Notes
It is readily verified that the constant arrival rate assumed in [1] can be generalized to be the level-dependent arrival rate.
References
Arita, C.: Queueing process with excluded-volume effect. Phys. Rev. E 80, 051119 (2009)
Baumann, H., Sandmann, W.: Numerical solution of level dependent quasi-birth-and-death processes. Procedia Comput. Sci. 1, 1561–1569 (2012)
Blythe, R.B., Evans, M.R.: Nonequilibrium steady states of matrix-product form: A solver’s guide. J. Phys. A: Math. Theor. 40, R333–R441 (2007)
Bright, L., Taylor, P.G.: Calculating the equilibrium distribution in level-dependent quasi-birth-and-death processes. Stoch. Models 11, 497–526 (1995)
Kelly, F.P.: Reversibility and Stochastic Networks. Wiley, Chichester, UK (1979)
Kimura, M., Takine, T.: Computing the conditional stationary distribution in Markov chains of level-dependent M/G/1-type. Stoch. Models 34, 207–238 (2018)
Kimura, M., Takine, T.: Characterization of the conditional stationary distribution in Markov chains via systems of linear inequalities. Adv. Appl. Prob. 52, 1241–1283 (2020)
Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. SIAM, Philadelphia, PA (1999)
Phung-Duc, T., Masuyama, H., Kasahara, S., Takahashi, Y.: A simple algorithm for the rate matrices of level-dependent QBD processes. In: Proceedings of QTNA 2010, July 24–26, Beijing, China (2010)
Takine, T.: Analysis and computation of the stationary distribution in a special class of Markov chains of level-dependent M/G/1-type and its application to BMAP/M/\(\infty \) and BMAP/M/c+M queues. Queueing Syst. 84, 49–77 (2016)
Xu, S.H., Gao, L., Ou, J.: Service performance analysis and improvement for a ticket queue with balking customers. Manage. Sci. 53, 971–990 (2007)
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This research was supported in part by JSPS KAKENHI Grant Number 19K11841.
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Takine, T. On level-dependent QBD processes with explosive state space. Queueing Syst 100, 353–355 (2022). https://doi.org/10.1007/s11134-022-09796-1
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DOI: https://doi.org/10.1007/s11134-022-09796-1