Cybernetics And Systems Analysis logo
Editorial Board Announcements Abstracts Authors Archive
KIBERNETYKA TA SYSTEMNYI ANALIZ
International Theoretical Science Journal
Volume 62 >>> № 4 JULY — AUGUST 2026
-->


DOI 10.34229/KCA2522-9664.26.4.10
UDC 519.872

A. Melikov
Baku Engineering University; Institute of Mathematics, Ministry of Science and Education, Baku, Azerbaijan,  amelikov@beu.edu.az

S. Navendhan
Puducherry Technological University, Puducherry, India,
 navendhan160600@gmail.com

G. Ayyappan
Puducherry Technological University, Puducherry, India,
 ayyappan@ptuniv.edu.in


ANALYSIS OF BMAP/PH/1/K QUEUEING MODEL WITH CONTROLLABLE GROUP
SERVICE, SETUP TIME AND BALKING

Abstract. In the finite capacity queueing system, a single server provides service in the form of groups, where customers utilize the Batch Markovian Arrival Process (BMAP) for entering the system. Group of various batch sizes are served, and it is expected that the service durations follow a phase-type distribution. The service rate accelerates to rapid service when the buffer size is greater than some threshold value, and it returns to its normal rate after the current service is finished. The batch arrival customer may balk at the idle time of the server. The server undergoes the setup process during the completion of each service. The matrix analytic method is used to analyze the steady-state conditions for this queueing model. Performance measures based on regular splitting are presented, and numerical examples are discussed.

Keywords: batch Markovian arrival process, phase-type distribution, setup process, balking, matrix analytic method, regular splitting.


