B alancing R educes A symptotic V ariance of O utputs

B alancing R educes A symptotic V ariance of O utputs

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B alancing R educes A symptotic V ariance of O utputs

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1. Balancing Reduces Asymptotic Variance of Outputs Yoni Nazarathy* EURANDOM, Eindhoven University of Technology,The Netherlands. Based on some joint works withAhmad Al Hanbali, Michel Mandjes, Gideon Weiss and Ward Whitt QTNA 2010, Beijing, July 26, 2010. *Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber

2. Overview • GI/G/1/K Queue (with or ) • number of customers served during • Asymptotic variance • Surprising results when Balancing Reduces Asymptotic Variance of Outputs

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

4. 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

5. Asymptotic Variance for (simple) After finite time, server busy forever… is approximately the same as when or

6. Intermediate Summary GI/G/1 GI/G/1/K ? ? M/M/1/K M/M/1 ? ?

7. Balancing Reduces Asymptotic Varianceof Outputs Theorem (Al Hanbali, Mandjes, N. , Whitt 2010):For the GI/G/1 queue with , under some further technical conditions: • Theorem (N. , Weiss 2008): For the M/M/1/K queue with : • Conjecture (N. , 2009):For the GI/G/1/K queue with , under furthertechnical conditions :

8. BRAVO Summary for GI/G/1/K For GI/G/1/K with : Proven: • : M/M/1/K • : * M/M/1 * Assuming finite forth moments: *M/G/1 *GI/NWU/1 (includes GI/M/1) *Any GI/G/1 with Numerically Conjectured: GI/G/1/K with light tails

9. Numerical Illustration: M/M/1/K

10. Numerical Illustration: M/M/1(finite T)

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

12. References • Yoni Nazarathy and Gideon Weiss, The asymptotic variance rate of the output process of finite capacity birth-death queues.Queueing Systems, 59(2):135-156, 2008. • Yoni Nazarathy, 2009, The variance of departure processes: Puzzling behavior and open problems. Preprint, EURANDOM Technical Report Series, 2009-045. • Ahmad Al-Hanbali, Michel Mandjes, Yoni Nazarathy and Ward Whitt. Preprint. The asymptotic variance of departures in critically loaded queues. Preprint, EURANDOM Technical Report Series, 2010-001.