Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-27T18:56:27.744Z Has data issue: false hasContentIssue false
  • English
  • Français

RETRIAL NETWORKS WITH FINITE BUFFERS AND THEIR APPLICATION TO INTERNET DATA TRAFFIC

Published online by Cambridge University Press:  25 September 2008

Kostia Avrachenkov  and
Uri Yechiali
Kostia Avrachenkov
Affiliation:
INRIA, Sophia Antipolis, France E-mail: k.avrachenkov@sophia.inria.fr
Uri Yechiali
Affiliation:
Tel Aviv University, Tel Aviv, Israel E-mail: uriy@post.tau.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

Data on the Internet is sent by packets that go through a network of routers. A router drops packets either when its buffer is full or when it uses the Active Queue Management. Currently, the majority of the Internet routers use a simple Drop Tail strategy. The rate at which a user injects the data into the network is determined by transmission control protocol (TCP). However, most connections in the Internet consist only of few packets, and TCP does not really have an opportunity to adjust the sending rate. Thus, the data flow generated by short TCP connections appears to be some uncontrolled stochastic process. In the present work we try to describe the interaction of the data flow generated by short TCP connections with a network of finite buffers. The framework of retrial queues and networks seems to be an adequate approach for this problem. The effect of packet retransmission becomes essential when the network congestion level is high. We consider several benchmark retrial network models. In some particular cases, an explicit analytic solution is possible. If the analytic solution is not available or too entangled, we suggest using a fixed-point approximation scheme. In particular, we consider a network of one or two tandem M/M/1/K-type queues with blocking and with an M/M/1/∞-type retrial (orbit) queue. We explicitly solve the models with particular choices of K, derive stability conditions for K≥1, and present several graphs based on numerical results.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 4 , October 2008 , pp. 519 - 536
Copyright
Copyright © Cambridge University Press 2008

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.Allman, M., Paxson, V. & Stevens, W. (1999). TCP congestion control. RFC 2581. Available at http://www.ietf.org/rfc/rfc2581.txt.Google Scholar
2.Artalejo, J.R. (1999). Accessible bibliography on retrial queues. Mathematical and Computer Modelling 30: 223–233.CrossRefGoogle Scholar
3.Artalejo, J.R. & Economou, A. (2005). On the non-existence of product-form solutions for queueing networks with retrials, Electronic Modeling 27: 13–19.Google Scholar
4.Artalejo, J.R. & Gómez-Corral, A. (1997). Steady state solution of a single-server queue with linear request repeated. Journal of Applied Probability 34: 223–233.CrossRefGoogle Scholar
5.Artalejo, J.R., Gómez-Corral, A. & Neuts, M.F. (2001). Analysis of multiserver queues with constant retrial rate. European Journal of Operational Research 135: 569–581.CrossRefGoogle Scholar
6.Cohen, J.W. (1957). Basic problems of telephone traffic theory and the influence of repeated calls. Philips Telecommunication Review 18: 49–100.Google Scholar
7.Dudin, A.N. & Klimenok, V.I. (1999). Multi-dimensional quasi-Toeplitz Markov chains. Journal of Applied Mathematics and Stochastic Analysis 12: 393–415.CrossRefGoogle Scholar
8.Elaydi, S.N. (1999). An introduction to difference equations, 2nd ed.New York: Springer-Verlag.CrossRefGoogle Scholar
9.Falin, G.I. (1990). A survey of retrial queues. Queueing Systems 7: 127–167.CrossRefGoogle Scholar
10.Falin, G.I. & Gómez-Corral, A. (2000). On a bivariate Markov process arising in the theory of single-server retrial queues. Statistica Neerlandica 54: 67–78.CrossRefGoogle Scholar
11.Falin, G.I. & Templeton, J.G.C. (1997). Retrial queues. Boca Raton, FL: CRC Press.Google Scholar
12.Fayolle, G. (1986) A simple telephone exchange with delayed feedbacks. In: Boxma, O.J., Cohen, J.W. and Tijms, H.C. (eds.), Teletraffic analysis and computer performance evaluation, pp. 245–253. Amsterdam: Elsevier Science Publishers.Google Scholar
13.Floyd, S. & Jacobson, V. (1993). Random early detection gateways for congestion avoidance. IEEE/ACM Transactions on Networking 1: 397–413.CrossRefGoogle Scholar
14.Jacobson, V. (1988). Congestion avoidance and control. In Proceedings of ACM SIGCOMM’88, pp. 314–329.CrossRefGoogle Scholar
15.Kelly, F.P., Maulloo, A. & Tan, D. (1998). Rate control for communication networks: Shadow prices, proportional fairness and stability. Journal of Operational Research Society 49: 237–252.CrossRefGoogle Scholar
16.Kim, C.S., Klimenok, V.I. & Orlovsky, D.S. (2004). The BMAP/PH/N/N+R retrial queuing system with different disciplines of retrials. In Proceedings of ASMTA’04.Google Scholar
17.Kunniyur, S. & Srikant, R. (2000). End-to-end congestion control schemes: Utility functions, random losses, ECN marks. In Proceedings of IEEE INFOCOM’00. An extended version has appeared in IEEE/ACM Transactions on Networking 11: 689–702, 2003.Google Scholar
18.Low, S.H. & Lapsley, D.E. (1999). Optimization flow control I: Basic algorithm and convergence. IEEE/ACM Transactions on Networking 7: 861–874.CrossRefGoogle Scholar
19.Moutzoukis, E. & Langaris, C. (2001). Two queues in tandem with retrial customers. Probability in the Engineering and Informational Sciences 15: 311–325.Google Scholar
20.Neuts, M. (1989). Structured stochastic matrices of M/G/1 type and their applications. New York: Marcel Dekker.Google Scholar
21.Paxson, V. (1996). End-to-end routing behavior in the Internet. In Proceedings of ACM SIGCOMM’96, pp. 25–38. An extended version appeared in IEEE/ACM Transactions on Networking 5: 601–615, 1997.Google Scholar
22.Pourbabai, B. (1990). Tandem behavior of a telecommunication system with finite buffers and repeated calls. Queueing Systems 6: 89–108.Google Scholar
23.Ramalhoto, M.F. & Gómez-Corral, A. (1998). Some decomposition formulae for M/M/r/r+d queues with constant retrial rate. Communications in Statistics—Stochastic Models 14: 123–145.CrossRefGoogle Scholar
24.Takahara, G.K. (1996). Fixed point approximations for retrial networks. Probability in the Engineering and Informational Sciences 10: 243–259.CrossRefGoogle Scholar