Showing papers on "Probability-generating function published in 2008"
TL;DR: The statistical distribution of capture times is obtained from Monte Carlo calculations and shows a crossover from power-law to exponential behavior, and predicts the distribution function for a lattice with perfect mixing.
168 citations
TL;DR: The (partial) probability generating functions of the number of customers present when the system is occupied with a U-task as well as when it acts as an M/M/∞ queue are derived and explicit expressions for the corresponding mean queue sizes are obtained.
Abstract: A system is operating as an M/M/∞ queue. However, when it becomes empty, it is assigned to perform another task, the duration U of which is random. Customers arriving while the system is unavailable for service (i.e., occupied with a U-task) become impatient: Each individual activates an “impatience timer” having random duration T such that if the system does not become available by the time the timer expires, the customer leaves the system never to return. When the system completes a U-task and there are waiting customers, each one is taken immediately into service. We analyze both multiple and single U-task scenarios and consider both exponentially and generally distributed task and impatience times. We derive the (partial) probability generating functions of the number of customers present when the system is occupied with a U-task as well as when it acts as an M/M/∞ queue and we obtain explicit expressions for the corresponding mean queue sizes. We further calculate the mean length of a busy period, the mean cycle time, and the quality of service measure: proportion of customers being served.
64 citations
TL;DR: In this paper, stress and strength are treated as discrete random variables, and a discrete SSI model is presented by using the universal generating function (UGF) method to demonstrate the validity of the discrete model in a variety of circumstances.
61 citations
19 Mar 2008
TL;DR: A Gaussian approximation is proposed that accurately models the information-outage probability for moderately small codes and it is shown that theInformation- outage probability is a useful predictor of achievable error rate.
Abstract: The performance of random error control codes approaches the Shannon capacity limit as the code length goes to infinity. When the code length is finite, then the code will be unable to achieve arbitrarily low error probability and a nonzero codeword error rate is inevitable. Information-theoretic bounds on codeword error rate may be found as a function of length through traditional methods such as sphere packing. Alternatively, the behavior of finite-length codes can be characterized in terms of an information-outage probability. The information- outage probability is the probability that the mutual information, which is a random variable, is less than the rate. In this paper, a Gaussian approximation is proposed that accurately models the information-outage probability for moderately small codes. The information-outage probability is related to several previously derived bounds, including Shannon's sphere-packing and random coding bounds, as well as a bound on maximal error probability known as Feinstein's lemma. It is shown that the information- outage probability is a useful predictor of achievable error rate.
46 citations
Journal Article•
01 Jan 2008-Nuclear Instruments & Methods in Physics Research Section B-beam Interactions With Materials and Atoms
TL;DR: In this article, a stochastic neutron transport theory with isotropic scatting was applied to a point model and an approximate solution for the generating function and the equations for moments of the probability distribution and their solutions were derived.
Abstract: A stochastic neutron transport theory,in which we consider the probability PN(r,t,uΩ) that the neutron densities Ni(i=1,2,…,n) emerge in the phase space point(r,uiΩi) at time t respectively,was given by means of the probability theory,and a set of non-linear integral-differential equations for the probability generating functions Fn(r,t,uΩ,S) was derived.The equation for one-order moment F1/S1 under some approximation is just the Boltzman equation for the average neutron number.One-velocity neutron stochastic theory with isotropic scatting was applied to a point model.An approximate solution for the generating function and the equations for moments of the probability distribution and their solutions were derived.It is shown that in a supercritical system,at t→∞,the probability appearing finite neutrons is zero,PN=0(0N∞),in other words,the system has no or infinite neutrons.A formula for standard deviation shows that the fluctuation of neutron number in the near critical(0λ1) system should be paid our attention when the fluctuation of initial neutron number ξ0 is larger and the initial neutron average number 0 is not large enough,or neutron source Q is weaker.
