##
**Queues and risk models with simultaneous arrivals.**
*(English)*
Zbl 1311.60103

Consider a multidimensional classical risk model. That is, claims occur according to a Poisson process, and the claim sizes are i.i.d. vectors independent of the claim arrivals. The premiums are earned linearly in time. By changing the monetary units, one can assume that the premium rate is one in each of the coordinates.

Using the arguments in [S. Asmussen and S. S. Petersen, Adv. Appl. Probab. 20, No. 4, 913–916 (1988; Zbl 0657.60111)], a dual version of a multivariate queueing model is obtained. In this way, ruin probabilities can be expressed as the stationary distribution in the queueing model. The Laplace transform of the stationary distribution is found. Using Rouché’s theorem, a unique zero of the characteristic equation can be identified. This zero has an interpretation connected to the distribution of the additional workload if the shortest queue becomes idle.

It is further assumed that at each arrival the service times are ordered. Thus, the first queue always gets the largest service time, the last queue the shortest. Considering an insurance surplus, this limits the applicability of the results in risk theory. In the queueing context the last queue gets idle first. The stationary workload can then be decomposed. The workload is the sum of independent workloads, where in the second sum the last server is idle, in the third sum, the last but one server is idle, etc. In the last sum, all servers but the first are idle. These sums have the following interpretations. Consider a queue where all servers empty at the time the last server becomes idle. The first sum is then the stationary distribution of this modified queue. The sum of the first two components gives the stationary distribution if the whole queue empties when the last but one server becomes idle. In the case of two servers, more explicit results are given.

Using the arguments in [S. Asmussen and S. S. Petersen, Adv. Appl. Probab. 20, No. 4, 913–916 (1988; Zbl 0657.60111)], a dual version of a multivariate queueing model is obtained. In this way, ruin probabilities can be expressed as the stationary distribution in the queueing model. The Laplace transform of the stationary distribution is found. Using Rouché’s theorem, a unique zero of the characteristic equation can be identified. This zero has an interpretation connected to the distribution of the additional workload if the shortest queue becomes idle.

It is further assumed that at each arrival the service times are ordered. Thus, the first queue always gets the largest service time, the last queue the shortest. Considering an insurance surplus, this limits the applicability of the results in risk theory. In the queueing context the last queue gets idle first. The stationary workload can then be decomposed. The workload is the sum of independent workloads, where in the second sum the last server is idle, in the third sum, the last but one server is idle, etc. In the last sum, all servers but the first are idle. These sums have the following interpretations. Consider a queue where all servers empty at the time the last server becomes idle. The first sum is then the stationary distribution of this modified queue. The sum of the first two components gives the stationary distribution if the whole queue empties when the last but one server becomes idle. In the case of two servers, more explicit results are given.

Reviewer: Hanspeter Schmidli (Köln)

### MSC:

60K25 | Queueing theory (aspects of probability theory) |

60J25 | Continuous-time Markov processes on general state spaces |

90B22 | Queues and service in operations research |

91B30 | Risk theory, insurance (MSC2010) |

### Keywords:

queues with simultaneous arrival; stationary distribution; stochastic decomposition; duality; multivariate risk model### Citations:

Zbl 0657.60111### References:

[1] | Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities , 2nd edn. World Scientific, Hackensack, NJ. · Zbl 1247.91080 |

[2] | Avram, F., Palmowski, Z. and Pistorius, M. (2008). A two-dimensional ruin problem on the positive quadrant. Insurance Math. Econom. 42, 227-234. · Zbl 1141.91482 · doi:10.1016/j.insmatheco.2007.02.004 |

[3] | 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. · Zbl 1163.60010 · doi:10.1214/08-AAP529 |

[4] | Baccelli, F. (1985). Two parallel queues created by arrivals with two demands: The M/G/\(2\) symmetrical case. Res. Rep. 426, INRIA-Rocquencourt. |

[5] | 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. · Zbl 0681.60096 · doi:10.2307/1427640 |

[6] | 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. · Zbl 1239.91073 · doi:10.1239/jap/1316796912 |

[7] | 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. · Zbl 1311.60103 |

[8] | 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. · Zbl 1055.91041 · doi:10.1016/S0167-6687(03)00115-X |

[9] | Cohen, J. W. (1988). Boundary value problems in queueing theory. Queueing Systems 3, 97-128. · Zbl 0662.60098 · doi:10.1007/BF01189045 |

[10] | Cohen, J. W. (1992). Analysis of Random Walks . IOS Press, Amsterdam. · Zbl 0809.60081 |

[11] | Cohen, J. W. and Boxma, O. J. (1983). Boundary Value Problems in Queueing System Analysis . North-Holland, Amsterdam. · Zbl 0515.60092 |

[12] | De Klein, S. J. (1988). Fredholm integral equations in queueing analysis . Doctoral Thesis, University of Utrecht. |

[13] | Fayolle, G. and Iasnogorodski, R. (1979). Two coupled processors: the reduction to a Riemann-Hilbert problem. Z. Wahrscheinlichkeitsth. 47, 325-351. · Zbl 0395.68032 · doi:10.1007/BF00535168 |

[14] | Fayolle, G., Iasnogorodski, R. and Malyshev, V. (1999). Random Walks in the Quarter-Plane . Springer, Berlin. · Zbl 0932.60002 |

[15] | 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.) · Zbl 0554.90041 · doi:10.1137/0144074 |

[16] | Frostig, E. (2004). Upper bounds on the expected time to ruin and on the expected recovery time. Adv. Appl. Prob. 36, 377-397. · Zbl 1123.91335 · doi:10.1239/aap/1086957577 |

[17] | Gakhov, F. D. (1990). Boundary Value Problems . Dover, New York. · Zbl 0830.30026 |

[18] | Kella, O. (1993). Parallel and tandem fluid networks with dependent Lévy inputs. Ann. Appl. Prob. 3, 682-695. · Zbl 0780.60072 · doi:10.1214/aoap/1177005358 |

[19] | 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. · Zbl 1201.60087 · doi:10.1007/s11134-010-9168-z |

[20] | Muskhelishvili, N. I. (2008). Singular Integral Equations , 2nd edn. Dover, Mineola, NY. · Zbl 0108.29203 |

[21] | Nelson, R. and Tantawi, A. N. (1987). Approximating task response times in fork/join queues. IBM Res. Rep. RC13012. |

[22] | Nelson, R. and Tantawi, A. N. (1988). Approximate analysis of fork/join synchronization in parallel queues. IEEE Trans. Comput. 37, 739-743. |

[23] | Rolski, T., Schmidli, H., Schmidt, V. and Teugels, J. (1999). Stochastic Processes for Insurance and Finance . John Wiley, Chichester. · Zbl 0940.60005 |

[24] | Siegmund, D. (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov Processes. Ann. Prob. 4, 914-924. · Zbl 0364.60109 · doi:10.1214/aop/1176995936 |

[25] | Wright, P. E. (1992). Two parallel processors with coupled inputs. Adv. Appl. Prob. 24, 986-1007. · Zbl 0760.60093 · doi:10.2307/1427722 |

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