Delay analysis of a two-class priority queue with external arrivals and correlated arrivals from another node
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Delay analysis of a two-class priority queue with external arrivals and correlated arrivals from another node Sofian De Clercq1 · Joris Walraevens1 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Calculation of the delay distribution of (low-priority) customers in priority queues is difficult, especially when the arrival process is correlated. In this paper, we find the generating functions of the delays of high- and low-priority customers in a discrete-time queue where the arrivals are generated by a superposition of two processes, an independent one and a correlated one that can bring an extra customer in the buffer during a slot. The latter process can be seen as a simplified model for the output process of another priority queue. As a consequence, our results can be used to estimate the delay in the second stage of a tandem queueing system with an extra exogenous arrival process. A tandem priority queue is not of product-form type and its analysis is known to be a hard problem. This paper can lead to a first approximate analysis that can later be extended to incorporate the correlation in the output process of a priority queue more accurately. Keywords Priority queue · Delay · Correlated arrivals · Generating functions
1 Introduction Priority scheduling is still one of the main scheduling types in network buffers to diversify delays of traffic streams with different delay requirements (see e.g. Chen et al. 2018; Gurvich and Van Mieghem 2018; Kafaei et al. 2018; Guo et al. 2018). When delay-sensitive highpriority packets (packets of voice and video streams, interactive gaming …) are present in the buffer, they are transmitted. Best-effort low-priority packets can thus only be transmitted when no high-priority traffic is present. Since priority scheduling is specifically implemented to affect and diversify delays, analysis of the delay distributions is essential to estimate its impact. Analysis of high-priority delay is usually much easier than that of low-priority delay, since high-priority packets are not (if the priority is preemptive) or only slightly (if the priority is non-preemptive) impacted
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Joris Walraevens [email protected] Sofian De Clercq [email protected]
1
Department of Telecommunications and Information Processing (EA07), Ghent University - UGent, Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium
123
Annals of Operations Research
by low-priority packets. Low-priority delay on the other hand is highly impacted by highpriority packets. In fact, the server is unavailable for low-priority packets when high-priority packets are being served. Therefore, low-priority delay analysis has quite some analogies with busy period analysis of a single-class queue (Stanford 1997). One element that they have in common is that their Laplace–Stieltjes Transforms (in case of continuous time) or Probability Generating Functions (in case of discrete time) cannot be calculated explicitly, and some implicit functions are still present in the final expression. The dela
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