754 research outputs found
Scaling limits via excursion theory: Interplay between Crump-Mode-Jagers branching processes and processor-sharing queues
We study the convergence of the processor-sharing, queue length
process in the heavy traffic regime, in the finite variance case. To do so, we
combine results pertaining to L\'{e}vy processes, branching processes and
queuing theory. These results yield the convergence of long excursions of the
queue length processes, toward excursions obtained from those of some reflected
Brownian motion with drift, after taking the image of their local time process
by the Lamperti transformation. We also show, via excursion theoretic
arguments, that this entails the convergence of the entire processes to some
(other) reflected Brownian motion with drift. Along the way, we prove various
invariance principles for homogeneous, binary Crump-Mode-Jagers processes. In
the last section we discuss potential implications of the state space collapse
property, well known in the queuing literature, to branching processes.Comment: Published in at http://dx.doi.org/10.1214/12-AAP904 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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
Control of the multiclass queue in the moderate deviation regime
A multi-class single-server system with general service time distributions is
studied in a moderate deviation heavy traffic regime. In the scaling limit, an
optimal control problem associated with the model is shown to be governed by a
differential game that can be explicitly solved. While the characterization of
the limit by a differential game is akin to results at the large deviation
scale, the analysis of the problem is closely related to the much studied area
of control in heavy traffic at the diffusion scale.Comment: Published in at http://dx.doi.org/10.1214/13-AAP971 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bits Through Bufferless Queues
This paper investigates the capacity of a channel in which information is
conveyed by the timing of consecutive packets passing through a queue with
independent and identically distributed service times. Such timing channels are
commonly studied under the assumption of a work-conserving queue. In contrast,
this paper studies the case of a bufferless queue that drops arriving packets
while a packet is in service. Under this bufferless model, the paper provides
upper bounds on the capacity of timing channels and establishes achievable
rates for the case of bufferless M/M/1 and M/G/1 queues. In particular, it is
shown that a bufferless M/M/1 queue at worst suffers less than 10% reduction in
capacity when compared to an M/M/1 work-conserving queue.Comment: 8 pages, 3 figures, accepted in 51st Annual Allerton Conference on
Communication, Control, and Computing, University of Illinois, Monticello,
Illinois, Oct 2-4, 201
Sample path large deviations for multiclass feedforward queueing networks in critical loading
We consider multiclass feedforward queueing networks with first in first out
and priority service disciplines at the nodes, and class dependent
deterministic routing between nodes. The random behavior of the network is
constructed from cumulative arrival and service time processes which are
assumed to satisfy an appropriate sample path large deviation principle. We
establish logarithmic asymptotics of large deviations for waiting time, idle
time, queue length, departure and sojourn-time processes in critical loading.
This transfers similar results from Puhalskii about single class queueing
networks with feedback to multiclass feedforward queueing networks, and
complements diffusion approximation results from Peterson. An example with
renewal inter arrival and service time processes yields the rate function of a
reflected Brownian motion. The model directly captures stationary situations.Comment: Published at http://dx.doi.org/10.1214/105051606000000439 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
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