## Advances in Applied Probability

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

### Characterizing heavy-tailed distributions induced by retransmissions

Predrag R. Jelenković and Jian Tan

#### 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*, *A*_{i}}_{i≥1} that is independent of *L*. During each
period of time that the channel becomes available, say *A*_{i}, we
attempt to transmit the data unit. If *L* <*A*_{i}, the transmission is
considered successful; otherwise, we wait for the next available period
*A*_{i+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*) ≈ n^{β}, 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*)) ≈ *n*^{γ}.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 15 March 2013

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1239/aap/1363354105

**Mathematical Reviews number (MathSciNet)**

MR3077543

**Zentralblatt MATH identifier**

1287.60045

**Subjects**

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

**Keywords**

Retransmission channel (systems) with failures restart origins of heavy tails (subexponentiality) Gaussian distribution exponential distribution Weibull distribution log-normal distribution power law

#### 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