# [e2e] Why do we need TCP flow control (rwnd)?

David P. Reed dpreed at reed.com
Mon Jul 14 18:09:07 PDT 2008

```Ted - you must not be reading my email very carefully.  The description
you give below *is* [subject to very poor writing style] a Poisson process.

The construction that is implied by "events with interarrival times
given by Y" is expressed via really lousy mathematical exposition -
where was the editor?  But since I can guess what the author(s?)
intended to say, I'd forgive them for not specifying that they are
suggesting an algorithm that takes as input  an ordered sequence of
samples taken from repeated experiments based on different
instantiations of the variable Y (computed from different instantiations
of a random variable X), and then converting the ordered sequence into a
particular event sequence that they call Poisson.

A careful mathematical exposition would describe this in terms of random
sequences Xbar (selected independently from a uniform distribution), and
Ybar, and the resulting sequence Zbar, which is the summation of
prefixes of Ybar.

Neither X nor Y are Poisson random variables, but the sequence thus
derived is an instance of a Poisson process.

A Poisson process is defined as an event-generating process that assigns
a specific probability to a particular count of events that occur in
each interval of the real line.

- Peace

Ted Faber wrote:
> On Sat, Jul 12, 2008 at 02:42:18AM -0400, David P. Reed wrote:
>
>> Actually, Ted, constructing a sequence of events that are Poisson
>> distributed in time *requires* a Poisson process.
>>
>
> Given a random variable X that's uniformly distributed on (0,1), Y= -a
> ln(1-X), for a > 0,  is exponentially distributed with parameter a.
> Events with interarrivaltimes given by Y are Poisson distributed with
> parameter a.
>
> Poisson distributed events from a uniform random variable.
>
> This isn't esoteric;  it's an example right out of my graduate
> probability text.  I'm old enough that it's not online but if your
> library has Kishor Trivedi's _Probability & Statistics With Reliability,
> Queueing, And Computer Science Applications_,  you can find the proof in
> the section on computing distributions of funtions of a random variable.
> In my 1982 edition, the proof is on page 140.
>
>
```