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# lec6 - Queueing Systems Lecture 2 Amedeo R Odoni Lecture...

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Queueing Systems: Lecture 2 Amedeo R. Odoni October 11, 2006 Lecture Outline M/M/m M/M/ M/M/1: finite system capacity, K M/M/m: finite system capacity, K M/M/m: finite system capacity, K=m Related observations and extensions M/E 2 /1 example M/G/1: epochs and transition probabilities Reference: Chapter 4, pp. 203-217

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M/M/m (infinite queue capacity) ( λ ) n μ P n = P 0 for n = 0, 1, 2, .... , m 1 n ! ( λ ) n μ P n = n m P 0 for n = m , m + 1, m + 2, .... m m ! Condition for steady state? 0 1 2 m-1 m m+1 λ λ λ λ λ λ λ 3 μ 2 μ μ (m-1) μ m μ m μ m μ …. M/M/ (infinite no. of servers) 0 1 2 m-1 m m+1 λ λ λ λ λ λ λ 3 μ 2 μ μ (m-1) μ m μ (m+1} μ (m+2) μ λ ( λ ) n e ( μ ) μ P n = for n = 0, 1, 2, ..... n ! N is Poisson distributed! L = λ / μ ; W = 1 / μ ; L q = 0; W q = 0 Many applications
M/M/1: finite system capacity, K; customers finding system full are lost λ λ λ λ λ 0 1 2 K-1 K μ μ μ μ μ P n = ρ n (1 K + ρ 1 ) for n = 0, 1, 2, ..... , K 1 ρ Steady state is always reached Be careful in applying Little’s Law! Must count only the

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• Fall '06
• hansman
• Markov process, Markov chain, Continuous-time Markov process, pn, transition probabilities, finite system capacity

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lec6 - Queueing Systems Lecture 2 Amedeo R Odoni Lecture...

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