1,131 research outputs found
Fixed points for multi-class queues
Burke's theorem can be seen as a fixed-point result for an exponential
single-server queue; when the arrival process is Poisson, the departure process
has the same distribution as the arrival process. We consider extensions of
this result to multi-type queues, in which different types of customer have
different levels of priority. We work with a model of a queueing server which
includes discrete-time and continuous-time M/M/1 queues as well as queues with
exponential or geometric service batches occurring in discrete time or at
points of a Poisson process. The fixed-point results are proved using
interchangeability properties for queues in tandem, which have previously been
established for one-type M/M/1 systems. Some of the fixed-point results have
previously been derived as a consequence of the construction of stationary
distributions for multi-type interacting particle systems, and we explain the
links between the two frameworks. The fixed points have interesting
"clustering" properties for lower-priority customers. An extreme case is an
example of a Brownian queue, in which lower-priority work only occurs at a set
of times of measure 0 (and corresponds to a local time process for the
queue-length process of higher priority work).Comment: 25 page
Lattice path counting and the theory of queues
In this paper we will show how recent advances in the combinatorics of lattice paths can be applied to solve interesting and nontrivial problems in the theory of queues. The problems we discuss range from classical ones like M^a/M^b/1 systems to open tandem systems with and without global blocking and to queueing models that are related to random walks in a quarter plane like the Flatto-Hahn model or systems with preemptive priorities. (author´s abstract)Series: Research Report Series / Department of Statistics and Mathematic
Waiting times in queueing networks with a single shared server
We study a queueing network with a single shared server that serves the
queues in a cyclic order. External customers arrive at the queues according to
independent Poisson processes. After completing service, a customer either
leaves the system or is routed to another queue. This model is very generic and
finds many applications in computer systems, communication networks,
manufacturing systems, and robotics. Special cases of the introduced network
include well-known polling models, tandem queues, systems with a waiting room,
multi-stage models with parallel queues, and many others. A complicating factor
of this model is that the internally rerouted customers do not arrive at the
various queues according to a Poisson process, causing standard techniques to
find waiting-time distributions to fail. In this paper we develop a new method
to obtain exact expressions for the Laplace-Stieltjes transforms of the
steady-state waiting-time distributions. This method can be applied to a wide
variety of models which lacked an analysis of the waiting-time distribution
until now
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
Separation of timescales in a two-layered network
We investigate a computer network consisting of two layers occurring in, for
example, application servers. The first layer incorporates the arrival of jobs
at a network of multi-server nodes, which we model as a many-server Jackson
network. At the second layer, active servers at these nodes act now as
customers who are served by a common CPU. Our main result shows a separation of
time scales in heavy traffic: the main source of randomness occurs at the
(aggregate) CPU layer; the interactions between different types of nodes at the
other layer is shown to converge to a fixed point at a faster time scale; this
also yields a state-space collapse property. Apart from these fundamental
insights, we also obtain an explicit approximation for the joint law of the
number of jobs in the system, which is provably accurate for heavily loaded
systems and performs numerically well for moderately loaded systems. The
obtained results for the model under consideration can be applied to
thread-pool dimensioning in application servers, while the technique seems
applicable to other layered systems too.Comment: 8 pages, 2 figures, 1 table, ITC 24 (2012
Stationary distributions of the multi-type ASEP
We give a recursive construction of the stationary distribution of multi-type
asymmetric simple exclusion processes on a finite ring or on the infinite line
. The construction can be interpreted in terms of "multi-line diagrams" or
systems of queues in tandem. Let be the asymmetry parameter of the system.
The queueing construction generalises the one previously known for the totally
asymmetric () case, by introducing queues in which each potential service
is unused with probability when the queue-length is . The analysis is
based on the matrix product representation of Prolhac, Evans and Mallick.
Consequences of the construction include: a simple method for sampling exactly
from the stationary distribution for the system on a ring; results on common
denominators of the stationary probabilities, expressed as rational functions
of with non-negative integer coefficients; and probabilistic descriptions
of "convoy formation" phenomena in large systems.Comment: 54 pages, 4 figure
Alternative proof and interpretations for a recent state-dependent importance sampling scheme
Recently, a state-dependent change of measure for simulating overflows in the two-node tandem queue was proposed by Dupuis et al. (Ann. Appl. Probab. 17(4):1306–1346, 2007), together with a proof of its asymptotic optimality. In the present paper, we present an alternative, shorter and simpler proof. As a side result, we obtain interpretations for several of the quantities involved in the change of measure in terms of likelihood ratios
Stationary distributions of multi-type totally asymmetric exclusion processes
We consider totally asymmetric simple exclusion processes with n types of
particle and holes (-TASEPs) on and on the cycle . Angel recently gave an elegant construction of the stationary measures
for the 2-TASEP, based on a pair of independent product measures. We show that
Angel's construction can be interpreted in terms of the operation of a
discrete-time queueing server; the two product measures correspond to
the arrival and service processes of the queue. We extend this construction to
represent the stationary measures of an n-TASEP in terms of a system of queues
in tandem. The proof of stationarity involves a system of n 1-TASEPs, whose
evolutions are coupled but whose distributions at any fixed time are
independent. Using the queueing representation, we give quantitative results
for stationary probabilities of states of the n-TASEP on , and
simple proofs of various independence and regeneration properties for systems
on .Comment: Published at http://dx.doi.org/10.1214/009117906000000944 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Detecting Markov Chain Instability: A Monte Carlo Approach
We devise a Monte Carlo based method for detecting whether a non-negative
Markov chain is stable for a given set of parameter values. More precisely, for
a given subset of the parameter space, we develop an algorithm that is capable
of deciding whether the set has a subset of positive Lebesgue measure for which
the Markov chain is unstable. The approach is based on a variant of simulated
annealing, and consequently only mild assumptions are needed to obtain
performance guarantees.
The theoretical underpinnings of our algorithm are based on a result stating
that the stability of a set of parameters can be phrased in terms of the
stability of a single Markov chain that searches the set for unstable
parameters. Our framework leads to a procedure that is capable of performing
statistically rigorous tests for instability, which has been extensively tested
using several examples of standard and non-standard queueing networks
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