238 research outputs found
A large-deviations analysis of the GI/GI/1 SRPT queue
We consider a GI/GI/1 queue with the shortest remaining processing time
discipline (SRPT) and light-tailed service times. Our interest is focused on
the tail behavior of the sojourn-time distribution. We obtain a general
expression for its large-deviations decay rate. The value of this decay rate
critically depends on whether there is mass in the endpoint of the service-time
distribution or not. An auxiliary priority queue, for which we obtain some new
results, plays an important role in our analysis. We apply our SRPT-results to
compare SRPT with FIFO from a large-deviations point of view.Comment: 22 page
Sojourn time asymptotics in processor sharing queues
This paper addresses the sojourn time asymptotics for a GI/GI/• queue operating under the
Processor Sharing (PS) discipline with stochastically varying service rate. Our focus is on the
logarithmic estimates of the tail of sojourn-time distribution, under the assumption that the jobsize
distribution has a light tail. Whereas upper bounds on the decay rate can be derived under
fairly general conditions, the establishment of the corresponding lower bounds requires that the
service process satisfies a samplepath large-deviation principle. We show that the class of
allowed service processes includes the case where the service rate is modulated by a Markov
process. Finally, we extend our results to a similar system operation under the Discriminatory
Processor Sharing (DPS) discipline. Our analysis relies predominantly on large-deviations
techniques
Asymptotic Expansions for the Conditional Sojourn Time Distribution in the -PS Queue
We consider the queue with processor sharing. We study the
conditional sojourn time distribution, conditioned on the customer's service
requirement, in various asymptotic limits. These include large time and/or
large service request, and heavy traffic, where the arrival rate is only
slightly less than the service rate. The asymptotic formulas relate to, and
extend, some results of Morrison \cite{MO} and Flatto \cite{FL}.Comment: 30 pages, 3 figures and 1 tabl
Proportional fairness and its relationship with multi-class queueing networks
We consider multi-class single-server queueing networks that have a product
form stationary distribution. A new limit result proves a sequence of such
networks converges weakly to a stochastic flow level model. The stochastic flow
level model found is insensitive. A large deviation principle for the
stationary distribution of these multi-class queueing networks is also found.
Its rate function has a dual form that coincides with proportional fairness. We
then give the first rigorous proof that the stationary throughput of a
multi-class single-server queueing network converges to a proportionally fair
allocation. This work combines classical queueing networks with more recent
work on stochastic flow level models and proportional fairness. One could view
these seemingly different models as the same system described at different
levels of granularity: a microscopic, queueing level description; a
macroscopic, flow level description and a teleological, optimization
description.Comment: Published in at http://dx.doi.org/10.1214/09-AAP612 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Tail asymptotics for processor sharing queues
The basic queueing system considered in this paper is the M/G/1 processor-sharing queue with or without impatience and with finite or infinite capacity. Under some mild assumptions, a criterion for the validity of the reduced-service-rate approximation is established when service times are heavy tailed. This result is applied to various models based on M/G/1 processor-sharing queues
PSBS: Practical Size-Based Scheduling
Size-based schedulers have very desirable performance properties: optimal or
near-optimal response time can be coupled with strong fairness guarantees.
Despite this, such systems are very rarely implemented in practical settings,
because they require knowing a priori the amount of work needed to complete
jobs: this assumption is very difficult to satisfy in concrete systems. It is
definitely more likely to inform the system with an estimate of the job sizes,
but existing studies point to somewhat pessimistic results if existing
scheduler policies are used based on imprecise job size estimations. We take
the goal of designing scheduling policies that are explicitly designed to deal
with inexact job sizes: first, we show that existing size-based schedulers can
have bad performance with inexact job size information when job sizes are
heavily skewed; we show that this issue, and the pessimistic results shown in
the literature, are due to problematic behavior when large jobs are
underestimated. Once the problem is identified, it is possible to amend
existing size-based schedulers to solve the issue. We generalize FSP -- a fair
and efficient size-based scheduling policy -- in order to solve the problem
highlighted above; in addition, our solution deals with different job weights
(that can be assigned to a job independently from its size). We provide an
efficient implementation of the resulting protocol, which we call Practical
Size-Based Scheduler (PSBS). Through simulations evaluated on synthetic and
real workloads, we show that PSBS has near-optimal performance in a large
variety of cases with inaccurate size information, that it performs fairly and
it handles correctly job weights. We believe that this work shows that PSBS is
indeed pratical, and we maintain that it could inspire the design of schedulers
in a wide array of real-world use cases.Comment: arXiv admin note: substantial text overlap with arXiv:1403.599
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