Abstract
In this chapter we consider quasi-birth and death processes with low rank downward and upward transitions. We show how such structure can be exploited to reduce the computational cost of the cyclic reduction iteration. The proposed algorithm saves computation by performing multiplications and inversions of matrices of small size (equal to the rank instead of to the phase space dimension) and inherits the stability property of the customary cyclic reduction. Numerical experiments show the gain of the new algorithm in terms of computational cost. Quasi-birth and death process, Low rank matrix, Cyclic reduction
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Acknowledgements
The authors wish to thank the anonymous referees, and Juan Pérez and Benny Van Houdt for providing the code to construct the matrices A i , \(i = -1,0,1\), for the overflow queueing model example.
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Bini, D.A., Favati, P., Meini, B. (2013). A Compressed Cyclic Reduction for QBD processes with Low-Rank Upper and Lower Transitions. In: Latouche, G., Ramaswami, V., Sethuraman, J., Sigman, K., Squillante, M., D. Yao, D. (eds) Matrix-Analytic Methods in Stochastic Models. Springer Proceedings in Mathematics & Statistics, vol 27. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4909-6_2
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DOI: https://doi.org/10.1007/978-1-4614-4909-6_2
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