In 1998 and 1999, research
within Stieltjes on heavy tails in queues
has focused on two topics.
The first topic is,
whether other service disciplines than FCFS may lead
to a less detrimental response time behaviour. For
the queue with the
Processor Sharing discipline
[14] and
the
queue with
the Last-Come-First-Served Preemptive
Resume discipline [8], important
disciplines that both play a key role in queueing networks of so-called product-form,
the response time tail behaviour
turns out to be regularly varying of index
iff
the service time tail behaviour is regularly varying of index
.
Hence these disciplines result in better tail behaviour of sojourn
times than FCFS.
In a series of studies (see, e.g., [5]) we have also
investigated
the workload tail behaviour of service systems
with Generalized Processor Sharing (GPS).
GPS-based scheduling algorithms, such as Weighted-Fair-Queueing,
have emerged as
an important mechanism for achieving
differentiated Quality-of-Service in networks
with integrated traffic services.
We have identified conditions under which, in GPS, traffic of one class does
not suffer from heavier-tailed traffic
of another class.
The second topic
is the heavy-traffic behaviour
of a queue with heavy-tailed service time distribution.
A queueing system is said to be in heavy traffic when its traffic load
.
This topic is of theoretical interest, since
in the traditional heavy-traffic limit theorems
it is assumed that the second moments of service and
interarrival times are finite, whereas (2)
with
leads to an infinite second moment.
The topic is also of practical interest,
since heavy-traffic limit theorems may give rise to
useful approximations [6] in situations with a reasonably
light traffic load.
One can identify a coefficient of contraction
such that
times the waiting time
has a proper limiting distribution for
.
One of the results obtained in [7] reads:
For the stable FCFS queue with regularly varying service time
distribution
with mean
and
of index
, as specified in (2),
the `contracted' waiting time
converges in
distribution for
. The limiting distribution is
specified by its Laplace-Stieltjes transform
,
,
and the coefficient of contraction
is the root
of the equation
,
with
, with the property that
for
.
This result strongly differs from the case of a service time
distribution with finite variance; there the coefficient of contraction
behaves like instead of
,
and the limiting distribution is the negative exponential distribution
(hence with Laplace-Stieltjes distribution
).
Many challenging unsolved problems remain. In particular, one would like to know the tail behaviour of the workload in the case of fluid queues fed by multiple sources, some of which have heavy-tailed on-periods; and the related problem of the asymptotics of multiserver queues with heavy-tailed service time distributions also remains open.