### Characterizing heavy-tailed distributions induced by retransmissions

#### Abstract

Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel availability dynamic is modeled as an independent and identically distributed sequence {A, Ai}i≥1 that is independent of L. During each period of time that the channel becomes available, say Ai, we attempt to transmit the data unit. If L <Ai, the transmission is considered successful; otherwise, we wait for the next available period Ai+1 and attempt to retransmit the data from the beginning. We investigate the asymptotic properties of the number of retransmissions N and the total transmission time T until the data is successfully transmitted. In the context of studying the completion times in systems with failures where jobs restart from the beginning, it was first recognized by Fiorini, Sheahan and Lipsky (2005) and Sheahan, Lipsky, Fiorini and Asmussen (2006) that this model results in power-law and, in general, heavy-tailed delays. The main objective of this paper is to uncover the detailed structure of this class of heavy-tailed distributions induced by retransmissions. More precisely, we study how the functional relationship ℙ[L>x]-1 ≈ Φ (ℙ[A>x]-1) impacts the distributions of N and T; the approximation ≈' will be appropriately defined in the paper based on the context. Depending on the growth rate of Φ(·), we discover several criticality points that separate classes of different functional behaviors of the distribution of N. For example, we show that if log(Φ(n)) is slowly varying then log(1/ℙ[N>n]) is essentially slowly varying as well. Interestingly, if log(Φ(n))$grows slower than e√(logn) then we have the asymptotic equivalence log(ℙ[N>n]) ≈ - log(Φ(n))$. However, if log(Φ(n)) grows faster than e√(logn), this asymptotic equivalence does not hold and admits a different functional form. Similarly, different types of distributional behavior are shown for moderately heavy tails (Weibull distributions) where log(ℙ[N>n]) ≈ -(logΦ(n))1/(β+1), assuming that log \Φ(n) ≈ nβ, as well as the nearly exponential ones of the form log(ℙ[N>n]) ≈ -n/(log n)1/γ, γ>0, when Φ(·) grows faster than two exponential scales log log (Φ(n)) ≈ nγ.

#### Article information

Source
Adv. in Appl. Probab., Volume 45, Number 1 (2013), 106-138.

Dates
First available in Project Euclid: 15 March 2013

https://projecteuclid.org/euclid.aap/1363354105

Digital Object Identifier
doi:10.1239/aap/1363354105

Mathematical Reviews number (MathSciNet)
MR3077543

Zentralblatt MATH identifier
1287.60045

#### Citation

Jelenković, Predrag R.; Tan, Jian. Characterizing heavy-tailed distributions induced by retransmissions. Adv. in Appl. Probab. 45 (2013), no. 1, 106--138. doi:10.1239/aap/1363354105. https://projecteuclid.org/euclid.aap/1363354105

#### References

• Asmussen, S. \et (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.
• 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. In ACM SIGMETRICS Performance Evaluation Rev. 33, 15–17.
• Jelenković, P. and Momčilović, P. (2003). Large deviation analysis of subexponential waiting times in a processor-sharing queue. Math. Operat. Res. 28, 587–608.
• Jelenković, P. R. and Momčilović, P. (2004). Large deviations of square root insensitive random sums. Math. Operat. Res. 29, 398–406.
• Jelenković, P. R. and Tan, J. (2007). Is ALOHA causing power law delays? In Managing Traffic Performance in Converged Networks (Lecture Notes Comput. 4516), Springer, Berlin, pp. 1149–1160.
• Jelenković, P. R. and Tan, J. (2007). Can retransmissions of superexponential documents cause subexponential delays? In Proc. IEEE INFOCOM 07(New York, May 2007), pp. 892–900.
• Jelenković, P. R. and Tan, J. (2007). Characterizing heavy-tailed distributions induced by retransmissions. Preprint. Available at http:// arxiv.org/abs/0709.1138v2.
• Kulkarni, V. G., Nicola, V. F. and Trivedi, K. S. (1987). The completion time of a job on a multimode system. Adv. Appl. Prob. 19, 932–954.
• Nagaev, S. V. (1979). Large deviations of sums of independent random variables. Ann. Prob. 7, 745–789.
• Ross, S. M. (2002). A First Course in Probability, 6th edn. Prentice Hall.
• 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. In ACM SIGMETRICS Performance Evaluation Rev. 34, 24–26.