Lecture10_2up

# Lecture10_2up - ECE-547 Introduction to Computer...

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1 Purdue University ECE ECE -547 547 Introduction to Computer Introduction to Computer Communication Networks Communication Networks Instructor: Instructor: Xiaojun Xiaojun Lin Lin Lecture 10 Lecture 10 Purdue University ¾ Another important quantity of interest: S n : time until the n th arrival. …. X 1 X 2 X 3 X n

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2 Purdue University ¾ S n can can be expressed as: ¾ The event {S n t}, “the n th arrival before time t” is equivalent to the event {N(t) n} “the number of arrivals occurring by time t is at least n”. = = n i i n X S 1 …. S n t S n : Time until the : Time until the n-th th Arrival Arrival Purdue University ¾ P{S n t} = P{N(t) n} ¾ Differentiation both sides with respect to t, we have ! ) ( } ) ( { j t e j t N P j n j t n j λ = = = = = )! 1 ( ) ( ) ( )! 1 ( ) ( ! ) ( ) ( 1 1 = + = = = n t e t f j t e j t e t f n t S j n j t j n j t S n n
3 Purdue University ¾ is the Gamma or Erlang distribution ¾ Another (less instructive) Derivation: Why? n n X S zt z n i i n t t t Then F G M e E t Define X S n ) ( )] ( [ ) ( : . . . ] [ ) ( : 1 1 = = = = = λ ψ )! 1 ( ) ( ) ( 1 = n t e t f n t S n Purdue University ¾ Answer: This is the M.G.F of an Erlang (Gamma) distribution. n n X n t X t X t X t X t X t X t X t X t S S t t e E e E e E e E e e e E e E e E t n n n i i n n ) ( )] ( [ ]] [ [ ] [ ... ] [ ] [ ] ... [ ] [ ] [ ) ( 1 1 2 1 2 1 1 ) ( = = = = = = = = (Independence)

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4 Purdue University Paradox of Residual Life Paradox of Residual Life ¾ Suppose that buses arrive at a stop according to a
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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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Lecture10_2up - ECE-547 Introduction to Computer...

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