Kybernetika 49 no. 6, 855-867, 2013

On the queue-size distribution in the multi-server system with bounded capacity and packet dropping

Oleg Tikhonenko and Wojciech M. Kempa

Abstract:

A multi-server $M/M/n$-type queueing system with a bounded total volume and finite queue size is considered. An AQM algorithm with the "accepting'' function is being used to control the arrival process of incoming packets. The stationary queue-size distribution and the loss probability are derived. Numerical examples illustrating theoretical results are attached as well.

Keywords:

AQM algorithms, loss probability, multi-server queueing system, queue-size distribution

Classification:

90B22, 60K25

paper.pdf

References:

  1. S. Athuraliya, S. H. Low, V. H. Li and Y. Qinghe: REM: active queue management. IEEE Network 15 (2001), 3, 48-53.   CrossRef
  2. J. Aweya, M. Ouellette, D. Y. Montuno and A. Chapman: A control theoretic approach to Active Queue Management.    CrossRef
  3. P. P. Bocharov, C. D\rq{Apice}, A. V. Pechinkin and S. Salerno: Queueing Theory. VSP, Utrecht-Boston, 2004.   CrossRef
  4. T. Bonald, M. May and J. Ch. Bolot: Analytic evaluation of RED performance. In: Proc. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies 3 (2000), pp. 1415-1424.   CrossRef
  5. L. Chrost, A. Brachman and A. Chydzinski: On the performance of AQM algoritms with small buffers. Comput. Network CCIS 39 (2009), 168-173.   CrossRef
  6. A. Chydzinski: Towards a stable AQM via dropping function shaping. In: Proc. Ninth International Conference on Networks (ICN) (2010), pp. 93-97.   CrossRef
  7. A. Chydzinski and L. Chrost: Analysis of AQM queues with queue size based packet dropping. Int. J. Appl. Math. Comput. Sci. 21 (2011), 3, 567-577.   CrossRef
  8. A. Chydzinski: Optimization problems in the theory of queues with dropping functions. HET-NETs (2011), 121-132.   CrossRef
  9. W. Feller: Introduction to Probability Theory and Its Applications. Wiley, 1971.   CrossRef
  10. S. Floyd and V. Jacobson: Random early detection gateways for congestion avoidance. IEEE ACM T. Network 1 (1993), 4, 397-412.   CrossRef
  11. S. Floyd: Recommendations on using the gentle variant of RED. \url{http://www.aciri.org/floyd/red/gentle.html}, March 2000.   CrossRef
  12. S. Floyd: Adaptive RED: an algorithm for increasing the robustness of RED's Active Queue Management. \url{http://www.aciri.org/floyd/papers/adaptiveRed.pdf}   CrossRef
  13. W. Hao and Y. Wei: An extended $GI^{X}/M/1/N$ queueing model for evaluating the performance of AQM algorithms with aggregate traffic. Lect. Notes Comput. Sci. 3619 (2005), 395-404.   CrossRef
  14. W. M. Kempa: On main characteristics of the $M/M/1/N$ queue with single and batch arrivals and the queue size controlled by AQM algorithms. Kybernetika 47 (2011), 6, 930-943.   CrossRef
  15. W. M. Kempa: A direct approach to transient queue-size distribution in a finite-buffer queue with AQM. Appl. Math. Inform. Sci. 7 (2013), 3, 909-915.   CrossRef
  16. S. Liu, T. Basar and R. Srikant: Exponential RED: A stabilizing AQM scheme for low- and high-speed TCP protocols. IEEE/ACM Trans. Newtorking 13 (2005), 5, 1068-1081.   CrossRef
  17. V. Rosolen, O. Bonaventure and G. Leduc: A RED discard strategy for ATM networks and its performance evaluation with TCP/IP traffic. Comput. Commun. Rew. 29 (1999), 3, 23-43.   CrossRef
  18. L. Sun and L. Wang: A novel RED scheme with preferential dynamic threshold deployment. In: Computational Intelligence and Security Workshops 2007, pp. 854-857.   CrossRef
  19. S. Suresh and O. Gol: Congestion management of self similar IP traffic - application of the RED scheme. In: Wireless and Optical Communications Networks, Second IFIP International Conference 2005, pp. 372-376.   CrossRef
  20. O. Tikhonenko: Queueing systems of a random length demands with restrictions. Automat. Remote Control 52 (1991), 10, 1431-1437.   CrossRef
  21. O. Tikhonenko: Generalized Erlang problem for service systems with finite total capacity. Probl. Inform. Transmission 41 (2005), 3, 243-253.   CrossRef
  22. O. Tikhonenko and W. M. Kempa: The generalization of AQM algorithms for queueing systems with bounded capacity. Lect. Notes Comput. Sci. 7204 (2012), 242-251.   CrossRef
  23. N. Xiong, Y. Yang, X. Defago and Y. He: LRC-RED: A self-tuning robust and adaptive AQM scheme. In: Sixth International Conference on Parallel and Distributed Computing Applications and Technologies 2005, pp. 655-659.   CrossRef
  24. K. Zhou, K. L. Yeung and V. O. K. Li: Nonlinear RED: A simple yet efficient active queue management scheme. Comput. Networks 50 (2006), 18, 3784-3794.   CrossRef