Lecture9

Lecture9 - ECE-547 Introduction to Computer Communication...

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Purdue University ECE ECE - - 547 547 Introduction to Computer Introduction to Computer Communication Networks Communication Networks Instructor: Instructor: Xiaojun Xiaojun Lin Lin Lecture 9 Lecture 9
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Purdue University Superposition of Poisson Superposition of Poisson ¾ Theorem: Let {L t , t 0} and {M t , t 0} be two independent Poisson Processes with rates λ and μ , respectively. For each ω∈Ω and t 0, let N t ( ω ) = L t ( ω ) + M t ( ω ) Then the resulting process {N t , t 0} is called the superposition of processes {L t , t 0} and {M t , t 0} and is a Poisson Process with rate λ + μ . L t M t N t
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Purdue University ¾ Proof: To show n! ] ) [( e n} P{N )t -( t n t μ λ + = = + k n k n k n k n k n t t t k n P k k n k Note k n k = + = = = = + + + = = = = = = = = = = = = = ) ( ) ( k)! - (n k! ! n! ] ) [( e k)! - (n ] [ e k! ] [ e } M { } L P{ }] M , {L } n N [{ : } M , L P{ n} P{N n 0 k t ) ( - t - n 0 k t - t t n 0 k 0 t t t t t n 0 k t U
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Purdue University Proof (Cont.) Proof (Cont.) • Finally: • Thus: n! ] ) [( e )t -( n t μ λ + = + n!
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This note was uploaded on 01/06/2010 for the course ECE 547 taught by Professor Xiaojunlin during the Spring '09 term at Purdue.

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Lecture9 - ECE-547 Introduction to Computer Communication...

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