The Variance of Production Counts over a Long Time Horizon

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Presentation transcript:

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

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

Asymptotic Variance Rate of Outputs For Renewal Processes: 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:

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

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

Our main focus: Overloaded and critically loaded systems

GI/G/1 Non-Stop Service

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

Re-entrant Line bottleneck In the stable case:

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

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

Shocking result* coming up… * at least for me

Back to Single Server (GI/G/1/K) What happens here? Balancing Reduces Asymptotic Variance of Outputs Note: the figure assumes

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

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

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

Questions?