WaitingLinesFormula_3Models

WaitingLinesFormula_3Models - Basic waiting line models...

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Unformatted text preview: Basic waiting line models Model 1. This is a M/M/1 queue. σ = 1/μ and using the Pollaczek-Khintchine formula, we get λ2 μ (μ − λ ) λ L= μ −λ Lq = Wq = λ μ (μ − λ ) 1 . μ −λ W= n ⎛ λ ⎞⎛ λ ⎞ p n = ⎜1 − ⎟⎜ ⎟ , ⎜ μ ⎟⎜ μ ⎟ ⎝ ⎠⎝ ⎠ n = 0,1,2,... Model 2. This is a M/D/1 queue. σ = 0 and the corresponding formula are: λ2 2μ ( μ − λ ) λ L = Lq + μ Lq = Wq = λ 2μ ( μ − λ ) W = Wq + 1 μ . ⎛ λ⎞ p 0 = ⎜1 − ⎟. ⎜ μ⎟ ⎝ ⎠ Model 3. M/M/s queue. 1 p0 = ⎡ s −1 (λ / μ ) n ⎤ (λ / μ ) s 1 + ∑ n! ⎥ ⎢ s! 1 − λ /( sμ ) ⎦ ⎣ n =0 ⎧ (λ / μ ) n p0 , 0 ≤ n ≤ s ⎪ ⎪ n! pn = ⎨ n ⎪ (λ / μ ) p 0 , n > s. ⎪ s! s n − s ⎩ (λ / μ ) s λ / sμ Lq = p0 . s! (1 − λ / sμ ) 2 , Note: The following is an important model, though it is not required in this course, you may encounter it in a lot of situations, in study or in work. It is a more general model than that of Model 1 and Model 2. (In other words, Model 1 and Model 2 are special cases of this model: The M/G/1 queue. G stands for general distribution.) For M/G/1 queue, when ρ = λ/μ < 1, we have the Pollaczek-Khintchine formula: Lq = λ 2σ 2 + ρ 2 , 2(1 − ρ ) p0 = 1 − λ . μ where σ2 is the variance of the service time. 1 ...
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This note was uploaded on 09/21/2011 for the course COMP 4631 taught by Professor Ding during the Fall '11 term at HKUST.

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