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<font face="Helvetica, Arial, sans-serif">As I noted, Little's theorem
(at least its intuitive conclusion) applies to a wide variety of
arrival processes and a wide variety of queues. Did I say that
Little's theorem is the only knowledge one needs to get from studying
queueing theory? Of course not.<br>
<br>
At least some of this discussion moved on to recognize that Little's
theorem does NOT apply very well to arrival processes that involve
closed-loop control of packet admission (such as TCP) do not follow
Little's theorem, and arrival processes that involve even more complex
control systems such as dynamic routing or packet scheduling involving
propagation in WLANs or scheduled admission in WWANs.<br>
<br>
So, as Detlef suggests, we should be careful in saying that Little's
theorem should be gospel everywhere, without understanding how it
derives from its assumptions. That is why one needs to *learn*
queueing theory, not as a cookbook, but as a way of thinking that one
can adapt to complex problems.<br>
<br>
I still remain shocked that the people who designed DOCSIS put many
*seconds* of buffering into the downlink and uplink, which get filled
(because the DOCSIS modem is the rate limiting device), and the result
is that except for FTP, which cares little about latency, DOCSIS modems
are *unusable* without implementing a "supervisory" layer on top of
them that refuses to use the buffers inserted into the end-to-end
path. Clearly this is due to bad decision making on the part of DOCSIS
modem designers and deployers - assuming they wanted to support
Internet service, rather than degrade it.<br>
</font><br>
On 06/21/2009 05:33 AM, Detlef Bosau wrote:
<blockquote cite="mid:4A3DFE7E.1040000@web.de" type="cite">David P.
Reed wrote:
<br>
<blockquote type="cite">Dave - This is variously known as Little's
Theorem or Little's Lemma. The general pattern is true for many
stochastic arrival processes into queues. It precedes Kleinrock, and
belongs to queueing theory.
<br>
</blockquote>
<br>
Little's Theorem can be easily applied in wired networks where a link's
capacity is easily expressed as "latency throghput product", often
referred to as "latency bandwidth product" which is in fact a bit
sloppy.
<br>
<br>
The situation becomes a bit more complicated in wireless networks,
particularly WWAN, where the preconditions for Little's Theorem may not
hold, particularly the service time may not be stationary or stable.
<br>
<br>
I sometimes wonder about papers who claim quite impressive "latency
bandwidth products" for wireless networks - and actually the authors
simply miss the fact that the transportation system is highly occupied
by local retransmissions and that we have a relationship between
average service, average throughput and the average amount of data
being in flight.
<br>
<br>
I even remember a paper which claims latency bandwidth products for
GPRS in the range of MBytes IIRC.
<br>
<br>
At a first glance, I wondered where this huge amount of data would fit
onto the air interface ;-)
<br>
<br>
So, we should be extremely careful in applying Little's Theorem on
WWAN. As a consequence, we should even reconsider approaches like
packet pair, packet train and the like and whether they really hold in
WWAN or similar networks with highly volatile line conditions.
<br>
<br>
Detlef
<br>
<br>
</blockquote>
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