662 research outputs found
Instability in Stochastic and Fluid Queueing Networks
The fluid model has proven to be one of the most effective tools for the
analysis of stochastic queueing networks, specifically for the analysis of
stability. It is known that stability of a fluid model implies positive
(Harris) recurrence (stability) of a corresponding stochastic queueing network,
and weak stability implies rate stability of a corresponding stochastic
network. These results have been established both for cases of specific
scheduling policies and for the class of all work conserving policies.
However, only partial converse results have been established and in certain
cases converse statements do not hold. In this paper we close one of the
existing gaps. For the case of networks with two stations we prove that if the
fluid model is not weakly stable under the class of all work conserving
policies, then a corresponding queueing network is not rate stable under the
class of all work conserving policies. We establish the result by building a
particular work conserving scheduling policy which makes the associated
stochastic process transient. An important corollary of our result is that the
condition , which was proven in \cite{daivan97} to be the exact
condition for global weak stability of the fluid model, is also the exact
global rate stability condition for an associated queueing network. Here
is a certain computable parameter of the network involving virtual
station and push start conditions.Comment: 30 pages, To appear in Annals of Applied Probabilit
The ODE method for stability of skip-free Markov chains with applications to MCMC
Fluid limit techniques have become a central tool to analyze queueing
networks over the last decade, with applications to performance analysis,
simulation and optimization. In this paper, some of these techniques are
extended to a general class of skip-free Markov chains. As in the case of
queueing models, a fluid approximation is obtained by scaling time, space and
the initial condition by a large constant. The resulting fluid limit is the
solution of an ordinary differential equation (ODE) in ``most'' of the state
space. Stability and finer ergodic properties for the stochastic model then
follow from stability of the set of fluid limits. Moreover, similarly to the
queueing context where fluid models are routinely used to design control
policies, the structure of the limiting ODE in this general setting provides an
understanding of the dynamics of the Markov chain. These results are
illustrated through application to Markov chain Monte Carlo methods.Comment: Published in at http://dx.doi.org/10.1214/07-AAP471 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Numerical Approach to Stability of Multiclass Queueing Networks
The Multi-class Queueing Network (McQN) arises as a natural multi-class
extension of the traditional (single-class) Jackson network. In a single-class
network subcriticality (i.e. subunitary nominal workload at every station)
entails stability, but this is no longer sufficient when jobs/customers of
different classes (i.e. with different service requirements and/or routing
scheme) visit the same server; therefore, analytical conditions for stability
of McQNs are lacking, in general. In this note we design a numerical
(simulation-based) method for determining the stability region of a McQN, in
terms of arrival rate(s). Our method exploits certain (stochastic) monotonicity
properties enjoyed by the associated Markovian queue-configuration process.
Stochastic monotonicity is a quite common feature of queueing models and can be
easily established in the single-class framework (Jackson networks); recently,
also for a wide class of McQNs, including first-come-first-serve (FCFS)
networks, monotonicity properties have been established. Here, we provide a
minimal set of conditions under which the method performs correctly.
Eventually, we illustrate the use of our numerical method by presenting a set
of numerical experiments, covering both single and multi-class networks
On the Stability of a Polling System with an Adaptive Service Mechanism
We consider a single-server cyclic polling system with three queues where the
server follows an adaptive rule: if it finds one of queues empty in a given
cycle, it decides not to visit that queue in the next cycle. In the case of
limited service policies, we prove stability and instability results under some
conditions which are sufficient but not necessary, in general. Then we discuss
open problems with identifying the exact stability region for models with
limited service disciplines: we conjecture that a necessary and sufficient
condition for the stability may depend on the whole distributions of the
primitive sequences, and illustrate that by examples. We conclude the paper
with a section on the stability analysis of a polling system with either gated
or exhaustive service disciplines.Comment: 16 page
Validity of heavy traffic steady-state approximations in generalized Jackson Networks
We consider a single class open queueing network, also known as a generalized
Jackson network (GJN). A classical result in heavy-traffic theory asserts that
the sequence of normalized queue length processes of the GJN converge weakly to
a reflected Brownian motion (RBM) in the orthant, as the traffic intensity
approaches unity. However, barring simple instances, it is still not known
whether the stationary distribution of RBM provides a valid approximation for
the steady-state of the original network. In this paper we resolve this open
problem by proving that the re-scaled stationary distribution of the GJN
converges to the stationary distribution of the RBM, thus validating a
so-called ``interchange-of-limits'' for this class of networks. Our method of
proof involves a combination of Lyapunov function techniques, strong
approximations and tail probability bounds that yield tightness of the sequence
of stationary distributions of the GJN.Comment: Published at http://dx.doi.org/10.1214/105051605000000638 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Queue-Based Random-Access Algorithms: Fluid Limits and Stability Issues
We use fluid limits to explore the (in)stability properties of wireless
networks with queue-based random-access algorithms. Queue-based random-access
schemes are simple and inherently distributed in nature, yet provide the
capability to match the optimal throughput performance of centralized
scheduling mechanisms in a wide range of scenarios. Unfortunately, the type of
activation rules for which throughput optimality has been established, may
result in excessive queue lengths and delays. The use of more
aggressive/persistent access schemes can improve the delay performance, but
does not offer any universal maximum-stability guarantees. In order to gain
qualitative insight and investigate the (in)stability properties of more
aggressive/persistent activation rules, we examine fluid limits where the
dynamics are scaled in space and time. In some situations, the fluid limits
have smooth deterministic features and maximum stability is maintained, while
in other scenarios they exhibit random oscillatory characteristics, giving rise
to major technical challenges. In the latter regime, more aggressive access
schemes continue to provide maximum stability in some networks, but may cause
instability in others. Simulation experiments are conducted to illustrate and
validate the analytical results
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