132B_1_Sec13_CTQS_090710

132B_1_Sec13_CTQS_090710 - Continuous-Time Queueing Systems...

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Continuous-Time Queueing Systems Professor Izhak Rubin Electrical Engineering Department UCLA rubin@ee.ucla.edu © 2010-2011 by Izhak Rubin
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© Prof. Izhak Rubin 2 M/M/1 Queueing System λ Server () For M/M/1 queueing system: 1. Messages arrive in accordance with a Poisson process with intensity [ /sec] Interarrival time distribution: 1 2. Message service time is exponentially dis t mess At e ut λ =− tributed with rate ; mean service time = 1/ [sec/mess]. Message service time distribution: 1 3. A single service channel is provided. 4. Unlimited storage (buffer) capacity. t Bt e μ
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© Prof. Izhak Rubin 3 System Size Process For M/M/1 queueing system: The system size process, { , 0}, {0,1,2,. ..} is a continuous time birth-and-death Markov chain, with: ,0 , ,1 . The traffic intensity is t i i XX t S i i λλ μμ λ ρ μ =≥ = = = total offered arrival rate / total service rate.
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© Prof. Izhak Rubin 4 M/M/1 Queueing System: System Size () 0 0 0 01 1 0 12 00 0 0, lim , From the CTBD analysis: . ... We have 1, , j 0 . ... 1 . lim i i t t i i ji i j j j j j i i ii i i t t a PX j Pj a Pj a a aa a a λλ λ λ ρ μμ μ ρρ = →∞ = = ∞∞ == = =∞ <∞ = ⎛⎞ = = ⎜⎟ ⎝⎠ <∞ ⇔ < =∞ ⇔ ∑∑ 1 j0 . 1, 1 j j ρρ ρ −<
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© Prof. Izhak Rubin 5 M/M/1 Queueing System: Average Message Waiting and Delay Times () ( ) ( ) 1 For 1, the steady-state mean queue size is: ,1 . 1 The steady-state mean message delay and waiting time \(using Little's Formula, noting that = for 1) are given as: 1 1 D EX ED EW ρ λλ λρ μρ μ < =< < == < =− 1 . 1
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© Prof. Izhak Rubin 6 M/M/1 Queueing System: Waiting Time Distribution () Steady-state waiting time distribution: lim ( ), t 0. We set: ( ) = P{an arriving message finds j messages ahead of itself in the queue}. We obtain ( ) ( n n Wt PW t j jP j π −>∞ =≤ = ) (1 ) , 0, by using a result that states that Poisson Arrivals See Time Averages (PASTA). To calculate the waiting time distribution, we assume a FCFS service policy. An arriving j j ρρ =− message that finds upon arrival that there are j messages ahead of itself, j 1, will have to wait until these messages are served before it is admitted into service. The services times of these j mes sages are statistically independent and identically distributed; this is also true for the message that is in the midst of its service when the message for which the wait time is computed arrives into the system. Clearly, the distribution of the sum of j independent service times (actually, the sum of the residual service time of the message in service plus the service times of the waiting j-1 messages) is equal to the j-th order convolution of the service time distribution. When j=0, the arriving message enters service without having to wait.
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© Prof. Izhak Rubin 7 M/M/1 Queueing System: Waiting Time Distribution (Cont.) () * 1 1 Hence, we write: (0) 1 ( ) (1 ) ( ) , t 0.
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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_Sec13_CTQS_090710 - Continuous-Time Queueing Systems...

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