full text

REFERENCES

    1. Crabill T.B. Optimal control of a service facility with variable exponential service times and constant arrival rate. Management Science. 1972. Vol. 18, N 9. P. 465–587. https://doi.org/10.1287/mnsc.18.9.560.
    2. Neuts M.F., Rao B.M. On the design of a finite-capacity queue with Phase-type service times and hysteretic control. European Journal of Operational Research. 1992. Vol. 62, Iss. 2. P. 221–240. https://doi.org/10.1016/0377-2217(92)90250-D.
    3. Chakravarthy S.R. Analysis of the queue with service control. Applied Stochastic Models and Data Analysis. 1996. Vol. 12, Iss. 3. P. 179–191. https://doi.org/10.1002/(SICI)1099-0747(199609)12:33.0.CO;2-U.
    4. Chakravarthy S.R. Analysis of a priority polling system with group services. Communications in Statistics. Stochastic Models. 1998. Vol. 14, Iss. 1–2. P. 25–49. https://doi.org/10.1080/15326349808807459.
    5. Chakravarthy S.R., Ozkar S.A. A queueing model with BMAP arrivals and heterogeneous phase type group services. Methodology and Computing in Applied Probability. 2024. Vol. 26. Article number 53. https://doi.org/10.1007/s11009-024-10122-w.
    6. Lucantoni D.M, Meier-Hellstern K.S., Neuts M.F. A single-server queue with server vacations and a class of non-renewal arrival process. Advances in Applied Probability. 1990. Vol. 22, Iss. 3. P. 676–705. https://doi.org/10.2307/1427464.
    7. Lucantoni D.M. New results on the single server queue with a batch Markovian arrival process. Communications in Statistics. Stochastic Models. 1991. Vol. 7, Iss. 1. P. 1–46. https://doi.org/10.1080/15326349108807174
    8. Neuts M.F. A versatile Markovian point process. Journal of Applied Probability. 1979. Vol. 16, Iss. 4. P. 764–779. https://doi.org/10.2307/3213143.
    9. Ghosh S., Banik A.D., Walraevens J., Bruneel H. A detailed note on the finite-buffer queueing system with correlated batch-arrivals and batch-size-phase-dependent bulk-service. 4OR. A Quarterly Jounal of Operations Research. 2022. Vol. 20. P. 241–272. https://doi.org/10.1007/s10288-021-00478-x.
    10. Dudin A., Chakravarthy S. Optimal hysteretic control for the system with single and group service modes. Annals of Operations Research. 2002. Vol. 112, Iss. 1–4. P. 153–169. https://doi.org/10.1023/A:1020985106453.
    11. Chakravarthy S.R. Introduction to matrix-analytic methods in queues 2: Analytical and simulation approach — Queues and simulation. New York: London and John Wiley and Sons, 2022. 415 p. https://doi.org/10.1002/9781394174201.
    12. Dario A.B., Chakravarthy S.R., Beatrice M. Control of the queue with group services. In: Advances in Algorithmic Methods for Stochastic Models. New Jersey: Notable Publications Inc., 2000. P. 57–72. URL: https://digitalcommons.kettering.edu/industrialmanuf_eng_conference/16.
    13. Ayyappan G., Gowthami R. A system with two types of heterogeneous service, setup, closedown, vacation, immediate feedback, breakdown and repair. International Journal of Mathematics in Operational Research. 2022. Vol. 22, N 3. P. 313–348. https://doi.org/10.1504/IJMOR.2022.124136.
    14. Kalyanaraman R., Janani G. Finite population and finite capacity single server batch service queues with compulsory working vacation. International Journal of Mathematics in Operational Research. 2024. Vol. 29, N 2. P. 163–190. https://doi.org/10.1504/IJMOR.2024.142117.
    15. Melikov A., Divya V., Aliyeva S. Analyses of feedback queue with positive server setup time and impatient calls. Communications in Computer and Information Science. 2021. Vol. 1391. P. 155–179. https://doi.org/10.1007/978-3-030-72247-0_12.
    16. Melikov A., Divya V., Aliyeva S. Markov model of queueing system with instantaneous feedback and server setup time. Journal of Modern Technology and Engineering. 2021. Vol. 6, N 1. P. 5–12. URL: https://jomardpublishing.com/UploadFiles/Files/journals/JTME/v6n1/MelikovA_et_al.pdf.
    17. Azhagappan A., Deepa T. Transient behavior of a single-server Markovian queue with balking and working vacation interruptions. Journal of the Operations Research Society of China. 2021. Vol. 9, Iss. 2. P. 321–341. https://doi.org/10.1007/s40305-019-00288-3.
    18. Melikov A., Ponomarenko L., Kuliyeva K. Numerical analysis of a queueing system with feedback. Cybernetics and Systems Analysis. 2015. Vol. 51, N 4. P. 566–573. https://doi.org/10.1007/s10559-015-9747-4.
    19. Banik A.D., Gupta U.C., Chaudhry M.L. Finite-buffer bulk service queue under Markovian service process: . Stochastic Analysis and Applications. 2009. Vol. 27, Iss. 3. P. 500–522. https://doi.org/10.1080/07362990902844157.
    20. Sundar R., Ganesan V., Rita S. Batch arrival Poisson queue with breakdown and repairs. International Journal of Mathematics in Operational Research. 2020. Vol. 17, N 3. P. 424–435. https://doi.org/10.1504/IJMOR.2020.110033.
    21. Melikov A., Aliyeva S., Sztrik J. Analysis of queueing system with delayed feedback. Mathematics. 2019. Vol. 7, Iss. 11. Article number 1128. https://doi.org/10.3390/math7111128.
    22. Majid S. Performance analysis of a Markovian queue with impatient customers and working vacation. Journal of the Operations Research Society of China. 2023. Vol. 11, Iss. 1. P. 133–156. https://doi.org/10.1007/s40305-021-00361-w.
    23. Chang S.H., Choi D.W. Modeling and performance analysis of a finite-buffer queue with batch arrivals, batch services, and setup times: The queue with setup times. INFORMS Journal on Computing. 2006. Vol. 18, N 2. P. 218–228. https://doi.org/10.1287/ijoc.1040.0117[12].
    24. Madhu J., Anshul K. Effect of disaster and balking on driven fluid queue with working vacation. International Journal of Mathematics in Operational Research. 2022. Vol. 23, N 3. P. 359–371. https://doi.org/10.1504/IJMOR.2022.127380.
    25. Nagarajan P., Kalyanaraman R. An queue with Bernoulli vacation and unreliable server. International Journal of Mathematics in Operational Research. 2023. Vol. 25, N 4. P. 492–510. https://doi.org/10.1504/IJMOR.2023.132836.
    26. Ayyappan G., Nithya S. A retrial queue with priority services, differentiate breakdown, delayed repair, Bernoulli feedback, balking, and working vacation. International Journal of Mathematics in Operational Research. 2024. Vol. 28, N 4. P. 491–513. https://doi.org/10.1504/IJMOR.2024.140347.
    27. Ushakumari P.V. On a two-server reliability system with one-server idle below a threshold. International Journal of Mathematics in Operational Research. 2021. Vol. 18, N 4. P. 528–544. https://doi.org/10.1504/IJMOR.2021.114209.
    28. Sivasamy R., Peter O.P. A c-server Poisson queue with customer impatience due to slow phase service. International Journal of Mathematics in Operational Research. 2021. Vol. 20, N 1. P. 85–98. https://doi.org/10.1504/IJMOR.2021.117632.
    29. Dudin A., Dudina O., Dudin S., Melikov A. Analysis of a queueing model with flexible priority, batch arrival, and impatient customers. Computation. 2025. Vol. 13, Iss. 3. Article number 77. https://doi.org/10.3390/computation13030077.
    30. Steeb W.-H., Hardy Y. Matrix calculus and Kronecker product: A practical approach to linear and multilinear algebra. Singapore: World Scientific Publishing Co. Pte. Ltd, 2011. 324 p.
    31. Donald J.R. Convergent regular splitting for singular M-matrices. SIAM Journal on Algebraic Discrete Methods. 1984. Vol. 5, Iss. 1. P. 133–144. https://doi.org/10.1137/0605015.
    32. Pierre S. Iterative algorithms for large stochastic matrices. Linear Algebra and its Application. 1991. Vol. 154–156. P. 65–103. URL: https://www.sciencedirect.com/journal/linear-algebra-and-its-applications/vol/154/suppl/C.
    33. Klimenok V., Dudin A. On the distribution of the number of consecutively lost customers in the system. Lecture Notes in Computer Science. 2022. Vol. 13766. P. 257–269. https://doi.org/10.1007/978-3-031-23207-7_20.
    34. Klimenok V.I., Dudin A.N., Samouylov K.E. Analysis of the queueing system with backup servers. Applied Mathematical Modelling. 2018. Vol. 57. P. 64–84. https://doi.org/10.1016/J.APM.2017.12.024.



Editorial Board Announcements Abstracts Authors Archive
about scope office editorial board instructions for authors subscriptions contacts
© 2026 Kibernetika.org. All rights reserved.