1,828 research outputs found
Human activity modeling and Barabasi's queueing systems
It has been shown by A.-L. Barabasi that the priority based scheduling rules
in single stage queuing systems (QS) generates fat tail behavior for the tasks
waiting time distributions (WTD). Such fat tails are due to the waiting times
of very low priority tasks which stay unserved almost forever as the task
priority indices (PI) are "frozen in time" (i.e. a task priority is assigned
once for all to each incoming task). Relaxing the "frozen in time" assumption,
this paper studies the new dynamic behavior expected when the priority of each
incoming tasks is time-dependent (i.e. "aging mechanisms" are allowed). For two
class of models, namely 1) a population type model with an age structure and 2)
a QS with deadlines assigned to the incoming tasks which is operated under the
"earliest-deadline-first" policy, we are able to analytically extract some
relevant characteristics of the the tasks waiting time distribution. As the
aging mechanism ultimately assign high priority to any long waiting tasks, fat
tails in the WTD cannot find their origin in the scheduling rule alone thus
showing a fundamental difference between the present and the A.-L. Barabasi's
class of models.Comment: 16 pages, 2 figure
Simple and explicit bounds for multi-server queues with (and sometimes better) scaling
We consider the FCFS queue, and prove the first simple and explicit
bounds that scale as (and sometimes better). Here
denotes the corresponding traffic intensity. Conceptually, our results can be
viewed as a multi-server analogue of Kingman's bound. Our main results are
bounds for the tail of the steady-state queue length and the steady-state
probability of delay. The strength of our bounds (e.g. in the form of tail
decay rate) is a function of how many moments of the inter-arrival and service
distributions are assumed finite. More formally, suppose that the inter-arrival
and service times (distributed as random variables and respectively)
have finite th moment for some Let (respectively )
denote (respectively ). Then
our bounds (also for higher moments) are simple and explicit functions of
, and
only. Our bounds scale gracefully even when the number of
servers grows large and the traffic intensity converges to unity
simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale
better than in certain asymptotic regimes. More precisely,
they scale as multiplied by an inverse polynomial in These results formalize the intuition that bounds should be tighter
in light traffic as well as certain heavy-traffic regimes (e.g. with
fixed and large). In these same asymptotic regimes we also prove bounds for
the tail of the steady-state number in service.
Our main proofs proceed by explicitly analyzing the bounding process which
arises in the stochastic comparison bounds of amarnik and Goldberg for
multi-server queues. Along the way we derive several novel results for suprema
of random walks and pooled renewal processes which may be of independent
interest. We also prove several additional bounds using drift arguments (which
have much smaller pre-factors), and make several conjectures which would imply
further related bounds and generalizations
Queue Length Asymptotics for Generalized Max-Weight Scheduling in the presence of Heavy-Tailed Traffic
We investigate the asymptotic behavior of the steady-state queue length
distribution under generalized max-weight scheduling in the presence of
heavy-tailed traffic. We consider a system consisting of two parallel queues,
served by a single server. One of the queues receives heavy-tailed traffic, and
the other receives light-tailed traffic. We study the class of throughput
optimal max-weight-alpha scheduling policies, and derive an exact asymptotic
characterization of the steady-state queue length distributions. In particular,
we show that the tail of the light queue distribution is heavier than a
power-law curve, whose tail coefficient we obtain explicitly. Our asymptotic
characterization also contains an intuitively surprising result - the
celebrated max-weight scheduling policy leads to the worst possible tail of the
light queue distribution, among all non-idling policies. Motivated by the above
negative result regarding the max-weight-alpha policy, we analyze a
log-max-weight (LMW) scheduling policy. We show that the LMW policy guarantees
an exponentially decaying light queue tail, while still being throughput
optimal
On the transition from heavy traffic to heavy tails for the M/G/1 queue: The regularly varying case
Two of the most popular approximations for the distribution of the
steady-state waiting time, , of the M/G/1 queue are the so-called
heavy-traffic approximation and heavy-tailed asymptotic, respectively. If the
traffic intensity, , is close to 1 and the processing times have finite
variance, the heavy-traffic approximation states that the distribution of
is roughly exponential at scale , while the
heavy tailed asymptotic describes power law decay in the tail of the
distribution of for a fixed traffic intensity. In this paper, we
assume a regularly varying processing time distribution and obtain a sharp
threshold in terms of the tail value, or equivalently in terms of ,
that describes the point at which the tail behavior transitions from the
heavy-traffic regime to the heavy-tailed asymptotic. We also provide new
approximations that are either uniform in the traffic intensity, or uniform on
the positive axis, that avoid the need to use different expressions on the two
regions defined by the threshold.Comment: Published in at http://dx.doi.org/10.1214/10-AAP707 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Customer sojourn time in GI/G/1 feedback queue in the presence of heavy tails
We consider a single-server GI/GI/1 queueing system with feedback. We assume
the service times distribution to be (intermediate) regularly varying. We find
the tail asymptotics for a customer's sojourn time in two regimes: the customer
arrives in an empty system, and the customer arrives in the system in the
stationary regime. In particular, in the case of Poisson input we use the
branching processes structure and provide more precise formulae. As auxiliary
results, we find the tail asymptotics for the busy period distribution in a
single-server queue with an intermediate varying service times distribution and
establish the principle-of-a-single-big-jump equivalences that characterise the
asymptotics.Comment: 34 pages, 4 figures, to appear in Journal of Statistical Physic
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