Advances in Applied Probability

Distribution of the number of retransmissions of bounded documents

Predrag R. Jelenković and Evangelia D. Skiani

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Retransmission-based failure recovery represents a primary approach in existing communication networks that guarantees data delivery in the presence of channel failures. Recent work has shown that, when data sizes have infinite support, retransmissions can cause long (-tailed) delays even if all traffic and network characteristics are light-tailed. In this paper we investigate the practically important case of bounded data units 0 ≤ L b b under the condition that the hazard functions of the distributions of data sizes and channel statistics are proportional. To this end, we provide an explicit and uniform characterization of the entire body of the retransmission distribution P[ N b > n] in both n and b. Our main discovery is that this distribution can be represented as the product of a power law and gamma distribution. This rigorous approximation clearly demonstrates the coupling of a power law distribution, dominating the main body, and the gamma distribution, determining the exponential tail. Our results are validated via simulation experiments and can be useful for designing retransmission-based systems with the required performance characteristics. From a broader perspective, this study applies to any other system, e.g. computing, where restart mechanisms are employed after a job processing failure.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 2 (2015), 425-449.

Dates
First available in Project Euclid: 25 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1435236982

Digital Object Identifier
doi:10.1239/aap/1435236982

Mathematical Reviews number (MathSciNet)
MR3360384

Zentralblatt MATH identifier
1337.60253

Subjects
Primary: 60F99: None of the above, but in this section
Secondary: 60F10: Large deviations 60G99: None of the above, but in this section

Keywords
Retransmission restart channel with failure truncated distribution power law gamma distribution heavy-tailed distribution light-tailed distribution

Citation

Jelenković, Predrag R.; Skiani, Evangelia D. Distribution of the number of retransmissions of bounded documents. Adv. in Appl. Probab. 47 (2015), no. 2, 425--449. doi:10.1239/aap/1435236982. https://projecteuclid.org/euclid.aap/1435236982


Export citation

References

  • Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, D.C.
  • Asmussen, S., Lipsky, L. and Thompson, S. (2014). Failure Recovery in Computing and Data Transmission. In Analytic and Stochastic Modelling Techniques and Applications. (Lecture Notes Comput. Sci. 8499). Springer, Berlin, pp. 253–272.
  • Asmussen, S. et al. (2008). Asymptotic behavior of total times for jobs that must start over if a failure occurs. Math. Operat. Res. 33, 932–944.
  • Bertsekas, D. P. and Gallager, R. (1992). Data Networks, 2nd edn. Prentice Hall, London.
  • Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.
  • Fiorini, P. M., Sheahan, R. and Lipsky, L. (2005). On unreliable computing systems when heavy-tails appear as a result of the recovery procedure. ACM SIGMETRICS Performance Evaluation Rev. 33, 15–17.
  • Jelenković, P. R. and Olvera-Cravioto, M. (2012). Implicit renewal theorem for trees with general weights. Stoch. Process. Appl. 122, 3209–3238.
  • Jelenković, P. R. and Skiani, E. D. (2012). Uniform approximation of the distribution for the number of retransmissions of bounded documents. In Proc. 12th ACM SIGMETRICS/PERFORMANCE Joint International Conference on Measurement and Modeling of Computer Systems, ACM, New York, pp. 101–112. June 2012.
  • Jelenković, P. R. and Skiani, E. D. (2012). Distribution of the number of retransmissions of bounded documents. Extended version. Available at http://arxiv.org/abs/1210.8421v1.
  • Jelenković, P. R. and Tan, J. (2007). Are end-to-end acknowlegements causing power law delays in large multi-hop networks? In 14th Informs Applied Probability Conference, APS, pp. 9–11.
  • Jelenković, P. R. and Tan, J. (2007). Can retransmissions of superexponential documents cause subexponential delays? In Proc. 26th IEEE INFOCOM'07, IEEE, pp. 892–900.
  • Jelenković, P. R. and Tan, J. (2007). Is ALOHA causing power law delays? In Managing Traffic Performance in Converged Networks (Lecture Notes Comput. Sci. 4516). Springer, Berlin, pp. 1149–1160.
  • Jelenković, P. R. and Tan, J. (2008). Dynamic packet fragmentation for wireless channels with failures. In Proc. 9th ACM International Symposium on Mobile Ad Hoc Networking and Computing, ACM, New York, pp. 73–82.
  • Jelenković, P. R. and Tan, J. (2009). Stability of finite population ALOHA with variable packets. Preprint. Available at http://arxiv.org/abs/0902.4481v2.
  • Jelenković, P. R. and Tan, J. (2010). Modulated branching processes, origins of power laws, and queueing duality. Math. Operat. Res. 35, 807–829.
  • Jelenković, P. R. and Tan, J. (2013). Characterizing heavy-tailed distributions induced by retransmissions. Adv. Appl. Prob. 45, 106–138. Extended version available at http://arxiv.org/pdf/0709.1138.pdf
  • Nair, J. et al. (2010). File fragmentation over an unreliable channel. In Proc. IEEE INFOCOM'10, IEEE, pp. 965–973.
  • Ross, S. M. (2002). A First Course in Probability, 6th edn. Prentice Hall, Upper Saddle River, NJ.
  • Sheahan, R., Lipsky, L., Fiorini, P. M. and Asmussen, S. (2006). On the completion time distribution for tasks that must restart from the beginning if a failure occurs. ACM SIGMETRICS Performance Evaluation Rev. 34, 24–26.
  • Tan, J. and Shroff, N. B. (2010). Transition from heavy to light tails in retransmission durations. In Proc. IEEE INFOCOM'10, IEEE, pp. 1334–1342.