Tight Analysis of Priority Queuing for Egress Traffic

  • Jun Kawahara
  • Koji M. Kobayashi
  • Tomotaka Maeda
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8881)


Recently, the problems of evaluating performances of switches and routers have been formulated as online problems, and a great amount of results have been presented. In this paper, we focus on managing outgoing packets (called egress traffic) on switches that support Quality of Service (QoS), and analyze the performance of one of the most fundamental scheduling policies Priority Queuing (\(PQ\)) using competitive analysis. We formulate the problem of managing egress queues as follows: An output interface is equipped with \(m\) queues, each of which has a buffer of size \(B\). The size of a packet is unit, and each buffer can store up to \(B\) packets simultaneously. Each packet is associated with one of \(m\) priority values \(\alpha _{j}\) (\(1 \le j \le m\)), where \(\alpha _{1} \le \alpha _{2} \le \cdots \le \alpha _{m}\), \(\alpha _{1} = 1\), and \(\alpha _{m} = \alpha \) and the task of an online algorithm is to select one of \(m\) queues at each scheduling step. The purpose of this problem is to maximize the sum of the values of the scheduled packets.

For any \(B\) and any \(m\), we show that the competitive ratio of \(PQ\) is exactly \(2 - \min _{x \in [1, m-1] } \{ \frac{ \alpha _{x+1} }{ \sum _{j = 1}^{x+1} \alpha _{j} } \}\). That is, we conduct a complete analysis of the performance of \(PQ\) using worst case analysis. Moreover, we show that no deterministic online algorithm can have a competitive ratio smaller than \(1 + \frac{ \alpha ^3 + \alpha ^2 + \alpha }{ \alpha ^4 + 4 \alpha ^3 + 3 \alpha ^2 + 4 \alpha + 1 }\).



We would like to deeply thank Associate Professor Shuichi Miyazaki for a lot of advice on an earlier version of this paper. This work was supported by JSPS KAKENHI Grant Number 26730008 and Cyber Physical System Integrated IT Platform project.


