A queueing system with a batch Markovian arrival process and varying priorities

  • Valentina I. Klimenok Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
DOI: https://doi.org/10.33581/2520-6508-2022-2-47-56

Abstract

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.

Author Biography

Valentina I. Klimenok, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

doctor of science (physics and mathematics), full professor; chief researcher at the laboratory of applied probabilistic analysis, faculty of applied mathematics and computer science

References

  1. Bilodeau B, Stanford DA. Average waiting times in the two-class M/G/1 delayed accumulating priority queue. arXiv:2001.06054v1 [Preprint]. 2020 [cited 2022 March 10]: [19 p.]. Available from: https://arxiv.org/abs/2001.06054v1.
  2. Fajardo VA, Drekic S. Waiting time distributions in the preemptive accumulating priority queue. Methodology and Computing in Applied Probability. 2017;19:255–284. DOI: 10.1007/s11009-015-9476-1.
  3. Mojalal M, Stanford DA, Caron RJ. The lower-class waiting time distribution in the delayed accumulating priority queue. INFOR: Information Systems and Operational Research. 2020;58(1):60–86. DOI: 10.1080/03155986.2019.1624473.
  4. Sharma KC, Sharma GC. A delay dependent queue without pre-emption with general linearly increasing priority function. Journal of the Operational Research Society. 1994;45(8):948–953. DOI: 10.1057/jors.1994.147.
  5. Stanford DA, Taylor P, Ziedins I. Waiting time distributions in the accumulating priority queue. Queueing Systems. 2014;77:297–330. DOI: 10.1007/s11134-013-9382-6.
  6. Qi-Ming He, Jingui Xie, Xiaobo Zhao. Stability conditions of a preemptive repeat priority MMAP[N]/PH[N]/S queue with customer transfers (short version). In: Proceedings of the 13th International conference advanced stochastic models and data analysis (ASMDA-2009); 2009 June 30 – July 3; Vilnius, Lithuania. p. 463–467.
  7. Qi-Ming He, Jingui Xie, Xiaobo Zhao. Priority queue with customer upgrades. Naval Research Logistics. 2012;59(5):362–375. DOI: 10.1002/nav.21494.
  8. Klimenok V, Dudin A, Dudina O, Kochetkova I. Queuing system with two types of customers and dynamic change of a priority. Mathematics. 2020;8(5):824. DOI: 10.3390/math8050824.
  9. Jingui Xie, Qi-Ming He, Xiaobo Zhao. On the stationary distribution of queue lengths in a multi-class priority queueing system with customer transfers. Queueing Systems. 2009;62:255–277. DOI: 10.1007/s11134-009-9130-0.
  10. Jingui Xie, Ping Cao, Boray Huang, Marcus Eng Hock Ong. Determining the conditions for reverse triage in emergency medical services using queuing theory. International Journal of Production Research. 2016;54(11):3347–3364.
  11. Jingui Xie, Taozeng Zhu, An-Kuo Chao, Shuaian Wang. Performance analysis of service systems with priority upgrades. Annals of Operations Research. 2017;253:683–705. DOI: 10.1007/s10479-016-2370-6.
  12. Ping Cao, Jingui Xie. Optimal control of a multiclass queueing system when customers can change types. Queueing Systems. 2016;82:285–313. DOI: 10.1007/s11134-015-9466-6.
  13. Jingui Xie, Qi-Ming He, Xiaobo Zhao. Stability of a priority queueing system with customer transfers. Operations Research Letters. 2008;36:705–709. DOI: 10.1016/j.orl.2008.06.007.
  14. Cildoz M, Ibarra A, Mallor F. Accumulating priority queues versus pure priority queues for managing patients in emergency departments. Operations Research for Health Care. 2019;23:100224. DOI: 10.1016/j.orhc.2019.100224.
  15. Brown L, Gans N, Mandelbaum A, Sakov A, Shen H, Zeltyn S, et al. Statistical analysis of a telephone call center: a queueingscience perspective. Journal of the American Statistical Association. 2005;100:36–50. DOI: 10.2307/27590517.
  16. Lucantoni DM. New results on the single server queue with a batch Markovian arrival process. Communications in Statistics. Stochastic Models. 1991;7(1):1–46. DOI: 10.1080/15326349108807174.
  17. Dudin AN, Klimenok VI, Vishnevsky VM. The theory of queuing systems with correlated flows. Berlin: Springer Nature; 2019. 410 p. DOI: 10.1007/978-3-030-32072-0.
  18. Neuts MF. Matrix-geometric solutions in stochastic models: an algorithmic approach. Baltimore: The Johns Hopkins University Press; 1981. 348 p.
  19. Graham A. Kronecker products and matrix calculus with applications. Cichester: Ellis Horwood; 1981. 130 p.
  20. Klimenok VI, Kim CS, Orlovsky DS, Dudin AN. Lack of invariant property of Erlang loss model in case of MAP input. Queueing Systems. 2005;49:187–213. DOI: 10.1007/s11134-005-6481-z.
Published
2022-07-27
Keywords: queueing system, finite buffer, batch Markovian arrival process, changing priorities, stationary distribution, performance characteristics
How to Cite
Klimenok, V. I. (2022). A queueing system with a batch Markovian arrival process and varying priorities. Journal of the Belarusian State University. Mathematics and Informatics, 2, 47-56. https://doi.org/10.33581/2520-6508-2022-2-47-56
Issue
No 2 (2022): Journal of the Belarusian State University. Mathematics and Informatics
Section
Probability Theory and Mathematical Statistics

Copyright (c) 2022 Journal of the Belarusian State University. Mathematics and Informatics

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

The authors who are published in this journal agree to the following:

  1. The authors retain copyright on the work and provide the journal with the right of first publication of the work on condition of license Creative Commons Attribution-NonCommercial. 4.0 International (CC BY-NC 4.0).
  2. The authors retain the right to enter into certain contractual agreements relating to the non-exclusive distribution of the published version of the work (e.g. post it on the institutional repository, publication in the book), with the reference to its original publication in this journal.
  3. The authors have the right to post their work on the Internet (e.g. on the institutional store or personal website) prior to and during the review process, conducted by the journal, as this may lead to a productive discussion and a large number of references to this work. (See The Effect of Open Access.)