Posso Processes
Summary The Posso Process Properes of he Posso Process Ierarrval mes Memoryless propery ad he resdual lfeme paradox Superposo of Posso processes Radom seleco of Posso Pos Bulk Arrvals ad Compoud Posso Processes 2
The Posso Coug Process Le he process {N()} whch cous he umber of eves ha have occurred he erval [0,). For ay 0 N 0 N N... N... k 0 2 k 6 4 2 0 2 3 k- k, N N N k k k k Process wh depede cremes: The radom varables N( ), N(, 2 ),, N( k-, k ), are muually depede. Process wh saoary depede cremes: The radom varable N( k-, k ), does o deped o k-, k bu oly o k - k- 3
The Posso Coug Process Assumpos: A mos oe eve ca occur a ay me sa (o wo or more eves ca occur a he same me) A process wh saoary depede cremes Pr N, Pr N k k k k Gve ha a process sasfes he above assumpos, fd P Pr N, 0,,2,... 4
The Posso Process Sep : Deerme Sarg from P 0 Pr N 0 PrN s0 Pr N 0 ad N, s 0 Pr N 0 Pr Ns0 Saoary depede cremes P s P P s 0 0 0 Lemma: Le g() be a dffereable fuco for all 0 such ha g(0)= ad g() for all >0. The for ay, s 0 g ( s) g gs g e for some λ>0 5
The Posso Process Therefore P 0 Pr N 0 e Sep 2: Deerme P 0 (Δ) for a small Δ. 2 3 Pr 0 N e... 2! 3! o. Sep 3: Deerme P (Δ) for a small Δ. For =2,3, sce by assumpo o wo eves ca occur a he same me As a resul, for = P PrN o P Pr N o 6
The Posso Process Sep 4: Deerme P ( (+Δ) for ay P PrN P k Pk k 0 0. o o P P P P o Movg erms bewee ee sdes, P P o P P Takg he lm as Δ 0 P P o.. dp d P P 7
The Posso Process Sep 4: Deerme P ( (+Δ) for ay P PrN P k Pk k 0 0. o o P P P P o Movg erms bewee ee sdes, P P o P P Takg he lm as Δ 0 P P o.. dp d P P 8
The Posso Process Sep 5: Solve he dffereal equao o oba P Pr N e, 0, 0,, 2,...!! Ths expresso s kow as he Posso dsrbuo ad full characerzes he sochasc process {N()} [0,) [,) uder he assumpos ha No wo eves ca occur a exacly he same me, ad Idepede saoary cremes You should verfy ha E N N ad N var () Parameer λ has he erpreao of he rae ha eves 9 arrve.
Properes of he Posso Process: Iereve Tmes Le k be he me whe he k- eve has occurred ad le k- V k deoe he (radom varable) ereve me bewee he kh ad k- eves. Wha s he cdf of V k, G k ()? Pr Pr PrN 0 G V V k k k Pr 0 arrvals he erval [, ) e G e k k Saoary depede cremes V k k- + N( k-, k- +)=0 Expoeal Dsrbuo 0 k-
Properes of he Posso Process: Expoeal Iereve Tmes The process {V k } k=,2,,,, ha correspods o he ereve mes of a Posso process s a d sochasc sequece wh cdf G Pr V e k Therefore, he Posso s also a reewal process The correspodg pdf s p g e, 0 Oe ca easly show ha EV k ad k 2 var V
Properes of he Posso Process: Memoryless Propery Le k be he me whe prevous eve has occurred ad le V deoe he me ul he ex eve. Assumg ha we have bee a he curre sae for z me us, le Y be he remag me ul he ex eve. V Wha s he cdf of Y? k k +z Y=V-z Y Pr Pr Pr V z ad V z Pr Pr V z Pr V z F Y V z V z G z Gz G z F e G Y z e z e e z V z z Memoryless! I does o maer ha we have already spe z me us a he 2 curre sae.
