Abstract
Motivated by applications in production and computer-communication systems, we study an N-queue polling system, consisting of an inner part and an outer part, and where products receive service in batches. Type-i products arrive at the outer system according to a renewal process and accumulate into a type-i batch. As soon as D i products have accumulated, the batch is forwarded to the inner system where the batch is processed. The service requirement of a type-i batch is independent of its size D i . For this model, we study the problem of determining the combination of batch sizes \({\vec{D}^{({\rm opt})} }\) that minimizes a weighted sum of the mean waiting times. This model does not allow for an exact analysis. Therefore, we propose a simple closed-form approximation for \({\vec{D}^{({\rm opt})}}\), and present a numerical approach, based on the recently proposed mean waiting-time approximation in Boon et al. (Perform Eval 68, 290–306, 2011). Extensive numerical experimentation shows that the numerical approach is slightly more accurate than the closed-form solution, while the latter provides explicit insights into the dependence of the optimal batch sizes on the system parameters and into the behavior of the system. As a by-product, we observe near-insensitivity properties of \({\vec{D}^{({\rm opt})}}\), e.g. to higher moments of the interarrival and switch-over time distributions.
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Acknowledgements
The authors wish to thank Marko Boon for placing parts of his polling system simulation program at their disposal, and for useful comments on earlier drafts of this paper.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Dorsman, J.L., Van der Mei, R.D. & Winands, E.M.M. Polling systems with batch service. OR Spectrum 34, 743–761 (2012). https://doi.org/10.1007/s00291-011-0275-y
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DOI: https://doi.org/10.1007/s00291-011-0275-y