Geodesic
A queueing system with a batch Markovian arrival process and varying priorities
Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2022), pp. 47-56.

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We consider herein a single-server queueing system with a finite buffer and a batch Markovian arrival process. Customers staying in the buffer may have a lower or higher priority. Immediately after arrival each of the customer is assigned the lowest priority and a timer is set for it, which is defined as a random variable distributed according to the phase law and having two absorbing states. After the timer enters one of the absorbing states, the customer may leave the system forever (get lost) or change its priority to the highest. When the timer enters another absorbing state, the customer is lost with some probability and the timer is set again with an additional probability. If a customer enters a completely full system, it is lost. Systems of this type can serve as mathematical models of many real-life medical care systems, contact centers, perishable food storage systems, etc. The operation of the system is described in terms of a multidimensional Markov chain, the stationary distribution and a number of performance characteristics of the system are calculated.
Keywords: Queueing system; finite buffer; batch Markovian arrival process; changing priorities; stationary distribution; performance characteristics. 
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V. I. Klimenok. A queueing system with a batch Markovian arrival process and varying priorities. Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2022), pp. 47-56. http://geodesic.mathdoc.fr/item/BGUMI_2022_2_a4/

[1] B. Bilodeau, D. A. Stanford, Average waiting times in the two-class M/G/1 delayed accumulating priority queue, 2020, 19 pp., arXiv: 2001.06054

[2] V. A. Fajardo, S. Drekic, “Waiting time distributions in the preemptive accumulating priority queue”, Methodology and Computing in Applied Probability, 19 (2017), 255–284 | DOI | MR | Zbl

[3] M. Mojalal, D. A. Stanford, R. J. Caron, “The lower-class waiting time distribution in the delayed accumulating priority queue”, Information Systems and Operational Research, 58:1 (2020), 60–86 | DOI | MR

[4] K. C. Sharma, G. C. Sharma, “A delay dependent queue without pre-emption with general linearly increasing priority function”, Journal of the Operational Research Society, 45:8 (1994), 948–953 | DOI | Zbl

[5] D. A. Stanford, P. Taylor, I. Ziedins, “Waiting time distributions in the accumulating priority queue”, Queueing Systems, 77 (2014), 297–330 | DOI | MR | Zbl

[6] H. e. Qi-Ming, Xie. Jingui, Zhao. Xiaobo, “Stability conditions of a preemptive repeat priority MMAP[N]/PH[N]/S queue with customer transfers”, Proceedings of the 13th International conference advanced stochastic models and data analysis, Lithuania, Vilnius, 2009, 463–467 | MR

[7] H. e. Qi-Ming, Xie. Jingui, Zhao. Xiaobo, “Priority queue with customer upgrades”, Naval Research Logistics, 59:5 (2012), 362–375 | DOI | MR | Zbl

[8] V. Klimenok, A. Dudin, O. Dudina, I. Kochetkova, “Queuing system with two types of customers and dynamic change of a priority”, Mathematics, 8(5) (2020), 824 | DOI

[9] Xie. Jingui, H. e. Qi-Ming, Zhao. Xiaobo, “On the stationary distribution of queue lengths in a multi-class priority queueing system with customer transfers”, Queueing Systems, 62 (2009), 255–277 | DOI | MR | Zbl

[10] Xie. Jingui, Cao. Ping, Huang. Boray, Eng. Marcus, “Determining the conditions for reverse triage in emergency medical services using queuing theory”, International Journal of Production Research, 54:11 (2016), 3347–3364 | DOI

[11] Xie. Jingui, Zhu. Taozeng, Chao. An-Kuo, Wang. Shuaian, “Performance analysis of service systems with priority upgrades”, Annals of Operations Research, 253 (2017), 683–705 | DOI | MR | Zbl

[12] Cao. Ping, Xie. Jingui, “Optimal control of a multiclass queueing system when customers can change types”, Queueing Systems, 82 (2016), 285–313 | DOI | MR | Zbl

[13] Xie. Jingui, H. e. Qi-Ming, Zhao. Xiaobo, “Stability of a priority queueing system with customer transfers”, Operations Research Letters, 36 (2008), 705–709 | DOI | MR | Zbl

[14] M. Cildoz, A. Ibarra, F. Mallor, “Accumulating priority queues versus pure priority queues for managing patients in emergency departments”, Operations Research for Health Care, 23 (2019), 100224 | DOI

[15] L. Brown, N. Gans, A. Mandelbaum, A. Sakov, H. Shen, S. Zeltyn, a. l. et, “Statistical analysis of a telephone call center: a queueingscience perspective”, Journal of the American Statistical Association, 100 (2005), 36–50 | DOI | MR | Zbl

[16] DM. Lucantoni, “New results on the single server queue with a batch Markovian arrival process”, Communications in Statistics. Stochastic Models, 7:1 (1991), 1–46 | DOI | MR | Zbl

[17] A. N. Dudin, V. I. Klimenok, V. M. Vishnevsky, The theory of queuing systems with correlated flows, Springer Nature, Berlin, 2019, 410 pp. | DOI | MR

[18] M. F. Neuts, Matrix-geometric solutions in stochastic models: an algorithmic approach, The Johns Hopkins University Press, Baltimore, 1981, 348 pp. | MR

[19] A. Graham, Kronecker products and matrix calculus with applications, Ellis Horwood, Cichester, 1981, 130 pp. | MR

[20] V. I. Klimenok, C. S. Kim, D. S. Orlovsky, A. N. Dudin, “Lack of invariant property of Erlang loss model in case of MAP input”, Queueing Systems, 49 (2005), 187–213 | DOI | MR | Zbl