#### For citation:

Shklennik M. A., Moiseev A. N. Method of Markovian summation for study the repeated flow in queueing tandem M|GI|∞ → GI|∞. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2021, vol. 21, iss. 1, pp. 125-137. DOI: 10.18500/1816-9791-2021-21-1-125-137

# Method of Markovian summation for study the repeated flow in queueing tandem M|GI|∞ → GI|∞

The paper presents a mathematical model of queueing tandem M|GI|∞ → GI|∞ with feedback. The service times at the first stage are independent and identically distributed (i.i.d.) with an arbitrary distribution function B1(x). Service times at the second stage are i.i.d. with an arbitrary distribution function B2(x). The problem is to determine the probability distribution of the number of repeated customers (r-flow) during fixed time period. To solve this problem, the Markov summation method was used, which is based on the consideration of Markov processes and the solution of the Kolmogorov equation. In the course of the solution, the so-called local r-flow was studied — the number of r-flow calls generated by one incoming customer received by the system. As a result, an expression is obtained for the characteristic probability distribution function of the number of calls in the local r-flow, which can be used to study queuing systems with a similar service discipline and non-Markov incoming flows. As a result of the study, an expression is obtained for the characteristic probability distribution function of the number of repeated calls to the system at a given time interval during non-stationary regime, which allows one to obtain the probability distribution of the number of calls in the flow under study, as well as its main probability characteristics.

- Morozova A. S., Moiseeva S. P., Nazarov A. A. Investigation of the economicmathematical model of discount for patrons influence on income of trading. Tomsk State University Journal, 2006, no. 293, pp. 49–52 (in Russian).
- Zhidkova L. A., Moiseeva S. P. A mathematical model of customer flows of a two-product trading company in the form of a queuing system with repeated calls to blocks. Izvestiya Tomskogo Politekhnicheskogo Universiteta, 2013, vol. 322, iss. 6, pp. 5–9 (in Russian).
- Ananina I. A. Mathematical model of the income change process of the trading company expanding the presence in the market. Tomsk State University Journal of Control and Computer Science, 2011, iss. 2, pp. 5–14 (in Russian).
- Nazarov A. A., Ananina I. A. A mathematical model of a life annuity procedure. Izvestiya Tomskogo Politekhnicheskogo Universiteta, 2011, vol. 318, iss. 5, pp. 160–165 (in Russian).
- Shklennik M., Moiseeva S., Moiseev A. Optimization of two-level discount values using queueing tandem model with feedback. In: A. Dudin, A. Nazarov, A. Moiseev, eds. Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2018, WRQ 2018. Communications in Computer and Information Science, vol. 912. Springer, Cham, 2018, pp. 321–332. https://doi.org/10.1007/978-3-319-97595-5_25
- Sevast’yanov B. A. An ergodic theorem for Markov processes and its application to telephone systems with fefusals. Theory of Probability & Its Applications, 1957, vol. 2, iss. 1, pp. 102–112. https://doi.org/10.1137/1102005
- Morozova A. S., Moiseeva S. P. Study of repeated customers flow in the queueing system with unlimited number of servers and feedback. Tomsk State University Journal, 2005, no. 287, pp. 46–51 (in Russian).
- Nazarov A. A., Moiseeva S. P., Morozova A. S. Investigation systems of service with repeated handling. Method of limit decomposition. Computational Technologies, 2005, vol. 13, Special iss. 5, pp. 88–92 (in Russian).
- Ananina I. A., Nazarov A. A., Moiseeva S. P. Research of streams in system M|GI|∞ with repeated references the method of limiting decomposition. Tomsk State University Journal of Control and Computer Science, 2009, iss. 3 (8), pp. 56–67 (in Russian).
- Ananina I. A. Analysis of total flow of customers in queueing tandem system with unlimited number of servers and feedback by the method of limiting decomposition. Nauchnoe tvorchestvo molodezhi: materialy XIV Vserossiyskoy nauchno-prakticheskoy konferentsii [Scientific creativity of youth: Materials of the XIV All-Russian Scientific and Practical Conference]. Tomsk, Izdatel’stvo Tomskogo gosudarstvennogo universiteta, 2010, pt. 1, pp. 3–5 (in Russian).
- Moiseeva S. P., Shklennik M. A., Nabokova O. O. Analysis of flows in queueing tandem with unlimited number of servers and feedback by method of limiting decomposition. Informatsionnye tekhnologii i matematicheskoe modelirovanie (ITMM-2017): materialy XVI Mezhdunarodnoi konferentsii imeni A. F. Terpugova [Information Technologies and Mathematical Modelling (ITMM-2017): Materials of the XVI International Conference named after A. F. Terpugov]. Tomsk, NTL, 2017, pt. 1, pp. 108–114 (in Russian).
- Zadiraniva L. A., Moiseeva S. P. Asymptotic analysis of the flow of repeated requests in system MMP P |M|∞ with repeated requests. Tomsk State University Journal of Control and Computer Science, 2015, iss. 2 (31), pp. 26–34 (in Russian). https://doi.org/10.17223/19988605/31/3
- Melikov A., Zadiranova L., Moiseev A. Two asymptotic conditions in queue with MMPP arrivals and feedback. In: V. Vishnevskiy, K. Samouylov, D. Kozyrev, eds. Distributed Computer and Communication Networks. DCCN 2016. Communications in Computer and Information Science, vol. 678. Springer, Cham, 2016, pp. 231–240. https://doi.org/10.1007/978-3-319-51917-3_21
- Zadiranova L. A. Analysis of the flow of repeated requests in system GI|M|∞ with repeated requests. In: Teoriya veroyatnostey, sluchaynye protsessy, matematicheskaya statistika i prilozheniya: materialy Mezhdunarodnoy nauchnoy konferentsii, posvyashchennoy 80-letiyu professora G. A. Medvedeva [Probability theory, random processes, mathematical statistics and applications: Materials of the International Scientific Conference dedicated to the 80th anniversary of prof. G. A. Medvedev]. Minsk, RIVSh, 2015, pp. 43–46.
- Nazarov A., Dammer D. Methods of limiting decomposition and Markovian summation in queueing system with infinite number of servers. In: A. Dudin, A. Nazarov, A. Moiseev, eds. Information Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2018, WRQ 2018. Communications in Computer and Information Science, vol. 912. Springer, Cham, 2018. pp. 71–82. https://doi.org/10.1007/978-3-319-97595-5_6
- Shklennik M. A., Moiseev A. N. Mathematical model of a system for physics experimental data processing with the need to reprocess data. Russian Physics Journal, 2019, vol. 62, iss. 3, pp. 553–560. https://doi.org/10.1007/s11182-019-01746-4

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