Balancing Reduces Asymptotic Variance of Outputs Yoni Nazarathy

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 of Erjen Lefeber

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

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 Poisson. Process( ): * Stationary M/M/1/1 with. D(t) is Renewal. Process(Erlang(2, )): Notes: * In general, for renewal process with : * The output process of most queueing systems is NOT renewal

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

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

B alancing R A educes 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 technical conditions : : , under further

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

Numerical Illustration: M/M/1/K

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

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

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.