Hostname: page-component-89b8bd64d-mmrw7 Total loading time: 0 Render date: 2026-05-06T17:59:36.970Z Has data issue: false hasContentIssue false
  • English
  • Français

PROCESSOR SHARING G-QUEUES WITH INERT CUSTOMERS AND CATASTROPHES: A MODEL FOR SERVER AGING AND REJUVENATION

Published online by Cambridge University Press:  24 April 2017

J.-M. Fourneau  and
Y. Ait El Majhoub
J.-M. Fourneau
Affiliation:
DAVID, UVSQ, Université Paris-Saclay, 45, Av. des États-Unis, 78035 Versailles, France E-mail: Jean-Michel.Fourneau@uvsq.fr
Y. Ait El Majhoub
Affiliation:
DAVID, UVSQ, Université Paris-Saclay, 45, Av. des États-Unis, 78035 Versailles, France
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider open networks of queues with Processor-Sharing discipline and signals. The signals deletes all the customers present in the queues and vanish instantaneously. The customers may be usual customers or inert customers. Inert customers do not receive service but the servers still try to share the service capacity between all the customers (inert or usual). Thus a part of the service capacity is wasted. We prove that such a model has a product-form steady-state distribution when the signal arrival rates are positive.

Keywords

agingcatastrophesG-networks with customers and signalsinert customersproduct – form steady-state distributionrejuvenation

Information

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 31 , Special Issue 4: G-Networks and their Applications , October 2017 , pp. 420 - 435
Copyright
Copyright © Cambridge University Press 2017 

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.)

Article purchase

Temporarily unavailable

References

1. Balsamo, S., Harrison, P.G., & Marin, A. (2010). An unifying approach to product-forms in networks with finite capacity constraints. In SIGMETRICS 2010, Proceedings of the 2010 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, USA. In Misra, V., Barford, P., & Squillante, M.S. (eds.), ACM, pp. 25–36.Google Scholar
2. Chao, X., Miyazawa, M., & Pinedo, M. (1999). Queueing networks customers, signals and product form solutions. Chichester, UK, John Wiley & Sons.Google Scholar
3. 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
4. Dudin, A.N. & Karolik, A.V. (2001). BMAP/SM/1 queue with Markovian input of disasters and non-instantaneous recovery. Performance Evaluation 45(1): 19–32.CrossRefGoogle 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., Ait El Mahjoub, Y., Quessette, F., & Vekris, D. (2016). Xborne 2016: A brief introduction. In Computer and Information Sciences – 31st International Symposium, ISCIS 2016, Krakow, Poland, vol. 659, Communications in Computer and Information Science. In Czachórski, T., Gelenbe, E., Grochla, K., & Lent, R. (eds.), Springer.Google 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, and Simulation of Computer and Telecommunication Systems, Washington, DC, USA, IEEE Computer Society, pp. 28–32 CrossRefGoogle Scholar
8. Fourneau, J.-M. & Quessette, F. (2006). Computing the steady-state distribution of G-networks with synchronized partial flushing. In Computer and Information Sciences – ISCIS 2006, 21th International Symposium, Istanbul, Turkey, volume 4263 of Lecture Notes in Computer Science. In Levi, A., Savas, E., Yenigün, H., Balcisoy, S., & Saygin, Y (eds.), Springer, pp. 887–896.Google Scholar
9. Garcia, C.D. & Zangwill, W.I. (1981). Pathways to solutions, fixed points, and equilibria. Englewood Cliffs, NJ: Prentice–Hall.Google Scholar
10. Gelenbe, E. (1991). Product-form queuing networks with negative and positive customers. Journal of Applied Probability 28: 656–663.Google Scholar
11. Gelenbe, E. (1993). G-networks with instantaneous customer movement. Journal of Applied Probability 30(3): 742–748.Google Scholar
12. Gelenbe, E. (1993). G-networks with signals and batch removal. Probability in the Engineering and Informational Sciences 7: 335–342.Google Scholar
13. Gelenbe, E. (1994). G-networks: An unifying model for queuing networks and neural networks. Annals of Operations Research 48(1–4): 433–461.Google Scholar
14. Gelenbe, E. (2014). Adaptive management of energy packets. In IEEE 38th Annual Computer Software and Applications Conference, COMPSAC Workshops, IEEE Computer Society, pp. 1–6.Google Scholar
15. Gelenbe, E. & Ceran, E.T. (2016). Energy packet networks with energy harvesting. IEEE Access 4: 1321–1331.Google Scholar
16. Gelenbe, E. & Fourneau, J.-M. (1999). Random neural networks with multiple classes of signals. Neural Computation 11(4): 953–963.Google Scholar
17. Gelenbe, E. & Fourneau, J.-M. (2002). G-networks with resets. Performance Evaluation 49(1–4): 179–191.Google Scholar
18. 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
19. Gelenbe, E., Lent, R. & Xu, Z. (2001). Design and performance of cognitive packet networks. Performance Evaluation 46(2–3): 155–176.Google Scholar
20. Gelenbe, E. & Mitrani, I. (2010). Analysis and synthesis of computer systems. London: Imperial College Press.Google Scholar
21. Gelenbe, E. & Schassberger, R. (1992). Stability of g-networks. Probability in the Engineering and Informational Sciences 6: 271–276.Google Scholar
22. Harrison, P.G. (2003). Turning back time in Markovian process algebra. Theoretical Computer Science 290(3): 1947–1986.Google Scholar
23. Harrison, P.G. (2004). Compositional reversed Markov processes, with applications to G-networks. Performance Evaluation 57(3): 379–408.CrossRefGoogle Scholar
24. Krishna Kumar, B. & Arivudainambi, D. (2000). Transient solution of an m/m/1 queue with catastrophes. Computers & Mathematics With Applications 40(10–11): 1233–1240.Google Scholar
25. 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