# qlec4 - Queueing Systems Lecture 4 Amedeo R Odoni Lecture...

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Unformatted text preview: October 20, 2004 Queueing Systems: Lecture 4 Amedeo R. Odoni Lecture Outline • M/G/1: a couple of examples • Introduction to systems with priorities • Representation of a priority queuing system • The M/G/1 non-preemptive priority system • An important optimization theorem • … and an important corollary • Brief mention of other priority systems • Bounds for G/G/1 systems Reference: Chapter 4, pp. 222-239 (just skim Sections 4.8.2 and 4.8.4) Expected values for M/G/1 ) 1 ( ) 1 ( 2 2 2 2 < − ρ ρ ρ ρ S L ) 1 ( 2 2 2 2 ρ ρ − = S q L ) 1 ( 2 1 2 2 2 ρ λ ρ µ − S W 2 ) 1 ( ) 1 ( 1 ) 1 ( 2 ) 1 ( ) 1 ( 2 2 2 2 2 2 2 S S S q C C W + ⋅ − ⋅ = − + = − = ρ ρ µ ρ λ ρ ρ λ ρ S S S C Note σ ⋅ = = ] [ : ⋅ + + = σ λ σ λ ⋅ + σ λ ⋅ + + = ⋅ + σ λ S E σ µ Runway Example • Single runway, mixed operations • E[S] = 75 seconds; σ S = 25 seconds µ = 3600 / 75 = 48 per hour • Assume demand is relatively constant for a sufficiently long period of time to have approximately steady-state conditions • Assume Poisson process is reasonable approximation for instants when demands occur Estimated expected queue length and expected waiting time λ (per hour) ρ L q L q (% change) W q (seconds) W q (% change) 30 0.625 0.58 69 30.3 0.63125 0.60 3.4% 71 2.9% 36 0.75 1.25 125 36.36 0.7575 1.31 4.8% 130 4% 42 0.875 3.40 292 42.42 0.88375 3.73 9.7% 317 8.6% 45 0.9375 7.81 625 45.45 0.946875 9.38 20.1% 743 18.9% Can also estimate PHCAP ≅ 40.9 per hour Background and observations • W, L, W q and L q are not affected as long as the queue discipline does not give priority to certain classes of customers • W FIFO = W SIRO = W LIFO (what about the corresponding variances?) • Things may change, however, in systems...
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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.

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qlec4 - Queueing Systems Lecture 4 Amedeo R Odoni Lecture...

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