Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2022110.22363/2312-9735-2018-26-4-303-320Research ArticleTowards the Analysis of the Queuing System Operating in the Random Environment with Resource AllocationZaryadovIvan S<p>Candidate of Physical and Mathematical Sciences, assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Senior Researcher of Institute of Informatics Problems of Federal Research Center “Computer Science and Control” Russian Academy of Sciences</p>zaryadov-is@rudn.ruTsurlukovVladimir V<p>master’s degree student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)</p>dober.vvt@gmail.comCarvalhoCravid Viana<p>master’s degree student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)</p>hilvianamat1@gmail.comZaytsevaAnna A<p>master’s degree student of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)</p>anna-z96@mail.ruMilovanovaTatiana A<p>Candidate of Physical and Mathematical Sciences, lecturer of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University)</p>milovanova-ta@rudn.ruPeoples’ Friendship University of Russia (RUDN University)Institute of Informatics Problems, FRC CSC RAS1512201826430332021122018Copyright © 2018, Zaryadov I.S., Tsurlukov V.V., Carvalho C.V., Zaytseva A.A., Milovanova T.A.2018<p>The mathematical model of the system, that consists of a storage device and several homogeneous servers and operates in a random environment, and provides incoming applications not only services, but also access to resources of the system, is being constructed. The random environment is represented by two independent Markov processes. The first of Markov processes controls the incoming flow of applications to the system and the size of resources required by each application. The incoming flow is a Poisson one, the rate of the flow and the amount of resources required for the application are determined by the state of the external Markov process. The service time for applications on servers is exponential distributed. The service rate and the maximum amount of system resources are determined by the state of the second external Markov process. When the application leaves the system, its resources are returned to the system. In the system under consideration, there may be failures in accepting incoming applications due to a lack of resources, as well as loss of the applications already accepted in the system, when the state of the external Markov process controlling the service and provision of resources changes. A random process describing the functioning of this system is constructed. The system of equations for the stationary probability distribution of the constructed random process is presented in scalar form. The main tasks for further research are formulated.</p>queuing systemrandom environmentMarkov modulated Poisson processMarkov modulated service processresource allocationMMPPсистема массового обслуживанияслучайное окружениепредоставление ресурсов[Queueing Theory / P. P. Bocharov, C. D’Apice, A. V. Pechinkin, S. Salerno. — Utrecht, Boston: VSP, 2004. — 446 p.][Basharin G. P., Gaidamaka Yu. V., Samouylov K. E. Mathematical Theory of Teletraffic and its Application to the Analysis of Multiservice Communication of Next Generation Networks // Automatic Control and Computer Sciences. — 2013. — Vol. 47, No 2. — Pp. 62–69. — DOI: 10.3103/S0146411613020028.][Trivedi Kishor S. Probability and Statistics with Reliability, Queuing, and Computer Science Applications, Second Edition. — John Wiley & Sons, 2016. — 830 p.][Chee-Hoc N., Boon-Hee S. Queueing Modelling Fundamentals with Applications in Communication Networks. 2nd Edition. — John Wiley & Sons, 2008. — 292 p.][Neuts M. F. A Versatile Markovian Point Process // Journal of Applied Probability. — 1979. — Vol. 16, No 4. — Pp. 764–779. — DOI: 10.2307/3213143.][Neuts M. F. Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach. — Baltimore: Johns Hopkins University Press, 1981. — 352 p.][Neuts M. F. Matrix-Analytic Methods in Queuing Theory // European Journal of Operational Research. — 1984. — Vol. 15, No 1. — Pp. 2–12. — DOI: 10.1016/03772217(84)90034-1.][Neuts M. F. Structured Stochastic Matrices of M/G/1 Type and Their Applications. — New York: Marcel Dekker Inc., 1989. — 512 p.][Fisher W., Meier-Hellstern K. S. The Markov-Modulated Poisson Process (MMPP) Cookbook // Performance Evaluation. — 1993. — Vol. 18, No 2. — Pp. 149–171. — DOI: 10.1016/0166-5316(93)90035-S.][Andersen A., Nielsen B. A Markovian Approach for Modelling Packet Traffic with Long-Range Dependence // IEEE Journal on Selected Areas in Communications. — 1998. — Vol. 16, No 5. — Pp. 719–732. — DOI: 10.1109/49.700908.][Markov Models of Internet Traffic and a New Hierarchical MMPP Model / L. Muscariello, M. Mellia, M. Meo et al. // Computer Communications. — 2005. — Vol. 28, No 16. — Pp. 1835–1852. — DOI: 10.1016/j.comcom.2005.02.012.][Andronov A. M., Vishnevsky V. M. Markov-Modulated Continuous Time Finite Markov chain as the Model of Hybrid Wireless Communication Channels Operation // Automatic Control and Computer Sciences. — 2016. — Vol. 50, No 3. — Pp. 125–132. — DOI: 10.3103/S0146411616030020.][Reliability-Centric Analysis of Offloaded Computation in Cooperative Wearable Applications / A. Ometov, D. Kozyrev, V. Rykov et al. // Wireless Communications and Mobile Computing. — 2017. — Vol. 2017. — P. 9625687. — DOI: 10.1155/2017/9625687.][Rykov V., Kozyrev D. Analysis of Renewable Reliability Systems by Markovization Method // Lecture Notes in Computer Science / Analytical and Computational Methods in Probability Theory, ACMPT 2017. — Germany, Heidelberg, SpringerVerlag, 2017. — Pp. 210–220. — DOI: 10.1007/978-3-319-71504-9 19.][Rykov V., Kozyrev D., Zaripova E. Modeling and Simulation of Reliability Function of a Homogeneous Hot Double Redundant Repairable System // ECMS 2017 Proceedings / European Council for Modeling and Simulation. — Budapest, Hungary, May 23–26: European Council for Modelling and Simulation, 2017. — Pp. 701–705. — DOI: 10.7148/2017-0701.][Pechinkin A. P., Razumchik R. Approach for Analysis of Finite M2—M2—1—R with Hysteric Policy for SIP Server Hop-by-Hop Overload Control // Proceedings – 27th European Conference on Modelling and Simulation, ECMS 2013 / European Council for Modelling and Simulation. — Alesund, Norway, May 27–30, 2013: European Council for Modelling and Simulation, 2013. — Pp. 573–579. — DOI: 10.7148/20130573.][Razumchik R. Analysis of Finite MAP|PH|1 Queue with Hysteretic Control of Arrivals // International Congress on Ultra Modern Telecommunications and Control Systems and Workshops, ICUMI-2016 / IEEE Computer Society. — Lisbon, Portugal, October 18–20: IEEE Computer Society, 2016. — Pp. 288–293. — DOI: 10.1109/ICUMT.2016.7765373.][Razumchik R., Telek M. Delay Analysis of a Queue with Re-sequencing Buffer and Markov Environment // Queueing Systems. — 2016. — Vol. 82, No 1–2. — Pp. 7–28. — DOI: 10.1007/s11134-015-9444-z.][Razumchik R., Telek M. Delay Analysis of Resequencing Buffer in Markov Environment with HOQ-FIFO-LIFO Policy // Lecture Notes in Computer Science / 14th European Workshop on Computer Performance Engineering, EPEW 2017. — Vol. 10497. — Berlin, Germany, September 7–8: Springer Verlag, 2017. — Pp. 53–68. — DOI: 10.1007/978-3-319-66583-2 4.][Retrial Tandem Queue with BMAP-input and Semi-Markovian Service Process / V. Klimenok, O. Dudina, V. Vishnevsky, K. Samouylov // Communications in Computer and Information Science, vol. 700 / 20th International Conference on Distributed Computer and Communication Networks, DCCN 2017. — Moscow, Russian Federation, September 25–29: Springer Verlag, 2017. — Pp. 159–173. — DOI: 10.1007/978-3-319-66836-9 14.][Analysis