Showing papers on "Probability-generating function published in 2000"

The probabilistic solutions to nonlinear random vibrations of multi-degree-of-freedom systems

Abstract: The probability density function of the responses of nonlinear random vibration of a multi-degree-of-freedom system is formulated in the defined domain as an exponential function of polynomials in state variables. The probability density function is assumed to be governed by Fokker-Planck-Kolmogorov (FPK) equation. Special measure is taken to satisfy the FPK equation in the average sense of integration with the assumed function and quadratic algebraic equations are obtained for determining the unknown probability density function. Two-degree-of-freedom systems are analyzed with the proposed method to validate the method for nonlinear multi-degree-of-freedom systems. The probability density functions obtained with the proposed method are compared with the obtainable exact and simulated ones. Numerical results showed that the probability density function solutions obtained with the presented method are much closer to the exact and simulated solutions even for highly nonlinear systems with both external and parametric excitations.

Discrete-time queues with general service times and general server interruptions

Abstract: In this contribution, we investigate a discrete-time single- server queue subjected to server interruptions. Server interruptions are modeled as an on/off process with geometrically distributed on-periods and generally distributed off-periods. As message lengths can exceed one time-slot, different operation modes are considered depending on whether service of an interrupted message continues, partially restarts or completely restarts after an interruption. For all alternatives, we establish expressions for the steady-state probability generating functions of the buffer contents at message departure time and at random slot boundaries. From these results, closed- form expressions for various performance measures, such as mean and variance of the buffer occupancy, can be established. As an application, we show that this model is able to assess performance of low-priority traffic in a two- priority HOL scheduling discipline. We then illustrate our approach with some numerical examples.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Random matrix theory over finite fields: a survey

Abstract: First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful. Connections are made with symmetric function theory, Markov chains, potential theory, Rogers-Ramanujan type identities, quivers, and various measures on partitions.

Waiting time problems in a two-state Markov chain

Abstract: Let F0 be the event that l0 0-runs of length k0 occur and F1 be the event that l1 1-runs of length k1 occur in a two-state Markov chain. In this paper using a combinatorial method and the Markov chain imbedding method, we obtained explicit formulas of the probability generating functions of the sooner and later waiting time between F0 and F1 by the non-overlapping, overlapping and "greater than or equal" enumeration scheme. These formulas are convenient for evaluating the distributions of the sooner and later waiting time problems.

Closed polling models with failing nodes

Abstract: Closed polling systems with station breakdowns, under the gated, exhaustive or globally gated services regimes, are studied and analyzed. Multi-dimensional sets of probability generating functions of the system’s state are derived. They are further utilized to obtain an approximate solution for the mean number of jobs residing in the system’s various queues at polling instants. The analysis is then concentrated on the case of cyclic Bernoulli polling. Explicit formulae for the mean number of jobs, as well as for the expected cycle duration and system utilization, are derived. Comparison of the throughputs of the three regimes concludes the paper.

General discrete-time queueing systems with correlated batch arrivals and departures

Abstract: Discrete time queueing systems have gained attention recently due to their applications in the performance analysis of ATM and other systems. In this paper, we analyze generic discrete-time queueing models with general distribution for batch arrivals and departures. Our models allow correlation and distributions with arbitrary rational probability generating functions. Our solution methodology is significantly different, compared with the traditional methods. We provide a state-space representation of the model resulting in an exact simple matrix geometric solution of the system probability vector. Our approach is algorithmic, numerically robust and efficient. Applications in ATM systems and classical (discrete) G/G/1 queues with semi-Markovian arrival and departure processes are immediate and are discussed.

CGTDEMO — educational software for the central limit theorem

Abstract: This paper describes the demo package for teaching and visualizing the Central limit theorem. The topic treated in this paper is of significant interest in undergraduate coverage of non-deterministic systems. Two types of random variables are considered: symmetric, highly concentrated about its mean value and nonsymmetrical random variables. The demo program is developed in MATLAB 5.2. The program gives the user step by step guidance. The user chooses the type of variable, the length of the sum N, and the corresponding parameters of a random variable. Successive plots of the sums of random variables and the estimations of the corresponding probability density functions are obtained. Finally the comparison with a normal variable is given.

Mathematic description about random variable with fuzzy density function (RVFDF)

Abstract: The random variable with fuzzy probability caused by fuzziness of probability density function was studied. The basic concepts/ definitions and calculating methods of the interval/fuzzy probability density function, the random variable with fuzzy density function (RVFDF) and its distribution function, mathematical expectation and variance are given and same theorems related to the RVFDF are proved.

Sooner and later waiting time problems based on a dependent sequence

Abstract: This paper introduces a new concept: a binary sequence of order (k,r), which is an extension of a binary sequence of order k and a Markov dependent sequence. The probability functions of the sooner and later waiting time random variables are derived in the binary sequence of order (k,r). The probability generating functions of the sooner and later waiting time distributions are also obtained. Extensions of these results to binary sequence of order (g,h) are also presented.

Analysis of buffers with Markov modulated server interruptions

Abstract: In this paper we study a discrete-time single-server queue with deterministic service times, where the server is subject to random server interruptions. We consider both the case where the service of a message can continue after an interruption and the case where the server has to restart the processing of the complete message after an interruption. For both alternatives, we derive closed-form expressions for the probability generating functions of the buffer contents (i.e., the number of messages in the system) and the unfinished work (i.e., the number of service time units required to empty the system). This eventually leads to explicit results for various performance measures such as the moments and the tail distribution of the system contents and the unfinished work.