30 citations
TL;DR: In this paper, the authors proposed a new estimation method by minimizing a suitable weighted L 2 -distance between the empirical and the theoretical probability generating functions, and provided a goodness-of-fit statistic based on the same distance.
Abstract: The discrete stable family constitutes an interesting two-parameter model of distributions on the non-negative integers with a Paretian tail The practical use of the discrete stable distribution is inhibited by the lack of an explicit expression for its probability function Moreover, the distribution does not possess moments of any order Therefore, the usual tools—such as the maximum-likelihood method or even the moment method—are not feasible for parameter estimation However, the probability generating function of the discrete stable distribution is available in a simple form Hence, we initially explore the application of some existing estimation procedures based on the empirical probability generating function Subsequently, we propose a new estimation method by minimizing a suitable weighted L 2-distance between the empirical and the theoretical probability generating functions In addition, we provide a goodness-of-fit statistic based on the same distance
24 citations
01 Jan 2008
TL;DR: In this article, some interesting applications of Dirac's delta function in statistics have been discussed and extended to the more than one variable case while focusing on the bivariate case of the Dirac delta function.
Abstract: In this paper, we discuss some interesting applications of Dirac's delta function in Statistics We have tried to extend some of the existing results to the more than one variable case While doing that, we particularly concentrate on the bivariate case
22 citations
01 Oct 2008
TL;DR: Several stationary integer-valued first-order autoregressive [INAR(1)] models with discrete semistable marginals and related distributions with corresponding first- order moving average processes are proposed.
Abstract: We propose several stationary integer-valued first-order autoregressive [INAR(1)] models with discrete semistable marginals and related distributions. The corresponding first-order moving average processes are also presented.
18 citations
07 Apr 2008
TL;DR: This paper considers a discrete-time queueing model with infinite storage capacity and one single output line, which can for instance be applied to study the traffic of a file server, where one file download by a user corresponds to one session.
Abstract: This paper considers a discrete-time queueing model with infinite storage capacity and one single output line. Users can start and end sessions during which they are active and send packets to the queueing system. Each active user generates a random but strictly positive number of packets per time slot: this results in a session-based arrival process of packets. This model can for instance be applied to study the traffic of a file server, where one file download by a user corresponds to one session. The steady-state probability generating functions of the number of active sessions, buffer occupancy (number of packets stored in the buffer) and packet delay are derived. We also derive an approximation for the tail probabilities of the buffer occupancy. This allows us to study the influence of the different system parameters: some examples are presented.
10 citations
TL;DR: In this article, the cumulative distribution function of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases.
Abstract: The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in the case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of “indicator” variables. A connection is studied between Y and order statistics of the set of arguments.
10 citations
TL;DR: It is shown that discrete random variables taking real values, nonnecessarily integer or rational, may be studied with probability generating functions and drawn attention to some practical ways of performing the necessary calculations.
Abstract: The probability generating function is a powerful technique for studying the law of finite sums of independent discrete random variables taking integer positive values. For real-valued discrete random variables, the well-known elementary theory of Dirichlet series and the symbolic computation packages available nowadays, such as Mathematica 5, allow us to extend this technique to general discrete random variables. Being so, the purpose of this work is twofold. First, we show that discrete random variables taking real values, nonnecessarily integer or rational, may be studied with probability generating functions. Second, we intend to draw attention to some practical ways of performing the necessary calculations.
TL;DR: In this paper, it was shown that a random function in the Hardy space is a non-cyclic vector for the backward shift operator almost surely, and the question of existence of a local pseudocontinuation for a random analytic function is also studied.
Abstract: We prove that a random function in the Hardy space $H^2$ is a non-cyclic vector for the backward shift operator almost surely. The question of existence of a local pseudocontinuation for a random analytic function is also studied.
TL;DR: Approaches of such random sums in risk treatment, backup, outsourcing and other operations of computer systems are established.
