## Advances in Applied Probability

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

### Distribution of the number of retransmissions of bounded documents

Predrag R. Jelenković and Evangelia D. Skiani

#### 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