Showing papers on "Probability-generating function published in 2002"

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

Abstract: Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

Exact sampling formulas for multi-type Galton-Watson processes.

Abstract: Exact formulas for the mean and variance of the proportion of different types in a fixed generation of a multi-type Galton-Watson process are derived. The formulas are given in terms of iterates of the probability generating function of the offspring distribution. It is also shown that the sequence of types backwards from a randomly sampled particle in a fixed generation is a non-homogeneous Markov chain where the transition probabilities can be given explicitly, again in terms of probability generating functions. Two biological applications are considered: mutations in mitochondrial DNA and the polymerase chain reaction.

Generalized waiting time problems associated with pattern in Polya's urn scheme

Abstract: Let X1, X2, ... be a sequence obtained by Polya's urn scheme. We consider a waiting time problem for the first occurrence of a pattern in the sequence X1, X2, ... , which is generalized by a notion “score”. The main part of our results is derived by the method of generalized probability generating functions. In Polya's urn scheme, the system of equations is composed of the infinite conditional probability generating functions, which can not be solved. Then, we present a new methodology to obtain the truncated probability generating function in a series up to an arbitrary order from the system of infinite equations. Numerical examples are also given in order to illustrate the feasibility of our results. Our results in this paper are not only new but also a first attempt to treat the system of infinite equations.

Discrete-time ATM Queues with Independent and Correlated Arrival Streams

Abstract: In this tutorial paper, we present a set of fundamental techniques of analysis for discrete-time queueing models with either independent or correlated arrival streams, deterministic service-time distribution and one or more equivalent servers. The main characteristic of our approach is that it is almost entirely analytical (except for a few minor numerical calculations) and that an extensive use of probability generating functions is being made. The analysis leads to simple and accurate (exact or approximate) formulas for a wide variety of performance measures of practical importance, such as mean and variance of buffer occupancies and cell delays, cell loss probabilities, delay jitter, etc. The theory developed in the paper is also applied to the detailed performance evaluation of ATM multiplexers and ATM switching elements with dedicated-buffer output queueing arrangements.

Queueing analysis of some continuous ARQ strategies with repeated transmissions

Abstract: The statistical analysis of a class of continuous automatic repeat request (ARQ) strategies is performed. In the considered class, an optimal (throughput-maximising) number of copies of each data block is sent contiguously instead of one single copy. Explicit formulas for the probability generating functions of the queue length and the delay time in the transmitter buffer are obtained.

Koenigs function and fractional iterates of probability generating functions

Abstract: The Koenigs function arises as the limit of an appropriately normalized sequence of iterates of holomorphic functions. On the other hand it is a solution of a certain functional equation and can be used for the definition of iterates of the original function. A description of the class of Koenigs functions corresponding to probability generating functions embeddable in a one-parameter group of fractional iterates is provided. The results obtained can be regarded as a test for the embeddability of a Galton-Watson process in a homogeneous Markov branching process.

Asymptotic analysis of a random walk with a history-dependent step length.

Abstract: We study an unbiased, discrete-time random walk on the nonnegative integers, with the origin absorbing, and a history-dependent step length. Letting y denote the maximum distance the walker has ever been from the origin, steps that do not change y have length v, while those that increase y (taking the walker to a site that has never been visited), have length n. The process serves as a simplified model of spreading in systems with an infinite number of absorbing configurations. Asymptotic analysis of the probability generating function shows that, for large t, the survival probability decays as S(t) ∼ t δ , with � = v/2n. Our expression for the decay exponent is in agreement with results obtained via numerical iteration of the transition matrix.

Patterns in a sequence of multistate trials

Abstract: Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

Stochastic quasi-gradient algorithms for maximization of the probability function. A new formula for the gradient of the probability function

Abstract: New methods for constructing statistical estimates of the gradient of the probability function (stochastic quasi-gradient) are developed for the problems where the probability function is defined by a nonlinear system of inequalities with discontinuous, Lipschitz, and smooth functions. These methods are based on the representations of the probability function as an expectation of discontinuous, Lipschitz, and smooth functions. A new formula for the gradient of the probability function is obtained in the form of a volume integral over the unit simplex. The methods proposed are utilized in a stochastic quasi-gradient algorithm for maximization of the probability function.

A semi-infinite random walk associated with the game of roulette

Abstract: This paper is concerned with a discrete time random walk on the integers 0,1,2,... which arises in the game of roulette. At each step either a unit displacement to the left with probability 1-p or a fixed multiple displacement to the right with probability p can occur. There is a partially absorbing barrier at the origin, the probabilities of reflection and absorption at?0 are ? and 1-?, respectively. Using generating functions and Lagrange's theorem for the expansion of a function as a power series, explicit expression for the probabilities of the player's capital at the nth step are deduced, as well as the probabilities of ultimate absorption at the origin.

