1. Balancing Reduces Asymptotic Variance of Outputs Yoni Nazarathy * EURANDOM, Eindhoven University of Technology, The Netherlands. Based on some joint works with Ahmad 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 * Load: * Squared coefficient of variation: * Assume

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

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

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

7. B alancing R educes A symptotic V ariance of O utputs 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 further technical 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. 01 K K-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.