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Conditional limit theorems for queues with Gaussian input, a weak convergence approach

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  • Dieker, A.B.

Abstract

We consider a buffered queueing system that is fed by a Gaussian source and drained at a constant rate. The fluid offered to the system in a time interval (0,t] is given by a separable continuous Gaussian process Y with stationary increments. The variance function of Y is assumed to be regularly varying with index 2H, for some 0 [infinity]. In addition, we study how a busy period longer than T typically occurs as T-->[infinity], and we find the logarithmic asymptotics for the probability of such a long busy period. The study relies on the weak convergence in an appropriate space of to a fractional Brownian motion with Hurst parameter H as [alpha]-->[infinity]. We prove this weak convergence under a fairly general condition on [sigma]2, sharpening recent results of Kozachenko et al. (Queueing Systems Theory Appl. 42 (2002) 113). The core of the proof consists of a new type of uniform convergence theorem for regularly varying functions with positive index.

Suggested Citation

  • Dieker, A.B., 2005. "Conditional limit theorems for queues with Gaussian input, a weak convergence approach," Stochastic Processes and their Applications, Elsevier, vol. 115(5), pages 849-873, May.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:5:p:849-873
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    Cited by:

    1. Debicki, K. & Kosinski, K.M. & Mandjes, M. & Rolski, T., 2010. "Extremes of multidimensional Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2289-2301, December.
    2. Mandjes, Michel & Mannersalo, Petteri & Norros, Ilkka & van Uitert, Miranda, 2006. "Large deviations of infinite intersections of events in Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1269-1293, September.
    3. Barbe, Ph. & McCormick, W.P., 2012. "Heavy-traffic approximations for fractionally integrated random walks in the domain of attraction of a non-Gaussian stable distribution," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1276-1303.

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