Abstract: The construction and investigation of discrete random sums are generally considered as very important activities of probability theory with significant applications in many fields. The paper concentrates on the construction of a random sum of integral part models. The paper also evaluates the probability generating function of the proposed random sum. Moreover, the paper establishes applications of such random sums in risk treatment, backup, outsourcing and other operations of computer systems.
01 Jan 2008
TL;DR: In this paper, an inequality of the Ostrowski type for a random variable whose probability density function belongs to Lp[a,b], in terms of the cumulative distribution function and expectation is established.
Abstract: We establish here an inequality of Ostrowski type for a random variable whose probability density function belongs to Lp[a,b], in terms of the cumulative distribution function and expectation The inequality is then applied to generalized beta random variable
TL;DR: The good agreement with simulation results for relatively small P2P networks makes the asymptotic formula for the probability density function useful for estimating the minimal number of peers to offer an acceptable quality.
Abstract: We model the weight (e.g., delay, distance, or cost) from an arbitrary node to the nearest (in weight) peer in a peer-to-peer (P2P) network. The exact probability generating function and an asymptotic analysis is presented for a random graph with independent and identically distributed exponential link weights. The asymptotic distribution function is a Fermi–Dirac distribution that frequently appears in statistical physics. The good agreement with simulation results for relatively small P2P networks makes the asymptotic formula for the probability density function useful for estimating the minimal number of peers to offer an acceptable quality (delay or latency).
TL;DR: In this article, the authors introduce a transformation which converts the probability generating function of an integral part model, based on a discrete random variable with values in the set of nonnegative integers and probability function having a unique mode at the point zero, into the probability generator function of a self-decomposable distribution.
Abstract: Stochastic interpretations in risk control operations for the class of discrete distributions corresponding to discrete random variables with values in the set of nonnegative integers and unique mode at the point zero, the class of discrete selfdecomposable distributions and the class of discrete distributions corresponding to integral part models are established by the paper. Moreover, the paper introduces a transformation which converts the probability generating function of an integral part model, based on a discrete random variable with values in the set of nonnegative integers and probability function having a unique mode at the point zero, into the probability generating function of a discrete random variable with values in the set of nonnegative integers and selfdecomposable distribution.
TL;DR: The behavior of a discrete-time multiserver buffer system with infinite buffer size is studied by means of an analytical technique based on probability generating functions (PGF’s) for the mean values, the variances and the tail distributions of the system contents and the packet delay.
Abstract: In this paper, we study the behavior of a discrete-time multiserver buffer system with infinite buffer size. Packets arrive at the system according to a two-state Markovian 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 mean values, the variances and the tail distributions of the system contents and the packet delay are calculated. Numerical examples are given to show the influence of various model parameters on the system behavior.
TL;DR: In this paper, a trinomial random walk, with one or two barriers, on the nonnegative integers is discussed, where the particle is either annihilated or reflects back to the system with respective probabilities 1 − ρ, ρ at the origin and 1 − ε, ω at L, 0 ≤ ε ≤ ρ ≤ 1.
Abstract: Trinomial random walk, with one or two barriers, on the non-negative integers is discussed. At the barriers, the particle is either annihilated or reflects back to the system with respective probabilities 1 − ρ, ρ at the origin and 1 − ω, ω at L, 0 ≤ ρ,ω ≤ 1. Theoretical formulae are given for the probability distribution, its generating function as well as the mean of the time taken before absorption. In the one-boundary case, very qualitatively different asymptotic forms for the result, depending on the relationship between transition probabilities and the annihilation probability, are obtained.
TL;DR: In this paper, an inequality of the Ostrowski type for a random variable whose probability density function belongs to Lp[a, b], in terms of the cumulative distribution function and expectation, is established.
Abstract: Abstract We establish here an inequality of Ostrowski type for a random variable whose probability density function belongs to Lp[a, b], in terms of the cumulative distribution function and expectation. The inequality is then applied to generalized beta random variable.
