Poisson Arrival Process

Transcription

1 Poisso Arrival Procss Arrivals occur i) i a mmylss mar ii) [ o arrival durig Δ ] = λδ + ( Δ ) P o [ o arrival durig Δ ] = λδ + ( Δ ) P o P j arrivals durig Δ = o Δ f j = 2,3, o Δ whr lim =. Δ Δ C C 2 C λ is rfrrd o as h arrival ra. 2 Thm. Poisso ad Epoial Disribuio L K do h umbr of arrivals durig h im irval,, ad l do h ir-arrival im: whr =. Th w claim h = =, 2, ( λ) λ ) K has a Poisso disribuio wih P[ K = ] = =,, 2, λ 2) is poially disribud wih f = λ Proof of (). Th umbr of arrivals has a Poisso disribuio. Proof will procd by iducio. Dfi P = P[ K = ]. F =, P P ( +Δ ) = o arrival durig (, +Δ ) Usig h Bays rul, o arrival durig (, ) P +Δ = P ( +Δ ) Po arrival durig, o arrival durig,

2 2 Usig h mmylss propry, ( +Δ ) Po arrival durig, o arrival durig, o arrival durig (, ) [ o arrival durig ay priod] λ o = P +Δ = P Δ = Δ + Δ W hav show λ P ( +Δ ) = Po arrival durig, Δ + o Δ ( λ ) = P () Δ + o Δ Dividig boh sids of q. 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 f som cosa c. As a auiliary codiio, w rquir ha P () =, which lads o c=. Fially w hav P () = = λ ( λ)! λ, which is Poisso F h gral, =,2, ( +Δ ) = arrivals durig (, +Δ ) P P = P arrivals durig, P o arrival durig, +Δ arrival durig, + P arrivals durig, P arrival durig, +Δ arrival durig, + P 2 arrivals durig, P 2 arrivals durig, +Δ 2 arrival durig, +

3 3 Usig h mmlss propry () λ P ( +Δ ) = P Δ + o Δ 2 () λ () Th rms coaiig o + P Δ + o Δ + P o Δ + ( Δ) ( λ ) λ will vaish. F simpliciy, w cosidr oly o-vaishig rms P ( +Δ ) = P Δ + P Δ f Dividig boh sids of q. f by Δ, P( +Δ) P + λp() = λp () Δ Taig h limi Δ, dp () + λp () = λp () f =,2,. f 2 d Now assum P is Poisso f, ha is, ( λ) ( ) λ P () =! f3 Subsiuig q. f 3 io q. f 2, ( λ) ( ) λ dp () + λp ( ) = λ ( f 4) d! λ Muliplyig o boh sids of q. f 4, dp () λ λ + λp = λ d which implis λ { } ( λ) ( ) ( λ) ( )! d P () = λ f 5 d! Th gral soluio of q. f 5 is λ ( λ) P () = + c'.

4 4 As a auiliary codiio, w rquir P () = f >, which lads o c ' =. Fially w should hav P () = ( λ) Ed of Proof. λ h Poisso Disribuio No ( hh ) P was dfid as h probabiliy of arrivals durig h im irval,. Howvr, du o h mmylss propry, h im irval ca b ay im irval of lgh, ha is,, + f ay h. No ( λδ) λδ P ( Δ ) =! λ Δ λ Δ = λδ λδ + + 2! 3! = λδ + o Δ also ( λδ) 2 λδ P2 ( Δ ) = 2! λ Δ λ Δ λ Δ = λδ ! 3! = o Δ which cofirms o h dfiiio of h Poisso arrival procss.

5 5 Proof of (2). Th ir-arrival im is poial. h : ir-arrival im, = F = P ( λ) = P a las o arrival durig, = P = = P a las o arrival durig, +! λ λ = d f = F d Ed of Proof. = λ λ f which is h Epoial Disribuio No Th ir-arrical im dos o dpd o h cusomr id. W will us i plac of. Summary. I a Poisso arrival procss wih arrival ra λ, ) Th umbr of arrivals durig ay im irval, ( λ) λ K has a Poisso disribuio wih P[ K = ] = =,, 2, 2) Th ir-arrival im bw ay adjac arrivals, is poially disribud wih f = λ λ No Eprims show h Poisso arrival procss modls may ral-wld physical radom procsss wll. I commuicaios sysms: Pac Arrivals Tlpho Call Amps

6 6 A mmylss pdf is a poial pdf. Assum is a ogaiv, coiuous radom variabl. Th probabiliy disribuio of is said o b mmylss if P + > = P f ay, >. Th pdf of mus b poial if h disribuio is mmylss. Proof Usig h Bays rul, [ ad ] P + > P < + P + > = = P > P > If is mmylss, [ < + ] P[ > ] P Eq. ca b wri as ( + ) F () F F = P () = F ( + ) = F F F F F { } ( + ) = F F F F Dividig boh sids by, ( + ) () = { F } F F F Noig F = f a ogaiv, coiuous radom variabl, ( + ) () = { F } F F F F Lig, { } = F F F Lig F = λ, λ F + F = λ λ Muliplyig boh sids by, λ λ λ λ λ F + F =.

7 7 λ { } d F = λ λ A gral soluio is F F = + C = implis C =. F = 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?. Gra poial radom umbrs 2. A Broulli radom procss ca approima a Poisso radom procss A h bus igi saio, w flip a coi vry Δ scods. Th coi falls dow hads wih probabiliy λδ. If hads, h a bus dpars immdialy.