For citation:
Polin E. P., Moiseeva S. P., Moiseev A. N. Heterogeneous Queueing System MR(S)/M(S)/∞ with Service Parameters Depending on the State of the Underlying Markov Chain. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2020, vol. 20, iss. 3, pp. 388-399. DOI: 10.18500/1816-9791-2020-20-3-388-399, EDN: GZPNTS
Heterogeneous Queueing System MR(S)/M(S)/∞ with Service Parameters Depending on the State of the Underlying Markov Chain
Data streams in information and communication systems include integrated heterogeneous streams, containing voice, text data and video. Since the service of different information units takes different time depending on their format, used protocols and so on, it is proposed to model such data transmission processes using heterogeneous queueing systems with services depending on the parameters of the incoming stream. In the paper, an infinite-server heterogeneous queueing system is considered. Arrivals are modeled as a Markov renewal process (MRP) with two states given by distribution functions of the interval lengths and by a transition probability matrix. The exponential distribution parameter of service time is determined by the state of the underlying Markov chain of the MRP at the moment when a customer arrives and does not change until the service completion. To study the system, the method of characteristic functions is used. Using their properties, analytical expressions are obtained for the initial moments of the first and the second order of the number of customers of each type present in the system in a steady-state regime. To analyze the relationship between the components of the process, a correlation moment is derived.
- Garayshina I. R., Nazarov A. A. Investigation of Russian Federation retirement fund insurance capital modification process mathematical model. Tomsk State University Journal, 2003, no. 280, pp. 109–111 (in Russian).
- Fokina N. P., Tananko I. E. A Method of Routing Control in Queueing Networks with Changing Topology. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2013, vol. 13, iss. 2, pt. 2, pp. 82–88 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2013-13-2-2-82-88
- Akhmedova D. D., Terpugov A. F. Mathematical model of an insurance company taking into account advertising costs. Izvestiya vuzov. Fizika, 2001, vol. 44, no. 1, pp. 25–29 (in Russian).
- Erlang A. K. The theory of probability and telephone conversations. Nyt Tidsskrift for Matematik, В, 1909, vol. 20, pp. 33–39.
- Pankratova Е. V. Investigation of the queuing system GI/GI/∞ with two types of arrivals. Informatsionnye tekhnologii i matematicheskoe modelirovanie (ITMM-2015): materialy XIV Mezhdunar. konf. im. A. F. Terpugova [Information Technology and Mathematical Modeling (ITMM-2015). Materials of the XIV Int. conf. named after A. F. Terpugov]. Tomsk, Izd-vo Tomskogo universiteta, 2015, pt. 1, pp. 152–157 (in Russian).
- Pankratova Е. V. Investigation of the queuing system MAP|M|∞ with heterogeneous service using the asymptotic analysis method under the condition of extremely rare changes in the state of the incoming stream. In: Distributed computer and communication networks: control, computation, communications (DCCN-2015). Moscow, Institut problem upravleniya RAS, 2015, pp. 585–592 (in Russian).
- Pankratova E. V., Moiseeva S. P. Queueing System GI/GI/∞ with n Types of Customers. Communications in Computer and Information Science. Switzerland, Springer, 2015, vol. 564, pp. 216–225.
- Moiseeva S. P., Pankratova Е. V., Ubonova E. G. Queuing system with renewal arrival process and two types of customers. Tomsk State University Journal of Control and Computer Science, 2016, no. 2 (35), pp. 46–53 (in Russian). DOI: https://doi.org/10.17223/19988605/35/5
- Moiseeva S. P., Sinyakova I. A. The method of moments for the study of the mathematical model of parallel servicing multiple claims of the Markov renewal process. Bulletin of the Tomsk Polytechnic University, 2012, vol. 321, no. 5, pp. 24–28 (in Russian).
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