150 research outputs found
Simple bounds for queueing systems with breakdowns
Computationally attractive and intuitively obvious simple bounds are proposed for finite service systems which are subject to random breakdowns. The services are assumed to be exponential. The up and down periods are allowed to be generally distributed. The bounds are based on product-form modifications and depend only on means. A formal proof is presented. This proof is of interest in itself. Numerical support indicates a potential usefulness for quick engineering and performance evaluation purposes
Queues with Congestion-dependent Feedback
This dissertation expands the theory of feedback queueing systems and applies a number of these models to a performance analysis of the Transmission Control Protocol, a flow control protocol commonly used in the Internet
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
Manufacturing flow line systems: a review of models and analytical results
The most important models and results of the manufacturing flow line literature are described. These include the major classes of models (asynchronous, synchronous, and continuous); the major features (blocking, processing times, failures and repairs); the major properties (conservation of flow, flow rate-idle time, reversibility, and others); and the relationships among different models. Exact and approximate methods for obtaining quantitative measures of performance are also reviewed. The exact methods are appropriate for small systems. The approximate methods, which are the only means available for large systems, are generally based on decomposition, and make use of the exact methods for small systems. Extensions are briefly discussed. Directions for future research are suggested.National Science Foundation (U.S.) (Grant DDM-8914277
Large deviations for acyclic networks of queues with correlated Gaussian inputs
We consider an acyclic network of single-server queues with heterogeneous
processing rates. It is assumed that each queue is fed by the superposition of
a large number of i.i.d. Gaussian processes with stationary increments and
positive drifts, which can be correlated across different queues. The flow of
work departing from each server is split deterministically and routed to its
neighbors according to a fixed routing matrix, with a fraction of it leaving
the network altogether.
We study the exponential decay rate of the probability that the steady-state
queue length at any given node in the network is above any fixed threshold,
also referred to as the "overflow probability". In particular, we first
leverage Schilder's sample-path large deviations theorem to obtain a general
lower bound for the limit of this exponential decay rate, as the number of
Gaussian processes goes to infinity. Then, we show that this lower bound is
tight under additional technical conditions. Finally, we show that if the input
processes to the different queues are non-negatively correlated, non
short-range dependent fractional Brownian motions, and if the processing rates
are large enough, then the asymptotic exponential decay rates of the queues
coincide with the ones of isolated queues with appropriate Gaussian inputs
Bounds Computation for Symmetric Nets
Monotonicity in Markov chains is the starting point for quantitative abstraction of complex probabilistic systems leading to (upper or lower) bounds for probabilities and mean values relevant to their analysis. While numerous case studies exist in the literature, there is no generic model for which monotonicity is directly derived from its structure. Here we propose such a model and formalize it as a subclass of Stochastic Symmetric (Petri) Nets (SSNs) called Stochastic Monotonic SNs (SMSNs). On this subclass the monotonicity is proven by coupling arguments that can be applied on an abstract description of the state (symbolic marking). Our class includes both process synchronizations and resource sharings and can be extended to model open or cyclic closed systems. Automatic methods for transforming a non monotonic system into a monotonic one matching the MSN pattern, or for transforming a monotonic system with large state space into one with reduced state space are presented. We illustrate the interest of the proposed method by expressing standard monotonic models and modelling a flexible manufacturing system case study
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