  1. 1.
    Aiello, W., Mansour, Y., Rajagopolan, S., Rosén, A.: Competitive queue policies for differentiated services. J. Algorithms 55(2), 113–141 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Al-Bawani, K., Souza, A.: Buffer overflow management with class segregation. Inf. Process. Lett. 113(4), 145–150 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Albers, S., Jacobs, T.: An experimental study of new and known online packet buffering algorithms. Algorithmica 57(4), 725–746 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Albers, S., Schmidt, M.: On the performance of greedy algorithms in packet buffering. SIAM J. Comput. 35(2), 278–304 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Andelman, N.: Randomized queue management for DiffServ. In: Proceedings of the 17th ACM Symposium on Parallel Algorithms and Architectures, pp. 1–10 (2005)Google Scholar
  6. 6.
    Andelman, N., Mansour, Y.: Competitive management of non-preemptive queues with multiple values. In: Fich, F.E. (ed.) DISC 2003. LNCS, vol. 2848, pp. 166–180. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Andelman, N., Mansour, Y., Zhu, A.: Competitive queueing policies for QoS switches. In: Proceedings of the 14th ACM-SIAM Symposium on Discrete Algorithms, pp. 761–770 (2003)Google Scholar
  8. 8.
    Azar, Y., Litichevskey, A.: Maximizing throughput in multi-queue switches. Algorithmica 45(1), 69–90 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Azar, Y., Richter, Y.: Management of multi-queue switches in QoS networks. Algorithmica 43(1–2), 81–96 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Azar, Y., Richter, Y.: An improved algorithm for CIOQ switches. ACM Trans. Algorithms 2(2), 282–295 (2006)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Azar, Y., Richter, Y.: The zero-one principle for switching networks. In: Proceedings of the 36th ACM Symposium on Theory of Computing, pp. 64–71 (2004)Google Scholar
  12. 12.
    Bar-Noy, A., Freund, A., Landa, S., Naor, J.: Competitive on-line switching policies. Algorithmica 36(3), 225–247 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Bienkowski, M., Mądry, A.: Geometric aspects of online packet buffering: an optimal randomized algorithm for two buffers. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 252–263. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Bienkowski, M.: An optimal lower bound for buffer management in multi-queue switches. Algorithmica 68(2), 426–447 (2014)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Blanke, S., Black, D., Carlson, M., Davies, E., Wang, Z., Weiss, W.: An architecture for differentiated services. RFC2475, IETF, December 1998Google Scholar
  16. 16.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  17. 17.
  18. 18.
    Cisco Systems Inc. Cisco Catalyst 2955 series switches data sheets (2014).
  19. 19.
    Cisco Systems Inc. Cisco Catalyst 6500 series switches data sheets (2014).
  20. 20.
    Demers, A., Keshav, S., Shenker, S.: Analysis and simulation of a fair queueing algorithm. J. Internetworking Res. Exper. 1(1), 3–26 (1990)Google Scholar
  21. 21.
    Englert, M., Westermann, M.: Lower and upper bounds on FIFO buffer management in QoS switches. Algorithmica 53(4), 523–548 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Goldwasser, M.: A survey of buffer management policies for packet switches. ACM SIGACT News 41(1), 100–128 (2010)CrossRefGoogle Scholar
  23. 23.
    Hahne, E., Kesselman, A., Mansour, Y.: Competitive buffer management for shared-memory switches. In: Proceedings of the 13th ACM Symposium on Parallel Algorithms and Architectures, pp. 53–58 (2001)Google Scholar
  24. 24.
    Katevenis, M., Sidiropopulos, S., Courcoubetis, C.: Weighted round-robin cell multiplexing in a general-purpose ATM switch chip. IEEE J. Sel. Area Commun. 9(8), 1265–1279 (1991)CrossRefGoogle Scholar
  25. 25.
    Kawahara, J., Kobayashi, K.M., Maeda, T.: Tight analysis of priority queuing policy for egress traffic (2012). arXiv:1207.5959 [cs.DS]
  26. 26.
    Kesselman, A., Lotker, Z., Mansour, Y., Patt-Shamir, B., Schieber, B., Sviridenko, M.: Buffer overflow management in QoS switches. SIAM J. Comput. 33(3), 563–583 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kesselman, A., Mansour, Y.: Harmonic buffer management policy for shared memory switches. Theoret. Comput. Sci. 324(2–3), 161–182 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Kesselman, A., Mansour, Y., van Stee, R.: Improved competitive guarantees for QoS buffering. Algorithimica 43(1–2), 63–80 (2005)CrossRefzbMATHGoogle Scholar
  29. 29.
    Kesselman, A., Rosén, A.: Scheduling policies for CIOQ switches. J. Algorithms 60(1), 60–83 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Kesselman, A., Rosén, A.: Controlling CIOQ switches with priority queuing and in multistage interconnection networks. J. Interconnection Netw. 9(1/2), 53–72 (2008)CrossRefGoogle Scholar
  31. 31.
    Kesselman, A., Kogan, K., Segal, M.: Packet mode and QoS algorithms for buffered crossbar switches with FIFO queuing. Distrib. Comput. 23(3), 163–175 (2010)CrossRefzbMATHGoogle Scholar
  32. 32.
    Kesselman, A., Kogan, K., Segal, M.: Best effort and priority queuing policies for buffered crossbar switches. Chicago J. Theor. Sci. 1–14 (2012)Google Scholar
  33. 33.
    Kesselman, A., Kogan, K., Segal, M.: Improved competitive performance bounds for CIOQ switches. Algorithmica 63(1–2), 411–424 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Kobayashi, K., Miyazaki, S., Okabe, Y.: A tight bound on online buffer management for two-port shared-memory switches. In: Proceedings of the 19th ACM Symposium on Parallel Algorithms and Architectures, pp. 358–364 (2007)Google Scholar
  35. 35.
    Kobayashi, K., Miyazaki, S., Okabe, Y.: A tight upper bound on online buffer management for multi-queue switches with bicodal buffers. IEICE Trans. Fund. Electron. Commun. Comput. Sci. E91–D(12), 2757–2769 (2008)Google Scholar
  36. 36.
    Kobayashi, K., Miyazaki, S., Okabe, Y.: Competitive buffer management for multi-queue switches in QoS networks using packet buffering algorithms. In: Proceedings of the 21st ACM Symposium on Parallel Algorithms and Architectures, pp. 328–336 (2009)Google Scholar
  37. 37.
    Kogan, K., Lopez-Ortiz, A., Nikolenko, S., Sirotkin, A.: Multi-queued network processors for packets with heterogeneous processing requirements. In: Proceedings of the 5th International Conference on Communication Systems and Networks, pp. 1–10 (2013)Google Scholar
  38. 38.
    Fleischer, R., Koga, H.: Balanced scheduling toward loss-free packet queuing and delay fairness. Algorithmica 38(2), 363–376 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Sleator, D., Tarjan, R.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Jun Kawahara
    • 1
  • Koji M. Kobayashi
    • 2
  • Tomotaka Maeda
    • 3
  1. 1.Nara Institute of Science and TechnologyNaraJapan
  2. 2.National Institute of InformaticsChiyodaJapan
  3. 3.Academic Center for Computing and Media StudiesKyoto UniversityKyotoJapan

Personalised recommendations