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Unformatted text preview: 1 ECE & CSE ECE-CSE 861: Introduction to Computer Communication Networks Ness B. Shroff ECE & CSE Lectures 12-13 ECE & CSE L= λ w Little’s Law L = expected number of packets in the system λ = arrival rate w = expected waiting time Little’s Law ECE & CSE Little’s Law (Proof) 4 3 1 2 5 6 t1 t2 t3 w 1 t4 w 2 w 3 w 4 T t5 t6 w 5 w 6 Cj Packets that arrive into system Time w j is defined as the waiting time in system for customer (packet) j c j is defined as the jth customer arriving into system t j is defined as the arrival time of customer c j Note: Queueing system is not FIFO (why?) ECE & CSE Little’s Law More Definitions: I j (t) = 1, c j is in system at time t (not departed) = 0, otherwise N t = number of customers in the system at time t 1 Hence, ( ) (1) Now ( ) (waiting time for customer ) and ( ) = # of arrivals (of customers) by time = max ( : ) t j j j j j N I t w I t dt j t t j t t ∞ = ∞ = = Λ ≤ ∑ ∫ 2 ECE & CSE Little’s Law In our system for j ≤ 4, ( ) ( ) (2) T j j j w I t dt I t dt ∞ = = ∫ ∫ ∫ ∫ > ≠ T T dt t I dt t I w 4 4 4 4 ) ( w fact, In ) ( Because all customers c j have departed by time T. Note that the second part of equation is not true if c j had not departed before time T because w 4 T ECE & CSE Little’s Law Now integrate N t (the number of customers in the system) ( ) 1 1 ( ) 1 0 ( ) 1 ( ) ( ) from(1) ( ) (interchanging thesumand theintegral) , for j 4 T T T t t j j j j T T j j T j j N dt I t dt I t dt I t dt w Λ ∞ = = Λ = Λ = = = = = ≤ ∑ ∑ ∫ ∫ ∫ ∑ ∫ ∑ ( ) 1 Error term(negative) t t u j j N du w Λ = = + ∑ ∫ Now for an arbitrary t ≥ 0, N t ≥ 0 ECE & CSE...
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This note was uploaded on 03/05/2011 for the course ECE 861 taught by Professor Shroff during the Winter '11 term at Ohio State.
- Winter '11