queue-1 - CDA6530: Performance Models of Computers and...

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CDA6530: Performance Models of Computers and Networks Chapter 6: Elementary Queuing Theory
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2 Definition Queuing system: a buffer (waiting room), service facility (one or more servers) a scheduling policy (first come first serve, etc.) We are interested in what happens when a stream of customers (jobs) arrive to such a system throughput, sojourn (response) time, Service time + waiting time number in system, server utilization, etc.
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3 Terminology A/B/c/K queue A - arrival process, interarrival time distr. B - service time distribution c - no. of servers K - capacity of buffer Does not specify scheduling policy
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4 Standard Values for A and B M - exponential distribution (M is for Markovian) D - deterministic (constant) GI; G - general distribution M/M/1: most simple queue M/D/1: expo. arrival, constant service time M/G/1: expo. arrival, general distr. service time
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5 Some Notations C n : custmer n, n=1,2, a n : arrival time of C n d n : departure time of C n α (t): no. of arrivals by time t (t): no. of departure by time t N(t): no. in system by time t N(t)= α (t)- (t)
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6 Average arrival rate (from t=0 to now): λ t = α (t)/t α (t) (t)
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7 Little’s Law (t): total time spent by all customers in system during interval (0, t) T t : average time spent in system during (0, t) by customers arriving in (0, t) T t = (t)/ α (t) N t : average no. of customers in system during (0, t) N t = (t)/t For a stable system, N t = λ t T t Remmeber λ t = α (t)/t For a long time and stable system N = λ T Regardless of distributions or scheduling policy
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8 Utilization Law for Single Server Queue X: service time, mean T=E[X] Y: server state, Y=1 busy, Y=0 idle ρ : server utilization, ρ = P(Y=1) Little’s Law: N = λ E[X] While: N = P(Y=1) · 1 + P(Y=0) ·
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This note was uploaded on 01/14/2012 for the course CDA 6530 taught by Professor Zou during the Fall '11 term at University of Central Florida.

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queue-1 - CDA6530: Performance Models of Computers and...

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