530 research outputs found
Computing stationary probability distributions and large deviation rates for constrained random walks. The undecidability results
Our model is a constrained homogeneous random walk in a nonnegative orthant
Z_+^d. The convergence to stationarity for such a random walk can often be
checked by constructing a Lyapunov function. The same Lyapunov function can
also be used for computing approximately the stationary distribution of this
random walk, using methods developed by Meyn and Tweedie. In this paper we show
that, for this type of random walks, computing the stationary probability
exactly is an undecidable problem: no algorithm can exist to achieve this task.
We then prove that computing large deviation rates for this model is also an
undecidable problem. We extend these results to a certain type of queueing
systems. The implication of these results is that no useful formulas for
computing stationary probabilities and large deviations rates can exist in
these systems
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
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
A Stochastic Resource-Sharing Network for Electric Vehicle Charging
We consider a distribution grid used to charge electric vehicles such that
voltage drops stay bounded. We model this as a class of resource-sharing
networks, known as bandwidth-sharing networks in the communication network
literature. We focus on resource-sharing networks that are driven by a class of
greedy control rules that can be implemented in a decentralized fashion. For a
large number of such control rules, we can characterize the performance of the
system by a fluid approximation. This leads to a set of dynamic equations that
take into account the stochastic behavior of EVs. We show that the invariant
point of these equations is unique and can be computed by solving a specific
ACOPF problem, which admits an exact convex relaxation. We illustrate our
findings with a case study using the SCE 47-bus network and several special
cases that allow for explicit computations.Comment: 13 pages, 8 figure
Geometric bounds for stationary distributions of infinite Markov chains via Lyapunov functions
Title from cover. "July, 1998."Includes bibliographical references (p. 18-20).Supported in part by a grant from the NSF. DMI-9610486 Supported in part by a grant from the ARO. DAAL-03-92-G-0115Dimitris Bertsimas, David Gamarnik, and John N. Tsitsiklis
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