Departure Process Variability of Queues and Networks

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Presentation transcript:

Departure Process Variability of Queues and Networks Yoni Nazarathy Swinburne University of Technology, Melbourne. Based on collaborations with Ahmad Al Hanbali, Daryl Daley, Michel Mandjes, Gideon Weiss and Ward Whitt IFORS 2011, Melbourne, July 15, 2011.

Problem Domain: Queueing Output Processes PLANT OUTPUT - Single Server Queue - Tandem Queue - Re-Entrant Line Desired over long term: High Throughput Low Variability Our focus: for large T

The GI/G/1/K Queue overflows * Assume * Load: * Squared coefficient of variation:

Variance of Outputs Asymptotic Variance Simple Examples: * Stationary stable M/M/1: D(t) is PoissonProcess( ): * Stationary M/M/1/1 with : D(t) is RenewalProcess(Erlang(2, )): Notes: * In general, for renewal process with * The output process of most queueing systems is NOT renewal

Asymptotic Variance for GI/G/1/K What happens here? Balancing Reduces Asymptotic Variance of Outputs Note: the figure assumes

BRAVO Effect (for M/M/1/K)

BRAVO Effect (illustration for M/M/1) More than a singular theoretic phenomenon

Balancing Reduces Asymptotic Variance of Outputs Theorem (Al Hanbali, Mandjes, N. , Whitt AAP 2011): For the GI/G/1 queue with , under some further technical conditions: Theorem (N. , Weiss QUESTA 2008): For the M/M/1/K queue with : Conjecture (N. , Daely, QUESTA To appear): For the GI/G/1/K queue with , under further technical conditions :

Additional Slides 4/28/2019Valuetools 2008

The Basic Loss-Less Stable Queueing System Q(t)

Some (partial) intuition for M/M/1/K Easy to see: 1 K K-1