Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-12T04:47:48.708Z Has data issue: false hasContentIssue false
  • English
  • Français

FINDING EXPECTED REVENUES IN G-NETWORK WITH MULTIPLE CLASSES OF POSITIVE AND NEGATIVE CUSTOMERS

Published online by Cambridge University Press:  14 February 2018

Mikhail Matalytski
Mikhail Matalytski*
Affiliation:
Institute of Mathematics, Czestochowa University of Technology, Czestochowa, Poland E-mail: m.matalytski@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Investigation of the G-network with multiple classes of positive and negative customers has been carried out in the article. The purpose of the investigation is to analyze such a network at a transient regime, finding expected revenues in the network systems depending on time. A negative customer arriving to the system and destroys a positive customer of its class. Streams of positive and negative customers arriving to each of the network systems are independent. Services of positive customers of all types occur in accordance with a random selection of them for service. For the expected revenues, a system of Kolmogorov's difference-differential equations has been derived. A method for their finding is proposed. It is based on the use of a modified method of successive approximations, combined with the method of series. A model example illustrating the finding of time-dependent expected revenues of network systems has been calculated, which shows that the expected revenues of network systems can be either increasing or decreasing time functions. The obtained results can be applied in forecasting losses in information and telecommunication systems and networks from the penetration of computer viruses into it and conducting computer attacks.

Keywords

applied probabilityqueueing theory
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 33 , Issue 1 , January 2019 , pp. 105 - 120
Copyright
Copyright © Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Crabil, T. (1972). Optimal control of a service facility with variable exponential service times and constant arrival rate. Manage. Sci. (18): 560–566.Google Scholar
2.Foschini, G. (1977). On heavy traffic diffusion analysis and dynamic routing in packet switched networks. Comput. Perform. (10): 499–514.Google Scholar
3.Stidham, S., Weber, R. (1993). A survey of Markov decision models for control of networks of queue. Queueing Syst. (3): 291–314.Google Scholar
4.Matalytski, M., Pankov, A. (2003). Analysis of the stochastic model of the changing of incomes in the open banking network. Comput. Sci. 3(5): 19–29.Google Scholar
5.Matalytski, M., Pankov, A. (2003). Incomes probabilistic models of the banking network. Sci. Res. Inst. Math. Comput. Sci. Czestochowa Univ. Technol. 1(2): 99–104.Google Scholar
6.Matalytski, M. (2009). On some results in analysis and optimization of Markov networks with incomes and their application. Autom. Remote Control 70(10): 1683–1697.Google Scholar
7.Howard, R. (1964). Dynamic programming and Markov processes. Moskow. Sov. Radio.Google Scholar
8.Gelenbe, E., Glynn, P., Sigman, K. (1991). Queues with negative arrivals. J. Appl. Probab. 28: 245–250.Google Scholar
9.Gelenbe, E. (1993). G-Networks with triggered customer movement. J. Appl. Probab. 30(3): 742–748.Google Scholar
10.Gelenbe, E. (1993). G-Networks with signals and batch removal. Probab. Eng. Inf. Sci. 7: 335–342.Google Scholar
11.Gelenbe, E. (1991). Product form queueing networks with negative and positive customers. J. Appl. Probab. 28: 656–663.Google Scholar
12.Matalytski, M. (2015). Analysis and forecasting of expected incomes in Markov networks with bounded waiting time for claims. Autom. Remote Control (6): 1005–1017.Google Scholar
13.Matalytski, M. (2015). Analysis and forecasting of expected incomes in Markov networks with unreliable servicing systems. Autom. Remote Control (12): 2179–2189.Google Scholar
14.Matalytski, M. (2017). Forecasting anticipated incomes in the Markov networks with positive and negative customers. Autom. Remote Control 78(5): 815–825.Google Scholar
15.Gelenbe, E., Schassberger, R. (1992). Stability of G-networks. Probab. Eng. Inf. Sci. 6(1): 271–276.Google Scholar
16.Gelenbe, E. (1994). G-networks: a unifying model for neural and queueing networks. Ann. Oper. Res. 48: 433–461.Google Scholar
17.Fourneau, J.N., Gelenbe, E., Suros, R. (1996). G-networks with multiple classes of negative and positive customers. Theor. Comput. Sci. 155: 141–156.Google Scholar
18.Gelenbe, E., Labed, A. (1998). G-networks with multiple classes of signals and positive customers. Eur. J. Oper. Res. 108(2): 293–305.Google Scholar
19.Matalytski, M. (2017). Analysis of G-network with multiple classes of customers at transient behavior. Probab. Eng. Inf. Sci.Google Scholar
20.Matalytski, M. (2017). Finding non-stationary state probabilities of G-network with signal and customers batch removal. Probab. Eng. Inf. Sci. 31(4): 396–412.Google Scholar
21.Korobeinik, Yu.F. (1970). Differential equations of infinite order and infinite systems of differential equations. Proc. USSR Acad. Sci. Math. Series 34(4): 881–922.Google Scholar
22.Valeev, K.G., Zhautykov, O.A. (1974). Infinite systems of differential equations. Alma-Ata: Sci. 415 p.Google Scholar
23.Matalytski, M., Naumenko, V. (2015). Simulation modeling of HM-networks with consideration of positive and negative messages. J. Appl. Math. Comput. Mech. 14(2): 49–60.Google Scholar