Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications

Through Laplace transforms, we study the extremes of a continuous-time Markov-additive process with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is ‘attained’. For this, we investigate discrete-time Markov-additive processes and use...

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Published in:	Methodology and computing in applied probability Vol. 13; no. 2; pp. 221 - 267
Main Authors:	Dieker, A. B., Mandjes, M.
Format:	Journal Article
Language:	English
Published:	Boston Springer US 01.06.2011 Springer Nature B.V
Subjects:	Buffers Business and Management Economics Electrical Engineering Laplace transforms Life Sciences Markov analysis Mathematical analysis Mathematical models Mathematics and Statistics Matrices Networks Nonlinearity Probability Queues Queuing Statistics Studies Secondary 90B05 Markov-additive processes Queueing networks Fluctuation theory 60G15 Primary 60K25
ISSN:	1387-5841, 1573-7713
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Abstract	Through Laplace transforms, we study the extremes of a continuous-time Markov-additive process with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is ‘attained’. For this, we investigate discrete-time Markov-additive processes and use an embedding to relate these to the continuous-time setting. The resulting Laplace transforms are given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. We also study networks of fluid queues, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product form. Fluid-driven priority systems also have this property.
AbstractList	Through Laplace transforms, we study the extremes of a continuous-time Markov-additive process with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is 'attained'. For this, we investigate discrete-time Markov-additive processes and use an embedding to relate these to the continuous-time setting. The resulting Laplace transforms are given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. We also study networks of fluid queues, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product form. Fluid-driven priority systems also have this property.[PUBLICATION ABSTRACT] Through Laplace transforms, we study the extremes of a continuous-time Markov-additive process with one-sided jumps and a finite-state background Markovian state-space, jointly with the epoch at which the extreme is 'attained'. For this, we investigate discrete-time Markov-additive processes and use an embedding to relate these to the continuous-time setting. The resulting Laplace transforms are given in terms of two matrices, which can be determined either through solving a nonlinear matrix equation or through a spectral method. Our results on extremes are first applied to determine the steady-state buffer-content distribution of several single-station queueing systems. We show that our framework comprises many models dealt with earlier, but, importantly, it also enables us to derive various new results. At the same time, our setup offers interesting insights into the connections between the approaches developed so far, including matrix-analytic techniques, martingale methods, the rate-conservation approach, and the occupation-measure method. We also study networks of fluid queues, and show how the results on single queues can be used to find the Laplace transform of the steady-state buffer-content vector; it has a matrix quasi-product form. Fluid-driven priority systems also have this property.
Author	Dieker, A. B. Mandjes, M.
Author_xml	– sequence: 1 givenname: A. B. surname: Dieker fullname: Dieker, A. B. organization: H. Milton Stewart School of Industrial and Systems Engineering – sequence: 2 givenname: M. surname: Mandjes fullname: Mandjes, M. email: m.r.h.mandjes@uva.nl organization: Korteweg-de Vries Institute for Mathematics, University of Amsterdam
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Snippet	Through Laplace transforms, we study the extremes of a continuous-time Markov-additive process with one-sided jumps and a finite-state background Markovian...
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SubjectTerms	Buffers Business and Management Economics Electrical Engineering Laplace transforms Life Sciences Markov analysis Mathematical analysis Mathematical models Mathematics and Statistics Matrices Networks Nonlinearity Probability Queues Queuing Statistics Studies
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Title	Extremes of Markov-additive Processes with One-sided Jumps, with Queueing Applications
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