132B_1_Recitation9_Queueing_Systems

132B_1_Recitation9_Queueing_Systems - EE132B - Recitation 9...

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EE132B - Recitation 9 Queueing Systems Prof. Izhak Rubin rubin@ee.ucla.edu Electrical Engineering Department UCLA
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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 2
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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 ii X ji P i j ij    0 1 2 3 λ 0 λ 1 λ 2 λ 3 μ 1 μ 2 μ 3 3
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Birth & Death Process (cont.) • Balanced equations – Flow into a node = flow out of a node 01 11 0 1 1 At node 0: (0) (1) At any other node : ( ) ( ) ( 1) ( Node 0: (0) (0) ( ( ), 0 j j j j j j PP j P j P j P j P j P j j    1 1 0 2 1 2 Node 1: ( ) (1) (0) (2) (2) P P P 4
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Birth & Death Process (cont.) 1 1 -2 -1 -2 -3 0 1 -2 -3 0 1 0 0 ... ( ) ( 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          -2 -3 0 0 1 1 1 00 0 ... (0) . . (0) . .. 1 ..... (0) 1, (0) ( ) , 0,1,2,. . j j j j j j jj j i i PP P a P a a P j j a      5
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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 6
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The Geom/Geom/1 QS Geom/Geom/1 – Inter-arrival times are geometrically distributed – Service times are geometrically distributed – 1 server is present in the system Recall that this system is discrete. That means that – Arrivals can only occur at the beginning of a time slot
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This note was uploaded on 01/12/2011 for the course EE 132B taught by Professor Izhakrubin during the Fall '09 term at UCLA.

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132B_1_Recitation9_Queueing_Systems - EE132B - Recitation 9...

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