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**Maintenance in single-server queues: a game-theoretic approach.**
*(English)*
Zbl 1190.90048

Summary: We use antagonistic stochastic games and fluctuation analysis to examine a single-server queue with bulk input and secondary work during server’s multiple vacations. When the buffer contents become exhausted the server leaves the system to perform some diagnostic service of a minimum of \(L\) jobs clustered in packets of random sizes (event A). The server is not supposed to stay longer than \(T\) units of time (event B). The server returns to the system when A or B occurs, whichever comes first. On the other hand, he may not break service of a packet in a middle even if A or B occurs. Furthermore, the server waits for batches of customers to arrive if upon his return the queue is still empty. We obtain a compact and explicit form functional for the queueing process in equilibrium.

### MSC:

90B22 | Queues and service in operations research |

90B25 | Reliability, availability, maintenance, inspection in operations research |

91A80 | Applications of game theory |

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\textit{N. Al-Matar} and \textit{J. H. Dshalalow}, Math. Probl. Eng. 2009, Article ID 857871, 23 p. (2009; Zbl 1190.90048)

### References:

[1] | T. S. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory, vol. 160 of Mathematics in Science and Engineering, Academic Press, Orlando, Fla, USA, 1982. · Zbl 0479.90085 |

[2] | M. H. Breitner, H. J. Pesch, and W. Grimm, “Complex differential games of pursuit-evasion type with state constraints. I. Necessary conditions for optimal open-loop strategies,” Journal of Optimization Theory and Applications, vol. 78, no. 3, pp. 419-441, 1993. · Zbl 0796.90078 |

[3] | A. Davidovitz and J. Shinar, “Two-target game model of an air combat with fire-and-forget all-aspect missiles,” Journal of Optimization Theory and Applications, vol. 63, no. 2, pp. 133-165, 1989. · Zbl 0662.90103 |

[4] | J. H. Dshalalow, “Random walk analysis in antagonistic stochastic games,” Stochastic Analysis and Applications, vol. 26, no. 4, pp. 738-783, 2008. · Zbl 1151.91330 |

[5] | J. H. Dshalalow and H.-J. Ke, “Multilayers in a modulated stochastic game,” Journal of Mathematical Analysis and Applications, vol. 353, no. 2, pp. 553-565, 2009. · Zbl 1173.91308 |

[6] | J. H. Dshalalow and A. Treerattrakoon, “Antagonistic games with an initial phase,” Nonlinear Dynamics and Systems Theory, vol. 9, no. 3, pp. 277-286, 2009. · Zbl 1301.91002 |

[7] | P. C. Fishburn, “Noncooperative stochastic dominance games,” International Journal of Game Theory, vol. 7, no. 1, pp. 51-61, 1978. · Zbl 0372.90133 |

[8] | R. Isaacs, Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization, Dover, New York, NY, USA, 1999. · Zbl 1233.91001 |

[9] | R. V. Konstantinov and E. S. Polovinkin, “Mathematical simulation of a dynamic game in the enterprise competition problem,” Cybernetics and Systems Analysis, vol. 40, no. 5, pp. 720-725, 2004. · Zbl 1132.91355 |

[10] | J. C. Perry and B. D. Roitberg, “Games among cannibals: competition to cannibalize and parent-offspring conflict lead to increased sibling cannibalism,” Journal of Evolutionary Biology, vol. 18, no. 6, pp. 1523-1533, 2005. |

[11] | V. N. Shashikhin, “Antagonistic game with interval payoff functions,” Cybernetics and Systems Analysis, vol. 40, no. 4, pp. 556-564, 2004. · Zbl 1132.91329 |

[12] | T. Shima, “Capture conditions in a pursuit-evasion game between players with biproper dynamics,” Journal of Optimization Theory and Applications, vol. 126, no. 3, pp. 503-528, 2005. · Zbl 1181.91036 |

[13] | M. J. Sobel, “Noncooperative stochastic games,” Annals of Mathematical Statistics, vol. 42, pp. 1930-1935, 1971. · Zbl 0229.90059 |

[14] | J. H. Dshalalow, “On exit times of multivariate random walk with some applications to finance,” Nonlinear Analysis: Theory, Methods & Applications, vol. 63, no. 5-7, pp. e569-e577, 2005. · Zbl 1159.60346 |

[15] | J. H. Dshalalow and W. Huang, “A stochastic game with a two-phase conflict,” in Advances in Nonlienar Analysis, chapter 18, pp. 201-209, Cambridge Academic Publishers, Cambridge, UK, 2008. |

[16] | J. H. Dshalalow and A. Liew, “On exit times of a multivariate random walk and its embedding in a quasi Poisson process,” Stochastic Analysis and Applications, vol. 24, no. 2, pp. 451-474, 2006. · Zbl 1105.60065 |

[17] | J. H. Dshalalow and A. Liew, “On fluctuations of a multivariate random walk with some applications to stock options trading and hedging,” Mathematical and Computer Modelling, vol. 44, no. 9-10, pp. 931-944, 2006. · Zbl 1133.91416 |

[18] | J. H. Dshalalow and W. Huang, “On noncooperative hybrid stochastic games,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 3, pp. 803-811, 2008. · Zbl 1213.91033 |

[19] | J. H. Dshalalow and W. Huang, “Sequential antagonistic games with initial phase (jointly with Weijun Huang),” in Functional Equations, Difference Inequalities and ULAM Stability Notions, chapter 2, pp. 15-36, Nova Science, New York, NY, USA, 2010. |

