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Characterizing Heavy-Tailed Distributions Induced by Retransmissions

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  04 January 2016

Predrag R. Jelenković  and
Jian Tan
Predrag R. Jelenković*
Affiliation:
Columbia University
Jian Tan*
Affiliation:
IBM T. J. Watson Research Center
*
∗ Postal address: Department of Electrical Engineering, Columbia University, New York, NY 10027, USA.
∗∗ Postal address: IBM T. J. Watson Research, Yorktown Heights, NY 10598, USA. Email address: [email protected]
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Abstract

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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]) ≈ -(log Φ(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.

Keywords

Retransmissionchannel (systems) with failuresrestartorigins of heavy tails (subexponentiality)Gaussian distributionexponential distributionWeibull distributionlog-normal distributionpower law

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 60F10: Large deviations 60G50: Sums of independent random variables; random walks 60G99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 106 - 138
Copyright
© Applied Probability Trust 

Footnotes

This work was supported by the NSF grant CCF-0915784.

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