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Analytic approximations of queues with lightly- and heavily-correlated autoregressive service times

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  • Dieter Fiems
  • Balakrishna Prabhu
  • Koen Turck

Abstract

We consider a single-server queueing system. The arrival process is modelled as a Poisson process while the service times of the consecutive customers constitute a sequence of autoregressive random variables. Our interest into autoregressive service times comes from the need to capture temporal correlation of the channel conditions on wireless network links. If these fluctuations are slow in comparison with the transmission times of the packets, transmission times of consecutive packets are correlated. Such correlation needs to be taken into account for an accurate performance assessment. By means of a transform approach, we obtain a functional equation for the joint transform of the queue content and the current service time at departure epochs in steady state. To the best of our knowledge, this functional equation cannot be solved by exact mathematical techniques, despite its simplicity. However, by means of a Taylor series expansion in the parameter of the autoregressive process, a “light-correlation” approximation is obtained for performance measures such as moments of the queue content and packet delay. We illustrate our approach by some numerical examples, thereby assessing the accuracy of our approximations by simulation. For the heavy correlation case, we give differential equation approximations based on the time-scale separation technique, and present numerical examples in support of this approximation. Copyright Springer Science+Business Media, LLC 2013

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  • Dieter Fiems & Balakrishna Prabhu & Koen Turck, 2013. "Analytic approximations of queues with lightly- and heavily-correlated autoregressive service times," Annals of Operations Research, Springer, vol. 202(1), pages 103-119, January.
    • Handle: RePEc:spr:annopr:v:202:y:2013:i:1:p:103-119:10.1007/s10479-011-0946-8
    DOI: 10.1007/s10479-011-0946-8
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    References listed on IDEAS

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    1. J. P. C. Blanc, 1992. "The Power-Series Algorithm Applied to the Shortest-Queue Model," Operations Research, INFORMS, vol. 40(1), pages 157-167, February.
    2. Blanc, J.P.C., 1998. "The power-series algorithm for polling systems with time limits," Other publications TiSEM 3366bad3-964d-4039-82cc-a, Tilburg University, School of Economics and Management.
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