Lecture23_2up

Lecture23_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 Lecture 23 23 Purdue University Jackson Jackson ’s Theorem s Theorem 1 (1) m is i k i k k qq λλ λ = =+ ¾ Outline of proof: W.L.O.G., we assume one source and one destination. Hence:
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2 Purdue University Jackson Jackson ’s Theorem s Theorem ¾ Example: . . . s s si q λ i 1 1( 2 ) m id ij j qq = + = −−− ¾ Similarly: ¾ Define: 1 12 1 1 ( ) { ,. .., } ( 1 ) { , ,..., , 1, ,..., } ( 1 1 ) { , ,. .., 1,. 1,. .. } m ii i i m ij i j m Pn n Pn n n n n n n n n −+ = −= −+ = + Purdue University Jackson Jackson ’s Theorem (Balance Equations) s Theorem (Balance Equations) ¾ State of the CTMC: (, , . . . , ) M nn n n = i i,j n 1 ,n 2 ,…,n m Two dimension One dimension m dimension: very cumbersome! } Rate arriving at n 11 1 Global balance equations: Rate leaving state n = Rate arriving into state n (n) ( 1 ) (1 ) is i i m id i i i pq p n qp n λμ μ == = ⎛⎞ + = ⎜⎟ ⎝⎠ + + ∑∑ (11 ) mm ji j i j ji n + + Rate leaving n (a) (b) (c) (3) when n i >= 1 for all i.
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3 Purdue University
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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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Lecture23_2up - ECE-547 Introduction to Computer...

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