Lecture14_2up

Lecture14_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 14 Lecture 14 Purdue University Little Little ’s Law (Proof) s Law (Proof) 4 3 0 1 2 5 6 t1 t2 t3 w 1 t4 w 2 w 3 w 4 Tt 5 t 6 w 5 w 6 Cj Packets that arrive into system Time ¾ w j is defined as the waiting time in system for customer (packet) j ¾ c j is defined as the j-th customer arriving into system ¾ t j is defined as the arrival time of customer c j ¾ Note: Queueing system is not FIFO (why?)
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2 Purdue University Little Little ’s Law s Law ¾ In our system for j 4, 00 () (2 ) T jj j wI t d t I t d t = = ∫∫ > T T dt t I dt t I w 0 4 4 0 4 4 ) ( w fact, In ) ( ¾ Because all customers c j have departed by time T. ¾ Note that the second part of equation is not true if c j had not departed before time T because w 4 T Purdue University Little Little ’s Law s Law ¾ Now integrate N t (the number of customers in the system) 11 1 0 1 ( ) ( ) from (1) ( ) (interchanging the sum and the integral) , for j 4 TT T t tj j T T j j T j j Ndt I t dt I t d t Itd t w Λ == 0 Λ = Λ = = = = = ∑∑
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Lecture14_2up - ECE-547 Introduction to Computer...

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