Abstract
A multi-server queueing system with various types of customers arriving in accordance with a Batch Marked Markov Arrival Process (BMMAP) is considered. The service times have exponential distribution with the rate corresponding to the type of customer. The system does not have a buffer. The customer acceptance discipline is assumed to be partial admission. The operation of the system is described by a multi-dimensional continuous time Markov chain. The stationary distribution of the states of the chain and the key system indicators (such as loss probabilities of customers of different types and the sojourn time distribution) are computed. The results are illustrated numerically.
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References
Chakravarthy, S.R.: The batch Markovian arrival process: a review and future work. In: Advances in Probability Theory and Stochastic Processes, pp. 21–29. Notable Publications Inc., New Jersey (2001)
Lucantoni, D.: New results on the single server queue with a batch Markovian arrival process. Commun. Stat. Stoch. Models 7, 1–46 (1991)
Dudin, A.N., Klimenok, V.I., Vishnevsky, V.M.: The Theory of Queuing Systems with Correlated Flows. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-32072-0
Vishnevskii, V.M., Dudin, A.N.: Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks. Autom. Remote. Control. 78(8), 1361–1403 (2017). https://doi.org/10.1134/S000511791708001X
Klimenok, V.I., Kim, C.S., Orlovsky, D.S., Dudin, A.N.: Lack of invariant property of Erlang \(BMAP/PH/N/0\) model. Queueing Syst. 49, 187–213 (2005)
He, Q.M.: Queues with marked customers. Adv. Appl. Probab. 28, 567–587 (1996)
Al-Begain, K., Dudin, A.N., Mushko, V.V.: Novel queueing model for multimedia over downlink in 3.5 G wireless network. J. Commun. Softw. Syst. 2, 68–80 (2006)
Kovalenko, I.N.: B.A. Sevastyanov’s famous theorem. Proc. Steklov Inst. Math. 282, 124–126 (2013). https://doi.org/10.1134/S0081543813060114
Ramaswami, V., Lucantoni, D.M.: Algorithms for the multi-server queue with phase type service. Stoch. Model. 1, 393–417 (1984)
Dudin, A., Kim, C., Dudina, O., Dudin, S.: Multi-server queueing system with a generalized phase-type service time distribution as a model of call center with a call-back option. Ann. Oper. Res. 239(2), 401–428 (2014). https://doi.org/10.1007/s10479-014-1626-2
Graham, A.: Kronecker Products and Matrix Calculus with Applications. Ellis Horwood, Cichester (1981)
Dudin, A., et al.: Analysis of single-server multi-class queue with unreliable service, batch correlated arrivals, customers impatience, and dynamical change of priorities. Mathematics 9(11), 1–17 (2021)
Acknowledgments
This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. NRF-2020R1A2C1006999) and by the RUDN University Strategic Academic Leadership Program.
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Kim, C., Dudin, A., Dudin, S., Dudina, O. (2021). Analysis of Multi-server Loss Queueing System with the Batch Marked Markov Arrival Process. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds) Distributed Computer and Communication Networks: Control, Computation, Communications. DCCN 2021. Lecture Notes in Computer Science(), vol 13144. Springer, Cham. https://doi.org/10.1007/978-3-030-92507-9_16
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