Technological developments have in recent years
led to dramatic changes in the way humans exchange information.
Examples are provided by
the Internet and mobile communications.
These new forms of communication
have, in their turn, given rise to exciting and challenging new problems
in the performance analysis of network congestion.
In this note we discuss one of the hottest topics in this area,
viz., the influence of heavy-tailed input processes on
network performance measures.
This is a topic that is also actively studied in Stieltjes.
We first present a global outline of the topic, and subsequently
some of our own research in this field.

In classical applications of teletraffic theory, the service time
distributions in queueing models were usually assumed to have a negative
exponential tail.
In modern applications of teletraffic theory, the assumption
of exponential tails often appears to be invalid.
Plots of traffic measurements
for traffic in Ethernet Local Area Networks [13],
Wide Area Networks [12] and Variable Bit Rate (VBR) video [1]
have shown a striking similarity
when one considers a time period of hours, minutes or milliseconds:
Bursty subperiods are alternated by less bursty subperiods on
each scale.
This scale-invariant or *self-similar* feature of traffic
and the related phenomenon of *long-range dependence*
(i.e., the integral over time
of the covariance of the traffic rate diverges)
were also convincingly demonstrated in [11] using
a careful statistical analysis - and they are
incompatible with exponential traffic assumptions.

A natural model for long-range dependence (LRD)
in a traffic process is a fluid queue fed by one or more
on/off sources (viz., sources that alternate
between active and silent periods), under the assumption that
the distribution of the on-period
of a source
has the following non-exponential behaviour:

(1)

with a positive constant and
(here
means that
as
).
When at least one of the sources exhibits such
*heavy-tail*
behaviour, the cumulative input process is LRD [9].
As observed in [13],
in many cases on-periods (or off-periods, which also give rise to LRD)
of actual traffic sources
do indeed exhibit such a heavy-tail behaviour.
The occurrence of heavy-tailed on- and/or off-periods of sources
seems to provide the most natural explanation of LRD and
self-similarity in aggregated packet traffic.
There are indeed many phenomena in modern
communication traffic that fit such a description,
like file sizes in Internet traffic (one can think
of postscript files sent by email) and lengthy
telephone connections via a modem.

These observations have triggered much research
on fluid queues with, in particular, heavy-tailed
on-period distributions (cf. the survey [9]).
In its turn, this leads to an interest
in the effect of heavy-tailed service time
distributions in *ordinary* single server queues.
There appears to be a considerable similarity
in behaviour of both types of queues.
In particular, the buffer content process at embedded moments
of the fluid queue with exponential off-periods can be expressed
as the waiting time in a certain queue.

An important class of heavy-tailed probability
distributions is the class of *regularly varying* [2]
distributions of index , .
When is a nonnegative
random variable, then
is said to be regularly varying at infinity of index when

(2)

with a slowly varying function at infinity, i.e.,

could for instance be a constant, or a logarithmic function.

An early result concerning regular variation in queueing
theory is due to Cohen [10].
He proved the following result for the queue
under the First-Come-First-Served (FCFS) discipline:
The tail of the waiting time distribution is regularly varying
of index if and only if the tail of the service time
distribution is regularly varying of index , .
This result has turned out to be very useful
in relating the regularly varying tail behaviour
of on-periods of on/off sources in fluid queues
to the buffer content [3, 4].

Cohen's result implies that, in the case of regular variation, the tail of the waiting and response time distributions is one degree heavier than that of the service time distribution. For , the mean of the service time distribution is finite, but the mean of the waiting time distribution is infinite.