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
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 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
Asymptotic Variance for (simple) After finite time, server busy forever… is approximately the same as when or
Intermediate Summary GI/G/1 GI/G/1/K ? ? M/M/1/K M/M/1 ? ?
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 :
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
K-1 K 0 1 Some (partial) intuition for M/M/1/K Easy to see:
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.