We study two families of error-correcting codes defined in terms of very sparse matrices "MN" (MacKay-Neal (1995)) codes are recently invented, and "Gallager codes" were first investigated in 1962, but appear to have been largely forgotten, in spite of their excellent properties The decoding of both codes can be tackled with a practical sum-product algorithm We prove that these codes are "very good", in that sequences of codes exist which, when optimally decoded, achieve information rates up to the Shannon limit This result holds not only for the binary-symmetric channel but also for any channel with symmetric stationary ergodic noise We give experimental results for binary-symmetric channels and Gaussian channels demonstrating that practical performance substantially better than that of standard convolutional and concatenated codes can be achieved; indeed, the performance of Gallager codes is almost as close to the Shannon limit as that of turbo codes

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Good error-correcting codes based on very sparse matrices

Download Introduction to Probability Models Sheldon M Download Pdf octave levenspiel solution manual pdf stochastic processes sheldon m ross pdf. Our nationwide network of sheldon m ross introduction to probability models solutions is dedicated to offering you the ideal service. With this kind of manual. MTL 106 (Introduction to Probability Theory and Stochastic Processes) 4 Credits Introduction to Probability Models, Sheldon M. Ross, Academic Press, ninth.

Introduction to Probability Models.

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Lectu re Notes in Economics and Mathematical Systems

Group representations in probability and statistics

We propose and axiomatize a model of preferences over acts such that the decision maker evaluates acts according to the expectation (over a set of probability measures) of an increasing transformation of an act's expected utility. This expectation is calculated using a subjective probability over the set of probability measures that the decision maker thinks are relevant given her subjective information. A key feature of our model is that it achieves a separation between ambiguity, identified as a characteristic of the decision maker's subjective information, and ambiguity attitude, a characteristic of the decision maker's tastes. We show that attitudes towards risk are characterized by the shape of the von Neumann-Morgenstern utility function, as usual, while attitudes towards ambiguity are characterized by the shape of the increasing transformation applied to expected utilities. We show that the negative exponential form of this transformation is the special case of constant ambiguity aversion. Ambiguity itself is defined behaviorally and is shown to be characterized by properties of the subjective set of measures. This characterization of ambiguity is formally related to the definitions of subjective ambiguity advanced by Epstein-Zhang (2001) and Ghirardato-Marinacci (2002). One advantage of this model is that the well-developed machinery for dealing with risk attitudes can be applied as well to ambiguity attitudes. The model is also distinct from many in the literature on ambiguity in that it allows smooth, rather than kinked, indifference curves. This leads to different behavior and improved tractability, while still sharing the main features (e.g., Ellsberg's Paradox, etc.). The Maxmin EU model (e.g., Gilboa and Schmeidler (1989)) with a given set of measures may be seen as an extreme case of our model with infinite ambiguity aversion. Two illustrative applications to portfolio choice are offered.

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A Smooth Model of Decision Making under Ambiguity

Introduction to the Theory of Queues.

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Combinatorial methods in the theory of stochastic processes

Let us suppose that customers arrive at a counter in accordance with a Poisson process of density λ. The customers are served by a single server in order of arrival. The service times are identically distributed, mutually independent, positive random variables with distribution function H(x). Suppose that after being served each customer either immediately joins the queue again with probability p or departs permanently with probability q (p + q =1). In this paper we shell determine for a stationary process the distribution of the queue size as well as the Laplace-Stieltjes transform and the first two moments of the distribution function of the total time spent in the system by a customer.

A single-server queue with feedback

Investigation of waiting time problems by reduction to Markov processes

Customers of different priorities are arriving at a counter in accordance with a Poisson process. The customers are served by a single server in order of priority and for each priority in order of arrival. Two cases are considered: i service with privileged interruptions and ii service without interruption. It is supposed that the service times are mutually independent random variables having a prescribed distribution for each priority and independent of the arrival times. In both cases i and ii a simple method is given for finding the Laplace-Stieltjes transform and the moments of the stationary distribution of the waiting time for each priority level.

Lajos Takács

Papers

Introduction to the Theory of Queues.

Combinatorial methods in the theory of stochastic processes

A single-server queue with feedback

Investigation of waiting time problems by reduction to Markov processes

Priority Queues