On Minimal-Moment Generating Functions

Abstract: A formula expressing the inverse cumulative distribution function of a non-negative random variable in terms of contour integrals of its minimal-moment generating function is proved as well as an analog of the classical continuity theorem for characteristic functions.

Waiting for the Pattern 123123 . . . 123 in Independent Trinomial Observations.

Abstract: The method for geometric transforms (probability generating functions) is used to study the expected number of observations until the pattern 123123 … 123 is obtained. These results provide a first generalization of similar problems considered by other authors.

Study of Optimal Fitting Method for Probability Density Function of Traffic Flow

Abstract: Probability density function of traffic flow varies with different survey data It is difficult to find a probability density function fitting for all kinds of traffic survey data Thus an ordinary method of generating probability density function is greatly important for traffic researcher and engineer In this paper , the authors analyzed the problem from the viewpoint of maximum entropy method and derived a practical formula based on least square approximation principle and its algorithm The two methods are proved to be really effective in theoretical simulation and practice examination

Randomly evolving trees I

Abstract: Generating function equation has been derived for the probability distribution of the number of nodes with $k \ge 0$ outgoing lines in randomly evolving special trees. The stochastic properties of end-nodes (k=0) have been analyzed, and it was shown that the relative variance of the number of end-nodes vs. time has a maximum when the evolution is either subcritical or supercritical. On the contrary, the time dependence of the relative dispersion of the number of dead end-nodes shows a minimum at the beginning of the evolution independently of its type. For the sake of better understanding of the evolution dynamics the survival probability of random trees has been investigated, and asymptotic expressions have been derived for this probability in the cases of subcritical, critical and supercritical evolutions. In critical evolution it was shown that the probability to find the tree lifetime larger than x, is decreasing to zero as 1/x, if x tends to infinity. Approaching the critical state it has been found the fluctuations of the tree lifetime to become extremely large, and so near the critical state the average lifetime could be hardly used for the characterization of the process.

[d1 확률모형] 이산시간 2단계 대기행렬시스템의 분석

Abstract: In this paper, we consider a discrete-time two-phase queueing system. We derive the PGF of the system size and show that it is decomposed into two probability generating functions (PGFs), one of which is the PGF of the system size in the standard Geo/G/1 queue. Based on this PGF, we present the performance measure of interest such as the mean number of customers in the system.

An Expansion of the Probability Mass Function that may be used to Improve Poisson Approximations

Abstract: Random variables defined on the natural numbers may often be approximated by Poisson variables. Just as normal approximations may be improved by the Edgeworth expansion, Poisson approximations may be substantially improved by a similar expansion. Both of these expansions require one to approximate a generating function. This paper will derive a separate expansion for improving Poisson approximations to tail probabilities by approximating the probability mass function directly. This new expansion is often useful for points in the tails of the distribution. Examples will be presented to illustrate the usefulness and accuracy of this new expansion.

A new class of variable-length uniform random number generators and their asymptotic properties

Abstract: We propose a new class of variable-length uniform random number generators (URNGs) and investigate their asymptotic properties. It is shown that (i) any variable-length URNG in the class is asymptotically optimal for any given general source, and (ii) the output length of such URNG per source symbol converges in probability to the self-information of the source per source symbol.

The role of the loss function in the probabilistic function approximation

Abstract: Generalising results on time series estimation, it is natural to consider function approximation with finite data observations in a probabilistic setting. The function is treated as a stochastic process where for each point in the functions domain, the function is a random variable. Equivalently the function can be considered as a single random variable whose range is a space of functions. In this paper, two results, well known within the the context of time series estimation and stochastic control, are generalised to probabilistic function approximation problems. Under mild conditions, on the space of functions, it is shown that the optimal function estimate corresponds, for all reasonable symmetrical loss functions, to the pointwise conditional expectation given the observed data. Further, in the case where the space of functions belongs to the class of Gaussian processes the optimal estimate is the conditional expectation even for asymmetric loss functions.

Probabilistic ostrowski type inequalities

Abstract: New very general univariate and multivariate probabilistic Ostrowski type inequalities are established, involving ‖·‖∞ and ‖·‖ p , p≥1 norms of probability density functions. Some of these inequalities provide pointwise estimates to the error of probability distribution function from the expectation of some simple function of the engaged random variable. Other inequalities give upper bounds for the expectation and variance of a random variable. All are done over finite domains. At the end are given applications, especially for the Beta random variable.