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Distribution of the Number of Retransmissions of Bounded Documents

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  22 February 2016

Predrag R. Jelenković  and
Evangelia D. Skiani
Predrag R. Jelenković*
Affiliation:
Columbia University
Evangelia D. Skiani*
Affiliation:
Columbia University
*
∗ Postal address: Department of Electrical Engineering, Columbia University, New York, NY 10027, USA.
∗ Postal address: Department of Electrical Engineering, Columbia University, New York, NY 10027, USA.
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Abstract

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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 ≤ Lb ≤ 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[Nb > 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.

Keywords

Retransmissionrestartchannel with failuretruncated distributionpower lawgamma distributionheavy-tailed distributionlight-tailed distribution

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 60F10: Large deviations 60G99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 425 - 449
Copyright
© Applied Probability Trust 

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