Investigation of G-networks with restart at a non-stationary mode and their application

• Autorzy: Matalytski Mikhail , Naumenko Victor , Kopats Dmitry

Identyfikatory

DOI

10.17512/jamcm.2019.2.04

• Języki publikacji: EN

Abstrakty

EN

This article discusses the question of restarting the script, when restart is used by many users of the information network, which can be modelled as a G-network. Negative claims simulate the crash of the script and the re-sending of the request. Investigation of an open queuing network (QN) with several types of positive customers with the phase type of distribution of their service time and one type of negative customers have been carried out. Negative customers are signals whose effect is to restart one customers in a queue. A technique is proposed for finding the probability of states. It is based on the use of a modified method of successive approximations, combined with the method of a series. The successive approximations converge with time to a stationary distribution of state probabilities, the form of which is indicated in the article, and the sequence of approximations converges to the solution of the difference-differential equations (DDE) system. The uniqueness of this solution is proved. Any successive approximation is representable in the form of a convergent power series with an infinite radius of convergence, the coefficients of which satisfy recurrence relations, which is convenient for computer calculations. A model example illustrating the finding of time-dependent probabilities of network states using the proposed technique is also presented.

Słowa kluczowe

EN

queueing network; G-network; positive and negative customers; non-stationary regime; multiple type customers; signals; restart; nonstationary state probabilities; successive approximation method

PL

sieć kolejkowa; sieć G; sygnały; metoda aproksymacji; aproksymacja; teoria kolejek; pozytywny klient; negatywny klient

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2019
• Tom: Vol. 18, nr 2
• Strony: 41--51
• Opis fizyczny: Bibliogr. 18 poz., rys.

Twórcy

autor

Matalytski Mikhail

• Institute of Mathematics, Czestochowa University of Technology Czestochowa, Poland

autor

Naumenko Victor

• Department of Fundamental and Applied Mathematics, Grodno State University of Yanka Kupala, Grodno, Belarus

autor

Kopats Dmitry

Bibliografia

• [1] Moorsel, P.A., & Wolter, K. (2004). Analysis and algorithms for restart. Quantitative Evaluation of Systems: Proc. 1st Intern. Conf., The Netherlands, 195-204.
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• [3] Moorsel, P.A., & Wolter, K. (2004). Optimal restart times for moments of completion time. IEEE Proceedings Software, 151(5), 219-223.
• [4] Marcel, F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Massachusetts: The Johns Hopkins University Press.
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• [6] Gelenbe, E. (1993). G-networks with triggered customer movement. Journal of Applied Probability, 30(3), 742-748.
• [7] Gelenbe, E. (1992). Stability of G-networks. Probability in the Engineering and Informational Sciences, 6(3), 271-276.
• [8] Chao, X., & Miyazawa, M., & Pinedo, M. (2001). Queueing networks customers, signals and product form solutions. Journal of Applied Mathematics and Stochastic Analysis, 14(2), 421-426.
• [9] Harrison, P.G. (2004). Compositional reversed Markov processes, with applications to G-networks. Perform. Eval., 57(3), 379-408.
• [10] Harrison, P.G. (2003). Turning back time in Markovian process algebra. Theoretical Computer Science, 290(3), 1947-1986.
• [11] Gelenbe, E. (1993). G-networks with instantaneous customer movement. Journal of Applied Probability, 30(3), 742-748.
• [12] Gelenbe, E. (1993). G-networks with signals and batch removal. Probability in the Engineering and Informational Sciences, 7(3), 335-342.
• [13] Gelenbe, E., & Fourneau, J.-M. (2002). G-networks with resets. Perform. Eval., 49(1), 179-191.
• [14] Thu-Ha, Dao-Thi, Fourneau, J.-M., & Minh-Anh (2011). Tran G-networks with synchronized arrivals. Perform. Eval., 68(4), 309-319.
• [15] Thu-Ha Dao Thi, Fourneau, J.-M., & Minh-Anh (2010). Tran networks of symmetric multi-class queues with signals changing classes. Analytical and Stochastic Modeling Techniques and Applications: Proc. 17th International Conference, Cardiff, Proceedings. Berlin: Ed. Khalid Al-Begain, 72-86.
• [16] Gelenbe, E., & Fourneau, J.M. (2002). G-Networks with resets. Performance Evaluation, 49, 179-191.
• [17] Gelenbe, E., & Fourneau, J.M. (2004). Flow equivalence and stochastic equivalence in G-networks. Computational Management Science, 1(2), 179-192.
• [18] Matalytski, M., & Kopats, D. (2017). Finding expected revenues in G-Network with signals and customers batch removal. Probability in the Engineering and Informational Sciences, 31(4), 561-575.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).