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lec8 (1)

# lec8 (1) - Queueing Systems Lecture 4 Amedeo R Odoni...

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Queueing Systems: Lecture 4 Amedeo R. Odoni October 20, 2006 Lecture Outline M/G/1: a simple example 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)

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Expected values for M/G/1 ρ 2 + λ 2 σ S 2 L = ρ + ( ρ < 1) 2(1 ρ ) 1 ρ 2 + λ 2 σ S 2 W = + μ 2 λ (1 ρ ) W q = ρ 2 + λ 2 σ S 2 = ρ 2 (1 + C S 2 ) = 1 ρ (1 + C S 2 ) 2 λ (1 ρ ) 2 λ (1 ρ ) μ (1 ρ ) 2 L q = ρ 2 2 + (1 λ 2 ρ σ ) S 2 Note : C S = E σ [ S S ] = μ σ S 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 L q L q W q W q λ (per hour) ρ (% change) (seconds) (% 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 Announcements Quiz #1: October 25, open book, in class; quiz begins at 10:00, ends at 12:30; pick any TWO hours (10-12 or 10:30-12:30) Odoni: Office hrs Tu 10-12 Old quiz problems and solutions: posted Quiz coverage for Chapter 4: Sections 4.0 – 4.6 (inclusive)

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Background and observations W, L, W q and L q are not affected by the order of service, as long as the queue discipline does not give priority to certain classes of
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