Showing papers on "Probability-generating function published in 2016"

Refined large deviations asymptotics for Markov-modulated infinite-server systems

Abstract: Many networking-related settings can be modeled by Markov-modulated infinite-server systems. In such models, the customers' arrival rates and service rates are modulated by a Markovian background process, additionally, there are infinitely many servers (and consequently the resulting model is often used as a proxy for the corresponding many-server model). The Markov-modulated infinite-server model hardly allows any explicit analysis, apart from results in terms of systems of (ordinary or partial) differential equations for the underlying probability generating functions, and recursions to obtain all moments. As a consequence, recent research efforts have pursued an asymptotic analysis in various limiting regimes, notably the central-limit regime (describing fluctuations around the average behavior) and the large-deviations regime (focusing on rare events). Many of these results use the property that the number of customers in the system obeys a Poisson distribution with a random parameter. The objective of this paper is to develop techniques to accurately approximate tail probabilities in the large-deviations regime. We consider the scaling in which the arrival rates are inflated by a factor $N$, and we are interested in the probability that the number of customers exceeds a given level $Na$. Where earlier contributions focused on so-called $logarithmic$ $asymptotics$ of this exceedance probability (which are inherently imprecise), the present paper improves upon those results in that $exact$ $asymptotics$ are established. These are found in two steps: first the distribution of the random parameter of the Poisson distribution is characterized, and then this knowledge is used to identify the exact asymptotics. The paper is concluded by a set of numerical experiments, in which the accuracy of the asymptotic results is assessed.

A discrete-time Geom/G/1 retrial queue with balking customers and second optional service

Abstract: In this paper, we discuss a discrete-time Geom/G/1 retrial queue with balking customers and second optional service where the retrial time follows a geometrical distribution. If an arriving customer finds the server is busy, he will leave the service area and go to the orbit with probability θ or leave the system with probability 1−θ; otherwise, he will begin his service immediately. In this model, after a customer finishes his first essential service, he may leave the system with probability 1−α or asks for a second optional service with probability α. Through studying the Markov chain underlying the model, we establish the probability generating functions of the orbit size and system size. Finally, some performance measures and numerical examples are presented.

July 2016 - P2 - Availability Assessment of Multi-State System by Hybrid Universal Generating Function and Probability Intervals

Abstract: Abstract: In this paper a non-repairable multi-state system with imprecise probabilities and performance rates are taken, representing imprecise probabilities by interval valued probabilities. These intervals are evaluated by computing bound of interval valued ordinary differential equation of the system. For imprecise performance rates random fuzzy variables are introduced. The proposed method is based on hybrid universal generating function (universal generating function representation of random fuzzy variable) and probability intervals to incorporate the uncertainty problem in availability assessment of the system. Finally, availability p-boxes of the system have been evaluated along with a numerical example.

Maximum Likelihood Analysis of the Ford---Fulkerson Method on Special Graphs

Abstract: We present original average-case results on the performance of the Ford---Fulkerson maxflow algorithm on grid graphs (sparse) and random geometric graphs (dense). The analysis technique combines experiments with probability generating functions, stochastic context free grammars and an application of the maximum likelihood principle enabling us to make statements about the performance, where a purely theoretical approach has little chance of success. The methods lends itself to automation allowing us to study more variations of the Ford---Fulkerson maxflow algorithm with different graph search strategies and several elementary operations. A simple depth-first search enhanced with random iterators provides the best performance on grid graphs. For random geometric graphs a simple priority-first search with a maximum-capacity heuristic provides the best performance. Notable is the observation that randomization improves the performance even when the inputs are created from a random process.

Probability generating function based Jeffrey's divergence for statistical inference

Abstract: Statistical inference procedures based on transforms such as characteristic function and probability generating function have been examined by many researchers because they are much simpler than probability density functions. Here, a probability generating function based Jeffrey's divergence measure is proposed for parameter estimation and goodness-of-fit test. Being a member of the M-estimators, the proposed estimator is consistent. Also, the proposed goodness-of-fit test has good statistical power. The proposed divergence measure shows improved performance over existing probability generating function based measures. Real data examples are given to illustrate the proposed parameter estimation method and goodness-of-fit test.

