How Many Tries to Send a Packet?

A radio packet gets through with probability p on each try, independently. The number of tries until the first success is geometric: usually one or two, occasionally many. Radios cap the retries, and the cap sets how often a packet is dropped.

How it works

The geometric distribution

In one sentence

If independent trials each succeed with probability $p$ , the trial number $X$ of the first success is geometric: $P(X=k) = (1-p)^{k-1}p$ for $k = 1, 2, \dots$ , with $E(X) = 1/p$ , $\operatorname{Var}(X) = (1-p)/p^2$ and $P(X > L) = (1-p)^L$ . It is memoryless: past failures don't change the chance of success on the next try.

General formula

P(X = {\htmlData{sym=1}{k}}) = (1-{\htmlData{sym=0}{p}})^{{\htmlData{sym=1}{k}}-1}{\htmlData{sym=0}{p}}, \qquad {\htmlData{sym=2}{E(X)}} = \frac{1}{\htmlData{sym=0}{{p}}}, \qquad P(X > {\htmlData{sym=3}{L}}) = (1-{\htmlData{sym=0}{p}})^{\htmlData{sym=3}{L}}

Each try is a fresh flip of a lopsided coin. Long runs of failure are unlikely, but always possible.

What the symbols mean

$p$
Success chance per try
The probability one transmission gets through.
$k$
Try number
The try on which the first success happens.
$E(X)$
Expected tries
The long-run average number of tries per packet.
$L$
Retry limit
How many tries the radio makes before giving up.

Each try is an independent trial with success probability $p$ . The first success on try $k$ needs $k-1$ failures first: $(1-p)^{k-1}p$ .
The bars shrink by the same factor $1-p$ at each step, a geometric sequence, which is where the name comes from.
On average it takes $E(X) = 1/p$ tries: a link that succeeds 25% of the time needs about four.
A radio that gives up after $L$ tries drops a packet with probability $(1-p)^L$ , the chance of $L$ failures in a row.

Learn the math

Geometric and Hypergeometric DistributionsStats · Lesson 16, Discrete Random Variables