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Unformatted text preview: October 18, 2004 Queueing Systems: Lecture 3 Amedeo R. Odoni Announcements • PS #3 due today • Office hours – Odoni: Tue. 1012 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) 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 K1 Can write system balance equations and obtain 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/ m1 (m1 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 • m1 m+1 (m1 (m+1} (m+2) ,..... 2 , 1 , 0 n for µ ; Many applications Variations and extensions of birthanddeath queueing systems • Huge number of extensions on the previous models • Most common is arrival rates and service rates that depend on state of the system; some lead to closedform expressions...
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This note was uploaded on 11/08/2011 for the course AERO 16.72 taught by Professor Hansman during the Fall '06 term at MIT.
 Fall '06
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