Analysis of random walks on a hexagonal lattice

Abstract: We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The convergence of the stochastic process to a 2-dimensional Brownian motion is also discussed. Furthermore, we obtain some results on its asymptotic behavior making use of large deviation theory. Finally, we investigate the first-passage-time problem of the random walk through a vertical straight-line. Under suitable symmetry assumptions we are able to determine the first-passage-time probabilities in a closed form, which deserve interest in applied fields.

Random walks on graphene: generating functions, state probabilities and asymptotic behavior

Abstract: We consider a discrete-time random walk on the nodes of a graphene-like graph, ie an unbounded hexagonal lattice We determine the probability generating functions, the transition probabilities and the relevant moments The convergence of the stochastic process to a 2-dimensional Brownian motion is also discussed Finally, we obtain some results on its asymptotic behavior making use of large deviation theory

Probabilistic Inference with Generating Functions for Poisson Latent Variable Models

Abstract: Graphical models with latent count variables arise in a number of fields. Standard exact inference techniques such as variable elimination and belief propagation do not apply to these models because the latent variables have countably infinite support. As a result, approximations such as truncation or MCMC are employed. We present the first exact inference algorithms for a class of models with latent count variables by developing a novel representation of countably infinite factors as probability generating functions, and then performing variable elimination with generating functions. Our approach is exact, runs in pseudo-polynomial time, and is much faster than existing approximate techniques. It leads to better parameter estimates for problems in population ecology by avoiding error introduced by approximate likelihood computations.

Bulk Arrival, Fixed Batch Service Queue with Unreliable Server and with Compulsory Vacation

Abstract: Objectives: To derive a steady state solution of bulk arrival queue with fixed batch service, in addition the server may break down and the server takes compulsory vacation at each service completion points. Methods/Statistical Analysis: Using supplementary variable technique and Laplace Stieltjes transform, the probability generating functions has been obtained for finding the steady state solution. Findings: Using the properties of probability generating function, some performance measures of the queuing model have been obtained at various server state. The model has been compared with existing once by assuming particular distributions to respective random variables. Application/Improvements: The results obtained has been comprehended by illustrating numerical examples.

Number of Jumps in Two-Sided First-Exit Problems for a Compound Poisson Process

Abstract: In this paper, we study the joint Laplace transform and probability generating functions of two pairs of random variables: (1) the two-sided first-exit time and the number of claims by this time; (2) the two-sided smooth exit-recovery time and its associated number of claims. The joint transforms are expressed in terms of the so-called doubly-killed scale function that is defined in this paper. We also find explicit expressions for the joint density function of the two-sided first-exit time and the number of claims by this time. Numerical examples are presented for exponential claims.

Estimation of Mutation Rates from Fluctuation Experiments via Probability Generating Functions

Abstract: This paper calculates probability distributions modeling the Luria-Delbruck experiment. We show that by thinking purely in terms of generating functions, and using a 'backwards in time' paradigm, that formulas describing various situations can be easily obtained. This includes a generating function for Haldane's probability distribution due to Ycart. We apply our formulas to both simulated and real data created by looking at yeast cells acquiring an immunization to the antibiotic canavanine. This paper is somewhat incomplete, having been last significantly modified in March 29, 2014. However the first author feels that this paper has some worthwhile ideas, and so is going to make this paper publicly available.

Quantile function of a fuzzy random variable and an expression for expectations

Abstract: We consider the quantile function of a fuzzy random variable and obtain expressions for some expectations related to fuzzy random variables via integrals of quantile functions.

The Linear Probability Density Function of Continuous Random Variables in the Real Number Field and Its Existence Proof

Abstract: For a continuous random variable in real number field, there must be a distribution and also a probability density function of this random variable. If there is a known function with this random variable as independent variable, its image is a smooth or piecewise smooth line, there must be at least one function that takes this random variable as its independent variable, these functions are bounded on the image of the first function. Any one of these functions conduct line integral operation to the line segment or arc length of the certain image of the first known function is the cumulative probability of this continuous random variable interval corresponding to the section of the image for line integral operation. A general designation for these functions are linear probability density function of continuous random variables. Conduct line integral operation to the linear probability density function and conduct integral operation to the probability density function have same results of the cumulative probability of continuous random variable. By the way, Line integration including curve integration. According to the uniqueness of the probability, the existence and the number of linear probability density function can be proved and calculated.

