ) = arrivals durig (, +Δ ) P P = P arrivals durig, P o arrival durig, +Δ arrival durig, + P 1 arrivals durig, P 1 arrival durig, +Δ 1 arrival durig, + P 2 arrivals durig, P 2 arrivals durig, +Δ 2 arrival durig, + Usig h mmorlss propry ( λ ) () λ P ( +Δ ) = P 1 Δ + o Δ 1 2 () () Th rms coaiig o + P Δ + o Δ + P o Δ + ( Δ) ( λ ) λ 1 will vaish. For simpliciy, w cosidr oly o-vaishig rms P ( +Δ ) = P ( ) 1 Δ + P ( ) Δ f1 Dividig boh sids of q. f 1 by Δ, P( +Δ) P( ) + λp() = λp 1() Δ Taig h limi Δ, dp () + λp () = λp 1 () for = 1,2,. f 2 d Now assum P ( ) is Poisso for 1, ha is, 1 ( λ) ( ) 1 ( λ) ( ) λ P 1() = f3 Subsiuig q. f 3 io q. f 2, λ dp () + λp ( ) = λ ( f 4) d λ Muliplyig o boh sids of q. f4,

4 4 dp () λ λ + λp ( ) = λ d which implis λ { } ( λ) ( ) ( λ) = + c'.! 1 1 ( λ) ( ) d P () = λ f 5 d Th gral soluio of q. f 5 is λ P () As a auiliary codiio, w rquir P () = for >, which lads o c ' =. Fially w should hav P () = Ed of Proof. ( λ)! λ h Poisso Disribuio No ( hh ) P ( ) was dfid as h probabiliy of arrivals durig h im irval,. Howvr, du o h mmorylss propry, h im irval ca b ay im irval of lgh, ha is,, + for ay h. No 1 ( λδ) λδ P1 ( Δ ) = λ Δ λ Δ = λδ 1 λδ + + 2! 3! = λδ + o Δ also

5 5 ( λδ) ( ) 2 λδ P2 ( Δ ) = 2! λ Δ λ Δ λ Δ = 1 λδ ! 3! = o Δ which cofirms o h dfiiio of h Poisso arrival procss.

6 6 Proof par 2: Th ir-arrival im is poial h : ir-arrival im, = 1 F ( ) = P ( λ) ( ) = P a las o arrival durig, = 1 P ( ) = 1 = Pa las o arrival durig 1, 1+! λ λ = 1 d f ( ) = F d Ed of Proof. = λ λ for which is h Epoial Disribuio No Th ir-arrical im dos o dpd o h cusomr id. W will us i plac of. No Eprims show h Poisso arrival procss modls may ral-world physical radom procsss wll. I commuicaios sysms: Pac Arrivals Tlpho Call Amps

7 7 A mmorylss pdf is a poial pdf. Assum is a ogaiv, coiuous radom variabl. is said o b mmorylss if P + > = P for ay, >. mus b a poial radom variabl if is mmorylss. Proof Usig h Bays rul, [ ad ] P + > P < + P + > = = P > P > If is mmorylss, [ < + ] P[ > ] P Eq. 1 ca b wri as or or ( + ) 1 F () F F = P (1) = F ( ) ( + ) = F F F F F { } ( + ) = 1 F F F F Dividig boh sids by, ( + ) () = { 1 F ( ) } F F F Noig F = for a ogaiv, coiuous radom variabl, ( + ) () ( ) = { 1 F ( ) } F F F F Lig, { } = ( ) 1 F F F Lig F = λ, λ F + F = λ.

8 8 λ Muliplyig boh sids by, or λ λ λ λ λ F + F = λ { } d F = λ A gral soluio is or F F = 1+ C λ = implis C = 1. F = 1 λ λ λ F = λ d+ C = + C λ λ λ f ( ) = λ Epoial Disribuio Ed of Proof. How do w gra a Poisso raffic? 1. Gra poial radom umbrs 2. A Broulli radom procss ca approima a Poisso radom procss A a bus sop, w flip a coi vry Δ scods. Th coi falls dow hads wih probabiliy λδ. If hads, h a bus dpars immdialy.