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Variance of Production Counts over Long Time Horizon: Workshop on Stochastic Models of Manufacturing Systems

This workshop focuses on queueing output processes in manufacturing systems, aiming for high throughput and low variability. The event discusses various queueing systems, re-entrant lines, and the BRAVO effect.

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Variance of Production Counts over Long Time Horizon: Workshop on Stochastic Models of Manufacturing Systems

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  1. The Variance of Production Counts over a Long Time Horizon Yoni Nazarathy* EURANDOM, TU/e Contains joint work with Ahmad Al-Hanbali, Yoav Kerner,Michel Mandjes, Gideon Weiss and Ward Whitt Workshop on Stochastic Models of Manufacturing Systems Eindhoven, June 2010 *Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber

  2. Problem Domain: Queueing Output Processes PLANT OUTPUT - Single Server Queues - Tandem Queues - Re-Entrant Lines • Desired over long term: • High Throughput • Low Variability Our focus: for large T

  3. Variance Curves Example: Stationary stable M/M/1, D(t) is PoissonProcess( ): Example: Stationary M/M/1/1 with . D(t) is RenewalProcess(Erlang(2, )): Asymptotic Variance Rate of Outputs For Renewal Processes:

  4. Asymptotic Variance Rate M/M/1 Non-Stop Service Burkes Theorem

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

  6. Our main focus:Overloaded and critically loaded systems

  7. GI/G/1 Non-Stop Service

  8. Queues in Tandem (with 1 bottleneck) Bottleneck Server Just as simple…

  9. Re-entrant Line bottleneck In the stable case:

  10. Overloaded case --> Infinite Supply Re-entrant Line Result:

  11. Overloaded case --> Infinite Supply Re-entrant Line 1 1 2 3 6 5 4 6 8 8 7 9 Result:

  12. Shocking result* coming up… * at least for me

  13. Back to Single Server (GI/G/1/K) What happens here? BalancingReducesAsymptoticVariance ofOutputs Note: the figure assumes

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

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

  16. K-1 K 0 1 Some (partial) intuition for M/M/1/K Easy to see:

  17. Questions?

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