### Queues and risk models with simultaneous arrivals

#### Abstract

We focus on a particular connection between queueing and risk models in a multidimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue), we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline of how the two-dimensional risk model with a general two-dimensional claim size distribution (hence, without ordering of claim sizes) is related to a known Riemann boundary-value problem.

#### Article information

Source
Adv. in Appl. Probab., Volume 46, Number 3 (2014), 812-831.

Dates
First available in Project Euclid: 29 August 2014

https://projecteuclid.org/euclid.aap/1409319561

Digital Object Identifier
doi:10.1239/aap/1409319561

Mathematical Reviews number (MathSciNet)
MR3254343

Zentralblatt MATH identifier
1311.60103

Subjects
Secondary: 91B30: Risk theory, insurance

#### Citation

Badila, E. S.; Boxma, O. J.; Resing, J. A. C.; Winands, E. M. M. Queues and risk models with simultaneous arrivals. Adv. in Appl. Probab. 46 (2014), no. 3, 812--831. doi:10.1239/aap/1409319561. https://projecteuclid.org/euclid.aap/1409319561

#### References

• Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.
• Avram, F., Palmowski, Z. and Pistorius, M. (2008). A two-dimensional ruin problem on the positive quadrant. Insurance Math. Econom. 42, 227–234.
• Avram, F., Palmowski, Z. and Pistorius, M. R. (2008). Exit problem of a two-dimensional risk process from the quadrant: exact and asymptotic results. Ann. Appl. Prob. 18, 2421–2449.
• Baccelli, F. (1985). Two parallel queues created by arrivals with two demands: The M/G/$2$ symmetrical case. Res. Rep. 426, INRIA-Rocquencourt.
• Baccelli, F., Makowski, A. M. and Shwartz, A. (1989). The fork-join queue and related systems with synchronization constraints: stochastic ordering and computable bounds. Adv. Appl. Prob. 21, 629–660.
• Badescu, A. L., Cheung, E. C. K. and Rabehasaina, L. (2011). A two-dimensional risk model with proportional reinsurance. J. Appl. Prob. 48, 749–765.
• Badila, E. S., Boxma, O. J., Resing, J. A. C. and Winands, E. M. M. (2012). Queues and risk models with simultaneous arrivals. Preprint. Available at http://arxiv.org/abs/1211.2193.
• Chan, W.-S., Yang, H. and Zhang, L. (2003). Some results on ruin probabilities in a two-dimensional risk model. Insurance Math. Econom. 32, 345–358.
• Cohen, J. W. (1988). Boundary value problems in queueing theory. Queueing Systems 3, 97–128.
• Cohen, J. W. (1992). Analysis of Random Walks. IOS Press, Amsterdam.
• Cohen, J. W. and Boxma, O. J. (1983). Boundary Value Problems in Queueing System Analysis. North-Holland, Amsterdam.
• De Klein, S. J. (1988). Fredholm integral equations in queueing analysis. Doctoral Thesis, University of Utrecht.
• Fayolle, G. and Iasnogorodski, R. (1979). Two coupled processors: the reduction to a Riemann–Hilbert problem. Z. Wahrscheinlichkeitsth. 47, 325–351.
• Fayolle, G., Iasnogorodski, R. and Malyshev, V. (1999). Random Walks in the Quarter-Plane. Springer, Berlin.
• Flatto, L. and Hahn, S. (1984). Two parallel queues created by arrivals with two demands. I. SIAM J. Appl. Math. 44, 1041–1053. (Erratum: 45 (1985), 168.)
• Frostig, E. (2004). Upper bounds on the expected time to ruin and on the expected recovery time. Adv. Appl. Prob. 36, 377–397.
• Gakhov, F. D. (1990). Boundary Value Problems. Dover, New York.
• Kella, O. (1993). Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Prob. 3, 682–695.
• Löpker, A. and Perry, D. (2010). The idle period of the finite ${G}/{M}/1$ queue with an interpretation in risk theory. Queueing Systems 64, 395–407.
• Muskhelishvili, N. I. (2008). Singular Integral Equations, 2nd edn. Dover, Mineola, NY.
• Nelson, R. and Tantawi, A. N. (1987). Approximating task response times in fork/join queues. IBM Res. Rep. RC13012.
• Nelson, R. and Tantawi, A. N. (1988). Approximate analysis of fork/join synchronization in parallel queues. IEEE Trans. Comput. 37, 739–743.
• Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.
• Siegmund, D. (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov Processes. Ann. Prob. 4, 914–924.
• Wright, P. E. (1992). Two parallel processors with coupled inputs. Adv. Appl. Prob. 24, 986–1007.