The asymptotic variance of departures in critically loaded queues

Al Hanbali, A.; Mandjes, M.; Nazarathy, Y.; Whitt, W.

Al Hanbali, A. ; Mandjes, M. ; Nazarathy, Y. ; Whitt, W.

Adv. in Appl. Probab., Tome 43 (2011) no. 1, p. 243-263 / Harvested from Project Euclid

Résumé

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case where the system load ϱ equals 1, and prove that the asymptotic variance rate satisfies lim_t→∞varD(t) / t = λ(1 - 2 / π)(c_a² + c_s²), where λ is the arrival rate, and c_a² and c_s² are squared coefficients of variation of the interarrival and service times, respectively. As a consequence, the departures variability has a remarkable singularity in the case in which ϱ equals 1, in line with the BRAVO (balancing reduces asymptotic variance of outputs) effect which was previously encountered in finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multiserver queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue, we present an explicit expression of the variance of D(t) for any t.

Publié le : 2011-03-15
Classification: GI/G/1 queue, critically loaded system, uniform integrability, departure process, renewal theory, Brownian bridge, multiserver queue, 90B22, 60G55

@article{1300198521,
     author = {Al Hanbali, A. and Mandjes, M. and Nazarathy, Y. and Whitt, W.},
     title = {The asymptotic variance of departures in critically loaded queues},
     journal = {Adv. in Appl. Probab.},
     volume = {43},
     number = {1},
     year = {2011},
     pages = { 243-263},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1300198521}
}

Al Hanbali, A.; Mandjes, M.; Nazarathy, Y.; Whitt, W. The asymptotic variance of departures in critically loaded queues. Adv. in Appl. Probab., Tome 43 (2011) no. 1, pp.  243-263. http://gdmltest.u-ga.fr/item/1300198521/