Showing papers on "Probability-generating function published in 2005"

Sequential Window Problems

Abstract: Window problems have an important application when there is concern about the clustering of nonconforming units or when there is an early surge of information that allows us to terminate an experiment sooner than usual. Motivated by these ideas, in this paper we use probability generating functions to study the stopping rules and their effects related to waiting time random variables. The use of windows introduces many new challenging probability questions, but it also requires basic tables that are not now available; some of these tables are included in the present paper. Since waiting-time random variables and scan statistics are closely related, the exact results and the computational method in this paper could also be useful in scan statistics. In the final section we apply the sequential-window concept to the possibility of changing the usual rules for a chess tournament and show that they could lead to a smaller expected number of games and to a higher chance of finding the better player. R...

A discrete-time HOL priority queue with multiple traffic classes

Abstract: Priority scheduling for packets is a hot topic, as interactive (voice,video) services are being integrated in existing data networks. In this paper, we consider a discrete-time queueing system with non-preemptive (or Head-Of-the-Line) priority scheduling and a general number of priority classes. Packets of variable length arrive in the queueing system. We derive expressions for the probability generating functions of the packet delays. From these functions, some performance measures (such as moments and approximate tail probabilities) are calculated. We apply the theoretical results to a queue that handles arriving multimedia traffic.

Mathematical expectation about discrete random variable with interval probability or fuzzy probability

Abstract: The character and an algorithm about DRVIP(discrete random variable with interval probability) and the second kind DRVFP (discrete random variable with crisp event-fuzzy probability) are researched. Using the fuzzy resolution theorem, the solving mathematical expectation of a DRVFP can be translated into solving mathematical expectation of a series of RVIP. It is obvious that solving mathematical expectation of a DRVIP is a typical linear programming problem. A very functional calculating formula for solving mathematical expectation of DRVIP was obtained by using the Dantzig’s simplex method. The example indicates that the result obtained by using the functional calculating formula fits together completely with the result obtained by using the linear programming method, but the process using the formula deduced is simpler.

The limit random variable W of a branching process

Abstract: A formal Taylor series approach for the generating function of the limit random variable W of a branching process is presented. The framework is applicable to any production distribution function for which all moments exist. The Taylor coefficients show an interesting relation to Gaussian polynomials. The application of the formal series approach to the Poisson production function leads to (a) a modular-like functional equation for the moment generating function of W and (b) two different series for the probability distribution function of W . 1 The Limit Random Variable W We consider a branching process in which each offspring produces a number of items independent from the others but with same distribution. Let Xk denote the total number of items produced in generation k and let the i.i.d. production in any generation be specified by the non-negative discrete random variable Y with E [Y ] = μ > 1. The set Xk describes the evolution of a branching process over the generations k. The scaled random variables {Wk}k≥1 defined by Wk = Xk μk constitute a martingale process with characteristic property that E [Wk] = E [X0] for all k. It is known [6] that the limit variable W = limk→∞Wk exists if μ > 1. In the sequel, we confine to the case where X0 = 1 and, mostly, μ > 1. The moment generating function (mgf) χW (t) = E £ e−tW ¤ obeys the functional equation for Re(t) ≥ 0, χW (t) = φY μ χW μ t μ ¶¶ (1) where φY (z) = E £ zY ¤ is production generating function. The limit t→∞ exists and limt→∞ χW (t) = Pr [W = 0] = π0 is the extinction probability which obeys, as follows from (1), the well-known equation π0 = φY (π0). The main motivation for this study was the computation of the limit random variable W that appeared in the distribution of the hopcount or distance between two arbitrary nodes in graphs with finite variance degree distributions [7, 12]. Although many theoretical results are available (see e.g. [6],[10],[3]), less effort has been devoted to compute the mgf χW (t) and the probability density function fW (x) of W . ∗Faculty of Electrical Engineering, Mathematics and Computer Science, P.O Box 5031, 2600 GA Delft, The Netherlands; email : P.VanMieghem@ewi.tudelft.nl

Approximating pseudo-Boolean functions on non-uniform domains

Abstract: In Machine Learning (ML) and Evolutionary Computation (EC), it is often beneficial to approximate a complicated function by a simpler one, such as a linear or quadratic function, for computational efficiency or feasibility reasons (cf. [Jin, 2005]). A complicated function (the target function in ML or the fitness function in EC) may require an exponential amount of computation to learn/evaluate, and thus approximations by simpler functions are needed. We consider the problem of approximating pseudo-Boolean functions by simpler (e.g., linear) functions when the instance space is associated with a probability distribution. We consider {0, 1}n as a sample space with a (possibly nonuniform) probability measure on it, thus making pseudo-Boolean functions into random variables. This is also in the spirit of the PAC learning framework of Valiant [Valiant, 1984] where the instance space has a probability distribution on it. The best approximation to a target function f is then defined as the function g (from all possible approximating functions of the simpler form) that minimizes the expected distance to f. In an example, we use methods from linear algebra to find, in this more general setting, the best approximation to a given pseudo-Boolean function by a linear function.

The Poisson approximation for the number of matches of values of a discrete function on segments of a sequence of random variables

Abstract: We find conditions which are sufficient for convergence of the distribution of the number of matches of values of a function considered on tuples of arguments taken from a sequence of independent identically distributed random variables to the Poisson law and estimate the convergence rate. We derive a series of corollaries of this result. In particular, in the equiprobable polynomial scheme we obtain Poisson limit theorems for the number of pairs of non-overlapping tuples with identical frequencies of occurrences of symbols and for the number of pairs of tuples with identical structure.