[20] | W. Huang and J. H. Dshalalow, “Tandem antagonistic games,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e259-e270, 2009. · Zbl 1238.91023 |

[21] | J. Medhi, Stochastic Models in Queueing Theory, Academic Press, Boston, Mass, USA, 1991. · Zbl 0743.60100 |

[22] | N. Tian and Z. G. Zhang, Vacation Queueing Models, International Series in Operations Research & Management Science, Springer, New York, NY, USA, 2006. |

[23] | J. L. Jain, S. G. Mohanty, and W. Böhm, A Course on Queueing Models, Statistics: Textbooks and Monographs, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2007. · Zbl 1120.60001 |

[24] | R. P. Agarwal and J. H. Dshalalow, “On multivariate delayed recurrent processes,” Panamerican Mathematical Journal, vol. 15, no. 3, pp. 35-49, 2005. · Zbl 1078.60036 |

[25] | J. H. Dshalalow, “On the level crossing of multidimensional delayed renewal processes,” Journal of Applied Mathematics and Stochastic Analysis, vol. 10, no. 4, pp. 355-361, 1997. · Zbl 0896.60056 |

[26] | J. Brandts and C. Solàc, “Reference points and negative reciprocity in simple sequential games,” Games and Economic Behavior, vol. 36, no. 2, pp. 138-157, 2001. · Zbl 1037.91505 |

[27] | I. Exman, “Solving sequential games with Boltzmann-learned tactics,” in Proceedings of the 1st Workshop on Parallel Problem Solving from Nature (PPSN ’90), vol. 496 of Lecture Notes In Computer Science, pp. 216-220, Springer, Dortmund, Germany, October 1990. |

[28] | T. Radzik and K. Szajowski, “Sequential games with random priority,” Sequential Analysis: Design Methods & Applications, vol. 9, no. 4, pp. 361-377, 1990. · Zbl 0745.62080 |

[29] | K. Siegrist and J. Steele, “Sequential games,” Journal of Applied Probability, vol. 38, no. 4, pp. 1006-1017, 2001. · Zbl 1002.60082 |

[30] | Q. Wen, “A folk theorem for repeated sequential games,” Review of Economic Studies, vol. 69, no. 2, pp. 493-512, 2002. · Zbl 1030.91008 |

[31] | D. Heyman, “The T-policy for the M/G/1 queue,” Management Science, vol. 23, pp. 775-778, 1977. · Zbl 0357.60022 |

[32] | H. W. Lee, S. S. Lee, J. O. Park, and K. C. Chae, “Analysis of the MX/G/1-policy and multiple vacations,” Journal of Applied Probability, vol. 31, no. 2, pp. 476-496, 1994. · Zbl 0804.60081 |

[33] | G. Alsmeyer, “On generalized renewal measures and certain first passage times,” The Annals of Probability, vol. 20, no. 3, pp. 1229-1247, 1992. · Zbl 0759.60088 |

[34] | N. H. Bingham, “Random walk and fluctuation theory,” in Stochastic Processes: Theory and Methods, D. N. Shanbhag and C. R. Rao, Eds., vol. 19 of Handbook of Statistics, pp. 171-213, Elsevier Science, Amsterdam, The Netherlands, 2001. · Zbl 0982.60038 |

[35] | L. Takács, “On fluctuations of sums of random variables,” in Studies in Probability and Ergodic Theory, G.-C. Rota, Ed., vol. 2 of Advances in Mathematics, Supplementary Studies, pp. 45-93, Academic Press, New York, NY, USA, 1978. · Zbl 0447.60053 |

[36] | L. Abolnikov, J. H. Dshalalow, and A. Treerattrakoon, “On a dual hybrid queueing systems,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 1, pp. 96-109, 2008. · Zbl 1157.93513 |

[37] | L. Takács, “On fluctuation problems in the theory of queues,” Advances in Applied Probability, vol. 8, no. 3, pp. 548-583, 1976. · Zbl 0357.60021 |

[38] | A. E. Kyprianou and M. R. Pistorius, “Perpetual options and Canadization through fluctuation theory,” The Annals of Applied Probability, vol. 13, no. 3, pp. 1077-1098, 2003. · Zbl 1039.60044 |

[39] | E. Mellander, A. Vredin, and A. Warne, “Stochastic trends and economic fluctuations in a small open economy,” Journal of Applied Economics, vol. 7, no. 4, pp. 369-394, 1992. |

[40] | J. F. Muzy, J. Delour, and E. Bacry, “Modelling fluctuations of financial time series: from cascade process to stochastic volatility model,” European Physical Journal B, vol. 17, no. 3, pp. 537-548, 2000. |

[41] | L. Garrido, Ed., “Fluctuations and stochastic phenomena in condensed matter,” in Proceedings of the Sitges Conference on Statistical Mechanics, vol. 268 of Lectures Notes in Physics, Springer, Barcelona, Spain, 1987. |

[42] | T. Hida, Ed., “Mathematical approach to fluctuations: astronomy, biology and quantum dynamics,” in Proceedings of the Iias Workshop, World Scientific, Kyoto, Japan, May 1992. · Zbl 0357.60022 |

[43] | W. Horsthemke and D. Kondepudi, Eds., Fluctuations and Sensitivity in Nonequilibrium Systems, Springer, New York, NY, USA, 1984. |

[44] | S. Redner, A Guide to First-Passage Processes, Cambridge University Press, Cambridge, UK, 2001. · Zbl 0980.60006 |

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.