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Poisson Arrival Process

Poisson Arrival Process - POISSON ARRIVAL RANDOM PROCESS...

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POISSON ARRIVAL RANDOM PROCESS Definition Arrivals occur 0 ) Memoryless ) P[one Arrival during ] ( ) P[no Arrival during ] 1 ( ) P[two or more Arrivals during ] ( ) ( ) where lim 0 t i ii t t o t t t o t t o t o t t λ λ ∆ → = + = + = = We call λ as the arrival rate, and 1 λ as the mean inter-arrival time. time t n t n+1 t n-1 t 0 t 1 . . . C n C n+1 C n-1 C 0 C 1 0

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Theorem Let t K denote the number of arrivals during the time interval ( ) 0, t Let X denote the inter-arrival time: 1 n n t t for any n . Then we can show: 1) t K is Poisson: ( ) ( ) P[ ] 0,1,2, ! k t k t t P t K k e k k λ λ == = 2) X is exponential: ( ) 0 x X f x e x λ λ = ( ) The theorem holds true even if we define as the number of arrivals during the time interval , for any t K h h t h +
proof. The number of arrivals is Poisson. proof by induction. ( ) 0 0 0 0 0 0 0 For 0, ( ) ( ) 1 using the memortless property. ( ) ( ) Dividing both sides by , ( ) ( ) Taking the limit 0, = ( ) k P t t P t t P t t P t t P t t dP t t P t dt λ λ λ = + ∆ = + ∆ = − ∆ → ( ) 0 0 0 0 General Solution: ( ) for any constant. Boundary Condition: we require that (0) 1, which leads to 1 Finally we have ( ) 0! t t t P t ce P c t e P t e λ λ λ λ = = = = =

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( ) 1 1 1 In general, we have ( ) ( ) 1 ( ) ( ) ( ) Dividing both sides by
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Poisson Arrival Process - POISSON ARRIVAL RANDOM PROCESS...

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