Discussion Notes 10

Discussion Notes 10 - EE132B Recitation 10 Queueing Systems...

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Unformatted text preview: EE132B - Recitation 10 Queueing Systems Prof. Izhak Rubin [email protected] Electrical Engineering Department UCLA Outline • Birth & Death Process • Discrete Time Queueing System – Geometric Distribution – The Geom/Geom/1 QS – Example • Continuous Time Queueing System – The M/M/1 QS – Example Birth & Death Process A process is said to be a birth & death process if the TPF is given by: , for 1 , for 1 ( , ) 1 , for 0, otherwise i i i i X j i j i P i j i j λ μ λ μ = + = - = -- = 1 2 3 λ λ 1 λ 2 λ 3 μ 1 μ 2 μ 3 Birth & Death Process (cont.) • Balanced equations – Flow into a node = flow out of a node 1 1 1 1 1 1 At node 0: (0) (1) At any other node : ( ) ( ) ( 1) ( 1) Node 0: (0) (1) (1) (0) ( 1) ( ), j j j j j j P P j P j P j P j P P P P P j P j j λ μ λ μ λ μ λ μ λ μ λ μ- + + = + =- + + = ⇒ = ∴ + = ≥ 1 1 2 1 2 Node 1: ( ) (1) (0) (2) (2) (1) P P P P P λ μ λ μ λ μ + = + ⇒ = Birth & Death Process (cont.) 1 1-2-1-2-3-1-1-1 1-1-2-3-1-1 1 ... ( ) ( 1) ( 2) .... (0) ..... ... Define ..... A unique stationary distribution exists for ( ) (0) j j j j j j j j j j j j j j j j j j j j j j P j P j P j P a a P j P λ λ λ λ λ λ λ μ μ μ μ μ μ μ λ λ λ λ μ μ μ μ-- ∞ = ∞ = =- =- = = < ∞ = ∑ ∑-1-2-3 1-1-1 1 1 ... (0) .. (0) ... 1 ..... (0) 1, (0) ( ) , 0,1,2,.. j j j j j j j j j j j i i P P P a P a a P j j a λ λ λ λ λ μ μ μ μ μ- ∞ ∞ = = ∞ = + + + + = ⇒ = = ∴ = = ∑ ∑ ∑ Discrete Time Queueing Systems • Example: Geom/Geom/1 • Geometric Distribution – Recall the following properties for geometric distribution • It is the only discrete memoryless distribution • Let p = probability of success • And 1-p = probability of failure • If T n is a geometrically distributed random variable, P(T n = j) = p(1-p) j-1 The Geom/Geom/1 QS...
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Discussion Notes 10 - EE132B Recitation 10 Queueing Systems...

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