Delay analysis of the go-back-n ARQ protocol over a time-varying channel

Abstract: In this paper we present an analytical technique, based on the use of probability generating functions, to analyze the throughput performance and the transmitter buffer behavior of a Go-Back-N ARQ system, with the notable complication that the errors in the channel are correlated in time. We model the transmitter buffer as a discrete-time queue with infinite storage capacity and independent and identically distributed packet arrivals. Arriving packets are stored in the queue until they are successfully transmitted over the channel. The probability of an erroneous transmission is modulated by a general Markov chain with M states, rather than assuming stationary channel errors. We find explicit expressions for the probability generating functions of the buffer content and packet delay. From these functions moments and tail probabilities can be derived. Numerical results illustrate the impact of the error process on the system performance.

Heterogeneous bursty traffic dispersion over multiple server clusters

Abstract: This letter presents performance analysis of multiple subnets, each representing a cluster of computing server nodes, that are introduced with non-uniformly distributed bursty packet arrivals. In particular, we study the case of a multi-state Markov-modulated arrival process, heterogeneously dispersed among designated queues. Cluster processing is modeled by employing a batch service discipline. The probability generating functions of the interarrival times distributions are utilized to derive closed-form expressions for each of the queue size distributions.

On the properties of asymptotic probability for random boolean expression values in binary bases

Abstract: The present paper deals with the problem of analyzing the value of a random Boolean expression The expressions are constructed of Boolean operations and constants chosen independently at random with given probabilities The dependence between the expression value probability and the constants' probabilities is investigated for different sets of operations The asymptotic behavior of this dependence is given by a probability function, explicitly obtained through analysis of generating functions for expressions Special attention is given to the case of binary Boolean operations The paper demonstrates some probability function properties and their connection with the properties of Boolean operations used in random expressions

Analysis of discrete-time multiserver queues with constant service times and correlated arrivals

Abstract: We investigate the behavior of a discrete-time multiserver buffer system with infinite buffer size. Packets arrive at the system according to a two-state correlated arrival process. The service times of the packets are assumed to be constant, equal to multiple slots. The behavior of the system is analyzed by means of an analytical technique based on probability generating functions (pgf’s). Explicit expressions are obtained for the pgf’s of the system contents and the packet delay. From these, the moments and the tail distributions of the system contents and the packet delay can be calculated. Numerical examples are given to show the influence of various model parameters on the system behavior.

Stochastic Theory of Rate Processes

Abstract: This chapter is written under the assumption that the reader has basic knowledge of probability theory such as needed in an elementary course in statistical mechanics, and at least an intuitive feeling for random numbers as those generated by tossing a coin or throwing a die. A random function is a function that assigns a random number to each value of its argument. Using this argument as an ordering parameter, each realization of this function is an ordered sequence of such random numbers. When the ordering parameter is time we have a time series of random variables, which is called a stochastic process. For example, the random function F(t) that assign to each time t the number of cars on a given highway segment is a random function of time, i.e., a stochastic process. Time is a continuous ordering parameter, however if observations of the random function z(t) are made at discrete time 0 < t1 < t2,…, < tn < T, then the sequence z(ti) is a discrete sample of the continuous function z(t).

On the two-plied oriented percolation on the square lattice and its critical probability function

Abstract: In this paper, we study the two-plied oriented percolation on the square lattice Z2. We prove that the corresponding critical probability function is monotone, symmetrical and continuous. In addition, we point out that the related Grimmett's conjecture follows from the strict concavity of this critical probability function.

Comment on: a hazard function approximation used in reliability theory

Abstract: In a recent paper, Keren & Mehrez have shown that in the case of a geometric probability mass function, the approximation to the cumulative hazard function converges to a finite value (1.606695), when the actual value of the cumulative hazard function should be diverging. In this commentary, we show that gives incorrect results, and that the corresponding results for the approximation are actually diverging.

Mathematical Expectation About Discrete Random Variable With Interval Probability(DRVIP) or Fuzzy Probability(DRVFP)

Abstract: The character and an algorithm about DRVIP(discrete randam variable with interval probability) and the second kind DRVFP (discrete random variable with crisp event-fuzzy probability) are researchedUsing the fuzzy resolution theorem,the solving mathematical expectation of a DRVFP can be translated into solving mathematical expectation of a series of RVIPIt is obvious that solving mathematical expectation of a DRVIP is typically solving a linear programming problemA very functional calculating formula solving mathematical expectation of DRVIP was obtained by using the Dantzig's simplex methodThe example indicates that the result obtained by using the functional calculate formula fits together completely with the result obtained by using the linear programming method,but the process using the formula deduced is simpler

A Note On Generating Systems Composed Of Binary And Ternary Random Variables

Abstract: We present some tips for generating multiple sequences of two- and three-valued random variables.

Seismic Behaviors of a Bridge System in the Stochastic Perspectives

Abstract: Semi-analytical methodology to examine the dynamic responses of a bridge is developed via the joint probability density function. The evolution of joint probability density function is evaluated by the semi-analytical procedure developed. The joint probability function of the bridge responses can be obtained by solving the path-integral solution of the Fokker-Planet equation corresponding to the stochastic differential equations of the system. The response characteristics are observed from the joint probability density function and the boundary of the envelope of the probability density function can provide the maxima ol the bridge responses.