MIT16_36s09_lec20

MIT16_36s09_lec20 - MIT OpenCourseWare http://ocw.mit.edu...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.36 Communication Systems Engineering Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 16.36: Communication Systems Engineering Lecture 20: Delay Models for Data Networks Part 2: Single Server Queues Eytan Modiano Eytan Modiano Slide 1 Single server queues buffer packet per second Server packet per second Service time = 1/ M/M/1 Poisson arrivals, exponential service times M/G/1 Poisson arrivals, general service times M/D/1 Poisson arrivals, deterministic service times (fixed) Eytan Modiano Slide 2 Markov Chain for M/M/1 system 0 1 2 k 1 State k k customers in the system P(i,j) = probability of transition from state I to state j As 0, we get: P(0,0) = 1 - , P(j,j+1) = P(j,j) = 1 - P(j,j-1) = P(i,j) = 0 for all other values of I,j. Birth-death chain: Transitions exist only between adjacent states , are flow rates between states Eytan Modiano Slide 3 Equilibrium analysis We want to obtain P(n) = the probability of being in state n At equilibrium P(n) = P(n+1) for all n P(n+1) = ( / )P(n) = P(n), = / It follows: P(n) = n P(0) Now by axiom of probability: P ( n ) = 1 i = ! " # $ n P (0) = P (0) 1 % $ = 1 i = ! " # P (0) = 1 %$ P ( n ) = $ n (1 % $ ) Eytan Modiano Slide 4 Average queue size N = nP ( n ) = n = ! " n # n (1 $ # ) = # 1 $ # n = ! " N = # 1 $ # = % / 1 $ % / = % $ % N = Average number of customers in the system The average amount of time that a customer spends in the system can be obtained from Littles formula (N=...
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MIT16_36s09_lec20 - MIT OpenCourseWare http://ocw.mit.edu...

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