The asymptotic variance of departures in critically loaded queues

Ahmad Al Hanbali, M.R.H. Mandjes, Y. Nazarathy, W. Whitt

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14 Citations (Scopus)

Abstract

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.
Original languageEnglish
Pages (from-to)243-263
JournalAdvances in applied probability
Volume43
Issue number1
DOIs
Publication statusPublished - 2011

Keywords

  • departure process
  • critically loaded system
  • uniform integrability
  • renewal theory
  • IR-76966
  • GI/G/1 queue
  • Multi-server queue
  • METIS-275143
  • Brownian bridge

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    Al Hanbali, A., Mandjes, M. R. H., Nazarathy, Y., & Whitt, W. (2011). The asymptotic variance of departures in critically loaded queues. Advances in applied probability, 43(1), 243-263. https://doi.org/10.1239/aap/1300198521