×

First passage of time-reversible spectrally negative Markov additive processes. (English) Zbl 1185.90201

Summary: We study the first passage process of a spectrally negative Markov additive process (MAP). The focus is on the background Markov chain at the times of the first passage. This process is a Markov chain itself with a transition rate matrix \(\varLambda \). Assuming time reversibility, we show that all the eigenvalues of \(\varLambda \) are real, with algebraic and geometric multiplicities being the same, which allows us to identify the Jordan normal form of \(\varLambda \). Furthermore, this fact simplifies the analysis of fluctuations of a MAP. We provide an illustrative example and show that our findings greatly reduce the computational efforts required to obtain \(\varLambda \) in the time-reversible case.

MSC:

90C40 Markov and semi-Markov decision processes
90B22 Queues and service in operations research
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Asmussen, S., Stationary distributions for fluid flow models with or without Brownian noise, Stochastic Models, 11, 21-49 (1995) · Zbl 0817.60086
[2] Asmussen, S., Applied probability and queues, Applications of Mathematics (2003), Springer-Verlag New York, Inc. · Zbl 1029.60001
[3] Asmussen, S.; Avram, F.; Pistorius, M., Russian and American put options under exponential phase-type Lévy models, Stochastic Processes and their Applications, 109, 79-111 (2004) · Zbl 1075.60037
[4] Asmussen, S.; Kella, O., A multi-dimensional martingale for Markov additive processes and its applications, Advances in Applied Probability, 32, 376-393 (2000) · Zbl 0961.60081
[5] Bellman, R., Introduction to Matrix Analysis (1960), McGraw-Hill: McGraw-Hill New York · Zbl 0124.01001
[6] Çinlar, E., Markov additive processes. I., Zeitschrift fur Wahrscheinlichkeitstheorie und Verw. Gebiete, 24, 85-93 (1972) · Zbl 0236.60047
[7] B. D’Auria, J. Ivanovs, O. Kella, M. Mandjes, First passage process of a Markov additive process, with applications to reflection problems (submitted for publication); B. D’Auria, J. Ivanovs, O. Kella, M. Mandjes, First passage process of a Markov additive process, with applications to reflection problems (submitted for publication)
[8] A.B. Dieker, M. Mandjes, Extremes of Markov-additive processes with one-sided jumps, with queueing applications, Methodology and Computing in Applied Probability (2009) (in press) http://www2.isye.gatech.edu/ adieker3/publications/modulatedfluid.pdf; A.B. Dieker, M. Mandjes, Extremes of Markov-additive processes with one-sided jumps, with queueing applications, Methodology and Computing in Applied Probability (2009) (in press) http://www2.isye.gatech.edu/ adieker3/publications/modulatedfluid.pdf · Zbl 1218.60077
[9] Elwalid, A.; Mitra, D., Effective bandwidth of general Markovian traffic sources and admission control of high speed networks, IEEE/ACM Transactions on Networking, 1, 329-343 (1993)
[10] Horn, R. A.; Johnson, C. A., Matrix Analysis (1985), Cambridge University Press · Zbl 0576.15001
[11] Hryniv, R.; Lancaster, P., On the perturbation of analytic matrix functions, Integral Equations Operator Theory, 34, 325-338 (1999) · Zbl 0940.47008
[12] J. Ivanovs, O. Boxma, M. Mandjes, Singularities of the generator of a Markov additive process with one-sided jumps Technical Report 2008-037, EURANDOM, http://www.eurandom.tue.nl/reports/2008/037-report.pdf; J. Ivanovs, O. Boxma, M. Mandjes, Singularities of the generator of a Markov additive process with one-sided jumps Technical Report 2008-037, EURANDOM, http://www.eurandom.tue.nl/reports/2008/037-report.pdf · Zbl 1197.60074
[13] Jobert, A.; Rogers, L. C.G., Option pricing with Markov-modulated dynamics, SIAM Journal on Control and Optimization, 44, 2063-2078 (2006) · Zbl 1158.91380
[14] Karandikar, R. L.; Kulkarni, V. G., Second-order fluid flow models: Reflected Brownian motion in a random environment, Operations Research, 43, 77-88 (1995) · Zbl 0821.60087
[15] Kella, O.; Stadje, W., A Brownian motion with two reflecting barriers and Markov-modulated speed, Journal of Applied Probability, 41, 1237-1242 (2004) · Zbl 1062.60101
[16] Kosten, L., Stochastic theory of data-handling systems with groups of multiple sources, (Performance of Computer-Communication Systems (1984), Elsevier: Elsevier Amsterdam), 321-331
[17] Kyprianou, A. E., (Introductory Lectures on Fluctuations of Lyévy Processes with Applications. Introductory Lectures on Fluctuations of Lyévy Processes with Applications, Universitext (2006), Springer-Verlag Berlin Heidelberg) · Zbl 1104.60001
[18] Kyprianou, A. E.; Palmowski, Z., Fluctuations of spectrally negative Markov additive processes, Séminaire de Probabilité, XLI, 121-135 (2008) · Zbl 1156.60060
[19] Prabhu, N. U., Stochastic storage processes, (Queues, Insurance Risk, Dams, and Data Communication. Queues, Insurance Risk, Dams, and Data Communication, Applications of Mathematics, vol. 15 (1998), Springer-Verlag: Springer-Verlag New York) · Zbl 0453.60094
[20] Prabhu, N. U.; Zhu, Y., Markov-modulated queueing systems, Queueing Systems, 5, 215-245 (1989) · Zbl 0694.60087
[21] Rogers, L. C.G., Fluid models in queueing theory and Wiener-Hopf factorization of Markov chains, Annals of Applied Probability, 4, 390-413 (1994) · Zbl 0806.60052
[22] Stern, T. E.; Elwalid, A. I., Analysis of separable Markov-modulated rate models for information-handling systems, Advances in Applied Probability, 23, 105-139 (1991) · Zbl 0716.60114
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.