Abstract
We consider the important decay parameter issue for a Markovian bulk-arrival and bulk-service queue which stops at reaching the first m states. The results obtained in this paper are a significant generalization of the ones obtained in Chen et al. (Queueing Syst 66:275–311, 2010). The exact value of the decay parameter is obtained. Based on the results obtained in this paper, we provide a practical method for calculating the decay parameter which is very effective. In some cases, the decay parameter can even be easily expressed explicitly. Some important key lemmas which also have their own interest are provided. The interesting and clear geometric interpretation of the decay parameter is explained. A few examples are provided to illustrate the results obtained in this paper.
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References
Anderson, W.: Continuous-Time Markov Chains: An Applications-Oriented Approach. Springer, New York (1991)
Chen, A.Y., Li, J.P., Hou, Z.T., Ng, K.W.: Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues. Queueing Syst. 66, 275–311 (2010)
Chen, A.Y., Wu, X.H., Zhang, J.: Markovian bulk-arrival and bulk-service queues with general state-dependent control. Queueing Syst. 95, 331–378 (2020)
Chen, M.: Eigenvalues, Inequalities, and Ergodic Theory. Probability and Its Applications. Springer, London (2005)
Chen, M.: Speed of stability for birth-death processes. Front. Math. China 5, 479–515 (2010)
Collet, P., Martinez, S., Martin, J.S.: Quasi-Stationary Distributions, Markov Chains, Diffusions and Dynamical Systems. Springer, New York (2013)
Darroch, J.N., Seneta, E.: On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Probab. 4, 192–196 (1967)
Ferrari, P., Kesten, H., Martinez, S., Picco, P.: Existence of quasi-stationary distributions: a renewal dynamical approach. Ann. Prob. 23, 501–521 (1995)
Flaspohler, D.C.: Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann. Inst. Stat. Math. 26, 351–356 (1974)
Kingman, J.F.C.: The exponential decay of Markov transition probability. Proc. Lond. Math. Soc. 13, 337–358 (1963)
Kelly, F.P.: Invariant measures and the generator. In: Kingman, J.F.C., Reuter, G.E. (eds.) Probability, Statistics, and Analysis. London Math. Soc. Lecture Notes Series 79, pp. 143–160. Cambridge University Press, Cambridge (1983)
Kijima, M.: Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous-time. J. Appl. Probab. 30, 509–517 (1963)
Li, J.P., Chen, A.Y.: Decay property of stopped Markovian bulk-arriving queues. Adv. Appl. Probab. 40, 95–121 (2008)
Nair, M.G., Pollett, P.K.: On the relationship between \(\mu \)-invariant measures and quasi-stationary distributions for continuous-time Markov chains. Adv. Appl. Probab. 25, 82–102 (1993)
Pollett, P.K.: Reversibility, invariance and mu-invariance. Adv. Appl. Probab. 20, 600–621 (1988)
Tweedie, R.L.: Some ergodic properties of the Feller minimal process. Q. J. Math. Oxford 25(2), 485–493 (1974)
Van Doorn, E.A.: Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. Appl. Probab. 17, 514–530 (1985)
Van Doorn, E.A.: Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes. Adv. Appl. Probab. 23, 683–700 (1991)
Van Doorn, E.A.: Representations for the rate of convergence of birth-death processes. Theory Probab. Math. Stat. 65, 37–43 (2002)
Acknowledgements
The authors would like to express their sincere thanks to the anonymous referees and the associate editor who have provided extremely helpful comments and suggestions, which have resulted in a much improved version of this paper. The work of Junping Li is substantially supported by the National Natural Sciences Foundations of China (No. 11771452, No. 11971486). The work of Jing Zhang is supported by the fellowship from SRIBD (Shenzhen Research Institute of Big Data) and the financial support by AIRS (Shenzhen Institute of Artificial Intelligence and Robotics for Society) Project (No. 2019-INT002) and NSFC Project (No. 12001460).
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Chen, A., Li, J., Wu, X. et al. Decay parameter for general stopped Markovian bulk-arrival and bulk-service queues. Queueing Syst 99, 305–344 (2021). https://doi.org/10.1007/s11134-021-09712-z
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DOI: https://doi.org/10.1007/s11134-021-09712-z