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07 MM1

# 07 MM1 - CS-350 Fundamentals of Computing Systems Lecture...

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CS-350: Fundamentals of Computing Systems Page 1 of 9 Lecture Notes © Azer Bestavros. All rights reserved. Reproduction or copying (electronic or otherwise) is expressly forbidden except for students enrolled in CS-350. Elementary Queuing Analysis Notation In discussing various server queues, it will be necessary to talk about various “random variables” associated with these queues. The Figure 1 below provides the notation used to describe these variables. Notice that often times, we will be interested in the expected values of random variables, thus unless specified otherwise, when we use these variables to denote the expected value of the random variable in question. Figure 1 Notation used to refer to various variables in a single server queue Analysis of an M/M/1 Queuing System M/M/1 Queues An M/M/1 queuing system is a single-queue single-server queuing system in which arrivals are Poisson and service time is exponential. The notation M/M/1 describes the "queue" in the system as having a M arkovian arrival process (i.e. Poisson) and a M arkovian (i.e. exponential) service discipline with 1 server.

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CS-350: Fundamentals of Computing Systems Page 2 of 9 Lecture Notes © Azer Bestavros. All rights reserved. Reproduction or copying (electronic or otherwise) is expressly forbidden except for students enrolled in CS-350. Figure 2 A single queue with Poisson arrivals and Exponential Service time is an M/M/1 system Birth and death probabilities for M/M/1 Consider a very small interval of time of length h . Assume that this interval of time ( h ) is so small that a maximum of one arrival can realistically occur in that period of time. Since the rate of arrival is λ requests per unit time, then it follows that the rate of arrival per interval h is λ h . For instance, if λ = 100 requests/second, and h = 1msec (i.e., 0.001 seconds), then the rate of arrivals per millisecond is λ h = 100*0.001 = 0.1 requests/milliseconds. During an interval h one of two things can happen: either no requests arrive during that small interval of time, or one request does arrive. We call the arrival of a request to the system a “ birth event. We know that the probability density of the Poisson distribution is: ,... 2 , 1 , 0 , ) ! ( ) ( = = x e x x f x λ λ Given that the rate of arrival per interval h is λ h , the probability of x arrivals per interval h is ,... 2 , 1 , 0 , ) ! ) ( ( ) ( = = x e x h x f h x λ λ According to the above equation, the probability that there will be no arrivals during a given interval h is f(0) = e −λ h . Now, since we assume h to be too small for anything more than one arrival (or births), we can say that the probability of more than one arrival is negligible. Thus, the probability of one or more arrivals in the interval h is a good approximation for the probability of exactly one arrival.
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07 MM1 - CS-350 Fundamentals of Computing Systems Lecture...

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