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Unformatted text preview: October 27, 2004 Queueing Systems: Lecture 5 Amedeo R. Odoni Lecture Outline • Bounds for G/G/1 systems • A fundamental result for queueing networks • State transition diagrams for Markovian queueing systems and networks: examples • Analysis of dynamic systems • Qualitative behavior of dynamic systems Reference: Sections 4.8.3, 4.10, 4.11 + material in handout A general upper bound for G/G/1 systems • • (1) X S • ) 1 ( ) 1 ( 2 ) ( 2 2 < − ⋅ + ⋅ ≤ ρ ρ σ S X q W A number of bounds are available for very general queueing systems (see Section 4.8) A good example is an upper bound for the waiting time at G/G/1 systems: where and are, respectively, the r.v.’s denoting inter arrival times and service times Under some fairly general conditions, such bounds can be tightened and perform extremely well σ λ Better bounds for a (not so) special case • t , λ 1 ]  [ 0 ≤ > − t X ) 1 ( ) 1 ( 2 ) ( 2 1 2 2 < − ⋅ + ⋅ ≤ + − ρ ρ σ λ ρ S X q B W B ( ) • at most, 1/ λ ρ increases! (2) For G/G/1 systems whose interarrival times have the property that for all nonnegative values of t X E it has been shown that: = ≤ σ λ what does this mean, intuitively? Note that the upper and lower bounds in (1) differ by, and that the percent difference between the upper and lower bounds decreases as A result which is important in analyses of queuing networks Let the arrival process at a M/M/m queuing parameter λ ....
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 Fall '06
 hansman
 Poisson Distribution, Queueing theory, Queue area, Attractor

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