5,404 research outputs found
Queues and risk models with simultaneous arrivals
We focus on a particular connection between queueing and risk models in a
multi-dimensional setting. We first consider the joint workload process in a
queueing model with parallel queues and simultaneous arrivals at the queues.
For the case that the service times are ordered (from largest in the first
queue to smallest in the last queue) we obtain the Laplace-Stieltjes transform
of the joint stationary workload distribution. Using a multivariate duality
argument between queueing and risk models, this also gives the Laplace
transform of the survival probability of all books in a multivariate risk model
with simultaneous claim arrivals and the same ordering between claim sizes.
Other features of the paper include a stochastic decomposition result for the
workload vector, and an outline how the two-dimensional risk model with a
general two-dimensional claim size distribution (hence without ordering of
claim sizes) is related to a known Riemann boundary value problem
Inference and Learning in Networks of Queues
Probabilistic models of the performance of computer systems are useful both for predicting system performance in new conditions, and for diagnosing past performance problems. The most popular performance models are networks of queues. However, no current methods exist for parameter estimation or inference in networks of queues with missing data. In this paper, we present a novel viewpoint that combines queueing networks and graphical models, allowing Markov chain Monte Carlo to be applied. We demonstrate the effectiveness of our sampler on real-world data from a benchmark Web application.
Perfect Simulation of Queues
In this paper we describe a perfect simulation algorithm for the stable
queue. Sigman (2011: Exact Simulation of the Stationary Distribution of
the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209--213) showed how
to build a dominated CFTP algorithm for perfect simulation of the super-stable
queue operating under First Come First Served discipline, with
dominating process provided by the corresponding queue (using Wolff's
sample path monotonicity, which applies when service durations are coupled in
order of initiation of service), and exploiting the fact that the workload
process for the queue remains the same under different queueing
disciplines, in particular under the Processor Sharing discipline, for which a
dynamic reversibility property holds. We generalize Sigman's construction to
the stable case by comparing the queue to a copy run under Random
Assignment. This allows us to produce a naive perfect simulation algorithm
based on running the dominating process back to the time it first empties. We
also construct a more efficient algorithm that uses sandwiching by lower and
upper processes constructed as coupled queues started respectively from
the empty state and the state of the queue under Random Assignment. A
careful analysis shows that appropriate ordering relationships can still be
maintained, so long as service durations continue to be coupled in order of
initiation of service. We summarize statistical checks of simulation output,
and demonstrate that the mean run-time is finite so long as the second moment
of the service duration distribution is finite.Comment: 28 pages, 5 figure
Bayesian inference for queueing networks and modeling of internet services
Modern Internet services, such as those at Google, Yahoo!, and Amazon, handle
billions of requests per day on clusters of thousands of computers. Because
these services operate under strict performance requirements, a statistical
understanding of their performance is of great practical interest. Such
services are modeled by networks of queues, where each queue models one of the
computers in the system. A key challenge is that the data are incomplete,
because recording detailed information about every request to a heavily used
system can require unacceptable overhead. In this paper we develop a Bayesian
perspective on queueing models in which the arrival and departure times that
are not observed are treated as latent variables. Underlying this viewpoint is
the observation that a queueing model defines a deterministic transformation
between the data and a set of independent variables called the service times.
With this viewpoint in hand, we sample from the posterior distribution over
missing data and model parameters using Markov chain Monte Carlo. We evaluate
our framework on data from a benchmark Web application. We also present a
simple technique for selection among nested queueing models. We are unaware of
any previous work that considers inference in networks of queues in the
presence of missing data.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS392 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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