The Annals of Applied Probability

Hitting probabilities in a Markov additive process with linear movements and upward jumps: Applications to risk and queueing processes

Masakiyo Miyazawa

Full-text: Open access

Abstract

Motivated by a risk process with positive and negative premium rates, we consider a real-valued Markov additive process with finitely many background states. This additive process linearly increases or decreases while the background state is unchanged, and may have upward jumps at the transition instants of the background state. It is known that the hitting probabilities of this additive process at lower levels have a matrix exponential form. We here study the hitting probabilities at upper levels, which do not have a matrix exponential form in general. These probabilities give the ruin probabilities in the terminology of the risk process. Our major interests are in their analytic expressions and their asymptotic behavior when the hitting level goes to infinity under light tail conditions on the jump sizes. To derive those results, we use a certain duality on the hitting probabilities, which may have an independent interest because it does not need any Markovian assumption.

Article information

Source
Ann. Appl. Probab., Volume 14, Number 2 (2004), 1029-1054.

Dates
First available in Project Euclid: 23 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1082737121

Digital Object Identifier
doi:10.1214/105051604000000206

Mathematical Reviews number (MathSciNet)
MR2052912

Zentralblatt MATH identifier
1057.60073

Subjects
Primary: 90B22: Queues and service [See also 60K25, 68M20] 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60K20: Applications of Markov renewal processes (reliability, queueing networks, etc.) [See also 90Bxx] 60G55: Point processes

Keywords
Risk process Markov additive process hitting probability decay rate Markov renewal theorem stationary marked point process duality

Citation

Miyazawa, Masakiyo. Hitting probabilities in a Markov additive process with linear movements and upward jumps: Applications to risk and queueing processes. Ann. Appl. Probab. 14 (2004), no. 2, 1029--1054. doi:10.1214/105051604000000206. https://projecteuclid.org/euclid.aoap/1082737121


Export citation

References

  • Arjas, E. and Speed, T. P. (1973). Symmetric Wiener--Hopf factorizations in Markov additive processes. Z. Wahrsch. Verw. Gebiete 26 105--118.
  • Asmussen, S. (1987). Applied Probability and Queues. Wiley, Chichester.
  • Asmussen, S. (1991). Ladder heights and the Markov-modulated $M/G/1$ queue. Stochastic Process. Appl. 37 313--326.
  • Asmussen, S. (1994). Busy period analysis, rare events and transient behavior in fluid flow models. J. Appl. Math. Stochastic Anal. 7 269--299.
  • Asmussen, S. (1995). Stationary distributions for fluid flow models with or without Brownian noise. Stoch. Models 11 21--49.
  • Asmussen, S. (2000). Ruin Probabilities. World Scientific, Singapore.
  • Asmussen, S. and Højgaard, B. (1996). Finite horizon ruin probabilities for Markov-modulated risk processes with heavy tails. Theory of Stochastic Processes 2 96--107.
  • Asmussen, S. and Klüppelberg, C. (1996). Large deviations results for subexponential tails, with applications to insurance risk. Stochastic Process. Appl. 64 103--125.
  • Baccelli, F. and Brémaud, P. (2002). Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences, 2nd ed. Springer, New York.
  • Çinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.
  • Kingman, J. F. C. (1961). A convexity property of positive matrices. Quart. J. Math. Oxford Ser. (2) 12 283--284.
  • Loynes, R. M. (1962). The stability of a queue with nonindependent inter-arrival and service times. Proc. Cambridge Philosophical Society 58 497--520.
  • Miyazawa, M. (2002). A Markov renewal approach to the asymptotic decay of the tail probabilities in risk and queueing processes. Probab. Eng. Inform. Sci. 16 139--150.
  • Miyazawa, M. and Takada, H. (2002). A matrix exponential form for hitting probabilities and its application to a Markov modulated fluid queue with downward jumps. J. Appl. Probab. 39 604--618.
  • Neuts, M. F. (1989). Structured Stochastic Matrices of $M/G/1$ Type and Their Applications. Dekker, New York.
  • Rogers, L. C. G. (1994). Fluid models in queueing theory and Wiener--Hopf factorization of Markov chains. Ann. Appl. Probab. 4 390--413.
  • Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. Wiley, Chichester.
  • Schmidli, H. (1995). Cramér--Lundberg approximations for ruin probabilities of risk processes perturbed by a diffusion. Insurance Math. Econom. 16 135--149.
  • Seneta, E. (1980). Nonnegative Matrices and Markov Chains, 2nd ed. Springer, New York.
  • Sengupta, B. (1989). Markov processes whose steady state distribution is matrix-exponential with an application to the $GI/PH/1$ queue. Adv. in Appl. Probab. 21 159--180.
  • Takada, H. (2001). Markov modulated fluid queues with batch fluid arrivals. J. Oper. Res. Soc. Japan 44 344--365.
  • Takine, T. (2001). A recent progress in algorithmic analysis of FIFO queues with Markovian arrival streams. J. Korean Math. Soc. 38 807--842.