Journal Article•
TL;DR: In this paper, a moment method is employed to calculate general failure probability for low and medium dimension problems with fuzzy failure state and fuzzy safety state, and the results of a number of numerical and engineering examples show that the proposed method possesses high precision for low-and medium-dimensional problems.
Abstract: For general reliability analysis with fuzzy failure state and fuzzy safety state, moment method is proposed to calculate general failure probability. In the integral operation of the general failure probability, the integral region is scattered according to the value of performance function. In the scattered integral region, the membership function of the performance function to the fuzzy failure region keeps approximately constant, with which the calculation of the general failure probability is transformed into that of the random failure probability. After the transformation from the general failure probability to the random failure probability is completed, moment method is employed to obtain the general failure probability directly. The concept and the implementation of the proposed method are described step by step, and the results of a number of numerical and engineering examples show that the proposed method possesses high precision for low and medium dimension problems.
01 Dec 2008
TL;DR: In this article, the authors give a sufficient condition for a non-negative random variable X being heavy tailed, which is based on the complex Tauberian theorem by Graham-Vaaler.
Abstract: In this paper, we give a sufficient condition for a non-negative random variable X being heavy tailed. A light tailed case is studied in. The sufficient condition is an analytic property of Laplace-Stieltjes transform of the probability distribution function of X. The proof is based on the complex Tauberian theorem by Graham-Vaaler.
TL;DR: In this article, the empirical mass function associated with a sequence of i.i.d. discrete random variables converges in l r at the ( n / log 2 n ) 1 / 2 rate, for all r ≥ 2.
TL;DR: In this paper, the authors considered approximations to carrier-borne epidemic processes, including when the immigration of carriers or susceptibles is allowed, and derived the partial differential equations for the probability generating functions of the approximating processes.
Abstract: Summary
This paper considers approximations to carrier-borne epidemic processes, including when the immigration of carriers or susceptibles is allowed. The partial differential equations for the probability generating functions of the approximating processes are derived, and their solutions are obtained.
01 Jan 2008
TL;DR: In this article, the authors introduce the notion of interval distribution function of random events on the set of elementary events and the interval function of the frequencies of these events, which enables one to define the probability of occurrence of random event as a result of the limit transition.
Abstract: We introduce the notion of interval distribution function of random events on the set of elementary events and the notion of interval function of the frequencies of these events. In the limiting case, the interval function turns into the ordinary distribution function and the interval function of frequencies (under certain conditions) turns into the density of distribution of random events. The case of discrete sets of elementary events is also covered. This enables one to introduce the notion of the probability of occurrence of random events as a result of the limit transition.
18 Oct 2008
TL;DR: It is proven that characteristic function for random fuzzy variable phi(t 1, t2) is differentiable in t and t, respectively, and partial derivatives of phi (t1, t2) are continuous.
Abstract: A new concept of characteristic function for random fuzzy variable is introduced and some properties of characteristic function are obtained. It is proven that characteristic function for random fuzzy variable phi(t1, t2) is differentiable in t1 and t2, respectively, and partial derivatives of phi(t1,t2) are continuous. Furthermore, the Taylor's expansion of phi(t1,t2) is given and the relation between chancedistribution and characteristic function such as inversion formulais obtained.
TL;DR: In this paper, the authors explore conditions for a class of functions defined via an integral representation to be a probability generating function of some positive integer valued random variable, and show that there exist probability generating functions with our integral representation which are not discrete SDFR, but when used as shock resistance probabilities can give rise to a SDFr survival distribution in continuous time.
Abstract: We explore conditions for a class of functions defined via an integral representation to be a probability generating function of some positive integer valued random variable Interest in and research on this question is motivated by an apparently surprising connection between a family of classic shock models due to Esary et al (1973) and the negatively aging nonparametric notion of ``strongly decreasing failure rate'' (SDFR) introduced by Bhattacharjee (2005) A counterexample shows that there exist probability generating functions with our integral representation which are not discrete SDFR, but when used as shock resistance probabilities can give rise to a SDFR survival distribution in continuous time