Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T16:08:18.476Z Has data issue: false hasContentIssue false
  • English
  • Français

A multi-dimensional martingale for Markov additive processes and its applications

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Søren Asmussen  and
Offer Kella
Søren Asmussen*
Affiliation:
University of Lund
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
*
∗ Postal address: Department of Mathematical Statistics, University of Lund, Box 118, S-221 00 Lund, Sweden. Email address: [email protected]
∗∗ Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish new multidimensional martingales for Markov additive processes and certain modifications of such processes (e.g., such processes with reflecting barriers). These results generalize corresponding one-dimensional martingale results for Lévy processes. This martingale is then applied to various storage processes, queues and Brownian motion models.

Keywords

Lévy processEOQ modelfluid queuesMarkov modulationmultivariate martingalereflected Brownian motionstorage process

MSC classification

Primary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 2 , June 2000 , pp. 376 - 393
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Supported in part by grant 794/97 from the Israel Science Foundation.

References

[1] Anick, D., Mitra, D. and Sondhi, M. M. (1982). Stochastic theory of a data-handling system with multiple sources. Bell System Tech. J. 61, 1871–1894.Google Scholar
[2] Asmussen, S. (1989). Risk theory in a Markovian environment. Scand. Act. J. 1989, 69–100.Google Scholar
[3] Asmussen, S. (1991). Ladder heights and the Markov-modulated M/G/1 queue. Stoch. Proc. Appl. 37, 313–326.Google Scholar
[4] Asmussen, S. (1994). Busy period analysis, rare events and transient behavior in fluid flow models. J. Appl. Math. Stoch. Anal. 7, 269–299.Google Scholar
[5] Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Stoch. Models 11, 21–49.Google Scholar
[6] Asmussen, S. (1995). Stationary distributions via first passage times. In Advances in Queueing: Models, Methods & Problems, ed. Dshalalow, J. CRC Press, Boca Raton, FL, pp. 79–102.Google Scholar
[7] Asmussen, S. (1998). Extreme value theory for queues via cycle maxima. Extremes 1, 137–168.Google Scholar
[8] Asmussen, S and Perry, D. (1992). On cycle maxima, first passage problems and extreme value theory for queues. Stoch. Models 8, 421–458.Google Scholar
[9] Asmussen, S and Perry, D. (1997). Operational calculus for matrix-exponential distributions, with applications to Brownian (q,Q) models. Math. Operat. Res. 23, 166–176.Google Scholar
[10] Baccelli, F. and Makowski, A. M. (1991). Exponential martingales for queues in a random environment: the M/G/1 case. Stoch. Proc. Appl. 38, 99–133.CrossRefGoogle Scholar
[11] Berman, A. and Plemmons, R. J. (1979). Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York.Google Scholar
[12] Çinlar, E., (1971). Markov additive processes I-II. Z. Wahrscheinlichkeitsth. 24, 85–121.Google Scholar
[13] Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory. Springer, New York.CrossRefGoogle Scholar
[14] Harrison, J. M. (1985). Brownian Motion and Stochastic Flow Systems. Wiley, New York.Google Scholar
[15] Heath, D., Resnick, S. and Samorodnitsky, G. (1998). Heavy tails and long range dependence in On/Off processes and associated fluid models. Math. Operat. Res. 23, 145–165.Google Scholar
[16] Kaspi, H. (1984). Storage processes with Markov additive input and output. Math. Operat. Res. 9, 424–440.Google Scholar
[17] Kella, O. and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.CrossRefGoogle Scholar
[18] Kella, O. and Whitt, W. (1996). Stability and structural properties of stochastic storage networks. J. Appl. Prob. 33, 1169–1180.Google Scholar
[19] Nahmias, S. (1989). Production and Operations Analysis. Irwin, Homewood, IL.Google Scholar
[20] Neuts, M. F. (1979). A versatile Markovian point process. J. Appl. Prob. 16, 764–779.Google Scholar
[21] Neveu, J. (1961). Une généralisation des processus à accroissementes positifs indépendantes. Abh. Math. Sem. Hamburg 23, 36–61.Google Scholar
[22] Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, New York.CrossRefGoogle Scholar
[23] Regterschot, G. J. K. and de Smit, J. H. A. (1986). The queue M/G/1 with Markov-modulated arrivals and services. Math. Operat. Res. 11, 465–483.Google Scholar
[24] Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener–Hopf factorization of Markov chains. Ann. Appl. Prob. 4, 390–413.Google Scholar