Performance analysis of Markovian queue with impatience and vacation times

Abstract: We consider an $M/M/1$ queueing system with impatient customers with multiple and single vacations. It is assumed that customers are impatient whenever the state of the server. We derive the probability generating functions of the number of customers in the system and we obtain some performance measures.

Computing exact bundle compliance control charts via probability generating functions

Abstract: Compliance to evidenced-base practices, individually and in ‘bundles’, remains an important focus of healthcare quality improvement for many clinical conditions. The exact probability distribution of composite bundle compliance measures used to develop corresponding control charts and other statistical tests is based on a fairly large convolution whose direct calculation can be computationally prohibitive. Various series expansions and other approximation approaches have been proposed, each with computational and accuracy tradeoffs, especially in the tails. This same probability distribution also arises in other important healthcare applications, such as for risk-adjusted outcomes and bed demand prediction, with the same computational difficulties. As an alternative, we use probability generating functions to rapidly obtain exact results and illustrate the improved accuracy and detection over other methods. Numerical testing across a wide range of applications demonstrates the computational efficiency and accuracy of this approach.

On a continuous/discrete time queueing

Abstract: This paper studies the behaviour of a first-come-first-served queueing network with arrivals in batches of variable size and a certain service time distribution. The arrivals and departures of customers can take place only at the transition time marks and the intertransition time obeys a general distribution. The Laplace transforms of the probability generating functions for the queue length are obtained in the two cases; (i) when departures are correlated; (ii) when departures are uncorrelated; and the steady state results are derived therefrom. It has also been shown that the steady state continuous time solution is the same as the steady state discrete time solution. CONTINUOUS AND DISCRETE TIME QUEUEING SYSTEM; BATCH ARRIVALS OF

Some Results for Poisson and Beta distributions

Abstract: Explicit expressions of probability functions and probability generating functions for mixed Poisson distributed discrete random variables are given corresponding to the following structure density functions: generalized gamma, generalized shifted gamma and generalized shifted beta. A discrete symmetric distribution corresponding to a stochastic process is approximated by a beta distribution in a more accurate manner. A generalized Beta-Poisson distribution is obtained. The results are useful in biological and economical problems. Special cases are also mentioned. Graphs are drawn for probability functions showing the modality for different values of the parameters. Transition intensities can be easily obtained for the various cases discussed in this paper. Finally, by utilizing the fact that probabilities sum to 1, we obtain some new results for generalized hypergeometric functions.

A Study On The Analysis Of Performance Measure Of Bulk Input Queue With N Types Of Additional Optional Service, Service Interruption And Deterministic Vacation

Abstract: We present a bulk arrival queueing system with two types of services. An added assumption of multi types of optional services is considered. After the service completion, the server may take a vacation or continue to stay in the system. Here the vacation time is deterministic. Bernoulli schedule server vacation is assumed. Moreover service interruption is considered as a major phenomenon followed by repair process. Service follows general distribution. Breakdown is exponentially distributed. Using supplementary variable technique the probability generating functions for the numbers of customers in the system and performance measures are derived. Some special cases are also discussed. egative impact caused.

incompletely observed random walks

Abstract: Partial observation of a random walk results in independent convolutions of i.i.d. variables. It is known that under a scheme of sufficiently frequent observation, moments of the random walk can be consistently estimated. In these cases, probability generating functions (p.g.f.'s) can be used to circumvent the difficulties posed by likelihood estimation involving convolutions. Asymptotic properties of the p.g.f. estimates are given, and a comparison is made with the method-of-moments estimator, which is also shown to be asymptotically normal. In a parametric context, the p.g.f. estimator is shown to be asymptotically efficient. Monte Carlo experiments demonstrate that there are advantages to using the p.g.f.-based estimate in small samples as well.