Memoryless Propery Ths s a uque propery p of he expoeal dsrbuo. If a process has he memoryless propery, he mus be expoeal,.e., Pr V z V z Pr V Pr V e Posso Process λ Expoeal Iereve Tmes G()=-e -λ Memoryless Propery 3
Superposo of Posso Processes Cosder a DES wh m eves each modeled as a Posso Process wh rae λ, =,,m. Wha s he resulg process? Suppose a me k we observe eve. Le Y be he me ul he ex eve. Is cdf s G ()=-exp{-λ }. Le Y,,Y m deoe he resdual me ul he ex occurrece of he correspodg eve. Ther cdfs are: e G e V = Y Memoryless Propery k Y j =V j -z j Le Y * be he me ul he ex eve (ay ype). Y Therefore, we eed o fd * m Y e j * Pr G * Y 4 Y e V j
Superposo of Posso Processes Idepedece Pr m Y Y * Pr G Pr Y Y * GY Pr m Pr Y,..., Ym m Pr Y e * e m e m where = The superposo of m Posso processes s also a Posso process wh rae equal o he sum of he raes of he dvdual processes 5
Superposo of Posso Processes Suppose ha a me k a eve has occurred. Wha s he probably ha he ex eve o occur s eve j? Whou loss of geeraly, le j= ad defe m Y =m{y: Y=m{Y : =2,,m}. Pr ex eve s j Pr Y Y y 0 0 ~-exp exp 2 y y e e dy dy 0 y y e e dy where = 6 m
Resdual Lfeme Paradox Suppose ha buses pass by he bus sao accordg o a Posso process wh rae λ. A passeger arrves a he bus sao a some radom po. V b k p b k+ How log does he passeger has Z Y o wa? Soluo : E[V]= /λ. Therefore, sce he passeger wll (o average) arrve he mddle of he erval, he has o wa for E[Y]=E[V]/2= /(2λ). Bu usg he memoryless propery, he me ul he ex bus s expoeally dsrbued wh rae λ, herefore E[Y]=/λ o /(2λ)! Soluo 2: Usg he memoryless propery, he me ul he ex bus s expoeally dsrbued wh rae λ, herefore E[Y]=/λ. Bu oe ha E[Z]= /λ herefore E[V]= E[Z]+E[Y]= 2/λ o /λ! 7
Radom seleco of Posso Pos Le, 2,,, represe radom arrval pos of a Posso process X() wh parameer λ. Assocaed wh each po, we defe a depede Beroull R.V. N where P{ N } p, P{ N 0} p q. Defe Y ( ) X ( ) N ; Z( ) X ( ) ( N 2 ) X ( ) Y ( ) We clam ha boh Y() ad Z() are depede Posso processes wh parameers λp ad λq respecvely. 8
Radom seleco of Posso Pos Proof: P ( Y () k ) P { Y () k X () } P { X () )}. ( ) P { X ( ) } e.! k k k P{ Y( ) k X( ) } p q, 0 k, k pe PY { ( ) k} e pq ( ) k k! k k ( ) k ( q) ( k)! k!! ( k)! k k! k ( q) k k e p ( p) ( p) e, k 0,, 2, k! k! ~ P( p). e q 9
Radom seleco of Posso Pos Smlarly: P{ Z ( ) m} ~ P( q). More geerally, P { Y ( ) k, Z ( ) m } P { Y ( ) k, X ( ) Y ( ) m } PY { ( ) k, X( ) km} P{ Y( ) k X( ) km} P{ X( ) k m} km ( ) ( ) ( ) k m k m p p q q pq e e e k ( k m)! k! m! P{ Y( ) k} P{ Z( ) m}, PY ( ( ) k) PZ ( ( ) m) 20
Bulk Arrvals ad Compoud Posso Processes Cosder a radom umber of eves C occur smulaeously a ay eve sa me of a Posso process. 2 C 3 C 2 2 C 4 2 (a) Posso Process (b) Compoud Posso Process Le p P{ C k}, k 0,, 2,, he k s a compoud Posso process, where N() s a ordary Posso process. N() X() C, PX { ( ) m } PX { ( ) m N ( ) PN } { ( ) } 0 2
Bulk Arrvals ad Compoud Posso Processes ( z ) E { z C } P { C k } z C X k 0 ( ) { X() } { ( ) } m0 z E z P X m z m 0 0 PX { () m N () PN } { () z } m ( P{ C m} z ) P{ N( ) } 0 m0 k C m m C ( Ez { }) PN { ( ) } ( Ez { } ) PN { ( ) } 0 0 ( ( )) C C z PN { ( ) } e 0 ( ( z)) m 22
Bulk Arrvals ad Compoud Posso Processes 2 k We ca wre ( ( ( z ) 2 z ) k z ) ( z) e e e X We ca wre where k pk, whch shows ha he compoud Posso process ca be expressed as he sum of eger-scaled depede Posso processes m ), m ( ),. Thus X ( ) k k m k ( ). ( 2 More geerally, every lear combao of depede Posso processes represes a compoud Posso process. 23