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 ot t t λ ∆→ ∆= ∆+ ∆ ∆=− ∆+ ∆ ∆= = We call as the arrival rate, and 1 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 nn tt for any n . Then we can show: 1) t K is Poisson: () ( ) P[ ] 0,1, 2, ! k t kt t Pt K k e k k λ ≡= = = 2) X is exponential: 0 x X fx e x = ( ) The theorem holds true even if we define as the number of arrivals during the time interval , for any t K hh t h +
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proof. The number of arrivals is Poisson. proof by induction. ( ) 00 0 0 0 For 0, ( ) ( ) 1 using the memortless property. ( ) () Dividing both sides by , ( ) Taking the limit 0, = ( ) k Pt t Pt t t t dP t t dt 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 Pc te Pt e = = = = =
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( ) 1 1 1 In general, we have ( ) ( ) 1 ( ) ( ) () Dividing both sides by
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This note was uploaded on 11/23/2010 for the course EE EE528 taught by Professor Majungsoo during the Spring '10 term at Korea Advanced Institute of Science and Technology.

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Poisson Arrival Process - POISSON ARRIVAL RANDOM PROCESS...

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