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Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.3/189594
- The asymptotic variance of departures in critically loaded queues
- Al Hanbali, A.; Mandjes, M.; Nazarathy, Y.; Whitt, W.
- 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 limt→∞varD(t) / t = λ(1 - 2 / π)(ca2 + cs2), where λ is the arrival rate, and ca2 and cs2 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.
- Publication type
- Journal article
- Advances in Applied Probability, Vol. 43, no. 1 (Mar 2011), pp. 243-263
- Publication year
- FOR Code(s)
- 0102 Applied Mathematics; 0104 Statistics
- Brownian bridge; Critically loaded system; Departure process; GT/G/1 queue; Multiserver queue; Renewal theory; Uniform integrability
- Applied Probability Trust
- Publisher URL
- Copyright © 2011.
- Peer reviewed