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G-NETWORKS OF UNRELIABLE NODES

Published online by Cambridge University Press:  19 May 2016

Jean-Michel Fourneau
Jean-Michel Fourneau*
Affiliation:
DAVID, Université de Versailles-St Quentin, 45, Av. des États-Unis, 78035 Versailles, France E-mail: jmf@prism.uvsq.fr, Jean-Michel.Fourneau@uvsq.fr
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Abstract

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We study G-networks with positive and negative customers and signals. We consider two types of signals: they can make a subnetwork of queues operational or down. As signals are sent by queues after a customer service completion, one can model the availability of a sub-network of queues controlled by another network of queues. We prove that under classical assumptions for G-networks and assumptions on the rerouting probabilities when a subnetwork is not operational, the steady-state distribution, if it exists, has a product form steady state distribution. Some examples are given.

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 3: Erol Gelenbe's 70th Birthday , July 2016 , pp. 361 - 378
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Cambridge University Press 2016

References

1.Balsamo, S., Harrison, P.G. & Marin, A. (2010). A unifying approach to product-forms in networks with finite capacity constraints. In Misra, Vishal, Barford, Paul, & Squillante, Mark S., (eds.), SIGMETRICS 2010, Proceedings of the 2010 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, New York: ACM, pp. 25–36.CrossRefGoogle Scholar
2.Chao, X., Miyazawa, M. & Pinedo, M. (1999). Queueing networks customers, signals and product form solutions. Chichester: John Wiley & Sons.Google Scholar
3.Dao-Thi, T.-H., Fourneau, J.-M. & Tran, M.-A. (2010). Networks of symmetric multi-class queues with signals changing classes. In Al-Begain, Khalid, Fiems, Dieter, & Knottenbelt, W.J. (eds.), Analytical and Stochastic Modeling Techniques and Applications, 17th International Conference, ASMTA 2010, , vol. 6148 of Lecture Notes in Computer Science, Cardiff, UK: Springer, pp. 72–86.CrossRefGoogle Scholar
4.Dao-Thi, T.-H., Fourneau, J.-M. & Tran, M.-A. (2013). Network of queues with inert customers and signals. In 7th International Conference on Performance Evaluation Methodologies and Tools, ValueTools ’13, Italy: ICST/ACM, pp. 155–164.Google Scholar
5.Fourneau, J.-M. (1991). Computing the steady-state distribution of networks with positive and negative customers. In 13th IMACS World Congress on Computation and Applied Mathematics, Dublin.Google Scholar
6.Fourneau, J.-M. & Gelenbe, E. (2004). Flow equivalence and stochastic equivalence in g-networks. Computational Management Science 1(2): 179–192.CrossRefGoogle Scholar
7.Fourneau, J.-M., Kloul, L. & Quessette, F. (1995). Multiple class G-Networks with jumps back to zero. In MASCOTS ’95: Proceedings of the 3rd International Workshop on Modeling, Analysis, & Simulation of Computer and Telecommunication Systems, Washington, DC, USA: IEEE Computer Society, pp. 28–32.CrossRefGoogle Scholar
8.Fourneau, J.-M., Kloul, L. & Quessette, F. (2000). Multiple class G-networks with iterated deletions. Performance Evaluation 42(1): 1–20.CrossRefGoogle Scholar
9.Gelenbe, E. (1991). Product-form queuing networks with negative and positive customers. Journal of Applied Probability 28: 656–663.CrossRefGoogle Scholar
10.Gelenbe, E. (1993). G-networks with instantaneous customer movement. Journal of Applied Probability 30(3): 742–748.CrossRefGoogle Scholar
11.Gelenbe, E. (1994). G-networks: An unifying model for queuing networks and neural networks. Annals of Operations Research 48(1–4): 433–461.CrossRefGoogle Scholar
12.Gelenbe, E. & Fourneau, J.-M. (1999). Random neural networks with multiple classes of signals. Neural Computation 11(4): 953–963.CrossRefGoogle ScholarPubMed
13.Gelenbe, E. & Fourneau, J.-M. (2002). G-networks with resets. Performance Evaluation 49(1–4): 179–191.CrossRefGoogle Scholar
14.Gelenbe, E. & Labed, A. (1998). G-networks with multiple classes of signals and positive customers. European Journal of Operations Research 108: 293–305.CrossRefGoogle Scholar
15.Gelenbe, E., Lent, R. & Xu, Z. (2001). Design and performance of cognitive packet networks. Performance Evaluation 46(2–3): 155–176.CrossRefGoogle Scholar
16.Gelenbe, E. & Mitrani, I. (2010). Analysis and synthesis of computer systems. London: Imperial College Press.CrossRefGoogle Scholar
17.Harrison, P.G. (2004). Compositional reversed Markov processes, with applications to G-networks. Performance Evaluation 57(3): 379–408.CrossRefGoogle Scholar
18.Harrison, P.G. (2003). Turning back time in Markovian process algebra. Theoretical Computer Science 290(3): 1947–1986.CrossRefGoogle Scholar
19.Hernandez, M. & Fourneau, J-M. (1993). Modelling defective parts in a fow system using G-Networks. In International Conference on performability, Mont Saint-Michel.Google Scholar
20.Kelly, F.P. (1987). Reversibility and stochastic networks. Chichester: Wiley.Google Scholar
21.Mohamed, S., Rubino, G. & Varela, M. (2004). Performance evaluation of real-time speech through a packet network: a random neural networks-based approach. Performance Evaluation 57(2): 141–161.CrossRefGoogle Scholar
22.Sauer, C. & Daduna, H. (2003). Availability formulas and performance measures for separable degradable networks. Economic Quality Control 18(2): 165–194.CrossRefGoogle Scholar
23.Trivedi, K.S. (2002). Probability and Statistic with Reliability, Queueing and Computer Science Applications, 2nd ed.New York: Wiley.Google Scholar