In this paper, we introduce and study the relations between parameters of the m-component mixture distributions which imply the stochastic and failure rate comparisons. Then we apply the failure rate and stochastic ordering techniques to construct the upper and lower bounds for the steady-state performance indexes in a multiserver queueing system with multicomponent exponential-Pareto mixture service time distribution. The uniform distance between multicomponent mixture distribution and its parent distribution is discussed. The obtained theoretical results are then illustrated by a few numerical examples based on the regenerative simulation multiserver queueing systems with mixture service time distributions.
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References
E. Amini-Seresht and Y. Zhang, “Stochastic comparisons on two finite mixture models,” Oper. Res. Let., 45, 475–480 (2017).
E. K. Al-Hussaini and K. S. Sultan, “Reliability and hazard based on finite mixture models,”, in: Advances in Reliability, Handbook of Statistics, Vol. 20 (2001), pp. 139–183.
S. Asmussen, Applied Probability and Queues, Springer-Verlag, New York (2003).
S. Asmussen and P. Glynn, Stochastic Simulation. Algorithms and Analysis, Springer-Verlag, New York (2007).
T. Aven and U. Jensen, Stochastic Models in Reliability, Springer-Veriag, New York (2013).
D. Batrakova and V. Korolev, “A new method for the probabilistic and statistical analysis of information flows in telecommunication networks,” Inform. Appl., 1, 40–53 (2007).
P. Embrechts, C. Klppelberg, and T.Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, New York (1997).
V.Yu. Korolev, V.A. Krylov, and V.Yu. Kuz’min, “Stability of finite mixtures of generalized Gamma-distributions with respect to disturbance of parameters,” Inform. Primen., 5, No. 1, 31–38 (2011).
G. I. Mclachlan and D. Peel, Peel Finite Mixture Models, Wiley (2001).
A. Marshall and I. Olkin, Life Distributions: Structure of Nonparametric, Semiparametric and Parametric Families, Springer, New York (2007).
E. Morozov, R. Nekrasova, I. Peshkova, and A. Rumyantsev, “A regeneration-based estimation of high performance multiserver systems,” Comm. Comput. Inform. Sci., 608, 271–282 (2016).
E. Morozov, M. Pagano, I. Peshkova, and A. Rumyantsev, “Sensitivity analysis and simulation of a multiserver queueing system with mixed service time distribution,” Mathematics, 8, 1277 (2020).
E. Morozov, I. Peshkova, and A. Rumyantsev, “On Failure Rate Comparison of Finite Multiserver Systems,” in: Lecture Notes in Computer Science, Vol. 11965, Springer (2019), pp. 419–431.
E. Morozov, I. Peshkova, and A. Rumyantsev, “On Regenerative Envelopes for Cluster Model Simulation,”in: Distributed Computer and Communication Networks: 19th International Conference, DCCN 2016, V. M. Vishnevskiy, K. E. Samouylov, and D. V. Kozyrev (eds.), Springer-Verlag (2016), pp. 222–230.
J. Navarro, “Stochastic comparisons of generalized mixtures and coherent systems,” TEST, 25, 150–169 (2016).
I. Peshkova, I.E. Morozov, and M. Maltseva, “On comparison of multiserver systems with Exponential-Pareto mixture distribution,” Comm. Comput. Inform. Sci., 1231, 141–152 (2020).
I. Peshkova, E. Morozov, and M. Maltseva, “On comparison of multiserver systems with two-component mixture distributions,” Comm. Comput. Inform. Sci., 1337, 340–352 (2020)
S. Ross, J. Shanthikumar, and Z. Zhu, “On increasing-failure-rate random variables,” J. Appl. Probab., 42, 797–809 (2005).
D. Sonderman, “Comparing multi-server queues with finite waiting rooms, I: Same number of servers,” Adv. Appl. Probab., 11, 439–447 (1979).
D.M. Titterington, A. F.M. Smith, and U. E. Makov, Statistical Analysis of Finite Mixture Distributions, John Wiley (1985).
W. Whitt, “Comparing counting processes and queues,” Adv. Appl. Probab., 13, No. 1, 207–220 (1981).
V.M. Zolotarev, Modern Theory of Summarizing of Independent Random Variables, Nauka, Moscow (1986).
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Proceedings of the XXXVI International Seminar on Stability Problems for Stochastic Models, Petrozavodsk, Russia, 22–26 June, 2020. Part I.
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Peshkova, I.V., Morozov, E.V. On Comparison of Multiserver Systems with Multicomponent Mixture Distributions. J Math Sci 267, 260–272 (2022). https://doi.org/10.1007/s10958-022-06132-z
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DOI: https://doi.org/10.1007/s10958-022-06132-z