Showing papers on "Probability-generating function published in 2017"

On Stein operators for discrete approximations

Abstract: In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, the Stein operators for certain compound distributions, where the random summand satisfies Panjer’s recurrence relation, are derived. A well-known perturbation approach for Stein’s method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approximation to sums of independent and dependent indicator random variables.

Minimum distance estimators for count data based on the probability generating function with applications

Abstract: This paper studies properties of parameter estimators obtained by minimizing a distance between the empirical probability generating function and the probability generating function of a model for count data. Specifically, it is shown that, under certain not restrictive conditions, the resulting estimators are consistent and, suitably normalized, asymptotically normal. These properties hold even if the model is misspecified. Three applications of the obtained results are considered. First, we revisit the goodness-of-fit problem for count data and propose a weighted bootstrap estimator of the null distribution of test statistics based on the above cited distance. Second, we give a probability generating function version of the model selection test problem for separate, overlapping and nested families of distributions. Finally, we provide an application to the problem of testing for separate families of distributions. All applications are illustrated with numerical examples.

Combinatorics of chemical reaction systems

Abstract: We propose a concise stochastic mechanics framework for chemical reaction systems that allows to formulate evolution equations for three general types of data: the probability generating functions, the exponential moment generating functions and the factorial moment generating functions. This formulation constitutes an intimate synergy between techniques of statistical physics and of combinatorics. We demonstrate how to analytically solve the evolution equations for all six elementary types of single-species chemical reactions by either combinatorial normal-ordering techniques, or, for the binary reactions, by means of Sobolev-Jacobi orthogonal polynomials. The former set of results in particular highlights the relationship between infinitesimal generators of stochastic evolution and parametric transformations fo probability distributions. Moreover, we present exact results for generic single-species non-binary reactions, hinting at a notion of compositionality of the analytic techniques.

Analysis of a discrete-time queue with general service demands and phase-type service capacities

Abstract: In this paper, we analyze a non-classical discrete-time queueing model where customers demand variable amounts of work from a server that is able to perform this work at a varying rate. The service demands of the customers are integer numbers of work units. They are assumed to be independent and identically distributed (i.i.d.) random variables. The service capacities, i.e., the numbers of work units that the server can process in the consecutive slots, are also assumed to be i.i.d. and their common probability generating function (pgf) is assumed to be rational. New customers arrive in the queueing system according to a general independent arrival process. For this queueing model we present an analysis method, which is based on complex contour integration. Expressions are obtained for the pgfs, the mean values and the tail probabilities of the customer delay and the system content in steady state. The analysis is illustrated by means of some numerical examples.

Generalized Moment Generating Functions of Random Variables and Their Probability Density Functions

Abstract: This paper seeks to develop a generalized method of generating the moments of random variables and their probability distributions. The Generalized Moment Generating Function is developed from the existing theory of moment generating function as the expected value of powers of the exponential constant. The methods were illustrated with the Beta and Gamma Family of Distributions and the Normal Distribution. The methods were found to be able to generate moments of powers of random variables enabling the generation of moments of not only integer powers but also real positive and negative powers. Unlike the traditional moment generating function, the generalized moment generating function has the ability to generate central moments and always exists for all continuous distribution but has not been developed for any discrete distribution.

Expectation thinning operators based on linear fractional probability generating functions

Abstract: We introduce a two-parameter expectation thinning operator based on a linear fractional probability generating function. The operator is then used to define a first-order integer-valued autoregressive \inar1 process. Distributional properties of the \inar1 process are described. We revisit the Bernoulli-geometric \inar1 process of Bourguignon and Wei{\ss} (2017) and we introduce a new stationary \inar1 process with a compound negative binomial distribution. Lastly, we show how a proper randomization of our operator leads to a generalized notion of monotonicity for distributions on \bzp.

Bulk queue with Bernoulli vacation and m-optional services under N-policy

Abstract: The present investigation deals with a stochastic model of queueing system wherein the customers join the system in bulk with different arrival rates. On arrival in the system, the customer may choose any of the available m-optional services and the server may avail the optional vacation after finishing any one of the service or stay in the system to provide the service to other customers. It is also assumed that the server starts the service, if at least N customers are there in the queue. The steady state behaviour of the system is discussed by using the supplementary variable technique. The mathematical model is analysed by using the probability generating functions of queue size distribution to obtain the performance indices of the system. The principle of maximum entropy is also employed to find the approximate values of waiting time of the system. To validate the results, the numerical illustration is presented.

Batch Arrival Queuing Models with Periodic Review

Abstract: In this paper, we propose a periodic review policy for the M/M /c type of queue The embedded Markov chain technique is used for the analysis of this system. To determine the mean queue length of mean job waiting times and higher moments of these quantities the probability generating functions are calculated (for the queue length) A comparison is made between constant and state dependent lengths of the review period.

Necessary Conditions for the Existence of Stationary Distribution in the Adaptive Control System of Conflict Flows

Abstract: A control process for conflict flows of nonhomogeneous arrivals is considered. A mathematical model of a control system with variable structure is constructed and studied. Recurrence relations are found for the states of the server and the queues length. Recurrence relations are also obtained for one-dimensional probability distributions for the vector Markovian sequence of the system states in one step and in the number of steps equal to the number of the server’s basic states. We propose an iterative-majorant method that allows to find easily verifiable necessary conditions for the stationary probability distribution existence.

On a Restricted Admissibility Compulsory Vacation Queue with Essential and Optional Services

Abstract: A single server queue with compulsory vacation has been considered. In addition the admission to queue is based on a Bernoulli process and also the server give two phase of essential services and an optional service. For this model the probability generating functions of number of customers in the queue for different server states are obtained using supplementary variable technique. Some performance measures are calculated. Particular models are deduced and some numerical examples are presented.

An M/G/1 Queue with K Phases of Vacation and with Second Optional Service and with State Dependent Arrival Rate

Abstract: We consider an M/G/1 queue with K-phase of vacation and with second optional service. The service policy is after completion of essential service, the customer chooses an optional service with probability p or leaves the system with probability (1-p). Both the essential service and optional service follows general distributions. In addition, after completion of essential service or second optional service, if there are no customers in the system, the server takes vacation consisting of K-phases. After completing the Kth phase of vacation, the server enters into the service station independent of the number of customers in the system. The vacation periods follows general distribution. For this model the supplementary variable technique has been applied to obtain the probability generating functions of number of customers in the queue at different server states. Some particular models are obtained, and a numerical study is also carried out.

Almost periodic random sequences in probability

Abstract: Abstract This paper is concerned with almost periodic random sequences in probability. Some basic and fundamental properties of such sequences are established.

General series expansion method by selecting the product of power function and exponential function as the basis function

Abstract: Abstract By using general series expansion method, this paper used the more complicated base function {(t - t0)m e−nt| m = 0,1, 2, ; n = 1, 2, } to study the differential equation V′(t) = 1 – V2(t), V(0) = 0. We could obtain a better result than the homotopy analysis method.

PDF evolution of a pulse train: A Characteristic function and adaptive spline approach

Abstract: By adaptively approximating the characteristic function to the required power by cubic spline functions, the probability density function of a sum of pulse functions with random amplitude and delay, can be approximated with good accuracy and over a wide range. The efficacy of the approach is shown by the determination of the probability density function evolution of a pulse train. Simulation results justify the approach.