Abstract
We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability β, and a customer of queue 2 is served with probability 1−β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z 1,z 2) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z 1,z 2) in β. The first coefficient of this power series corresponds to the priority case β=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Padé approximation.
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J. Walraevens is a postdoctoral fellow with the Fund for Scientific Research, Flanders (FWO-Vlaanderen). His research was done during a stay of the author at the EURANDOM Research institute, and was supported by a travel grant of the FWO-Vlaanderen.
J.S.H. van Leeuwaarden was supported by an NWO VENI grant.
Part of the research of O.J. Boxma was done in the framework of the European Network of Excellence Euro-FGI and the Dutch BRICKS project.
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Walraevens, J., van Leeuwaarden, J.S.H. & Boxma, O.J. Power series approximations for two-class generalized processor sharing systems. Queueing Syst 66, 107–130 (2010). https://doi.org/10.1007/s11134-010-9188-8
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DOI: https://doi.org/10.1007/s11134-010-9188-8