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qlec3 - Queueing Systems Lecture 3 Amedeo R Odoni...

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October 18, 2004 Queueing Systems: Lecture 3 Amedeo R. Odoni Announcements PS #3 due today Office hours – Odoni: Tue. 10-12 AM or send me a message for an appointment Quiz #1: October 25, open book, in class Old quiz problems and solutions: posted Quiz coverage for Chapter 4: Sections 4.0 – 4.6 (inclusive)
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Lecture Outline M/M/m: finite system capacity, K M/M/m: finite system capacity, K=m M/G/1: epochs and transition probabilities M/G/1: derivation of L Why M/G/m, G/G/1, etc. are difficult Reference: Sections 4.7 and 4.8.1 M/M/m: finite system capacity, K; customers finding system full are lost 0 1 2 m λ λ λ λ λ λ 3 µ 2 µ µ m µ m µ m µ K λ m µ m µ λ …… …… closed form expressions for P n , L , W , L q , W q Often useful in practice m+1 K-1 Can write system balance equations and obtain
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M/M/m: finite system capacity, m; special case! …… 0 1 2 m λ λ λ λ λ 3 µ 2 µ µ ) µ m µ m i n P m i i n n ! ) ( ! ) ( 0 = = = µ λ µ λ P m P n of M/ m-1 (m-1 n for ,... 2 , 1 , 0 Probability of full system, , is “Erlang’s loss formula” Exactly same expression for G/m system with K=m M/M/ (infinite no. of servers) 0 1 2 m λ λ λ λ λ λ λ 3 µ 2 µ µ ) µ m µ µ µ ! ) ( ) ( = = n e P n n µ λ µ λ N is Poisson distributed! L = λ / µ ; W = 1 / L q = 0; W q = 0 m-1 m+1 (m-1 (m+1} (m+2) , ..... 2 , 1 , 0 n for µ ; Many applications
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Variations and extensions of birth-and-death queueing systems Huge number of extensions on the previous models Most common is arrival rates and service
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