of a Retrial Queue with Limited Processor Sharing Operating in the Random Environment / S. Dudin, A. Dudin, O. Dudina, K. Samouylov // Lecture Notes in Computer Science, LNCS / 15th International Conference on Wired/Wireless Internet Communications, WWIC 2017. — Vol. 10372. — St. Petersburg, Russian Federation, June 21–23: Springer Verlag, 2017. — Pp. 38–49. — DOI: 10.1007/978-3-319-613826 4.][The Survey on Markov-Modulated Arrival Processes and Their Application to the Analysis of Active Queue Management Algorithms / I. Zaryadov, A. Korolkova, D. Kulyabov et al. // Communications in Computer and Information Science / 20th International Conference on Distributed Computer and Communication Networks, DCCN 2017. — Vol. 700. — Moscow, Russian Federation, September 25–29: Springer Verlag, 2017. — Pp. 417–430. — DOI: 10.1007/978-3-319-66836-9 35.][LTE Performance Analysis Using Queuing Systems with Finite Resources and Random Requirements / V. Naumov, K. Samouylov, N. Yarkina et al. // 7th International Congress on Ultra Modern Telecommunications and Control Systems ICUMT-2015 / IEEE Computer Society. — Brno, Czech Republic, October 6–8: IEEE Computer Society, 2015. — Pp. 100–103. — DOI: 10.1109/ICUMT.2015.7382412.][Convolution Algorithm for Normalization Constant Evaluation in Queuing System with Random Requirements / K. Samouylov, E. Sopin, O. Vikhrova, S. Shorgin // AIP Conference Proceedings / International Conference of Numerical Analysis and Applied Mathematics 2016, ICNAAM 2016. — Vol. 1863. — Rodos Palace HotelRhodes, Greece, September 19–25: American Institute of Physics Inc., 2017. — P. 090004. — DOI: 10.1063/1.4992269.][Naumov V. A., Samuilov K. E., Samuilov A. K. On the Total Amount of Resources Occupied by Serviced Customers // Automation and Remote Control. — 2016. — Vol. 77, No 8. — Pp. 1419–1427. — DOI: 10.1134/S0005117916080087.][Samouylov K. E., Sopin E. S., Shorgin S. Ya. Queuing Systems with Resources and Signals and Their Application for Performance Evaluation of Wireless Networks // Informatika i ee Primeneniya. — 2017. — Vol. 11, No 3. — Pp. 99–105. — DOI: 10.14357/19922264170311.][Sopin E., Vikhrova O., Samouylov K. LTE Network Model with Signals and Random Resource Requirements // 9th International Congress on Ultra Modern Telecommunications and Control Systems and Workshops. — Munich, Germany, November 6–8: IEEE Computer Society, 2017. — Pp. 101–106. — DOI: 10.1109/ICUMT.2017.8255155.][Naumov V., Samouylov K. Analysis of Multi-resource Loss System with State-dependent Arrival and Service Rates // Probability in the Engineering and Informational Sciences. — 2017. — Vol. 31, No 4. — Pp. 413–419. — DOI: 10.1017/S0269964817000079.][Sopin E. S., Vikhrova O. G. Probability Characteristics Evaluation in Queueing System with Random Requirements // CEUR Workshop Proceedings, vol. 1995 / 7th International Conference “Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”, ITTMM 2017. — Peoples’ Friendship University of Russia (RUDN University) Moscow, Russian Federation: CEUR-WS, 2017. — Pp. 85–92.][Tsurlukov V. V. The Mathematical Model of the System with Resources and Random Environment // Proceedings of 2nd International School on Applied Probability Theory and Communications Technologies, Peoples’ Friendship University of Russia (RUDN University), Moscow, Russian Federation, October 23—27 / 2nd International School on Applied Probability Theory and Communications Technologies. — Moscow: RUDN University, 2017. — С. 318–320.][Chakravarthy S., Alfa A. S. Matrix-Analytic Methods in Stochastic Models. — CRC Press, 1996. — P. 396.][Breuer L., Baum D. An Introduction to Queueing Theory and Matrix-Analytic Methods. — Dordrecht: Springer Netherlands, 2005. — P. 286.][Qi-Ming H. Fundamentals of Matrix-Analytic Methods. — New-York: Springer-Verlag New York, 2014. — P. 364.]